#  High-$T_{c}$ superconductor candidates proposed by machine learning

Siwoo Lee  Department of Chemistry, University of Toronto, St. George campus,
Toronto, ON, Canada  Acceleration Consortium, University of Toronto, St.
George campus, Toronto, ON, Canada  Jason Hattrick-Simpers
<EMAIL_ADDRESS>Acceleration Consortium, University of
Toronto, St. George campus, Toronto, ON, Canada  Department of Materials
Science & Engineering, University of Toronto, St. George campus, Toronto, ON,
Canada  Young-June Kim<EMAIL_ADDRESS>Department of Physics,
University of Toronto, St. George campus, Toronto, ON, Canada  O. Anatole von
Lilienfeld<EMAIL_ADDRESS>Department of Chemistry,
University of Toronto, St. George campus, Toronto, ON, Canada  Acceleration
Consortium, University of Toronto, St. George campus, Toronto, ON, Canada
Department of Materials Science & Engineering, University of Toronto, St.
George campus, Toronto, ON, Canada  Department of Physics, University of
Toronto, St. George campus, Toronto, ON, Canada  Vector Institute for
Artificial Intelligence, Toronto, ON, Canada

###### Abstract

We cast the relation between chemical compositions of solid-state materials
and their superconducting critical temperature ($T_{c}$) in terms of a
statistical learning problem with reduced complexity. Training of query-aware
similarity-based ridge regression models on experimental SuperCon data with
(implicit) and without (ambient) high pressure entries achieves average
$T_{c}$ prediction errors of $\pm$10 K for unseen out-of-sample materials.
Subsequent utilization of the approach to scan $\sim$153k materials in the
Materials Project enables the ranking of candidates by $T_{c}$ while taking
into account thermodynamic stability and small band gap. Stable top three
high-$T_{c}$ candidate materials with large band gaps for implicit and ambient
pressures are predicted to be Cs2Sn(H2N)6 (324 K), CsH5N2 (315K), Rb2Sn(H2N)6
(305 K), and H15IrBr3N5 (189 K), H12OsN5Cl3O (161 K), B10H13I (151 K),
respectively. Stable top three high-$T_{c}$ candidate materials with small
band gaps for implicit and ambient pressures are predicted to be RbLiH12Se3N4
(255 K), CeH14Cl3O7 (246 K), Li(H3N)4 (234 K), and ReH30Ru2(NCl)10 (127 K),
AlH18Ru(NF)6 (120 K), Sr(Li2P)2 (117 K), respectively.

††preprint: APS/123-QED

##  I Introduction

(a)  Figure 1:  Superconductor discovery using our similarity based ML
methodology. From left to right: Query candidates (structural depictions drawn
from the Materials Project 1) are used to select closest instances (in terms
of $d$) from the SuperCon data set 2 with known experimentally-measured
$T_{c}$’s for training ridge regression models. Materials representations do
not require any structural information and are based only on features derived
from chemical compositions. The application of this similarity-based machine
learning to $\sim$153k materials in the Materials Project enriches the $T_{c}$
distribution and includes multiple promising high-$T_{c}$ superconductors not
present in SuperCon.

A major unsolved question in the physical sciences is whether there exists at
ambient pressure a superconductor with its superconducting critical
temperature, $T_{c}$, at or above room temperature3. Under these conditions,
such a material must exhibit the hallmark properties of superconductivity 4,
5, i.e. the persistent absence of electrical resistance and the expulsion of
magnetic fields (the Meissner effect) below a critical magnetic field strength
($H_{c}$). Its discovery would be of immense significance to various
scientific and industrial applications, encompassing energy storage and
electrical power transmission6, 7, 8, 9, nuclear fusion energy10, 11, 12,
computing13, 14, 15, 16, medicine17, 18, public transport19, 20, 21, to
particle physics22, 23, 24, 25. Unfortunately, the identification of
candidates is difficult because the overwhelming majority of known
superconductors have $T_{c}$’s near 0 K2, making it challenging to learn from
the properties of high-$T_{c}$ samples and thereby propose novel candidates
with higher $T_{c}$’s. While the Bardeen-Cooper-Schrieffer (BCS) theory26, 27
can successfully explain superconductivity at low temperatures as a phonon-
mediated process to form a Cooper pair, there currently exists no complete
microscopic theory of high-$T_{c}$ superconductivity28. Therefore, approaches
to designing new higher-$T_{c}$ materials have historically been largely
empirical without the guidance of theory29. While ab initio electronic
structures methods could be more efficient than experiments, the lack of
predictive $T_{c}$ modeling also impedes the computational design and
discovery. This poses a severe bottleneck on the search for new
superconductors within the vast materials compound space estimated by some to
be populated by $\sim$$10^{100}$ plausible materials30.

In this work, we describe a similarity-based machine learning (ML) approach
that is capable of extrapolating beyond the distribution of experimentally-
measured $T_{c}$ values found in the SuperCon data set2 to potentially
identify materials with $T_{c}$’s greater than the currently known highest
value, which is 250 K for LaH1031. Recently, there have been many efforts to
predict $T_{c}$’s directly from the crystal structures and chemical
compositions of materials using ML32, 33, 34, 35, 36, 37, 38, 39. However, as
far as we are aware, none of these works has successfully addressed the out-
of-domain (OOD) problem40, 41 that is inherent to attempts of making accurate
predictions for samples with label values found beyond the range that the ML
models were trained on. This is, in fact, a well-known issue in the ML
field42, 43, and it is imperative that it is tackled if ML models are to be
used to propose novel high-$T_{c}$ candidates.

We have tackled this problem using the procedure outlined in Figure 1. First,
for a given material, its $n$-nearest neighbors in the SuperCon 2, 44 training
data sets are queried. These $n$-nearest neighbors are then used to train a
ridge regression model, from which the test sample’s $T_{c}$ is predicted.
Leave-one-out prediction tests of the resulting models trained on sets drawn
either from all of the $\sim$13.6k samples in SuperCon 2 (implicit model), or
from its subset exclusively containing entries at ambient pressures (ambient
model), suggest good predictive performance throughout the full range of
measured $T_{c}$’s. Subsequently, we have applied this approach to predict the
$T_{c}$’s of the $\sim$153k materials for which calculated density functional
theory (DFT) property results are recorded in the Materials Project1. After
filtering for thermodynamic stability, we have ranked all remaining materials
by our ML $T_{c}$ estimates. The implicit and ambient pressure model
identifies thirty-five and six candidate materials exhibiting predicted
$T_{c}$’s above 250 K and 135 K (the current record at ambient pressures45),
respectively. Filtering the ranked list also for DFT band gaps smaller than 1
eV, the top ranking materials of the implicit and ambient pressure models
exhibit predicted $T_{c}$’s of 255 K and 127 K, respectively.

##  II Methodology

###  II.1 Superconductors Data Set

Figure 2:  Distribution of 13,661 $T_{c}$ measurements in the curated SuperCon
data set, including samples with $T_{c}$’s measured under ambient and applied
pressures. Inset shows the distribution of samples with $T_{c}$’s measured to
be greater than 70 K and four exemplary crystal structures.

In this work, the SuperCon data set2 was obtained from the Materials Data
Repository, which is maintained by the Japanese National Institute for
Materials Science (NIMS). During the time which this work was conducted,
SuperCon listed 26,321 materials, each with their associated chemical
compositions, experimentally-measured $T_{c}$’s (Kelvin), and article
references. It does not include atomic coordinates required to construct unit
cells and does not mention whether the $T_{c}$ measurements were performed
while external pressures were applied to the samples. We cleaned the data set
by assigning to stoichiometries with multiple $T_{c}$ measurements their mean
values. Some contentious high-$T_{c}$ measurements 46, 47, 48 were also
removed. As it was deemed to be prohibitively time-consuming to verify via
human-effort the repute and accuracy of each measurement in SuperCon, it was
assumed that the issue of the inclusion of measurements from untrustworthy
experiments could be largely neglected in the low-$T_{c}$ regime where the
overwhelming majority of samples lie. Further, samples consisting of one or
ten different element types were removed, as were those specified by arbitrary
doping concentrations (e.g. Hg2Ba2YCu2O_8-x).

The final, cleaned data set includes 13,661 unique stoichiometries with
$T_{c}$’s ranging 0–250 K, with a median value of 10 K (Figure 2). Using the
definition of high-temperature superconductors as those with $T_{c}$’s greater
than the boiling point of liquid nitrogen, 77 K49, there exists only 1,372
such samples in the data set, constituting 10 $\%$ of its total size –
clearly, the distribution is highly right-skewed. To illustrate, the four
samples in the data set with the highest $T_{c}$’s are LaH10, H2S, H3S,
Hg_0.66Pb_0.34Ba2Ca_1.98Cu_2.9O_8.4, with corresponding $T_{c}$’s of 25031,
20350, 14751, and 14352 K, respectively. It may be of interest to note that
experimental studies suggest the superconducting phase of H2S which occurs
under high pressures (gigapascals) actually exists as H3S due to
decomposition50. While this mechanism may make redundant the inclusion in the
data set of both H2S and H3S, they were retained for chemical diversity in
this low-data regime.

The SuperCon2 data set44 – which was compiled by autonomously scraping the
literature to extract data related to superconductors – was then used to
determine the samples in SuperCon with $T_{c}$ measurements performed under
applied pressure. This analysis suggests that thirty-seven samples were
measured under applied pressures. These were removed from SuperCon to create a
separate data set containing only samples with $T_{c}$’s measured under
ambient conditions (0–135 K).

###  II.2 Similarity-Based Machine Learning

To estimate the $T_{c}$ of a given test sample via similarity-based ML53, the
training data set is first queried to find its $n$-nearest neighbors. The
distance metric we employ is the Euclidean distance, calculated as

$d(\textbf{x}_{q},\textbf{x}_{t})=\lVert\textbf{x}_{q}-\textbf{x}_{t}\rVert$
(1)

where $\textbf{x}_{q}$ is the feature vector of the query test sample and
$\textbf{x}_{t}$ is the feature vector of $t$’th training set sample.

The $n$ training samples with the smallest norm values are then chosen to
train a ridge regression model with the optimal $\alpha$ hyperparameter54
selected as the one yielding the best performance on the training set
(smallest mean squared error), evaluated by $n$-fold cross-validation. A
prediction for a given test sample is made as

$T_{c}^{\rm est}=\textbf{x}_{q}\textbf{w}_{q}+c_{q}$ (2)

where $c_{q}$ is the intercept and $\textbf{w}_{q}$ is the vector of weight
coefficients obtained from the training set as the closed-form solution of

$\textbf{w}_{q}=\left(\textbf{X}\textbf{X}^{\top}+\alpha\textbf{I}\right)^{-1}\textbf{X}^{\top}\textbf{y}$
(3)

for X the $n$-sample feature matrix and y the corresponding $n$-dimensional
labels vector ($T_{c}$). We perform all our computations with the scikit-
learn55, 56, 57 library in Python and perform the matrix inversion using the
Cholesky decomposition54. The absolute values of predictions are used for the
final estimates to avoid non-physical negative $T_{c}$ values.

###  II.3 Machine Learning Representations

Since only chemical compositions are provided in SuperCon, they form the basis
of the representations used to train our ML models. Despite their apparent
simplicity, chemical compositions have been shown to be sufficient in
accurately learning various properties of materials58, 59, 60, 61, 62. They
are also less restrictive in suggesting interesting stoichiometries to
experimentalists, as the predictions derived from them are not specific to
particular crystal structures contained in their convex hulls63.

147 features were generated for each sample from its composition using the
materials informatics Python library Matminer64. The minimum, maximum, range,
mean, average deviation, and mode were calculated from the following atomic
properties, with weights given by stoichiometric coefficients: atomic number;
Mendeleev number; atomic weight; melting temperature of elemental solids;
periodic table group number; periodic table period number; covalent radius;
electronegativity; number of filled s, p, d, f orbitals; number of valence
electrons; number of unfilled s, p, d, f orbitals; number of unfilled valence
orbitals; volume of elemental solid; band gap of elemental solid; magnetic
moment of elemental solids; and space group number of elemental solids.
Matminer was used to additionally calculate transition metal fractions;
stoichiometric 0-, 2-, 3-, 5-, 7-, and 10-norms; average number of valence
electrons in each of s, p, d, f orbitals; and the fractions of valence
electrons in each of s, p, d, f orbitals.

###  II.4 Materials Project

We apply our similarity-based ML method to $\sim$153k samples listed in the
Materials Project database to screen for potential novel superconductors with
high-$T_{c}$’s. For each material represented as described in subsection II.3,
predictions are made at implicit/ambient pressure by selecting $n$ training
samples from the cleaned implicit/ambient pressure SuperCon data set and
training a ridge regression model, as described in subsection II.2.

##  III Results and Discussion

###  III.1 SuperCon based validation of our approach

Figure 3:  Implicit pressure (left) and ambient pressure (right) model
learning curves. Prediction errors for $T_{c}$’s of unseen out-of-sample test
materials vs. training on $n$ nearest neighbors selected from variable
training pool sizes (indicated in the legend). The shading quantifies standard
deviation in the predictions of a fixed test set at each $n$, obtained by five
random selections of the training pool from the full data set.

The predictive performances of similarity-based ridge regression models in
mapping chemical compositions to their superconducting $T_{c}$’s in SuperCon
are first assessed with learning curves65. Due to their characteristic shape,
learning curves are useful for evaluating not only the data-efficiency of ML
models but also the learnability of the problem. In Figure 3, the learning
curves show the prediction error (mean absolute error, MAE) on the test set
after training as a function of $n$ nearest neighbors. Different learning
curves are plotted where $n$ training samples are selected from different
pool-sizes of the full implicit/ambient pressure data set. The shadings
indicate the standard deviations in the predictions obtained by training five
different models on $n$-nearest neighbors selected from five random pools of
the full data set. For example, in Figure 3, one curve is labelled “256” to
indicate that 256 samples were randomly designated as the training data set
from the full implicit/ambient pressure data set. From this pool of 256
samples, the model performances after training on different sized subsets, for
$n\leq 256$, are evaluated on the hold-out test set. The random pool selection
and the training on $n$ samples is performed five times at each point of the
learning curves. Note that the hold-out test set selected from the
implicit/ambient pressure data set was held constant throughout the generation
of all the learning curves.

It is clear from each learning curve that increasing $n$ produces systematic
decay in the MAE’s of the predicted $T_{c}$’s. However, contrary to noise-free
labels and representations, the performance of our model saturates at an error
of only $\sim$10 K after training on 3000 samples for the implicit pressure
models and after training on 6000 samples for the ambient pressure models.
While this residual error is similar or only slightly larger than prediction
errors of $T_{c}$ reported previously by others 32, 33, 34, 35, 36, 37, 38, we
stress again that our models, by contrast, are not incapable of extrapolation.
The deviation from the expected linear decrease in the error with $n$65
suggests that either the choice of composition-based representations to train
the ML models is not sufficiently unique for each sample, or that the labels
are noisy, or both. It might also imply that the complex (non-linear) mapping
from the features space to the labels space cannot be fully learned with
linear predictor functions. However, some studies also suggest that the
performance saturation may occur without regards to the selection of the
learning algorithm. For instance, the use of kernel ridge regression in
similarity-based learning of the atomization energies of molecules also
exhibits the levelling in prediction errors53. Nonetheless, our similarity-
based ML method reaches prediction errors on the order of $\sim$10 K, which,
in the context of the large range of experimentally-measured $T_{c}$ values
spanned by different materials in SuperCon, we assume in the following to be
sufficiently accurate and transferable for obtaining robust rankings among
high-$T_{c}$ candidate materials.

(a)

(b)

(c)

Figure 4:  (a) Leave-one-out $T_{c}$ predictions under implicit/ambient
pressure for 7,629/6,183 samples in the implicit/ambient pressure SuperCon
data set with spreads in their index-wise feature-weight products of less than
51/42 K, achieving MAE $\sim$5/6 K and $R^{2}$ $\sim$0.79/0.69. Insets display
these samples’ distributions of index-wise feature-weight products and signed-
errors. (b) Distributions of the number of different elements and (c) and
element types in the compositions of the SuperCon samples with absolute
prediction errors greater than 10 K (without filtering for spreads in feature-
weight products), compared against all SuperCon samples.

With the similarity-based ML approach, leave-one-out $T_{c}$ predictions under
unknown/ambient pressure are made for each of the 13,661/13,624 materials in
the implicit/ambient pressure data set. The implicit pressure prediction for
each sample was made by training a model on its 3000 nearest neighbors in the
implicit pressure data set. $n=3000$ was selected since, as can be seen from
Figure 3, it already reaches an MAE of $\sim$10 K. Hence, we can avoid the
increased computational cost of the matrix inversion involved in ridge
regression that would be incurred from training on a larger number of samples.
For the same reasoning we selected $n=6000$ for the ambient pressure models
(See Fig. Figure 3 (Right)). Note that overall, however, the execution of this
method is extremely efficient due to the relatively small size of $n$ and the
low cost associated to evaluate inner products as similarity measures. To
illustrate, the total wall time required to search for $n$-nearest neighbors,
tune for the $\alpha$ hyperparameter, train on the $n$ training samples, and
predict for the test sample is about a second on a laptop (12th gen. Intel
i7-1260P, 12 cores, 2.10 GHz clock rate). Consequently, scanning a materials
library for $T_{c}$ requires 1 s per material and node.

Upon examination of the distribution of the index-wise products in the dot
product between a query sample’s feature vector, $\textbf{x}_{q}$, and the
vector of weight coefficients, $\textbf{w}_{q}$

$\textbf{x}_{q}\textbf{w}_{q}=\sum_{i=1}^{147}x_{q,i}w_{q,i}$ (4)

it emerges that some samples have large standard deviations in their product
values (Figure 4a). This suggests that the $\alpha$ parameter, obtained from
the training set, is not generalizable to the test sample as it fails to
penalize the large coefficients that produce large $\lvert
x_{q,i}w_{q,i}\rvert$ values. The removal of such implicit/ambient pressure
samples with large spreads, defined as those greater than the mean spread of
51/42 K, improves predictive performances in terms of leave-one-out
predictions on SuperCon (Figure 4a), as the MAE can be reduced from $\sim$9/10
K to $\sim$5/6K. Although there are few samples with large prediction errors,
similarity-based ML appears to achieve relatively good performances across the
full range of $T_{c}$ values. For instance, LaH10 with $T_{c}$ of 250 K is
accurately predicted and no value greater than $\sim$250/135 K is predicted
for the implicit/ambient pressure SuperCon materials, which corroborates the
robustness of the models. The signed-errors distribution is symmetrical but
not normally distributed. The former suggests that the models’ predictions do
not carry a systematic error. The non-normal distribution with large tails is
due to the unphysical statistical nature of the outliers (particularly
prominent in the scarce data regime the similarity-based ML model is operating
in), and has previously already been observed and discussed 66.

An examination of the samples with absolute prediction errors greater than 10
K reveals they mostly comprise of compositions consisting of five different
elements and that the error-rate proportion increases with greater number of
differing elements in the system (Figure 4b). This may not be unexpected since
the different number of environments an element can experience in its crystal
system increases with increasing number of differing elements. However, in
SuperCon, the number of samples as a function of differing elements does not
appear to increase sufficiently for the different environments to be
adequately accounted for in our ML models. We also note that large errors are
associated with materials consisting of oxygen, copper, or barium (Figure 4c).
This may be the effect of a combination of a bias incurred from intense
research by the superconductor community into yttrium barium copper oxide and
related compounds, and that oxygen as a strong oxidant complicates the
chemistry involved in these systems.

The weights obtained from the training of ridge regression models were also
inspected to better understand which features made the most significant
contributions to the predictions of $T_{c}$ labels. On average for the samples
in the implicit pressure data set, the five most significant features, in
decreasing order of importance as calculated by the magnitude of the absolute
value of the product of the weight with its corresponding feature value
$\lvert x_{q,i}w_{q,i}\rvert$, are the average atomic number, average covalent
radius, average atomic weight, average deviation of atomic numbers, and
average deviation of atomic weights. Correspondingly, in the ambient pressure
data set, the five most important features are the average periodic table
period number, average atomic number, average atomic weight, average deviation
of atomic numbers, and average deviation of atomic weights. Clearly, the
$T_{c}$ predictions are greatly influenced by the masses and sizes of the
elemental species in a given material’s stoichiometry. These results are
logical considering the observations of the isotope effect in both
conventional and unconventional superconductors67, 68, 69, 70, in which their
$T_{c}$’s are inversely proportional to the square-root of the atomic masses
of the isotopes in their compositions as lighter ones produce higher phonon
frequencies.

###  III.2 Rediscovery of Known Superconductors

We further evaluated our ML method by predicting $T_{c}$’s of published
superconducting materials that are not part of the SuperCon data sets.
Specifically, the two compounds Ni(TePd)239/(SrCa)10Cu17O2971 were
experimentally-measured in 2023/2000 to exhibit $T_{c}$’s of $\sim$1/75 K. Our
ambient pressure model predicts $T_{c}$ for these samples to correspond to 13
K and 68 K, respectively, which are very close or even within the $\sim$10 K
error expected from our learning curves (Figure 3) and leave-one-out
predictions (Figure 4a). These results represent an entirely independent test
set and further corroborate our assumption that the approach is sufficiently
robust for the identification of high-$T_{c}$ material candidates.

###  III.3 Application to the materials project

Figure 5:  Distribution of implicit/ambient pressure $T_{c}$ predictions for
Materials Project samples with energies above convex hulls of less than 0.030
eV/atom (38,539/27,821 samples), with subsets filtered for band gaps of less
than 1.000 eV (23,402/25,186 samples). Inset shows the distributions of
samples with predicted $T_{c}$’s greater than 50 K and representative crystal
structures.

Our similarity-based ML approach has been used to estimate $T_{c}$’s of
$\sim$153k materials listed in the Materials Project. The Materials Project
was queried because it provides DFT-computed estimates of various materials
properties, such as stability and band gap. Due to the computational
efficiency of our approach, the entire scan consumed only $\sim$42 node hours.
We note that due to the generality of our approach, any other materials
library (e.g. with experimental stability and band gap values) could have been
used just as well.

Each sample’s predictions under implicit pressure ($n=3000$) and ambient
pressure ($n=6000$) were made by training on its nearest neighbors found in
the implicit and ambient pressure SuperCon data sets, respectively. Samples
with implicit and ambient pressure predictions associated with standard
deviations larger than 51 K and 42 K, respectively, are disregarded. Those
with computed energies above their convex hulls of greater than 0.030 eV/atom
are also disregarded as being thermodynamically unstable.

(a)

(b)

Figure 6:  Crystal structure and band structure of material in Table 1 with highest predicted $T_{c}$. (a) Crystal structure of Cs2Sn(H2N)6, and (b) its electronic band structures and density of states. The Fermi level is set to 0 eV, indicated by the dashed line. All data downloaded from materials project 72, 73.  Table 1:  Compositions of materials in the Materials Project with the highest predicted $T_{c}$’s under implicit pressures, that lie $<0.030$ ev/atom above their convex hulls.  ID | Composition |  $T_{c}^{\rm est}$ (K)
---|---|---
mp-505233 |  Cs2Sn(H2N)6 | 324
mp-1184037, mp-1198227 |  CsH5N2 | 315
mp-643359 |  Rb2Sn(H2N)6 | 305
mp-643371 |  K2Sn(H2N)6 | 301
mp-643158 |  PH8IN4 | 297
mp-1193046 |  CsMgAs(H6O5)2 | 297
mp-974267 |  KH8N3 | 288
mp-721084 |  H7IN2 | 287
mp-1202629 |  CdH20N6OF2 | 286
Table 2:  Compositions of materials in the Materials Project with the highest predicted $T_{c}$’s under ambient pressure, that lie $<0.030$ ev/atom above their convex hulls.  ID | Composition |  $T_{c}^{\rm est}$ (K)
---|---|---
mp-1205028 |  H15IrBr3N5 | 189
mp-24461 |  H12OsN5Cl3O | 161
mp-1199051, mp-1199255 |  B10H13I | 151
mp-30977 |  B5H6Br | 149
mp-1197561 |  B10H13Br | 143
mp-634446 |  CsAl(H2N)4 | 132
mp-1199374 |  TaH9N3F5 | 132
mp-1204398 |  CoMoH24N6ClO7 | 129
mp-767240 |  CsPH4(NO)2 | 129

The resulting distributions in $T_{c}$ predictions is shown in Figure 5. This
analysis suggests that at unknown, potentially very high pressures, thirty-
five materials may have $T_{c}$’s greater than 250 K. Of these materials,
fourteen may transition to their superconducting phase at temperatures greater
than 273 K (see Table 1 for representative samples), with the highest $T_{c}$
predicted to be 324 K for Cs2Sn(H2N)672, 73 (Figure 6a). When filtering for
metal-like materials with small band gaps (less than 1.000 eV), only one
sample of RbLiH12Se3N4 is predicted to have a $T_{c}$ (255 K) greater than 250
K. At ambient pressure, six materials are predicted to have $T_{c}$’s greater
than 135 K (143–189 K) (see Table 2), while metal-like materials are predicted
to reach only up to 127 K.

It is believed that one of the identifying characteristics of high-$T_{c}$
superconductors are strongly correlated bands that allow for unorthodox Cooper
pair formations74, 75. For instance, from Figure 6b, it appears that
Cs2Sn(H2N)6 has a pair of bands at its Fermi level that are relatively flat
and narrow with a maximum bandwidth of $\sim$160 meV. By the assumptions about
band flatness correlating with superconductivity, this result may suggest that
this material can transition to its superconducting phase at higher
temperatures. However, band structure calculations for any given material
often involve severe approximations and produce results that can deviate
significantly (usually as underestimates) from experimental results76, 77.
Therefore, it would be of great interest to verify the superconducting
properties and electronic band structures of Cs2Sn(H2N)6 via experimental
synthesis and measurements. Moreover, it would be interesting to conduct
experiments of the thermodynamic stability of Cs2Sn(H2N)6 because the
calculations conducted by the Materials Project state that its crystal
structure is not the most stable configuration as it lies $\sim$0.017 ev/atom
above this stoichiometry’s convex hull72, 73. After all, it would be of very
limited practical usefulness if its superconducting phase occurs only under
extremely high pressures. It appears that there currently has been no efforts
to conduct such experiments, as this material is not reported in SuperCon and,
to the best of our knowledge, it has not been studied in the literature.

##  IV Conclusion

We have introduced a data-efficient similarity-based ML approach to estimating
the superconducting critical temperatures of materials at both implicit and
ambient pressures. Predictions for novel materials require training a query
aware ridge regression model ‘on the fly’ using only the $n$ nearest
neighbours in the training data. This is feasible thanks to the extremely low
computational cost required by our model ($\sim$1 second/material on a twelve-
core CPU). When this simple method was evaluated via leave-one-out predictions
on the full SuperCon data set, it was found to be relatively robust in making
accurate predictions across the full range of $T_{c}$ values, 0–250 K.
Moreover, it was observed that one may be able to identify predictions with
large uncertainties as samples with large standard deviations in their index-
wise products contained in their feature-weights vectors dot products. The
analysis of the weight coefficients revealed that material properties related
to the atomic weights and radii of the elements in the stoichiometries were
the most significant contributors to the predictions of $T_{c}$’s, suggesting
that the ML models were able to at least capture the basic physical principles
involved in superconductivity.

We have used the model to rank the entire Materials Project data base of
$\sim$153k materials by the estimated $T_{c}$. Several materials were
identified as potentially being able to exist as superconductors near room-
temperature, albeit under unknown pressure, with Cs2Sn(H2N)6, despite its
large band gap ($\sim$2.5 eV according to DFT), being of particular interest
with an estimated $T_{c}$ of 324 K at a pressure that would remain to be
determined. Its electronic band structures records in the Materials Project
indicate, however, that there is a flat band near its Fermi level, which
possibly could be leveraged upon doping. As superconductivity experiments
appear to have not yet been performed for Cs2Sn(H2N)6, nor for the other
high-$T_{c}$ materials identified, they might be valuable targets for future
research.

Our results also exemplify the usefulness of similarity-based ML in
accelerating the virtual design and discovery of novel materials and molecules
with interesting properties. We note this may be facilitated by the specific
choice of a representation which is more or less agnostic (depending on use
case). For example, as done in this study, features only derived from chemical
composition may be advantageous in allowing for the screening of a much
greater number of materials as it would be trivial to create new
stoichiometries by combinatorics of element types and stoichiometric
coefficients. Further important criteria, such as measures of synthesizability
or stability, could be added as constraints later on.

##  V Data and Code Availability

The supplementary information contains a discussion, using toy problems, on
why we chose to develop our ML models with ridge regression, rather than other
learning algorithms. It displays a table listing the SuperCon samples that we
have identified as having $T_{c}$’s measured under applied pressure, and a
table ranking the relative importance of each of the feature descriptors used
in our ML models. It also displays two tables ranking the one-hundred highest
$T_{c}$ materials in the Materials Project identified by our implicit/ambient
pressure models, as well as two additional tables obtained after subsequent
filtering for band gaps lower than 1 eV.

Refer to https://zenodo.org/records/11255989 for: Python code to generate ML
features and to implement our similarity-based ML models; chemical
compositions, $T_{c}$’s, pressures, ML features, and implicit/ambient pressure
$T_{c}$ predictions for our SuperCon samples; and chemical compositions,
identifiers, energies above convex hulls, band gaps, ML features, and
implicit/ambient pressure $T_{c}$ predictions for materials in the Materials
Project.

##  VI Acknowledgments

We acknowledge the support of the Natural Sciences and Engineering Research
Council of Canada (NSERC), [funding reference number RGPIN-2023-04853]. Cette
recherche a été financée par le Conseil de recherches en sciences naturelles
et en génie du Canada (CRSNG), [numéro de référence RGPIN-2023-04853]. This
research was undertaken thanks in part to funding provided to the University
of Toronto’s Acceleration Consortium from the Canada First Research Excellence
Fund, grant number: CFREF-2022-00042. O.A.v.L. has received support as the Ed
Clark Chair of Advanced Materials and as a Canada CIFAR AI Chair.

##  VII Author Contributions

S.L. and O.A.v.L. conceived the idea. S.L. developed and implemented the
methodology and performed all experiments, with guidance from O.A.v.L. All
authors discussed the results, and made comments and edits to the manuscript
written by S.L. and O.A.v.L.

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##  VIII Supplementary Information

###  VIII.1 Toy Problem

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 7:  A simple toy problem of (a) $y=2x$ with the addition of Gaussian
noise, where the label values of the training set are less than those of the
test set. Shown are the predictions of three models on the training and test
sets, developed with the following three popular machine learning algorithms:
(b) ridge regression (c) kernel ridge regression and (d) eXtreme Gradient
Boosting regression. Similarly, the exercise is repeated for (e)
$y=0.2(x-5)^{2}$, with model predictions obtained using (f) ridge regression
(g) kernel ridge regression (h) and eXtreme Gradient Boosting regression.

The selection of ridge regression over other learning algorithms in our ML
models is motivated by its ability to potentially make accurate predictions
for samples with labels found beyond the distribution of the training set.
Consider the toy example of $y=2x$ with the addition of Gaussian noise in
Figure 7a. Here, the training set contains feature and label values that are
both less than those found in the test set. Thus, with this OOD transfer
learning problem, the performances of ML models trained with different
learning algorithms (hyperparameters selected as those returning the lowest
mean absolute error, MAE, upon five-fold cross-validation on the training set)
are evaluated by their abilities to effectively extrapolate beyond their
training set.

Ridge regression (Figure 7b) learns the linearity of the problem since for
greater values of $x$, it correctly predicts correspondingly greater values of
$y$ in both the training and test sets, although its accuracy could be
improved. Models developed using two other popular and powerful ML algorithms
of kernel ridge regression78 with the Laplacian kernel (Figure 7c) and eXtreme
Gradient Boosting79 (XGBoost) regression with the tree-based gbtree booster
(Figure 7d) exhibit, relative to the ridge regression model, improved
performances on the training set. However, regarding the test set, both fail
to make even a single prediction of $y$ that is greater than the maximum value
found in the training set. This is, perhaps, an unsurprising result. After
all, the objective of a kernel-based regression algorithm is to find a non-
parametric mapping between the domain of feature vectors, X, and their labels,
Y, where the estimated function outputs, $\hat{\textbf{Y}}$, are obtained as
kernel-weighted local averages78, 80, 81, 82. Thus, this fundamentally
restricts the value that a given $\hat{\textnormal{Y}}$ can take to within the
range of values found in $\textbf{Y}_{\textnormal{train}}$. The same
conclusion results from a tree-based regression algorithm, which returns the
weighted-average of the predictions obtained from an ensemble of decision
trees83, 84, 85.

In contrast, ridge regression weights a sample’s feature values, which thus
does not limit the model’s output to within the distribution of
$\textbf{Y}_{train}$. Therefore, ridge regression appears to be suitable for
the objective of predicting label values beyond those found in the training
set. Of course, before applying it to the prediction of out-of-distribution
test samples, care must be taken to ensure that its estimates are robust and
accurate since it requires the assumption that there exists some linear
relationship between the features and labels space. If this condition is not
fulfilled, then it can be seen from a quadratic toy problem of
$y=0.2(x-5)^{2}$ (Figure 7e) that ridge regression exhibits poor predictive
power in both the train and test regimes due to the non-linearity of the
problem. The fundamental limits on the labels space imposed by the different
learning algorithms are observed again, since ridge regression can predict
$y_{test}$ values greater than those found in training, albeit poorly in this
case; kernel ridge regression (Figure 7g) and XGBoost regression (Figure 7h)
exhibit good performances on the training set but they fail once more to
predict a single value of $y_{test}$ beyond $\textbf{Y}_{train}$.

###  VIII.2 SuperCon Pressure Samples

Table 3:  Compositions of materials in SuperCon that we have identified as having $T_{c}$’s measured under applied pressure.  Composition |  $T_{c}$ (K)
---|---
IrU |  0
SnTe |  0
KC8 |  0
Zn2Zr |  0
Ge2U |  0
LaC2 |  1
Te2U |  1
PdTe2 |  1
TiSe2 |  1
As2Cd3 |  3
OSn |  3
S2Ta |  3
BaBi3 |  6
B12Zr |  6
C6Yb |  6
Au2Pb |  6
Te2W |  7
Sb2Te3 |  7
NaAlSi |  7
Mo3Sb7 |  7
MoTe2 |  8
B2Nb |  9
C6Ca |  11
H4Si |  17
NNb |  17
C3Y2 |  18
LiFeAs |  18
Nb3Sn |  18
GeNb3 |  22
NaFeAs |  26
B2Mg |  40
FeSe |  80
Hg_0.75Ba_2.07Ca_2.07Cu_3.11O_8.208 |  135
Hg_0.66Pb_0.34Ba2Ca_1.98Cu_2.9O_8.4 |  143
H3S |  147
H2S |  203
LaH10 |  250

###  VIII.3 Features Rankings

Table 4:  Ranking of the relative average contribution of each ML feature to the $T_{c}$ predictions in the SuperCon implicit/ambient pressure models, from greatest to least.  |  |
---|---|---
Ranking |  Implicit pressure |  Ambient pressure
1 |  Atomic number, mean |  Row number, mean
2 |  Covalent radius, mean |  Atomic number, mean
3 |  Atomic weight, mean |  Atomic weight, mean
4 |  Atomic number, avg. dev. |  Atomic number, avg. dev.
5 |  Atomic weight, avg. dev. |  Atomic weight, avg. dev.
6 |  Mendeleev number, mean |  Atomic number, max.
7 |  Atomic number, mode |  Atomic weight, max.
8 |  Atomic weight, mode |  2-norm
9 |  Column number, mean |  Electronegativity, mean
10 |  Atomic number, max. |  Covalent radius, mean
11 |  Atomic weight, max. |  Column number, mean
12 |  Row number, mean |  Column number, mode
13 |  Elemental solid volume, mean |  Mendeleev number, mean
14 |  Covalent radius, max. |  Covalent radius, max.
15 |  Covalent radius, mode |  7-norm
16 |  Column number, max. |  5-norm
17 |  Column number, mode |  3-norm
18 |  Mendeleev number, max. |  Atomic number, mode
19 |  Atomic weight, range |  10-norm
20 |  Electronegativity, mean |  Row number, mode
21 |  Atomic number, range |  Mendeleev number, mode
22 |  Electronegativity, mode |  Mendeleev number, max.
23 |  Mendeleev number, mode |  Column number, avg. dev.
24 |  Covalent radius, min. |  Number of valence electrons, mean
25 |  Row number, max. |  Atomic weight, mode
26 |  Row number, mode |  Atomic weight, range
27 |  Number of valence electrons, mean |  Column number, max.
28 |  Column number, range |  Atomic number, range
29 |  Number of unfilled valence orbitals, mean |  Number of unfilled valence orbitals, mean
30 |  Space group number, mean |  Atomic number, min.
31 |  Melting temperature, mean |  Number of s valence electrons, mean
32 |  Covalent radius, range |  Number of filled s valence orbitals, mean
33 |  Column number, avg. dev. |  Covalent radius, range
34 |  Elemental solid volume, mode |  Electronegativity, max.
35 |  Atomic number, min. |  Space group number, max.
36 |  Space group number, max. |  Electronegativity, mode
37 |  Mendeleev number, avg. dev. |  Atomic weight, min.
38 |  0-norm |  Elemental solid volume, mean
39 |  Covalent radius, avg. dev. |  Column number, range
40 |  Number of valence electrons, mode |  Covalent radius, mode
41 |  Number of s valence electrons, mean |  Number of filled p valence orbitals, max.
42 |  Number of filled s valence orbitals, mean |  Number of valence electrons, mode
43 |  Atomic weight, min. |  Mendeleev number, avg. dev.
44 |  Melting temperature, avg. dev. |  Number of filled s valence orbitals, max.
45 |  Elemental solid volume, max. |  Elemental solid volume, max.
46 |  Electronegativity, max. |  Number of filled s valence orbitals, mode
47 |  Melting temperature, max. |  Electronegativity, avg. dev.
48 |  2-norm |  Space group number, mean
49 |  Mendeleev number, range |  Covalent radius, min.
50 |  Row number, min. |  0-norm
51 |  Number of unfilled valence orbitals, mode |  Row number, min.
52 |  Number of filled p valence orbitals, avg. dev. |  Melting temperature, max.
53 |  Space group number, avg. dev. |  Frac. d valence electrons
54 |  Melting temperature, range |  Row number, avg. dev.
55 |  Melting temperature, mode |  Elemental solid volume, mode
56 |  Number of filled d valence orbitals, max. |  Mendeleev number, range
57 |  Number of filled d valence orbitals, mean |  Row number, max.
58 |  Number of d valence electrons, mean |  Number of filled p valence orbitals, range
59 |  Row number, range |  Number of filled p valence orbitals, avg. dev.
60 |  Elemental solid volume, avg. dev. |  Covalent radius, avg. dev.
61 |  Number of filled p valence orbitals, max. |  Electronegativity, min.
62 |  Number of filled s valence orbitals, mode |  Number of unfilled valence orbitals, mode
63 |  3-norm |  Frac. s valence electrons
64 |  Row number, avg. dev. |  Row number, range
65 |  Electronegativity, min. |  Number of unfilled valence orbitals, max.
66 |  Elemental solid volume, range |  Elemental solid volume, range
67 |  Column number, min. |  Number of unfilled d valence orbitals, avg. dev.
68 |  Number of valence electrons, max. |  Number of filled d valence orbitals, avg. dev.
69 |  Number of filled d valence orbitals, avg. dev. |  Number of valence electrons, max.
70 |  10-norm |  Frac. p valence electrons
71 |  Number of valence electrons, avg. dev. |  Elemental solid volume, min.
72 |  Number of filled d valence orbitals, range |  Melting temperature, avg. dev.
73 |  Number of unfilled valence orbitals, max. |  Space group number, range
74 |  Number of filled p valence orbitals, range |  Melting temperature, mean
75 |  Space group number, range |  Number of filled p valence orbitals, mean
76 |  7-norm |  Number of p valence electrons, mean
77 |  Frac. d valence electrons |  Electronegativity, range
78 |  Space group number, mode |  Column number, min.
79 |  Number of unfilled p valence orbitals, avg. dev. |  Number of unfilled p valence orbitals, mean
80 |  Number of unfilled p valence orbitals, mean |  Number of d valence electrons, mean
81 |  Number of filled s valence orbitals, max. |  Number of filled d valence orbitals, mean
82 |  Elemental solid volume, min. |  Number of unfilled d valence orbitals, mean
83 |  5-norm |  Number of unfilled p valence orbitals, max.
84 |  Electronegativity, avg. dev. |  Space group number, mode
85 |  Mendeleev number, min. |  Number of unfilled valence orbitals, avg. dev.
86 |  Number of unfilled d valence orbitals, mean |  Mendeleev number, min.
87 |  Number of unfilled valence orbitals, avg. dev. |  Melting temperature, mode
88 |  Number of unfilled d valence orbitals, avg. dev. |  Melting temperature, range
89 |  Space group number, min. |  Number of valence electrons, avg. dev.
90 |  Number of valence electrons, range |  Number of valence electrons, range
91 |  Number of filled p valence orbitals, mean |  Number of filled d valence orbitals, range
92 |  Number of p valence electrons, mean |  Number of filled s valence orbitals, min.
93 |  Electronegativity, range |  Number of filled d valence orbitals, max.
94 |  Number of filled s valence orbitals, min. |  Number of unfilled p valence orbitals, range
95 |  Number of unfilled valence orbitals, range |  Number of unfilled s valence orbitals, avg. dev.
96 |  Melting temperature, min. |  Space group number, avg. dev.
97 |  Number of unfilled p valence orbitals, max. |  Number of unfilled valence orbitals, range
98 |  Frac. s valence electrons |  Number of unfilled d valence orbitals, max.
99 |  Number of unfilled p valence orbitals, range |  Number of filled p valence orbitals, mode
100 |  Number of unfilled d valence orbitals, max. |  MagpieData avg_dev GSvolume_pa
101 |  Number of filled p valence orbitals, mode |  Number of unfilled s valence orbitals, range
102 |  Number of valence electrons, min. |  Number of unfilled p valence orbitals, avg. dev.
103 |  Number of unfilled p valence orbitals, mode |  Number of valence electrons, min.
104 |  Number of unfilled d valence orbitals, range |  Number of unfilled p valence orbitals, mode
105 |  Transition metal fraction |  Number of filled s valence orbitals, range
106 |  Frac. p valence electrons |  Space group number, min.
107 |  Number of filled s valence orbitals, range |  Number of filled s valence orbitals, avg. dev.
108 |  Number of unfilled s valence orbitals, range |  Number of unfilled s valence orbitals, max.
109 |  Number of unfilled s valence orbitals, avg. dev. |  Melting temperature, min.
110 |  Number of filled f valence orbitals, max. |  Number of filled d valence orbitals, mode
111 |  Number of unfilled s valence orbitals, max. |  Frac. f valence electrons
112 |  Number of filled s valence orbitals, avg. dev. |  Transition metal fraction
113 |  Number of filled f valence orbitals, avg. dev. |  Number of unfilled s valence orbitals, mean
114 |  Number of filled d valence orbitals, mode |  Number of unfilled d valence orbitals, range
115 |  Number of f valence electrons, mean |  Elemental solid band gap, max.
116 |  Number of filled f valence orbitals, mean |  Number of filled f valence orbitals, avg. dev.
117 |  Number of unfilled s valence orbitals, mean |  Elemental solid band gap, range
118 |  Number of filled f valence orbitals, range |  Number of filled f valence orbitals, max.
119 |  Frac. f valence electrons |  Number of filled f valence orbitals, range
120 |  Number of unfilled valence orbitals, min. |  Number of unfilled d valence orbitals, mode
121 |  Elemental solid band gap, mean |  Elemental solid band gap, mean
122 |  Elemental solid band gap, avg. dev. |  Elemental solid band gap, avg. dev.
123 |  Elemental solid band gap, max. |  Number of unfilled valence orbitals, min.
124 |  Elemental solid band gap, range |  Number of f valence electrons, mean
125 |  Number of unfilled d valence orbitals, mode |  Number of filled f valence orbitals, mean
126 |  Elemental solid magnetic moment, avg. dev. |  Elemental solid magnetic moment, avg. dev.
127 |  Number of unfilled f valence orbitals, avg. dev. |  Elemental solid magnetic moment, mean
128 |  Number of unfilled f valence orbitals, mean |  Number of unfilled f valence orbitals, avg. dev.
129 |  Elemental solid magnetic moment, mean |  Number of filled d valence orbitals, min.
130 |  Number of filled d valence orbitals, min. |  Number of unfilled f valence orbitals, mean
131 |  Number of unfilled f valence orbitals, range |  Elemental solid band gap, mode
132 |  Number of unfilled f valence orbitals, max. |  Number of unfilled s valence orbitals, mode
133 |  Elemental solid band gap, mode |  Elemental solid magnetic moment, max.
134 |  Elemental solid magnetic moment, range |  Elemental solid magnetic moment, range
135 |  Elemental solid magnetic moment, max. |  Number of unfilled f valence orbitals, max.
136 |  Number of unfilled s valence orbitals, mode |  Number of unfilled f valence orbitals, range
137 |  Number of unfilled d valence orbitals, min. |  Number of unfilled d valence orbitals, min.
138 |  Number of filled f valence orbitals, mode |  Number of filled f valence orbitals, mode
139 |  Number of unfilled p valence orbitals, min. |  Number of filled p valence orbitals, min.
140 |  Number of filled p valence orbitals, min. |  Number of unfilled p valence orbitals, min.
141 |  Elemental solid magnetic moment, mode |  Elemental solid magnetic moment, mode
142 |  Number of filled f valence orbitals, min. |  Number of filled f valence orbitals, min.
143 |  Number of unfilled s valence orbitals, min. |  Number of unfilled s valence orbitals, min.
144 |  Number of unfilled f valence orbitals, mode |  Number of unfilled f valence orbitals, mode
145 |  Elemental solid band gap, min. |  Elemental solid band gap, min.
146 |  Elemental solid magnetic moment, min. |  Elemental solid magnetic moment, min.
147 |  Number of unfilled f valence orbitals, min. |  Number of unfilled f valence orbitals, min.
Table 4: (Continued)

###  VIII.4 Materials Project, Implicit Pressure

Table 5:  Top 100 materials in the Materials Project with highest $T_{c}$ predictions under unknown (potentially very high) pressure, dot product spreads of less than 51 K, and energies above their convex hulls of less than 0.030 eV/atom.  |  |  |  |  |
---|---|---|---|---|---
|  ID |  Composition |  Predicted $T_{c}$ (K) |  Dot product spread (K) |  Hull energy (eV/atom)
1 |  mp-505233 |  Cs2Sn(H2N)6 |  324 |  48 |  0.017
2 |  mp-1184037 |  CsH5N2 |  315 |  47 |  0.011
3 |  mp-1198227 |  CsH5N2 |  315 |  47 |  0.000
4 |  mp-643359 |  Rb2Sn(H2N)6 |  305 |  45 |  0.028
5 |  mp-643371 |  K2Sn(H2N)6 |  301 |  43 |  0.000
6 |  mp-1193046 |  CsMgAs(H6O5)2 |  297 |  35 |  0.000
7 |  mp-643158 |  PH8IN4 |  297 |  46 |  0.000
8 |  mp-974267 |  KH8N3 |  288 |  35 |  0.000
9 |  mp-721084 |  H7IN2 |  287 |  47 |  0.000
10 |  mp-1202629 |  CdH20N6OF2 |  286 |  45 |  0.000
11 |  mp-1204788 |  InH12(NF2)3 |  279 |  44 |  0.000
12 |  mp-738629 |  CsGaH24(SO10)2 |  277 |  46 |  0.000
13 |  mp-1198073 |  CsH4NF2 |  275 |  48 |  0.004
14 |  mp-767240 |  CsPH4(NO)2 |  274 |  47 |  0.000
15 |  mp-24113 |  SnH8(NF3)2 |  273 |  45 |  0.000
16 |  mp-1196740 |  Cs3H12N4F3 |  271 |  48 |  0.000
17 |  mp-863000 |  SnH6(NF2)2 |  270 |  45 |  0.000
18 |  mp-758953 |  P2H12BrN7 |  269 |  40 |  0.008
19 |  mp-642740 |  CsLi(H2N)2 |  269 |  45 |  0.000
20 |  mp-1194837 |  Ba3As2H34O25 |  268 |  34 |  0.024
21 |  mp-1197059 |  LiSn(H2N)3 |  266 |  44 |  0.000
22 |  mp-1198991 |  Rb3H12N5 |  265 |  40 |  0.000
23 |  mp-677248 |  Ca2AlH10IO8 |  265 |  37 |  0.000
24 |  mp-643902 |  SnH4(NF)2 |  262 |  46 |  0.000
25 |  mp-733457 |  CsNa2(H5O3)3 |  261 |  44 |  0.003
26 |  mp-733932 |  InH5(NF)2 |  260 |  45 |  0.000
27 |  mp-706544 |  NaGa(H2N)4 |  258 |  38 |  0.006
28 |  mp-28204 |  CsH7O4 |  257 |  46 |  0.000
29 |  mp-1179724 |  RbLi2(H2N)3 |  256 |  37 |  0.006
30 |  mp-722455 |  RbLi2(H2N)3 |  256 |  37 |  0.006
31 |  mp-643394 |  NaInH8(NF3)2 |  255 |  44 |  0.028
32 |  mp-866716 |  RbLiH12Se3N4 |  255 |  42 |  0.000
33 |  mp-849391 |  InH9(NCl)3 |  254 |  45 |  0.000
34 |  mp-28892 |  PH4N3 |  253 |  36 |  0.000
35 |  mp-7714 |  Li4UO5 |  251 |  48 |  0.000
36 |  mp-559463 |  RbMgAs(H6O5)2 |  250 |  31 |  0.000
37 |  mp-34381 |  H4IN |  248 |  50 |  0.006
38 |  mp-643062 |  H4IN |  248 |  50 |  0.000
39 |  mp-505786 |  CeH14Cl3O7 |  246 |  37 |  0.011
40 |  mp-1195832 |  SiH12(NF)4 |  246 |  41 |  0.012
41 |  mp-1204167 |  CeH14Cl3O7 |  246 |  37 |  0.014
42 |  mp-1198095 |  CsAlH24(SO10)2 |  245 |  35 |  0.000
43 |  mp-556009 |  MgTlAs(H6O5)2 |  244 |  41 |  0.000
44 |  mp-1192659 |  LiGa(H2N)4 |  243 |  37 |  0.002
45 |  mp-626263 |  Ba(H8O5)2 |  243 |  32 |  0.001
46 |  mp-626268 |  Ba(H8O5)2 |  243 |  32 |  0.000
47 |  mp-626264 |  Ba(H8O5)2 |  243 |  32 |  0.026
48 |  mp-626297 |  Ba(H8O5)2 |  243 |  32 |  0.028
49 |  mp-1353537 |  Ba(H8O5)2 |  243 |  32 |  0.001
50 |  mp-1194674 |  LiGa(H2N)4 |  243 |  37 |  0.000
51 |  mp-722774 |  Ba3As2H14S8O7 |  243 |  37 |  0.013
52 |  mp-1182486 |  Ba(H8O5)2 |  243 |  32 |  0.024
53 |  mp-1202018 |  CsH4NO |  242 |  48 |  0.000
54 |  mp-1201190 |  GaH9(NF)3 |  241 |  40 |  0.000
55 |  mp-510073 |  RbLi(H2N)2 |  241 |  38 |  0.000
56 |  mp-707956 |  KLi7(H2N)8 |  240 |  32 |  0.022
57 |  mp-773582 |  LiTe(H3N)4 |  240 |  37 |  0.000
58 |  mp-1202913 |  GaH15N5Cl3 |  240 |  40 |  0.000
59 |  mp-1202249 |  Na3Np(H7O6)2 |  240 |  44 |  0.016
60 |  mp-1195064 |  Sn2H22S6N4O3 |  239 |  45 |  0.000
61 |  mp-703559 |  KSrAs(H4O3)4 |  238 |  31 |  0.000
62 |  mp-1196329 |  K2AsH13O10 |  237 |  30 |  0.000
63 |  mp-1202324 |  P2H10SN6 |  237 |  36 |  0.004
64 |  mp-761178 |  CsMgP(H6O5)2 |  237 |  35 |  0.000
65 |  mp-541005 |  CsH5O3 |  236 |  47 |  0.000
66 |  mp-1194535 |  CaH16(IO4)2 |  236 |  37 |  0.010
67 |  mp-1195969 |  CaH14I2O7 |  236 |  37 |  0.012
68 |  mp-8609 |  Li6UO6 |  236 |  50 |  0.000
69 |  mp-28587 |  BaH8O5 |  236 |  30 |  0.000
70 |  mp-1196878 |  P2H10N6O |  236 |  36 |  0.015
71 |  mp-1212119 |  KCaAs(H4O3)4 |  236 |  32 |  0.005
72 |  mp-758356 |  P2H12N7Cl |  236 |  37 |  0.009
73 |  mp-24523 |  CsAlH24(SeO10)2 |  235 |  35 |  0.000
74 |  mp-634446 |  CsAl(H2N)4 |  235 |  31 |  0.000
75 |  mp-1195610 |  Ca2H26I4O13 |  234 |  37 |  0.003
76 |  mp-707454 |  Li(H3N)4 |  234 |  34 |  0.028
77 |  mp-760046 |  LiPH21S3N7 |  233 |  36 |  0.000
78 |  mp-1224176 |  KMg2As2H31O23 |  233 |  30 |  0.017
79 |  mp-781992 |  PH8N4Cl |  232 |  37 |  0.005
80 |  mp-1194478 |  CsP2H3N |  232 |  48 |  0.023
81 |  mp-758253 |  CaH12(IO6)2 |  232 |  39 |  0.024
82 |  mp-1226730 |  CdH6(NCl)2 |  230 |  47 |  0.000
83 |  mp-760697 |  GaH7(NF2)2 |  230 |  40 |  0.019
84 |  mp-722502 |  Li3P11(H3N)17 |  230 |  35 |  0.007
85 |  mp-1200428 |  KMgAs(H6O5)2 |  230 |  30 |  0.013
86 |  mp-865095 |  NaGaH8(NF3)2 |  229 |  39 |  0.008
87 |  mp-23675 |  H4BrN |  229 |  42 |  0.000
88 |  mp-36248 |  H4BrN |  229 |  42 |  0.001
89 |  mp-1224894 |  GaH6N2F3 |  229 |  40 |  0.000
90 |  mp-695316 |  Ca2AlH10BrO8 |  228 |  31 |  0.000
91 |  mp-1195558 |  Zn(H15N8)2 |  228 |  28 |  0.000
92 |  mp-1200443 |  CaSnP6(HO)12 |  228 |  39 |  0.013
93 |  mp-721171 |  CsMgH12(ClO2)3 |  228 |  35 |  0.000
94 |  mp-1227507 |  Ca2AlH10BrO8 |  228 |  31 |  0.001
95 |  mp-696275 |  SnH8(NCl3)2 |  227 |  47 |  0.007
96 |  mp-1195052 |  CaNp(HO)9 |  227 |  42 |  0.000
97 |  mp-23763 |  SnH8(NCl3)2 |  227 |  47 |  0.000
98 |  mp-707324 |  PH6SN3 |  226 |  36 |  0.000
99 |  mp-706979 |  PH6N3O |  226 |  36 |  0.007
100 |  mp-759242 |  InH10N2Cl5O |  226 |  46 |  0.011
Table 5: (Continued)

###  VIII.5 Materials Project, Implicit Pressure, Small Band Gaps

Table 6:  Top 100 materials in the Materials Project with the highest $T_{c}$ predictions under unknown (potentially very high) pressure, dot product spreads of less than 51 K, energies above their convex hulls of less than 0.030 eV/atom, and band gaps of less than 1.000 eV.  |  |  |  |  |  |
---|---|---|---|---|---|---
|  ID |  Composition |  Predicted $T_{c}$ (K) |  Dot product spread (K) |  Hull energy (eV/atom) |  Band gap (eV)
1 |  mp-866716 |  RbLiH12Se3N4 |  255 |  42 |  0.000 |  0.801
2 |  mp-505786 |  CeH14Cl3O7 |  246 |  37 |  0.011 |  0.023
3 |  mp-1204167 |  CeH14Cl3O7 |  246 |  37 |  0.014 |  0.328
4 |  mp-707454 |  Li(H3N)4 |  234 |  34 |  0.028 |  0.000
5 |  mp-1198097 |  Sr2SnH18(SeO3)4 |  212 |  34 |  0.000 |  0.819
6 |  mp-721650 |  CaH14I10O7 |  198 |  45 |  0.022 |  0.678
7 |  mp-570931 |  SrLi3MnN3 |  192 |  47 |  0.000 |  0.000
8 |  mp-1202248 |  ReH30Ru2(NCl)10 |  180 |  38 |  0.029 |  0.031
9 |  mp-643738 |  Eu(MgH3)2 |  176 |  34 |  0.000 |  0.000
10 |  mp-1181729 |  Eu2Mg3H10 |  174 |  35 |  0.020 |  0.442
11 |  mp-541365 |  LiEuH3 |  172 |  38 |  0.000 |  0.000
12 |  mp-697962 |  Eu6Mg7H26 |  169 |  35 |  0.014 |  0.427
13 |  mp-1245450 |  Ca8(MnN3)3 |  168 |  46 |  0.020 |  0.000
14 |  mp-697677 |  CeMg2H7 |  167 |  32 |  0.000 |  0.000
15 |  mp-643756 |  EuMgH4 |  164 |  36 |  0.000 |  0.171
16 |  mp-696588 |  PH3 |  158 |  36 |  0.029 |  0.000
17 |  mp-973064 |  LaH3 |  154 |  45 |  0.003 |  0.471
18 |  mp-1018144 |  LaH3 |  154 |  45 |  0.000 |  0.000
19 |  mp-569112 |  Li3CaMnN3 |  152 |  47 |  0.000 |  0.073
20 |  mp-754586 |  Li8BiO6 |  148 |  48 |  0.029 |  0.000
21 |  mp-1177487 |  Li47(CoO4)8 |  144 |  49 |  0.023 |  0.828
22 |  mp-543090 |  Li17(CoO4)3 |  140 |  48 |  0.019 |  0.000
23 |  mp-543091 |  Li16(CoO4)3 |  140 |  45 |  0.025 |  0.000
24 |  mp-1172980 |  Li17(CoO4)3 |  140 |  48 |  0.015 |  0.345
25 |  mp-1185524 |  Li95Mn16O64 |  138 |  31 |  0.013 |  0.000
26 |  mp-677406 |  CuH12N2(Cl2O)2 |  138 |  40 |  0.024 |  0.694
27 |  mp-604996 |  CuH12N2(Cl2O)2 |  138 |  40 |  0.014 |  0.655
28 |  mp-721415 |  CuH12N2(Cl2O)2 |  138 |  40 |  0.018 |  0.533
29 |  mp-989535 |  Cs2AlInH6 |  137 |  36 |  0.000 |  0.611
30 |  mp-769483 |  Li11(CoO4)2 |  135 |  44 |  0.024 |  0.289
31 |  mp-1188721 |  IN4 |  135 |  50 |  0.000 |  0.000
32 |  mp-699498 |  U2CuP2(HO)24 |  132 |  50 |  0.000 |  0.918
33 |  mp-1202929 |  AgH8C2S2N4Cl |  131 |  35 |  0.024 |  0.000
34 |  mp-1102496 |  EuH2 |  130 |  39 |  0.000 |  0.000
35 |  mp-1229294 |  AlZnH10O7F5 |  129 |  27 |  0.022 |  0.990
36 |  mp-1183713 |  CeH3 |  126 |  35 |  0.003 |  0.000
37 |  mp-698125 |  NpH16C4N8ClO6 |  126 |  37 |  0.010 |  0.429
38 |  mp-22950 |  CsNiCl3 |  125 |  45 |  0.000 |  0.805
39 |  mp-1245354 |  Li7CrN4 |  123 |  45 |  0.002 |  0.000
40 |  mp-1226890 |  Ce4H11 |  122 |  36 |  0.000 |  0.000
41 |  mp-695843 |  SbH7(Br2O)3 |  121 |  35 |  0.000 |  0.873
42 |  mp-1569254 |  Li2Ni2BiO6 |  120 |  35 |  0.024 |  0.831
43 |  mp-22552 |  Ni(AuF4)2 |  120 |  45 |  0.000 |  0.000
44 |  mp-1203875 |  Zn2Te15(H3N)8 |  120 |  50 |  0.000 |  0.619
45 |  mp-1409292 |  MgNiF6 |  119 |  23 |  0.000 |  0.000
46 |  mp-1206799 |  Ba(GaH)2 |  119 |  32 |  0.000 |  0.000
47 |  mp-1203248 |  TiH24C6I3(N2O)6 |  118 |  33 |  0.030 |  0.000
48 |  mp-569603 |  BaCa4(CuN2)2 |  117 |  37 |  0.000 |  0.219
49 |  mp-978854 |  Sr(GaH)2 |  117 |  30 |  0.000 |  0.000
50 |  mp-1200028 |  Al3PH29(SO14)2 |  116 |  26 |  0.029 |  0.062
51 |  mp-1104579 |  Ce2H5 |  115 |  37 |  0.012 |  0.000
52 |  mp-643570 |  NiH2SO5 |  115 |  50 |  0.000 |  0.000
53 |  mp-1229281 |  Ba6Ca15Cu18Hg3O43 |  114 |  48 |  0.026 |  0.000
54 |  mp-1185277 |  K8Li31Al8O32 |  113 |  26 |  0.015 |  0.000
55 |  mp-1229082 |  Ba6Ca12Cu15Hg3O37 |  113 |  50 |  0.025 |  0.000
56 |  mp-720912 |  KZnH4Br3O2 |  112 |  38 |  0.016 |  0.000
57 |  mp-570771 |  BaLi2(MgSi)2 |  112 |  42 |  0.000 |  0.000
58 |  mp-1189297 |  Cs3NiCl5 |  112 |  48 |  0.022 |  0.087
59 |  mp-15885 |  Li2US3 |  111 |  43 |  0.000 |  0.000
60 |  mp-561000 |  Cs4KLiFe2F12 |  111 |  38 |  0.000 |  0.000
61 |  mp-505569 |  CeH2 |  110 |  46 |  0.017 |  0.000
62 |  mp-1228579 |  Ba2Ca3Cu4HgO10 |  110 |  50 |  0.020 |  0.000
63 |  mp-14763 |  Ca3MnN3 |  108 |  50 |  0.000 |  0.000
64 |  mp-1226365 |  Cs2Cu3NiF10 |  107 |  38 |  0.008 |  0.605
65 |  mp-22601 |  Ba2Ca2Cu3HgO8 |  106 |  48 |  0.018 |  0.000
66 |  mp-989559 |  Cs2NaLiF6 |  106 |  48 |  0.000 |  0.000
67 |  mp-764501 |  Na8FeO6 |  105 |  49 |  0.019 |  0.017
68 |  mp-703531 |  CuSiH8(O2F3)2 |  105 |  50 |  0.004 |  0.640
69 |  mp-558211 |  Cs2NaCoF6 |  104 |  36 |  0.000 |  0.902
70 |  mp-1229126 |  Ba2Ca4TlCu5O13 |  104 |  49 |  0.025 |  0.000
71 |  mp-570185 |  Ba38Na58Li26N |  104 |  26 |  0.011 |  0.024
72 |  mp-755306 |  Li5Ni3(SnO5)2 |  103 |  30 |  0.012 |  0.066
73 |  mp-1289276 |  Li3Ni2SnO6 |  103 |  30 |  0.030 |  0.344
74 |  mp-1292604 |  Li3Ni2SnO6 |  103 |  30 |  0.029 |  0.357
75 |  mp-1312304 |  Li5Ni3(SnO5)2 |  103 |  30 |  0.011 |  0.619
76 |  mp-865625 |  Na2MgSn |  102 |  50 |  0.000 |  0.000
77 |  mp-29720 |  Li21Si5 |  101 |  31 |  0.000 |  0.000
78 |  mp-1228589 |  Ba2Ca3TlCu4O11 |  100 |  47 |  0.025 |  0.000
79 |  mp-5077 |  NaLi2Sb |  100 |  50 |  0.000 |  0.680
80 |  mp-14794 |  K6CoS4 |  100 |  27 |  0.028 |  0.438
81 |  mp-5515 |  Li7MnN4 |  100 |  43 |  0.000 |  0.656
82 |  mp-505213 |  Ba6Na16N |  99 |  26 |  0.019 |  0.000
83 |  mp-989568 |  Cs2NaMgF6 |  99 |  43 |  0.000 |  0.000
84 |  mp-569849 |  Li15Si4 |  99 |  47 |  0.000 |  0.000
85 |  mp-1222798 |  Li14MgSi4 |  99 |  48 |  0.000 |  0.099
86 |  mp-865964 |  Li2CaSn |  98 |  35 |  0.000 |  0.000
87 |  mp-1246000 |  LiMnN2 |  98 |  42 |  0.010 |  0.000
88 |  mp-1225961 |  CsScCuF6 |  98 |  43 |  0.013 |  0.000
89 |  mp-556086 |  Na5FeS4 |  97 |  49 |  0.005 |  0.797
90 |  mp-697560 |  CoPH6NO5 |  97 |  46 |  0.008 |  0.707
91 |  mp-1203051 |  K5Li2EuF10 |  97 |  34 |  0.013 |  0.133
92 |  mp-1184919 |  K3Na |  97 |  42 |  0.027 |  0.000
93 |  mp-1184893 |  K3Na |  97 |  42 |  0.027 |  0.000
94 |  mp-1184844 |  K3Na |  97 |  42 |  0.023 |  0.000
95 |  mp-865890 |  Li2CaIn |  96 |  37 |  0.000 |  0.000
96 |  mp-1229139 |  Ba10Ca5Cu10Hg5O31 |  96 |  49 |  0.028 |  0.000
97 |  mp-1226362 |  Cu6Te2Mo2H2Cl4O15 |  96 |  39 |  0.029 |  0.000
98 |  mp-849362 |  Na14Co2O9 |  96 |  47 |  0.009 |  0.892
99 |  mp-6879 |  Ba2CaCu2HgO6 |  96 |  49 |  0.018 |  0.000
100 |  mp-1227794 |  BaSrCa2Tl(CuO3)3 |  95 |  47 |  0.023 |  0.000
Table 6: (Continued)

###  VIII.6 Materials Project, Ambient Pressure

Table 7:  Top 100 materials in the Materials Project with the highest $T_{c}$ predictions using ML model trained on ambient pressure data only, dot product spreads of less than 42 K, and energies above their convex hulls of less than 0.030 eV/atom.  |  |  |  |  |
---|---|---|---|---|---
|  ID |  Composition |  Predicted $T_{c}$ (K) |  Dot product spread (K) |  Hull energy (eV/atom)
1 |  mp-1205028 |  H15IrBr3N5 |  189 |  33 |  0.000
2 |  mp-24461 |  H12OsN5Cl3O |  161 |  25 |  0.023
3 |  mp-1199051 |  B10H13I |  151 |  40 |  0.000
4 |  mp-1199255 |  B10H13I |  151 |  40 |  0.002
5 |  mp-30977 |  B5H6Br |  149 |  40 |  0.000
6 |  mp-1197561 |  B10H13Br |  143 |  39 |  0.000
7 |  mp-634446 |  CsAl(H2N)4 |  132 |  39 |  0.000
8 |  mp-1199374 |  TaH9N3F5 |  132 |  24 |  0.000
9 |  mp-1204398 |  CoMoH24N6ClO7 |  129 |  26 |  0.015
10 |  mp-767240 |  CsPH4(NO)2 |  129 |  37 |  0.000
11 |  mp-1202248 |  ReH30Ru2(NCl)10 |  127 |  24 |  0.029
12 |  mp-1198073 |  CsH4NF2 |  125 |  40 |  0.004
13 |  mp-1200630 |  B8H4PbO15 |  123 |  38 |  0.000
14 |  mp-642740 |  CsLi(H2N)2 |  123 |  39 |  0.000
15 |  mp-721697 |  CsNa2H4Cl3O2 |  122 |  37 |  0.020
16 |  mp-728329 |  MoH12Pd(NO)4 |  122 |  28 |  0.004
17 |  mp-733457 |  CsNa2(H5O3)3 |  121 |  39 |  0.003
18 |  mp-24135 |  AlH18Ru(NF)6 |  120 |  31 |  0.018
19 |  mp-17718 |  CsKNa2Li12(SiO4)4 |  120 |  37 |  0.000
20 |  mp-17125 |  CsNa3Li12(GeO4)4 |  119 |  35 |  0.001
21 |  mp-738629 |  CsGaH24(SO10)2 |  118 |  39 |  0.000
22 |  mp-570097 |  Sr(Li2P)2 |  117 |  36 |  0.000
23 |  mp-23835 |  HgH12C2(Br2N3)2 |  117 |  27 |  0.000
24 |  mp-15845 |  SrLi4N2 |  116 |  37 |  0.000
25 |  mp-643359 |  Rb2Sn(H2N)6 |  115 |  40 |  0.028
26 |  mp-1190754 |  Cd(BH4)2 |  115 |  36 |  0.000
27 |  mp-1191186 |  CsAlH4(OF2)2 |  114 |  31 |  0.000
28 |  mp-28204 |  CsH7O4 |  113 |  41 |  0.000
29 |  mp-722455 |  RbLi2(H2N)3 |  112 |  34 |  0.006
30 |  mp-1179724 |  RbLi2(H2N)3 |  112 |  34 |  0.006
31 |  mp-755322 |  BaNa6O4 |  112 |  40 |  0.018
32 |  mp-1179387 |  ReH12N3Cl4O3 |  111 |  24 |  0.001
33 |  mp-1193778 |  RbAl(H2N)4 |  111 |  38 |  0.000
34 |  mp-1251539 |  RbAl(H2N)4 |  111 |  38 |  0.002
35 |  mp-1202291 |  BiB4(HO3)3 |  109 |  41 |  0.008
36 |  mp-643371 |  K2Sn(H2N)6 |  109 |  39 |  0.000
37 |  mp-643905 |  Sr(H2N)2 |  108 |  39 |  0.000
38 |  mp-8611 |  Li8CeO6 |  108 |  41 |  0.002
39 |  mp-556177 |  Cs2Cu2Si8O19 |  108 |  40 |  0.000
40 |  mp-696969 |  Sr(H2N)2 |  108 |  39 |  0.001
41 |  mp-707501 |  CdH24C4(BrN2)6 |  108 |  21 |  0.002
42 |  mp-24523 |  CsAlH24(SeO10)2 |  107 |  30 |  0.000
43 |  mp-1212424 |  HgH12C2(I2N3)2 |  107 |  33 |  0.000
44 |  mp-1198273 |  SrP11H28N9 |  107 |  40 |  0.000
45 |  mp-604315 |  Zn(BH4)2 |  106 |  36 |  0.029
46 |  mp-1193046 |  CsMgAs(H6O5)2 |  106 |  39 |  0.000
47 |  mp-1191586 |  Zn(BH4)2 |  106 |  36 |  0.000
48 |  mp-1201403 |  ZnH13RuN5(Cl2O)2 |  106 |  34 |  0.018
49 |  mp-510073 |  RbLi(H2N)2 |  105 |  35 |  0.000
50 |  mp-1198095 |  CsAlH24(SO10)2 |  104 |  30 |  0.000
51 |  mp-696329 |  RbCa(H2N)3 |  104 |  39 |  0.000
52 |  mp-1016214 |  Bi2H18C3(IN)9 |  103 |  38 |  0.000
53 |  mp-1203928 |  NbH8N2OF5 |  103 |  27 |  0.000
54 |  mp-553901 |  H6RuN3Cl3O |  103 |  33 |  0.009
55 |  mp-974267 |  KH8N3 |  103 |  39 |  0.000
56 |  mp-695172 |  CsMg12Al25Si29O108 |  103 |  39 |  0.022
57 |  mp-695133 |  CsMg4Al9(SiO4)9 |  102 |  39 |  0.019
58 |  mp-8407 |  Li3LaP2 |  102 |  41 |  0.000
59 |  mp-574928 |  CsPH3O3F |  101 |  38 |  0.011
60 |  mp-1226440 |  CsAlSi5O12 |  101 |  40 |  0.015
61 |  mp-973335 |  Hg2H6CN3Cl5 |  101 |  33 |  0.001
62 |  mp-707900 |  PrH20S4N5O16 |  100 |  36 |  0.009
63 |  mp-707956 |  KLi7(H2N)8 |  100 |  32 |  0.022
64 |  mp-695891 |  CdH6C(BrN)3 |  100 |  26 |  0.000
65 |  mp-1205148 |  PrH20S4N5O16 |  100 |  36 |  0.005
66 |  mp-505791 |  ReH8(Br3N)2 |  100 |  37 |  0.000
67 |  mp-1204099 |  BiH14C4N8Cl5O2 |  99 |  21 |  0.001
68 |  mp-632724 |  ReH8(NCl3)2 |  99 |  27 |  0.000
69 |  mp-2646923 |  Lu(BH4)3 |  99 |  35 |  0.000
70 |  mp-2646940 |  Lu(BH4)3 |  99 |  35 |  0.003
71 |  mp-1220119 |  PrH18(BrO6)3 |  99 |  39 |  0.000
72 |  mp-1201394 |  K2Zn(H2N)4 |  99 |  28 |  0.004
73 |  mp-562920 |  CsAl(SiO3)2 |  99 |  41 |  0.000
74 |  mp-722979 |  K2Zn(H2N)4 |  99 |  28 |  0.000
75 |  mp-1195558 |  Zn(H15N8)2 |  98 |  27 |  0.000
76 |  mp-1215828 |  Zn8B4H3O15F |  98 |  41 |  0.005
77 |  mp-1211490 |  KMoH8(NF3)2 |  98 |  29 |  0.028
78 |  mp-1195877 |  B10H13Cl |  98 |  39 |  0.001
79 |  mp-1198991 |  Rb3H12N5 |  98 |  39 |  0.000
80 |  mp-569827 |  HgH8C2Br3N |  98 |  32 |  0.019
81 |  mp-676956 |  Zn8B4H3O15F |  98 |  41 |  0.017
82 |  mp-1212542 |  H9C3IN6O |  97 |  18 |  0.012
83 |  mp-866716 |  RbLiH12Se3N4 |  97 |  39 |  0.000
84 |  mp-1196579 |  Cs2CuSi5O12 |  97 |  41 |  0.006
85 |  mp-697558 |  SbH12C2N6F5 |  97 |  18 |  0.000
86 |  mp-761178 |  CsMgP(H6O5)2 |  97 |  37 |  0.000
87 |  mp-758775 |  CsMnH4(OF2)2 |  97 |  39 |  0.000
88 |  mp-1198673 |  SnH12C2(NCl)6 |  96 |  21 |  0.000
89 |  mp-2646967 |  La(BH4)3 |  96 |  36 |  0.000
90 |  mp-17240 |  RbNa3Li12(SiO4)4 |  96 |  22 |  0.000
91 |  mp-1199487 |  MoH8(S2N)2 |  96 |  29 |  0.000
92 |  mp-1196086 |  CaH24(BrN4)2 |  96 |  31 |  0.000
93 |  mp-761870 |  H7C3N6Cl |  95 |  18 |  0.000
94 |  mp-697905 |  LaMg2NiH7 |  95 |  38 |  0.005
95 |  mp-1199443 |  SnH6C(NCl)3 |  95 |  22 |  0.000
96 |  mp-721171 |  CsMgH12(ClO2)3 |  95 |  38 |  0.000
97 |  mp-554483 |  KCd3H8Cl7O4 |  95 |  39 |  0.014
98 |  mp-1201495 |  Cs2Al2Si3O10 |  94 |  41 |  0.011
99 |  mp-653631 |  BaNa4Cu3F12 |  94 |  40 |  0.014
100 |  mp-722866 |  SbH6CN3F4 |  94 |  20 |  0.000
Table 7: (Continued)

###  VIII.7 Materials Project, Ambient Pressure, Small Band Gaps

Table 8:  Top 100 materials in the Materials Project with the highest $T_{c}$ predictions using ML model trained on ambient pressure data only, dot product spreads of less than 42 K, and energies above their convex hulls of less than 0.030 ev/atom, and band gaps of less than 1.000 eV.  |  |  |  |  |  |
---|---|---|---|---|---|---
|  ID |  Composition |  Predicted $T_{c}$ (K) |  Dot product spread (K) |  Hull energy (eV/atom) |  Band gap (eV)
1 |  mp-1202248 |  ReH30Ru2(NCl)10 |  127 |  24 |  0.029 |  0.031
2 |  mp-24135 |  AlH18Ru(NF)6 |  120 |  31 |  0.018 |  0.000
3 |  mp-570097 |  Sr(Li2P)2 |  117 |  36 |  0.000 |  0.976
4 |  mp-15845 |  SrLi4N2 |  116 |  37 |  0.000 |  0.887
5 |  mp-556177 |  Cs2Cu2Si8O19 |  108 |  40 |  0.000 |  0.755
6 |  mp-8407 |  Li3LaP2 |  102 |  41 |  0.000 |  0.612
7 |  mp-505791 |  ReH8(Br3N)2 |  100 |  37 |  0.000 |  0.813
8 |  mp-866716 |  RbLiH12Se3N4 |  97 |  39 |  0.000 |  0.801
9 |  mp-1196579 |  Cs2CuSi5O12 |  97 |  41 |  0.006 |  0.180
10 |  mp-697905 |  LaMg2NiH7 |  95 |  38 |  0.005 |  0.951
11 |  mp-653631 |  BaNa4Cu3F12 |  94 |  40 |  0.014 |  0.160
12 |  mp-2646929 |  Ce(BH4)3 |  90 |  38 |  0.000 |  0.344
13 |  mp-1777 |  Li15Ge4 |  89 |  41 |  0.000 |  0.000
14 |  mp-18713 |  Ba2Li(CuO2)3 |  88 |  41 |  0.008 |  0.000
15 |  mp-1218762 |  Sr2Li7CuN4 |  88 |  27 |  0.000 |  0.568
16 |  mp-28994 |  KLi2As |  87 |  34 |  0.000 |  0.688
17 |  mp-1226362 |  Cu6Te2Mo2H2Cl4O15 |  87 |  37 |  0.029 |  0.000
18 |  mp-29630 |  Li7Ge2 |  87 |  41 |  0.011 |  0.000
19 |  mp-865964 |  Li2CaSn |  86 |  35 |  0.000 |  0.000
20 |  mp-736 |  Li3P |  85 |  37 |  0.000 |  0.700
21 |  mp-867342 |  Li3Ge |  84 |  39 |  0.009 |  0.000
22 |  mp-638252 |  Cs2Cu3(P2O7)2 |  84 |  40 |  0.000 |  0.755
23 |  mp-505089 |  Li4TbF8 |  83 |  29 |  0.012 |  0.000
24 |  mp-2251 |  Li3N |  83 |  36 |  0.000 |  0.984
25 |  mp-849362 |  Na14Co2O9 |  83 |  34 |  0.009 |  0.892
26 |  mp-6127 |  BaCu(Si2O5)2 |  82 |  40 |  0.013 |  0.855
27 |  mp-1203248 |  TiH24C6I3(N2O)6 |  82 |  19 |  0.030 |  0.000
28 |  mp-6496 |  Ba2Na(CuO2)3 |  82 |  41 |  0.000 |  0.000
29 |  mp-759174 |  Na6CoO4 |  82 |  33 |  0.019 |  0.790
30 |  mp-14885 |  Rb2Cu2Si8O19 |  81 |  39 |  0.002 |  0.659
31 |  mp-757 |  Li3As |  81 |  40 |  0.000 |  0.636
32 |  mp-27454 |  BaLi2Si |  81 |  31 |  0.000 |  0.000
33 |  mp-1185028 |  Li2CaAl |  81 |  19 |  0.020 |  0.000
34 |  mp-1195515 |  K2U(Si2O5)3 |  81 |  41 |  0.008 |  0.016
35 |  mp-867171 |  SrLi2Sn |  81 |  38 |  0.000 |  0.000
36 |  mp-1194845 |  K2U(Si2O5)3 |  81 |  41 |  0.009 |  0.116
37 |  mp-865890 |  Li2CaIn |  81 |  35 |  0.000 |  0.000
38 |  mp-1228881 |  CsMnCuF6 |  80 |  34 |  0.009 |  0.000
39 |  mp-20067 |  Na5OsO6 |  80 |  32 |  0.000 |  0.000
40 |  mp-505786 |  CeH14Cl3O7 |  79 |  41 |  0.011 |  0.023
41 |  mp-8406 |  Li3NdAs2 |  79 |  38 |  0.000 |  0.036
42 |  mp-1204167 |  CeH14Cl3O7 |  79 |  41 |  0.014 |  0.328
43 |  mp-1194910 |  Mn(BH4)2 |  78 |  33 |  0.013 |  0.000
44 |  mp-698125 |  NpH16C4N8ClO6 |  78 |  20 |  0.010 |  0.429
45 |  mp-1038816 |  CeMg |  78 |  41 |  0.016 |  0.000
46 |  mp-865986 |  Li2CaGe |  77 |  22 |  0.003 |  0.101
47 |  mp-1182662 |  BaCu(SiO3)2 |  77 |  41 |  0.022 |  0.613
48 |  mp-555332 |  BaCu(SiO3)2 |  77 |  41 |  0.021 |  0.000
49 |  mp-1197716 |  BaCu(SiO3)2 |  77 |  41 |  0.011 |  0.640
50 |  mp-1200513 |  BaCu2Ge3(HO5)2 |  77 |  41 |  0.021 |  0.579
51 |  mp-676620 |  Li27As10 |  77 |  39 |  0.025 |  0.000
52 |  mp-1198097 |  Sr2SnH18(SeO3)4 |  77 |  38 |  0.000 |  0.819
53 |  mp-764501 |  Na8FeO6 |  76 |  34 |  0.019 |  0.017
54 |  mp-1598 |  Na3P |  76 |  38 |  0.000 |  0.403
55 |  mp-1209376 |  RbAlCuF6 |  76 |  40 |  0.000 |  0.000
56 |  mp-19026 |  Na4FeO3 |  76 |  32 |  0.015 |  0.763
57 |  mp-1226046 |  CsInCu(PO3)6 |  76 |  40 |  0.000 |  0.590
58 |  mp-1196302 |  MnTe4(H3N)6 |  76 |  41 |  0.002 |  0.142
59 |  mp-764233 |  Na8CoO6 |  76 |  34 |  0.000 |  0.951
60 |  mp-5077 |  NaLi2Sb |  76 |  36 |  0.000 |  0.680
61 |  mp-1187184 |  SrLi2In |  76 |  40 |  0.025 |  0.000
62 |  mp-8469 |  Li8IrO6 |  76 |  29 |  0.005 |  0.000
63 |  mp-3988 |  BaCu3O4 |  75 |  41 |  0.014 |  0.000
64 |  mp-27932 |  Li9Ge4 |  74 |  37 |  0.003 |  0.000
65 |  mp-2362521 |  Li9Ge4 |  74 |  37 |  0.000 |  0.000
66 |  mp-1203793 |  CeB4H2ClO8 |  74 |  41 |  0.000 |  0.450
67 |  mp-697156 |  BaCu2P2H2O9 |  74 |  40 |  0.009 |  0.008
68 |  mp-1202929 |  AgH8C2S2N4Cl |  74 |  19 |  0.024 |  0.000
69 |  mp-1227989 |  BaSrAl20(CuO17)2 |  74 |  40 |  0.017 |  0.000
70 |  mp-1194032 |  Pu(SO6)2 |  73 |  39 |  0.019 |  0.000
71 |  mp-6960 |  Rb2SiAs2 |  73 |  26 |  0.000 |  0.955
72 |  mp-1196467 |  Rb2Cu3P4(O3F)4 |  71 |  35 |  0.000 |  0.527
73 |  mp-1228431 |  Ba2Cu3BrClO4 |  71 |  39 |  0.002 |  0.000
74 |  mp-570771 |  BaLi2(MgSi)2 |  71 |  32 |  0.000 |  0.000
75 |  mp-1226351 |  CsMn2(PO3)6 |  71 |  41 |  0.000 |  0.000
76 |  mp-1225984 |  CsMn2(PO3)6 |  71 |  41 |  0.002 |  0.000
77 |  mp-531553 |  Li5GeN3 |  71 |  33 |  0.000 |  0.001
78 |  mp-654359 |  BaCu2(PO4)2 |  71 |  41 |  0.012 |  0.326
79 |  mp-1213081 |  CsMn2(PO3)6 |  71 |  41 |  0.003 |  0.000
80 |  mp-1223676 |  K2Na4Co2O5 |  70 |  37 |  0.000 |  0.076
81 |  mp-1207213 |  K2LiCuF6 |  70 |  36 |  0.000 |  0.925
82 |  mp-1147739 |  Ba(CuO)4 |  70 |  39 |  0.026 |  0.000
83 |  mp-1018766 |  Li3LaAs2 |  70 |  40 |  0.000 |  0.502
84 |  mp-756356 |  Li8FeO6 |  70 |  30 |  0.029 |  0.000
85 |  mp-763305 |  Li8FeO6 |  70 |  30 |  0.025 |  0.000
86 |  mp-504863 |  KNa2FeO3 |  69 |  37 |  0.010 |  0.502
87 |  mp-653840 |  KCu7TeS5ClO24 |  69 |  40 |  0.000 |  0.000
88 |  mp-1186135 |  NaLi2Sn |  69 |  38 |  0.025 |  0.000
89 |  mp-1177487 |  Li47(CoO4)8 |  69 |  39 |  0.023 |  0.828
90 |  mp-542450 |  Cs3Mn4(PO3)12 |  69 |  41 |  0.000 |  0.000
91 |  mp-1218363 |  SrCa(CuO2)2 |  68 |  36 |  0.027 |  0.000
92 |  mp-23129 |  Ba2Cu3(BrO2)2 |  68 |  39 |  0.000 |  0.000
93 |  mp-1178147 |  K3TlPCO7 |  68 |  33 |  0.021 |  0.000
94 |  mp-707454 |  Li(H3N)4 |  68 |  23 |  0.028 |  0.000
95 |  mp-1218417 |  SrCa(CuO2)2 |  68 |  36 |  0.026 |  0.000
96 |  mp-976023 |  Li3Ga |  68 |  38 |  0.021 |  0.000
97 |  mp-1147551 |  KCuTeO6 |  68 |  41 |  0.001 |  0.780
98 |  mp-551283 |  Na2Cu2TeO6 |  68 |  38 |  0.005 |  0.066
99 |  mp-18124 |  KAlCuF6 |  67 |  39 |  0.003 |  0.081
100 |  mp-1196923 |  Li2TbF6 |  67 |  31 |  0.015 |  0.000
Table 8: (Continued)