2022

[1]Chris Wolverton

[1]Department of Materials Science and Engineering, Northwestern University,
2220 Campus Drive, Evanston, 60208, IL, USA

# Identification of High-Dielectric Constant Compounds from Statistical Design

Abhijith Gopakumar<EMAIL_ADDRESS>Koushik Pal
<EMAIL_ADDRESS><EMAIL_ADDRESS>*

###### Abstract

The discovery of high-dielectric materials is crucial to increasing the
efficiency of electronic devices and batteries. Here, we report three
previously unexplored materials with very high dielectric constants (69 $<$
$\epsilon$ $<$ 101) and large band gaps (2.9$<$ $E_{\text{g}}$(eV) $<$ 5.5)
obtained by screening materials databases using statistical optimization
algorithms aided by artificial neural networks (ANN). Two of these new
dielectrics are mixed-anion compounds (Eu5SiCl6O4 and HoClO), and are shown to
be thermodynamically stable against common semiconductors via phase-diagram
analysis. We also uncovered four other materials with relatively large
dielectric constants (20$<$$\epsilon$$<$40) and band gaps
(2.3$<$$E_{\text{g}}$(eV)$<$2.7). While the ANN training data is obtained from
Materials Project, the search-space consists of materials from Open Quantum
Materials Database (OQMD) - demonstrating a successful implementation of
cross-database materials design. Overall, we report dielectric properties of
17 materials calculated using ab-initio calculations, that were selected in
our design workflow. The dielectric materials with high dielectric properties
predicted in this work open up further experimental research opportunities.

##  1 Introduction

Dielectric materials are among the most vital components for microelectronic
device manufacturing. They are used in memory devices, capacitor based energy
storage, field effect transistors, etcortiz2009high ; wang2018high ;
kingon2000alternative . The dielectric constant (denoted here as $\epsilon$),
more commonly referred to as the relative permittivity, is the factor by which
the electric field strength decreases inside a material compared to the vacuum
when it is placed near a finite electric charge. The $\epsilon$ values of
commonly used dielectric materials range between 20 and 30ortiz2009high ;
shevlin2005ab ; delugas2007cation  \- for example, Ta2O5 ($\epsilon$ $\sim$
23-27, $E_{\text{g}}$=4.2 eV)ortiz2009high ; iino2003organic ; wang2018high ;
kukli1995properties  and TiO2 ($\epsilon$=27, $E_{\text{g}}$=3.5
eV)ortiz2009high ; ramajothi2008performance ; wang2018high . There is a high
demand to find novel materials with high $\epsilon$ to increase the device
performance and reliability. Typically, $\epsilon$ and $E_{\text{g}}$ are
inversely relatedwang2018high ; lee2018high  in a compound. As a result,
although several materials are reported to have an even larger $\epsilon$
values, they often have a small $E_{\text{g}}$wilk2001high ; lee2018high ;
petretto2018high ; petousis2017high , making the dielectric vulnerable to
leakage currents under exposure to large electric fieldsortiz2009high ;
wang2018high . Therefore, compounds with high $\epsilon$ and large band gaps
are preferred while designing charge storage applications and microelectronic
devices.

One of the methods to find high-$\epsilon$ compounds is to calculate the
dielectric constants and band gaps of a large number of compounds that are
available in large materials databases such as the Open Quantum Materials
Database (OQMD)saal2013materials ; kirklin2015open , Materials Project
(MP)Jain2013 , etc using ab-initio methods such as density functional theory
(DFT). However, since the accurate calculation of dielectric properties using
density functional perturbation theorygiannozzi2005density  (DFPT) is
computationally very expensive, it would be practically unfeasible to estimate
the dielectric constants of tens of thousands of materials available in those
database using high-throughput methods. In this work, we employ an advanced
screening strategy to identify compounds with better dielectric properties.
Thus, the goal of this work is to find dielectric materials with large values
for both $\epsilon$ and $E_{\text{g}}$ by screening materials databases but at
the expense of conducting as few DFPT calculations as possible. To accomplish
this task, we have employed a materials design strategy comprised of
statistical optimization models and DFPT calculations on a small set of
compounds. While our training set consists of a small amount of data
(dielectric constants) from the MP, the search-space contains a vast set of
compounds available in the OQMD.

Several online data repositories exist today that are dedicated to hosting
large sets of open-sourced inorganic crystal structure data generated from
high throughput (HT) DFT calculations such as the MPJain2013 ,
OQMDsaal2013materials ; kirklin2015open , and AFLOWLibcurtarolo2012aflowlib
among othersdraxl2019nomad ; choudhary2020high . The design and discovery of
novel materials using statistical modelling has become an active research area
pyzer2015learning ; saal2020machine ; park2020developing  in recent times,
largely attributed to the availability of such HT datasets. Recently, multiple
studies have reported HT-generation of dielectric data and subsequent
analysislee2018high ; umeda2019prediction ; qu2020high . For example, Morita
et al.reportedmorita2020modeling  machine learning modeling of data from MP
petousis2017high ; petretto2018high ; Jain2013  to assess the reliability of
the theoretical models currently available to describe the dielectric
properties of crystals.

In this work, we use the MP-dataset of 1864 dielectric tensorspetretto2018high
; petousis2017high  to train statistical models and subsequently identify
dielectrics from the set of stable materials in the OQMD. Thus the MP-data
forms the training-data and the set of materials from OQMD forms the search-
space for the materials design. This work is a successful demonstration of the
scenario where the data obtained from the multiple sources can be utilized to
discover new compounds. The negligible difference found between the
representation vectors, which are also called as feature vectors in machine
learning, generated for equivalent materials in MP and OQMD made the cross-
database design possible in this work. Overall, we conducted three design
cycles which required us to perform dielectric calculations for just 17
materials using DFPT. We report the dielectric constant values of all the 17
materials among which three of them (HoClO, Eu5SiCl6O4, and Tl3PbBr5) have
very large $\epsilon$ (69 $<\epsilon<$101) and $E_{\text{g}}$ (2.9 eV
$<E_{\text{g}}<$5.5 eV) values making them part of the Pareto-front of the
known data, and four other materials (Sr2LuBiO6, Bi5IO7, Bi3ClO4, and Bi3BrO4)
have moderately large $\epsilon$ (20 $<\epsilon<$40) and $E_{\text{g}}$ (2.3
eV $<E_{\text{g}}<$2.7 eV) values.

##  2 Results

###  2.1 Materials design strategy

Our objective is to find large band gap materials with optimal dielectric
constants. Since, the dielectric tensor of a compound has nine components, the
optimization of all nine components leads to a nine-objective optimization
problem which is difficult to solve with training-data of size $\sim$ 2000\.
Thus, we specifically optimize the largest eigenvalue of the dielectric
tensor, referred to from here onward as $\epsilon$, via statistical modelling
through the materials design workflow, as depicted in Figure 1. The workflow
is similar to the strategies that have been previously reported in
literaturebalachandran2016adaptive ; gopakumar2018multi , where each design
cycle consists of three steps - data processing, statistical modeling, and ab-
initio DFPT calculations. The largest eigenvalue of the total dielectric
tensor is chosen as the property to be optimized because that is the highest
possible dielectric behavior from a single crystal when it is aligned
perfectly along the corresponding direction between two metallic plates. The
total dielectric tensor is calculated as the sum of ionic and electronic
dielectric tensors. The good agreement between dielectric tensor eigenvalues
obtained from MP’s DFPT high throughput framework and experimentally measured
dielectric constant values was reported by Petousis et
al.petousis2016benchmarking . We preferred the largest eigenvalue over the
average of eigenvalues because the latter value may severely underestimate the
highest possible dielectric behavior from a single-crystal (Supplementary
Figure 1), even though it is a popular choice to estimate the polycrystalline
dielectric constantpetousis2016benchmarking ; petousis2017high . The new data
produced from DFPT calculations at the end of each cycle is fed into the next
design cycle. In the first step, we collected the relevant data from the MP
database (training-data) and OQMD (search-space). All materials in the
training-data have a known value for $\epsilon$ and $E_{\text{g}}$, while the
materials in search-space have known values of $E_{\text{g}}$ but their
$\epsilon$ values are unknown. In the second step, Modeling, we created an
ensemble of artificial neural network (ANN)jain1996artificial  models, fit on
the training-data, which learn to predict the $\epsilon$ value of materials
when their crystal structures and $E_{\text{g}}$ values are known. Using this
ANN-ensemble, we predicted the $\epsilon$ of each material in the search-
space. Since the prediction was done from an ensemble, the results were a
distribution of $\epsilon$ values for each material, contrary to the usage of
a single ANN-model where a single prediction-value is obtained. The trained
ANN ensemble was used to predict the $\epsilon$-distributions of 11,102 stable
non-metallic materials in the search-space, obtained from the OQMD.

Further, the predicted distribution of $\epsilon$ was input into the Efficient
Global Optimization (EGO)balachandran2016adaptive  algorithm. EGO takes into
account of the distribution’s mean and standard-deviation to rank the
materials in search-space based on their potential to increase the chances of
finding high-$\epsilon$ materials in this workflow within as few design cycles
as possible. In this work, the optimization in dielectrics refers to the
identification of dielectrics with large $\epsilon$ values. The reason for
employing an EGO-algorithm to explore the search-space is to account for the
uncertainty in ANN model predictions when the available training-data may not
have sampled the material-space uniformly. The advantages of EGO-based
optimization in materials design was first reported and benchmarked by
Balachandran et al.balachandran2016adaptive ; balachandran2017learning ;
balachandran2018predictions . In this work, we used the EGO algorithm to
select the best candidates that are either predicted to have high $\epsilon$
value or have a large uncertainty in their ANN-ensemble predictions. Materials
that belong to the latter category are from the regions of materials yet to be
sampled by the training-data. The DFPT characterization of such materials is
expected to increase the reliability of ANN-ensemble predictions after each
design cycle and eventually lead to better optimization of dielectrics during
the course of this work.

The metric that is used to rank the materials is called Expected Improvement,
or $E(I)$. More details on how the $E(I)$ is calculated, is provided in the
Methods section. A few (5-6) materials were selected in this step with highest
values of $E(I)$ and carried onto the next step - DFPT calculations. In this
final step, the dielectric tensor of the selected materials were calculated
using DFPT calculations. If DFPT results show that any of the materials have a
high value of $E_{\text{g}}$ and $\epsilon$, we stop the design workflow at
that point. Otherwise, a new design cycle is started after transferring the
newly computed $\epsilon$ values and the corresponding materials to the
training-data from the search-space. With an increased size of training-data,
the ANN-ensemble is expected to have less uncertainty in $\epsilon$
predictions in the new design cycle. The design cycle was repeated with
feedback three times in total in this work until three materials with very
large values for $E_{\text{g}}$ and $\epsilon$ were found.

###  2.2 Data

A dataset containing information about crystal structures, chemical
compositions, band-gap energy values and dielectric-tensors of 1864 stable
materials was obtained from the MPpetousis2017high ; petretto2018high ;
Jain2013  data repository. This dataset was used to generate the training-
data. The target-property, $\epsilon$, was obtained for each material in this
database from its calculated dielectric-tensor. Another dataset consisting of
11,102 stable, non-metallic materials containing information about crystal
structures, chemical compositions, band-gap energy values was obtained from
OQMDsaal2013materials ; kirklin2015open . This OQMD dataset was used to
generate the search-space in which the search to find dielectrics was
conducted. The dielectric tensor data of all crystals included in the search-
space were unknown at the beginning of this work.

The materials need to be represented as vectors of uniform length in order to
be input into a statistical model. We generated the material representations
using the Magpie ward2016general  crystal property generator tool. Magpie
generates a set of physical features (such as the mean electronegativity of
constituent atoms, average coordination number inside unit cell, etc) from a
given chemical composition and crystal structure. Within Magpie, the crystal’s
structure-related features are generated by building Voronoi tessellations
inside the crystal and finding nearest neighbors of each individual
atomward2017including . Magpie generated 271 input features that include 145
composition-based, and 126 structure-based features to represent each
material. In addition to these, the material’s DFT $E_{\text{g}}$ value was
also added as an extra feature to the representation-vector since it is
already known for all materials in both MP and OQMD datasets. Addition of
$E_{\text{g}}$ increased the size of representation-vector to 272, which was
generated for each material in training-data and search-space. The input
feature-vector size was further reduced to 100 using the widely-used feature
reduction techniques such as principal component analysis and model-based
selection, implemented in the Scikit-learn python libraryscikit-learn . The
set of material representation-vectors of training-data and the search-space,
in addition to the target values associated with the training-data, completes
the first step of materials design as depicted in Figure 1. The size of the
training-dataset increases after each design cycle as a result of conducting
DFPT calculations on new materials from the search-space.

Statistical modelling utilizing data from multiple computational material
databases is prone to error arising from the differences in the DFT parameters
used at each database’s high-throughput calculation strategy. Here, we have
investigated the difference in Magpie-generated features for equivalent
materials in OQMD and MP, cross referenced based on their associated Inorganic
Crystal Structure Databasebelsky2002new  (ICSD) Collection Codes. In total,
1717 out of 1864 materials in training data had an ICSD Collection Code
associated with them. The crystal structures from OQMD corresponding to all
the 1717 ICSD materials were obtained, and their Magpie-generated features
were compared against that of the structures obtained from MP as a part of the
training data. The results, as plotted in Figure 2a, shows negligible ($\leq
2\%$) relative difference in 263 out of a total of 271 Magpie features, while
the other eight features have low relative differences ($\leq 7\%$). All 145
composition-based features are computed to be identical across the databases,
as expected. The finite difference in some of the structure-based features
originate because of the difference in accuracy of crystal structural
minimization across databases. Band gap, which joins the Magpie features to
form the final material representation-vector, was also compared between OQMD
and MP for the 1717 equivalent materials, as shown in Figure 2b. Band gap
values showed a mean and median absolute deviation of 0.1 eV and 0.0 eV
respectively, pointing toward a negligible difference between the calculations
of band gap for materials included in the training data across OQMD and MP.
Overall, the materials representation-vector considered in this design is
generated in a cross-comparable manner across OQMD and MP structures with very
low errors.

The $\epsilon$ values in the training-data obtained from MP are predominantly
concentrated in the range of 0 to 25, making it difficult to model the data
reliably for materials with large $\epsilon$ due to a possible bias toward
smaller values. Less than 5% of the materials in the training-data have
$\epsilon$ $>$ 50\. The median of $\epsilon$ values in MP-dataset is 12.2
while the mean and standard deviation are 20.2 and 42.8 respectively. The
distribution of $\epsilon$ in training-data is shown in Supplementary Figure
2. The large spread of $\epsilon$ values is decreased upon a log-scale
transformation, as shown in Figure 3a. A smaller spread of target values helps
stabilize the machine learning model during the training by reducing the
probability of excessive changes in internal parameters, such as the weights
in an ANN. We also analyzed the correlation between $\epsilon$ and
$E_{\text{g}}$ values for the materials in the training data, and it is given
in Supplementary Figure 3.

The original dataset downloaded from MP listed BeO (MP ID: mp-1794) as having
large ab-initio computed values for $\epsilon$(=312) and $E_{\text{g}}$(=8.2
eV). This large value of $\epsilon$ is possibly caused by the improper
relaxation of the primitive cell of BeO in MP that leads to a large volume
change. Hence, the succeeding calculations on this compound such as DFPT may
be incorrect. We conducted a separate DFT cell-relaxation and DFPT calculation
for BeO using VASP starting with the MP’s initial structure and find that the
computed $\epsilon$ value for the correctly relaxed structure is 4 - well in
agreement with the previously reported values in literaturegroh2009first .
This compound was removed from the training-data before proceeding further. We
looked up other materials in training-data with very high $\epsilon$ and
smaller $E_{\text{g}}$ individually and confirmed that they did not have a
large cell-volume change upon relaxation in MP.

###  2.3 Statistical modeling

The predictions from trained machine learning models, such as ANNs, are often
prone to errors arising from the insufficient sampling of material space by
training-data. We needed to quantify the uncertainty associated with the
$\epsilon$ value predictions even though the available ANN algorithms
explicitly do not provide that value from a single ANN model. So we created an
ensemble of ANNs, each of which was trained on a randomly chosen subset of the
training-data, and has different architectures and internal parameters. An
ANN-ensemble containing 2000 independent ANN models was created and trained at
each design cycle. Each ANN in the ensemble predicted a single $\epsilon$
value upon inputting a material-representation vector, resulting in a
distribution of 2000 predicted $\epsilon$ values for each material in the
search-space. The standard deviation of each of the predicted
$\epsilon$-distribution was defined as the uncertainty of ANN modeling for the
corresponding material.

Further, a statistical single-objective optimization algorithm, called EGO
xue2016accelerated ; balachandran2016adaptive ; jones1998efficient ;
solomou2018multi ; talapatra2018autonomous , was used in this work to evaluate
the $\epsilon$-distribution and quantify a measure of probable optimization
associated with each material in the search-space. EGO is not a method to
model the data and predict $\epsilon$. Instead, EGO is an algorithm to select
the best candidates from a given search-space, based on their
$\epsilon$-distributions predicted by the ANN-ensemble, in order to discover
as many high-$\epsilon$ materials from as few design cycles as possible. Here,
the desired optimization is the maximization of $\epsilon$ among all the
materials in search-space. The quantified measure of predicted optimization in
EGO is called expected improvement, denoted as $E(I)$. Conceptually, the
$E(I)$ of a material in search-space is the quantified probability with which
a DFPT calculation of $\epsilon$ for that material will lead to a
identification of high-$\epsilon$ material in the design workflow within as
few design cycles as possible. Figure 3a shows the results from an ANN model
validation as a part of model training during the second design cycle. The
values of $E(I)$ computed for the same validation data split from the
training-data, is shown in Figure 3b. A simplified illustration of $E(I)$ with
the help an example is given below.

###  2.4 Example Illustration of E(I)

Suppose the predicted $\epsilon$-distribution belonging to a material $M_{1}$
in the search-space has a large standard deviation. Then it is highly probable
that the material $M_{1}$ belongs to a part of material representation-vector
space which was not sampled very well in the training set. Computing the
$\epsilon$ of $M_{1}$ using DFPT and feeding back that information to the
training-data will lead to better ANN modeling in the subsequent design
cycles. Thus, $M_{1}$ will have a large value of $E(I)$. Now consider another
material $M_{2}$ in search-space with a large mean and a small standard
deviation for its predicted $\epsilon$-distribution. The material $M_{2}$
belongs to a part of the material representation-vector space that was
sufficiently sampled by the training-data. So it is highly probable that
$M_{2}$ will turn out to be a high-$\epsilon$ material upon DFPT calculations.
Because of that, $M_{2}$ will also have a large value of $E(I)$.

In EGO, the calculation of $E(I)$ for a general optimization problem proceeds
as follows (also shown in Figure 4):
Let $Y$ be the target property to be maximized and $\varphi(Y)$ be the
predicted distribution of $Y$ for a given search-space material. The value,
$\varphi(Y=y)$ is the probability when the value of $Y$ is $y$. The largest
value of target property in the training-data is denoted as $y_{t}^{max}$. The
EGO algorithm, as formulated by Jones et al.jones1998efficient , computes the
expected improvement, $E(I)$, as

$E(I)=\int_{y_{t}^{max}}^{\infty}(y-y_{t}^{max})\;\varphi(Y=y)\;dy$ (1)

As mentioned in Balachandran et al.balachandran2016adaptive , if the predicted
distribution is approximated as a normal (i.e., Gaussian) distribution with a
mean $\mu$ and a standard deviation $\sigma$, the above equation can be re-
written as,

$E(I)=\sigma[\phi(z)+z\Phi(z)]$ (2)

where, $z=\frac{\mu-y_{t}^{max}}{\sigma}$, $\phi$ is the probability density
function, and $\Phi$ is the cumulative distribution functionjones1998efficient
of the normal distribution, $\varphi(Y)$.

For dielectric design, $Y$ is the dielectric constant ($\epsilon$) of a
candidate material, and $y_{t}^{max}$ is the highest value of $\epsilon$ in
the training-data obtained from DFPT calculations. In the MP-dataset, the
largest $\epsilon$ value is for TiO2 with $\epsilon$=988 and
$E_{\text{g}}$=1.8 eV. But our goal in this work is to find materials with
large $\epsilon$’s, not necessarily higher than 988 as long as the
$E_{\text{g}}$’s are greater than 1.8 eV. Thus the $y_{t}^{max}$ in this work
was set at 100.0 for all design cycles, instead of setting it at 988.0, to
consider the search-space materials whose $\epsilon$ values are predicted to
be sufficiently high. The $\varphi(Y)$ is approximated to be a normal
distribution with the same mean, $\mu$, and standard deviation, $\sigma$, as
that of the original $\epsilon$-distribution predicted by the ANN-ensemble for
each search-space material.

###  2.5 Design cycles with feedback

The $\epsilon$ values of a few materials selected from the statistical
modeling are computed from DFPT calculations, as shown in the final segment of
a design cycle in Figure 1. The results from the DFPT calculations are used to
determine whether to conduct any further design cycles. In this work, we
conducted the design cycles until at least one high-$\epsilon$ dielectric with
a large $E_{\text{g}}$ is identified. When no such materials are found during
a design cycle, all the selected materials along with their newly DFPT-
estimated $\epsilon$ values are transferred from search-space to training-
data, resulting in a feedback of information prior to the beginning of next
design cycle. The feedback is one of the most crucial parts of our material
design workflow, because it results in a better sampling of material
representation-vector space by training-data and thus, more reliable ANN model
predictions during the next design cycle. The advantage of the feedback
mechanism is prominent during the quantification of uncertainty which is used
directly by the EGO algorithm to identify the best candidates for the next set
of DFPT calculations. After the end of a design cycle, the uncertainty on
predicting the $\epsilon$ values is decreased for the set of materials which
are similar to the materials whose $\epsilon$ values were calculated using
DFPT in the given cycle.

In addition to the feedback mechanism, another factor that influenced the
candidate selection in the design workflow is the minimum cutoff imposed on
the band gap values of materials when they are included on the search-space.
The reason for implementing a cutoff is to externally introduce a character of
multi-objective optimization in this work. Without explicitly setting a
minimum band gap limit, the candidate selection process that is dictated by
the EGO algorithm tries to optimize only a single objective, which is the
$\epsilon$ value. We conducted three design cycles sequentially with feedback
of the newly calculated data into training-data after each cycle. In the first
design cycle, we set no band gap minimum cutoffs to allow the full exploration
of the search-space that consists of 11,102 non-metals from OQMD. In the
second design cycle, a minimum cutoff of 2.25 eV was set, leaving 6191
materials in the search-space. In the final cycle, the minimum cutoff was
increased to 5 eV to limit the candidate selection only to the materials with
very high-$E_{\text{g}}$. Hence, the search-space size in the final cycle was
reduced to 1046 materials. The workflow that we adopted in this work deviates
from the ideal situation where a dedicated multi-objective optimization
statistical algorithm will be used to find a material with high $\epsilon$ and
large $E_{\text{g}}$ values. Since the band gap values are already available
for all materials in the search-space, the best approach here was to implement
a statistical optimization algorithm to quickly find high-$\epsilon$ materials
while the preference for large band gap values is achieved by manually setting
a minimum cutoff. This work stands as an example for the modifications
required to practically implement the statistical algorithms that are often
benchmarked on idealistic scenarios.

###  2.6 New dielectric materials

The materials that are part of Pareto-front of MP-data is listed in Table 1,
while the Pareto-front of training-data at each design cycle is plotted in
Figure 5. Since the maximization of $\epsilon$ and $E_{\text{g}}$ values are
considered as optimal in this study, each material in the Pareto-front has a
higher value of either $\epsilon$ or $E_{\text{g}}$ than any other material in
the corresponding training-data. Therefore, the modification of the training-
data’s Pareto-front by any of the newly calculated dielectric constants after
each design cycle may indicate the identification of suitable, high-dielectric
materials.

During the first design cycle, the EGO algorithm picked out the five most
promising candidates with largest $E(I)$ values in the search-space. The
$\epsilon$ values of these five selected materials were calculated using DFPT.
Two materials among them turned out have very high $\epsilon$ values ($\sim$
370) but very low $E_{\text{g}}$ ($\sim$ 0.5 eV). The low $E_{\text{g}}$
values are not unexpected since the EGO algorithm implemented in this work
aims to maximize only the $\epsilon$ values. None of the materials selected in
this cycle modified the Pareto-front of MP dataset, as shown in Figure 5a. The
$\epsilon$ values of these five materials were appended to the training-data
prior to starting the next design cycle.

Five materials were selected in the second cycle and their dielectric
constants were calculated. Our calculations predict a large dielectric
constant for one of the five new materials - tetragonal Tl3PbBr5
($\epsilon$=101, $E_{\text{g}}$=2.9 eV). Tl3PbBr5 joined the Pareto-front, as
shown in Figure 5b. Three other new materials - Bi5IO7 ($\epsilon$=36,
$E_{\text{g}}$=2.7 eV), Bi3ClO4 ($\epsilon$=39, $E_{\text{g}}$=2.3 eV), and
Bi3BrO4 ($\epsilon$=39, $E_{\text{g}}$=2.3 eV), have moderately large
$\epsilon$ values, even though they did not improve the existing Pareto-front.
All the five new materials were appended into the training-data before
proceeding to begin the third design cycle.

During the third and final design cycle consisting of only materials with very
large $E_{\text{g}}$ in search-space, seven new candidate materials were
selected to do DFPT calculations. Two among them - Eu5SiCl6O4 ($\epsilon$=69,
$E_{\text{g}}$=5.5 eV) and HoClO ($\epsilon$=75, $E_{\text{g}}$=5.2 eV) joined
the Pareto-front due to their large $\epsilon$ and $E_{\text{g}}$ values, as
shown in Figure 5c. In total, three new dielectric materials in the Pareto-
front were discovered after three design cycles and 17 new DFPT calculations
were performed in the entire workflow. No further design cycles were conducted
since we have already identified multiple compounds with high $\epsilon$ and
$E_{\text{g}}$, which remained unexplored experimentally.

The $\epsilon$ values of all 17 materials which were obtained in this work are
given in Table 9. The $\epsilon$ and $E_{\text{g}}$ of all materials belonging
to the Pareto-front of MP-dataset is listed in Table 1 for comparison. Among
all the newly discovered dielectrics with large $\epsilon$ values, tetragonal
HoClO and monoclinic Eu5SiCl6O4 stand out because of their very large DFT-
calculated band gap energies (5.2 eV and 5.5 eV respectively). These two rare
earth oxychlorides are reported to have been experimentally synthesized
templeton1953crystal ; holsa2002stability ; basiev2005hydration ;
jacobsen1994synthesis  but their dielectric properties remained unstudied to
the extent of our knowledge. Both of these compounds are mixed-anionic
inorganic compounds - a class of emerging functional
materialskageyama2018expanding . Interestingly, the monoclinic Eu5SiCl6O4 has
32 atoms in its primitive unit cell which often exceeds the maximum cutoff on
number of atomic sites in high throughput studies involving computationally
expensive material propertiespetretto2018high ; choudhary2020high .

Thermodynamic stability of a dielectric when in contact with Si or other
semiconductors is an important requirement for it to be used in electronic
applications. Several of the high-$\epsilon$ dielectrics identified in the
published literature were shown to be unstable while forming an interface with
Si in subsequent experimental studies conducted at or above the room
temperature. The formation of SiOx and other undesired metal oxides were
reported at the interface between Si and the popular high-$\epsilon$
dielectrics such as Ta2O3atanassova1998x ; schlom2002thermodynamic ;
alers1998intermixing , TiO2perego2008energy ; mccurdy2004investigation ,
BaTiO3george2013preferentially  and SrTiO3hu2003interface ;
goncharova2006interface . The thermodynamic stability between two compounds
can be assessed from the phase diagram involving those compounds. In this
work, the phase diagram is constructed by computing the convex
hullbarber1996quickhull  of formation energies of all the materials that
belong to a given phase space spanned by their constituent elements. Each of
the compounds that form the convex hull not only has the lowest formation
energy at its composition but also has lower energy than any linear
combination of other materials in that phase space. The difference between the
formation energy of a compound and energy at the convex hull for the same
composition is called as hull-distance ($E_{\text{hd}}$). By definition, each
material that is on the convex hull has a hull distance of zero (i.e.,
$E_{\text{hd}}$ = 0) and is considered to be stable. On the other hand, every
material that falls above the convex hull is considered as metastable (0 $<$
$E_{\text{hd}}$ $\leq$ 50 meV per atom) or unstable ( $E_{\text{hd}}$ $>$ 50
meV per atom) depending on the magnitude of $E_{\text{hd}}$ according to the
heuristic conventions adopted in literature sun2016thermodynamic ;
balachandran2018predictions ; wu2013first ; zakutayev2013theoretical ;
pal2021accelerated . The presence of a tie line between two compounds in a
convex hull phase diagram indicate that they are thermodynamically stable
phases when in contact with each other. Our thermodynamic stability analysis
on Ta2O3, TiO2, BaTiO3 and SrTiO3 in OQMD using the qmpy APIkirklin2015open
showed no tie-lines connecting any of them to Si, indicating they are unstable
when in contact with Si. This is consistent with the published results
atanassova1998x ; schlom2002thermodynamic ; alers1998intermixing ;
perego2008energy ; mccurdy2004investigation ; george2013preferentially ;
hu2003interface ; goncharova2006interface . We also analyzed Gd2O3, a high
$\epsilon$ ($\sim$ 20zhou2004properties ) that is proven to be stable against
Sikwo2001properties , and found that a tie-line does exist between Si and
Gd2O3. These phase diagram plots are provided in Supplementary Figure 6. In
Figure 6, we report a phase diagram to assess the stability of newly
discovered high-$\epsilon$ dielectrics - HoClO and Eu5SiCl6O4. The phase
diagram shows that both these materials are thermodynamically stable with the
semiconductors such as Si, Ge, GaAs, GaN, and SiC at 0K, a requirement for
them to be used in microelectronic devices where an interface with one of the
common semiconductors is often necessaryrobertson2005high . The next most
promising candidate, tetragonal Tl3PbBr5, has a very large $\epsilon$ (101)
but possesses a relatively smaller band gap (2.9 eV) and is computed to be
thermodynamically metastable at 0K ($E_{\text{hd}}$ = 16 meV per atom)
according to the data obtained from the OQMD. Tl3PbBr5 is also reported in
literature to have been experimentally synthesizedkeller1981darstellung ;
denysyuk2013electronic ; ferrier2006tl3pbbr5 , without any mention of its
dielectric properties.

##  3 Discussion

We report the identification of three dielectric materials that contain a
combination of high dielectric constant and large band gap -
HoClO($\epsilon$=75, $E_{\text{g}}$=5.2 eV), Eu5SiCl6O4($\epsilon$=69,
$E_{\text{g}}$=5.5 eV), and Tl3PbBr5($\epsilon$=101, $E_{\text{g}}$=2.9 eV).
These compounds modify the Pareto-front of previously known high-throughput
dielectric constants data available from the MP database. Our screening
strategy also uncovers four other dielectric materials with large
$E_{\text{g}}$ and moderately large $\epsilon$ \- Sr2LuBiO6($\epsilon$=24,
$E_{\text{g}}$=2.4 eV), Bi5IO7($\epsilon$=36, $E_{\text{g}}$=2.7 eV),
Bi3ClO4($\epsilon$=39, $E_{\text{g}}$=2.3 eV), and Bi3BrO4($\epsilon$=39,
$E_{\text{g}}$=2.3 eV) - at the cost of conducting only 17 DFPT calculations
overall. We utilize the data available in the open-source databases (OQMD, MP)
to build a statistical optimization model and use it to select the best
candidates after searching among 11,102 stable non-metals that are available
in the OQMD. Among the newly discovered dielectrics, two mixed-anionic
materials - HoClO and Eu5SiCl6O4 are shown to have tie-lines with multiple,
commonly used semiconductors on their phase diagrams, that indicate their
thermodynamic equilibrium.

The presence of rare-earth elements such as Ho and Eu in dielectrics can be a
challenge for their use in practical applications. However, the ongoing
efforts toward increasing their availability such as efficient recycling of
rare-earth materialsqiu2019economic ; amato2019sustainability  can result in a
sufficient supply of elements for mass production of small electronic
components. In particular, Ho is an underutilized element in the
industrythornton2015homely  even though it is more abundant in the earth’s
crust than other widely mined elements such as Mo, Bi, and precious
metalsyaroshevsky2006abundances . Eu is more abundant on earth’s crust than Ho
and some of the heavily mined elements such as W and
Asyaroshevsky2006abundances . Hence, an active exploration of cheaper and
easier extraction methods for rare earth elements may make it feasible to
include them in mass-produced electronics in the near future. The presence of
toxic elements such as Pb and Tl can stand as a barrier against including
Tl3PbBr5 in consumer electronics. Since mixed-anionic materials are an
emerging class of functional materials, our identification of promising
dielectric materials in this family opens up further research opportunities on
rational design of high-performance dielectrics and their experimental
characterizations.

We also assessed the thermodynamic stability of the new dielectrics by
creating a large convex hull diagram containing the best two new dielectrics
(HoClO and Eu5SiCl6O4) and several commonly used materials in electronics. The
relevance of this analysis is also provided in detail along with examples of
previously reported high-$\epsilon$ dielectricsatanassova1998x ;
schlom2002thermodynamic ; alers1998intermixing ; perego2008energy ;
mccurdy2004investigation ; george2013preferentially ; hu2003interface ;
goncharova2006interface  that were later found out to be unstable when in
contact with common electronic component materials such as SiO2. Our convex
hull analysis indicates that both HoClO and Eu5SiCl6O4 are stable against the
common electronic materials that we considered.

To understand what features of HoClO, Eu5SiCl6O4, and Tl3PbBr5 make them the
best dielectric candidates in this study, we have calculated their electronic
structures and partial density of states (Supplementary Figure 5). Our
analysis shows that the top of the valence bands and bottom of the conduction
bands in these compounds consists of primarily the contributions from the
anions (Cl, Br) and cations (Ho, Eu, Tl), respectively. This analysis
indicates that having lighter anions (such as Cl, Br) is advantageous as their
valence orbitals making up the valence band edge in those compounds will have
lower energies, hence, a relatively larger band gap that is desired in
high-$\epsilon$ materials.

In addition to the identification of high-dielectrics, we successfully
demonstrated an implementation of cross-database statistical design for
computational materials selection. Datasets from the MP and OQMD repositories
are used in this work as training data and search-space, respectively. The
successful identification of new materials from such a workflow is a another
motivation for actively moving toward the interoperability of materials
databases, which is one of the four pillars of FAIR data
principleswilkinson2016fair  in the scientific data management. Therefore,
better interoperability across databases amplifies the flexibility in
utilizing materials data while solving a complex materials problem.

Lastly, this work also stands as an example of practical implementation of a
computational design strategy for property optimization via data-informed
material selection. A multi-objective optimization problem (maximizing
$\epsilon$ and $E_{\text{g}}$) is converted into a single objective
optimization using statistical methods (maximizing $\epsilon$) combined with
explicit constraining of band gap values (higher $E_{\text{g}}$) among
materials since $E_{\text{g}}$ is already available for all materials in the
search-space. The deviation from the ideal, statistically benchmarked multi-
objective optimization workflowsgopakumar2018multi  enabled the efficient
utilization of resources and resulted in the identification of three
high-$\epsilon$ dielectrics at the cost of just 17 new DFPT calculations.

##  4 Methods

###  4.1 ANN modeling

The individual models in the ANN-ensemble consisted of a single hidden layer
with the number of neurons in the range of $10^{2}$. The exact number of
neurons varied randomly within a small range (10-30) to avoid any bias that
may arise from model architecture since the subset of training-data for each
ANN was randomly sampled. Each ANN-ensemble consisted of 2000 independent
ANNs. Thus, the $\epsilon$-distribution for each material consisted of 2000
independent $\epsilon$ predictions. A new ANN-ensemble was created and trained
for each new design cycle to learn the incremented training-data. The Nadam
optimizer is used for network optimization during the training. Both L2 layer
regularization and early-stopping callback as implemented in
Keraschollet2015keras , are implemented for each ANN in the ensemble to
prevent the over-fitting. On average, it took between 300 to 400 epochs to
reach the local minimum of the loss function. Each epoch is a full iteration
of fitting the training-data to update the internal weights of an ANN.
Validation details of one of the randomly chosen ANN models from the ensemble
are plotted in Figure 3a for reference. Feature dimensional reduction prior to
the training of ANNs was done using principal component analysis algorithm
implemented in scikit-learnscikit-learn . Model validation during the training
of one of the 2000 ANN models in the second design cycle is plotted in Figure
3a.

###  4.2 DFPT calculations

We performed all DFT calculations calculations using the Vienna Ab-initio
Simulation Package (VASP)kresse1996efficiency ; kresse1996efficient  with
potentials derived using the projector-augmented wave (PAW)kresse1999ultrasoft
; blochl1994projector  method. We calculated the total dielectric constant
(sum of electronic and ionic components) values for selected materials using
DFPT as implemented in VASP. All the compounds were fully relaxed before the
dielectric calculations. We used an energy cut-off of 520 eV, k-mesh of 6000
k-points per reciprocal atom (KPPRA), and an energy-threshold of $10^{-8}$ eV
during the self-consistent calculations. The forces on the atoms after
structural relaxations were less than 10-3 eV Å-1. We used the generalized
gradient approximationperdew1996generalized  (GGA) to approximate the
exchange-correlation energies of the electrons. A detailed discussion on DFPT
calculations is provided in the Supplementary Methods section included within
the Supplementary Material. We did DFPT calculations on a set of well-known
dielectrics and a few rare-earth compounds, and benchmarked the results
against previously reported results in literature. These results indicate the
reliability of our calculated $\epsilon$ values, which are provided in
Supplementary Table 2. Specifically, two rare-earth oxides (EuO and Ho2O3) and
one rare-earth halide (EuF2) were benchmarked to test the accuracy of the
standard DFPT calculations in modelling these compounds. Furthermore, our
calculations reveal that no imaginary phonon modes appear in HoClO,
Eu5SiCl6O4, and Tl3PbBr5, the best high-$\epsilon$ materials identified in
this work. More details are provided in Supplementary Table 1 and
Supplementary Figure 4.

##  5 Data Availability

The data used in building statistical design models are open-sourced and
available via OQMD and Materials Project databases.

Other data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  6 Acknowledgements

This work was funded by the SAMSUNG Global Research Outreach Program, and the
U.S. Department of Commerce, National Institute of Standards and Technology as
part of the Center for Hierarchical Materials Design (CHiMaD) award
70NANB14H012. We acknowledge the computing resources provided by (1) the
National Energy Research Scientific Computing Center (NERSC), a U.S.
Department of Energy Office of Science User Facility operated under Contract
No. DE-AC02-05CH11231, (2) Quest high-performance computing facility at
Northwestern University which is jointly supported by the Office of the
Provost, the Office for Research, and Northwestern University Information
Technology, and (3) the Extreme Science and Engineering Discovery Environment
(National Science Foundation Contract ACI-1548562)

##  7 Author contributions

A.M.G. devised computational strategies, wrote the manuscript and conducted
the calculations. KP provided important hands-on guidance in calculations and
theoretical understanding. A.M.G. and C.W modeled the project and analyzed the
results. All authors have reviewed the manuscript.

##  8 Competing Interests

The authors declare no competing financial or non-financial interests.

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##  9 Table Titles and Captions

Table 1: The Pareto-front of dielectric materials dataset from Materials
Project

MP ID | Material |  $E_{g}$ (eV) | $\epsilon$
---|---|---|---
mp-1138 | LiF | 8.7 | 9.3
mp-13948 | Cs2HfF6 | 7.2 | 9.3
mp-13947 | Rb2HfF6 | 7.1 | 9.4
mp-7104 | CsCaF3 | 6.9 | 9.7
mp-5347 | KAlF4 | 6.8 | 10.6
mp-10250 | BaLiF3 | 6.6 | 14.8
mp-3654 | RbCaF3 | 6.4 | 18.9
mp-8455 | CsF | 5.9 | 20.3
mp-28243 | RbLiCl2 | 5.1 | 54.5
mp-5606 | AlTlF4 | 4.2 | 96.9
mp-23092 | Ba2TaBiO6 | 2.6 | 99.9
mp-27832 | Tl2SnCl6 | 2.5 | 100.8
mp-3614 | KTaO3 | 2.1 | 639.9
mp-2657 | TiO2 | 1.8 | 988.0

00footnotetext: MP ID corresponds to the unique ID of material in the
repository, $E_{\text{g}}$ is the band gap energy, and $\epsilon$ corresponds
to the largest eigenvalue in the dielectric constant tensor

Table 2: Dielectric constants of 17 materials calculated using DFT in this work OQMD |  |  |  |  |  |
---|---|---|---|---|---|---
ID | Material |  $E_{g}$ |  |  |  |
| $\epsilon_{x}$ | $\epsilon_{y}$ | $\epsilon_{z}$ | Cycle |  |
681780 | CaVO3 | 0.4 | 4.7 | 4.5 | 4.5 | 1
14476 | Sr2VN3 | 1.8 | 28.8 | 16.5 | 16.0 | 1
13450 | BaZrN2 | 1.2 | 31.2 | 31.2 | 21.7 | 1
1104204 | HoN | 0.4 | 376.9 | 373.0 | 372.7 | 1
649584 | Bi2SeO2 | 0.5 | 377.3 | 371.8 | 118.2 | 1
19571 | Sr2LuBiO6 | 2.4 | 24.1 | 19.4 | 18.7 | 2
5958 | Bi5IO7 | 2.7 | 35.8 | 28.2 | 23.1 | 2
24994 | Bi3ClO4 | 2.3 | 38.9 | 24.2 | 25.7 | 2
22697 | Bi3BrO4 | 2.3 | 39.0 | 23.7 | 22.1 | 2
118234 | Tl3PbBr5 | 2.9 | 100.8 | 36.4 | 36.4 | 2
11916 | Eu4Cl6O | 5.3 | 7.4 | 7.3 | 5.5 | 3
18953 | EuClF | 5.6 | 11.1 | 11.1 | 10.4 | 3
646321 | Rb2PrCl5 | 5.1 | 12.2 | 11.0 | 8.9 | 3
15191 | Cs2NaCeCl6 | 5.1 | 13.2 | 13.2 | 13.2 | 3
4063 | EuCl2 | 5.2 | 15.6 | 12.9 | 11.8 | 3
24611 | Eu5SiCl6O4 | 5.5 | 69.3 | 15.1 | 12.9 | 3
13689 | HoClO | 5.2 | 75.1 | 37.9 | 15.2 | 3
00footnotetext: OQMD ID refers to the materials’ unique entry ID in the OQMD
database, $E_{\text{g}}$ refers to the band gap energy in eV,
$\epsilon_{x,y,z}$ refers to the three eigenvalues (xx, yy, zz) of the of
dielectric constant tensor, and the Cycle mentions the design cycle when the
material was selected for the calculations of dielectric constant using DFPT.
The values $\epsilon_{x,y,z}$ are ordered in such a way that
$\epsilon_{x}>\epsilon_{y}>\epsilon_{z}$. The best seven materials found in
this work are highlighted in bold letters.

##  10 Figure Captions

Figure 1: Materials Design workflow used in this work. The three parts of the
design workflow shown here together completes a single design cycles. The
newly computed DFPT results from a design cycle is fed back into the training-
data for the next design cycle. Since the dielectric tensor of a crystal is of
shape 3$\times$3, optimizing all nine components of the dielectric tensor
leads to a nine-objective optimization problem. Thus, we specifically optimize
the largest value of dielectric constant ($\epsilon$) among all
crystallographic directions, also referred as the target-property hereinafter.
This scalar value is quantified as the largest eigenvalue of the total
dielectric tensor. The training-data in this work consists of nearly 2000
compounds from the MP for which $\epsilon$ and $E_{\text{g}}$ were known and
the training-data came from the OQMD, for which only $E_{\text{g}}$ were known
at the beginning of this work. Figure 2: Comparison of material representation
vectors between the OQMD and MP structures. Difference in material
representation vectors of the structures obtained from OQMD and MP for 1717
materials. a) Mean absolute difference in the Magpie-generated
representational feature vectors on structures obtained from the MP and OQMD
for 1717 materials in the training-data. Crystal structures of all 1717
materials were first obtained from MP as a part of generating training-data,
and further cross referenced to find their equivalent structures in OQMD based
on their ICSD Collection Codes. ICSD Collection Codes were not available for
the rest of the 143 materials in the MP training-data. b) Difference in band
gap values of 1717 materials from training-data that have a corresponding
structure entry in MP and OQMD, which are cross-referenced based on their ICSD
Collection Codes. Figure 3: Results from statistical modeling. (a) ANN model
validation on a test set of 373 materials split from the training-data. We
used an ensemble of ANNs to predict a distribution of values for each
material. This particular model-fit plot is taken from a single ANN-model that
was part of the ensemble in design cycle 2. The 373 materials plotted here
were not seen by this particular ANN model at any stage during the training.
These predictions are made only for this particular ANN model to show its
learning capabilities, and it is not part of the design workflow that we
created. In the design workflow, each ANN model in the ensemble is exposed
only to a unique subset of the full MP training data, excluding 373 randomly
chosen materials. Further, in the design workflow, this trained ANN model is
used to predict the dielectric values of only the search-space materials from
OQMD, not the 373 unseen materials from MP dataset. The model was trained to
predict $\log_{2}(\epsilon)$ because the $\epsilon$ values were highly non-
uniform in the training-data with most of the values were below 25, making
some of the very large values as outliers. A log-scale transformation of
$\epsilon$ reduced the numerical difference between the largest $\epsilon$
value and the median, making the former less of an outlier in ANN modeling.
The model fit shown in this plot has an $R^{2}$ score of 70%, and a Spearman’s
rank correlation of 85%. (b) This plot shows the predicted
$\epsilon$-distributions and corresponding $E(I)$ values on the same test
dataset consisting of 373 materials split from the training-data. The error
bars represent the standard deviation in ANN-ensemble predictions which is
quantified as the uncertainty of ANN modeling. For a clearer perspective, the
radius and color of the circles represent the same quantity - the Expected
Improvement,$E(I)$, value calculated using EGO algorithm. A point without an
outer circle around it represents a material with a negligible ($<10^{-3}$)
value for $E(I)$. In this figure, only 25 materials have an $E(I)$ value that
is greater than $10^{-3}$. Figure 4: The optimization algorithm. The value
$y_{t}^{max}$ represents the currently available highest value of $\epsilon$
among all materials in the training-data. $\mu$ and $\sigma$ represents the
mean and standard deviation of the ANN-ensemble predicted distribution of
$\epsilon$ for a material (blue dot) in the search-space. Within the predicted
distribution, which is assumed as a normalized Gaussian function here, the
region above $y_{t}^{max}$ represents the region of improvement - as shown in
green. If the ab-initio DFPT calculation determines that the material’s
$\epsilon$ value exists within the green-shaded region, it will be considered
as an improvement over the current best value $y_{t}^{max}$ in training-data.
Figure 5: Evolution of the Pareto-front with design cycles. The $\epsilon$ and
$E_{\text{g}}$ values of the training-data and newly characterized dielectric
materials are plotted for (a) design cycle 1, (b) design cycle 2 and (c)
design cycle 3. All the data shown in these plots originated from DFPT
calculations. Plot (a) shows the original Pareto-front of the dataset from
materials project (MP), because none of the materials measured in cycle 1
became part of the Pareto-front - predominantly owing to their low band gap
values. Assigning no restrictions on the band gap of search-space materials
during the first cycle directed the design algorithm to pick two materials
without any preference for large band gaps. The numerical values of the
materials in Pareto-front of MP-dataset is given in Table 1. In both (b) and
(c), only the materials with $E_{\text{g}}$ values greater than 2.0 eV are
plotted to highlight the area where some of the newly discovered dielectrics
in their corresponding cycles joined the Pareto-front. Due to this cropping,
two materials from MP-dataset which are actually in the Pareto-front in plots
(b) and (c) with very high $\epsilon$ values - tetragonal TiO2
($\epsilon$=988, $E_{\text{g}}$=1.8 eV) and cubic KTaO3 ($\epsilon$=640,
$E_{\text{g}}$=2.1 eV), are not shown here. Figure 6: Phase diagram of all
stable compounds in Ho-Cl-O-Eu-Si-Ge-Ga-As-C-N phase-space from OQMD (as of
January, 2022). The two new most promising dielectrics, HoClO and Eu5SiCl6O4
are plotted in large green circles in the center. The elements (Ho, Eu, Si,
Cl, Ge, Ga, As, C, N and O) and semiconductors of interest (Si, Ge, GaAs, SiC,
and GaN) are plotted in the middle layer in medium-sized yellow circles. All
other stable compounds in the phase diagram are plotted in small dark circles
in the outermost layer. Tie-lines between the new dielectrics and the
semiconductors or elements are shown as thick red lines. Other tie-lines from
the dielectrics to rest of the stable materials in outer layer are drawn as
narrow gray lines. Another 2326 tie-lines exist in this phase-diagram that do
not include either of the dielectrics, and are not shown in this network-plot
for better visibility of relevant information. The elements and compounds
without any visible tie-lines in the outermost layer do not have tie-lines
with HoClO or Eu5SiCl6O4, but they have tie-lines with some of the other
materials in the outer layer - making them part of this phase diagram. There
exist a tie-line from each dielectric material to each semiconductor that is
considered here for comparison. This indicates that the both HoClO and
Eu5SiCl6O4 are in thermodynamic equilibrium with Si, Ge, GaAs, GaN, and SiC at
0K. The thermodynamic stability is a requirement for dielectrics that needs to
form a stable interface with semiconductors in electronic
applicationsrobertson2005high .