# Automated Discovery of Wurtzite Solid Solutions with Enhanced Piezoelectric
Response

Drew Behrendt  Department of Chemistry, University of Pennsylvania,
Philadelphia, Pennsylvania, 19104-6323, USA  Sayan Banerjee  Department of
Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania, 19104-6323,
USA  Jiahao Zhang  Department of Chemistry, University of Pennsylvania,
Philadelphia, Pennsylvania, 19104-6323, USA  Andrew M. Rappe  Department of
Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania, 19104-6323,
USA<EMAIL_ADDRESS>
## Abstract:

While many piezoelectric materials are known, there is still great potential
to improve on the figures of merit of existing materials through compositional
doping, forming solid solutions. Specifically, it has been shown that doping
and alloying wurtzite-structured materials can improve the piezoelectric
response; however, a vast compositional space has remained unexplored. In this
work, we apply a multi-level screening protocol combining machine learning,
chemical intuition, and thermodynamics to systematically discover dopant
combinations in wurtzite material space that improve the desired piezoelectric
response. Through our protocol, we use computationally inexpensive screening
calculations to consider more than 3000 possible ternary wurtzite solid
solutions from 9 different wurtzite base systems: AlN, BeO, CdS, CdSe, GaN,
ZnO, ZnS, ZnSe, and AgI. Finally, based on thermodynamic analysis and explicit
piezoelectric response calculations, we predict 11 materials with improved
piezoelectric response, due to the incorporation of electropositive dopants.

## Main Text:

Piezoelectrics are non-centrosymmetric materials that are capable of
interconverting mechanical and electrical energy for a variety of
applications.1 Piezoelectrics provide the basis for microelectronic energy
harvesting, acoustic wave devices, actuators, and other devices that are
widely used in research, industry, and military applications. 2, 3, 4 The
energy efficiency and power output of these materials increases with
increasing relative piezoelectric response, but decreases with increasing
dielectric constant. Perovskite ferroelectrics are prototypical
piezoelectrics; however, while perovskites have a very high piezoelectric
response, they also possess high dielectric constants.5, 4 Furthermore, these
materials often lose their beneficial properties at high temperatures.5 An
alternate material class is wurtzite; though wurtzites have smaller
piezoelectric constants than their perovskite counterparts, they are known for
their low dielectric constants and high-temperature performance. 6, 7, 8
Furthermore, these materials are highly compatible with complementary metal-
oxide-semiconductor (CMOS) technology, which makes them highly attractive for
use in piezoelectric applications. 2, 3 Doping has long been used to improve
the electronic properties of host materials; for example, perovskite lead
magnesium niobate (PMN) is alloyed with lead titanate (PT) to enhance the
temperature stability and magnitude of piezoelectricity. Adding elements into
common wurtzite materials, such as doping aluminum nitride with scandium or
zinc oxide with magnesium, has been shown to appreciably increase the
piezoelectric response and even induce ferroelectricity, which is of interest
for many additional applications.7, 9, 10, 11, 12, 13

With the recent growth of machine learning (ML) applications in materials
science, many methods have been proposed to speed up materials discovery using
ML-based methods.14, 15, 16, 17, 18, 19, 20, 21, 22 Feature selection, where
one finds which input features are most correlated to a target output, is a
primary challenge of current ML applications in big data. Choosing features is
at the heart of this challenge, since it is inherently subjective, and the
only way to find meaningful correlations is to have representative and
meaningful features. Once chosen and selected, however, features can be
particularly useful in rational materials design because of their role in
providing an interpretative picture of the underlying physics. Feature
selection approaches have been shown to be successful for fields spanning from
materials discovery18, 23 to chemical reaction development using
homogeneous24, 25, 26, 27 and heterogeneous28, 29, 30, 22, 31 catalysts. Many
recent works have identified a "material gene" to perform advanced analysis of
material structure-property relations, where the material gene corresponds to
the most important feature(s) that correlate with and therefore influencing a
target property.29, 32, 33, 34, 35, 36, 37, 38, 39 The features can be broadly
classified into two different types: primitive and calculated. Primitive
features, or primitives, are purely based on summary statistics of atomic data
such mass, location on the periodic table, electronegativity, and charges.37
Calculated features, reffered to as proxies, involve calculations such as _ab
initio_ density functional theory calculations, and these include ionic
charges, bond lengths, and bond angles. It is worth noting that, to be most
useful, calculated features of a material should be simpler and less time
consuming to calculate than the target property of interest. For example,
calculation of the lattice parameters requires much less computational time
than piezoelectric tensor calculations. Our multi-level screening protocol is
based on the combination of these two types of featurization.

In the current work, we have investigated nine different base wurtzite
materials and their solid solutions with every relevant metal and metalloid
element, over 3000 possible candidate materials. We narrow down this space to
30 solid solutions which are predicted to improve the piezoelectric response.
Finally, we perform thermodynamic analysis to predict 11 best candidates for
future experimental verification. An overview of this workflow is shown in Fig
1. The first level of screening is designed to find one screening proxy to use
in place of expensive piezoelectric calculations. The second level of the
screening serves to reduce the space of all examined ternary solid solutions
to just a few suggested high-value materials using automated machine learning
candidate selection and cost of the suggested dopants.

Figure 1: Multi-level screening workflow to explore the compositional phase
space of the nine different wurtzite base materials to maximize the
piezoelectric tensor component $e_{33}$. The grey, blue, and green boxes
represent data sets, methods, and deliverables, respectively.

We consider a 2 $\times$ 2 $\times$ 1 wurtzite supercell consisting of 8 metal
and 8 non-metal sites. For doped materials, 2 of the 8 cation sites have been
replaced. The relative positions of the dopants within the unit cell has been
shown to affect the piezoelectric response, 7 so it is kept constant
throughout this entire study to compare elemental doping effects on an equal
footing. The calculations for the $e_{33}$ component of the piezoelectric
tensor are performed using the Quantum Espresso 40 software package with
optimized, norm-conserving pseudopoentials generated by OPIUM. 41, 42 To do
this, the system is strained along the $c$ axis for five different values
($-$1.0 to +1.0 percent strain), and then atomic positions are allowed to
relax. The polarization is then calculated using the Berry’s phase method.43,
44 The slope of polarization _vs._ strain was calculated and taken to be the
$e_{33}$ component45 of the piezoelectric tensor for the calculated material.
For doping, all combinations of metallic elements that would preserve charge
neutrality are considered for each system. For example, combinations of
(+3,+3) or (+2,+4) dopants are considered for AlN, whereas for ZnO (+1,+3) or
(+2,+2) are screened. Five different machine learning (ML) methods, linear
regression, least absolute shrinkage and selection operator (LASSO), ridge,
recursive feature elimination (RFE), and random forest (RF), from python’s
sklearn are employed.46 By training multiple machine learning algorithms on
each round of the active learning, we can ensure faster sampling of a larger
candidate space since the algorithms will select different types of materials
from the limited initial database, and ultimately we can ensure that if any
algorithm finds a candidate to be viable then it is screened. Additionally,
after training these different methods we can compare the relative
effectiveness of each in predicting the material proxy from the primary
features.

To identify the best screening proxy, we started with a moderate dataset of
additive elements in the most commonly used wurtzite, aluminium nitride (AlN),
to improve the piezoelectric response (Level 1 in Fig. 1). There have been
multiple studies showing the effects of co-doping into AlN. Various material
descriptors, especiallly the lattice $\frac{c}{a}$ ratio, have been identified
as key properties that correlate with the piezoelectric response. 47, 8, 7,
48, 10, 49 In total, 53 materials (see supplementary materials) are included
in our initial dataset, and the results for the piezoelectric response are
summarized in Table 1. Notably, AlN doped with boron and scandium together
shows a slight increase in the piezoelectric constant relative to AlN with
scandium alone, which is currently attracting great interest.2, 3

Dopants | $e_{33}$
---|---
None | 1.46
Sc | 1.86
Sc,B | 1.90
Mg,Hf | 1.84
Mg,Ti | 1.60
Table 1: Notable piezoelectric response for dopants in AlN, from among the
initial dataset of 53 materials.

For each material in the dataset, we used 15 possible candidate proxy
properties based on previous work on material genes and wurtzite
characterization (supplementary material). We then used machine learning
feature selection to find the material proxy that is, on average, most
correlated to $e_{33}$, and the results are shown in Figure 2. Note that the
feature importance for each feature is averaged across all of the ML methods
for proxy selection. We find the lattice $\frac{c}{a}$ ratio to be the most
important feature (Fig. 2a), which has also been reported in previous
literature for host materials alone and based on manual manipulation of the
lattice parameters for sample systems.8, 9, 10. Therefore, a lower lattice
$\frac{c}{a}$ ratio for wurtzite solid solutions corresponds to a higher value
of $e_{33}$ (Fig. 2b), in accordance with previous works.8, 9, 10 We use the
$\frac{c}{a}$ ratio as a screening proxy for our automated workflow (Fig. 2b).

Figure 2: Feature selection in the initial dataset of 53 co-doped AlN
materials: (a) ML feature selection to identify the most important material
proxy. (b) Scatter plot showing the correlation between lattice $\frac{c}{a}$
and $e_{33}$ response. Abbreviated proxies in (a) are shown in full in the
supplementary material. All of the important proxies represent optimized bond
lengths ($u$ parameter), angles, lattice parameters, or simple electronic
properties.

Based on the lattice $\frac{c}{a}$ ratio proxy, we generated a set of over
3000 possible candidate solid solutions from 9 different wurtzite base
systems: AlN, BeO, CdS, CdSe, GaN, ZnO, ZnS, ZnSe, and AgI. Starting from the
initial 53-material dataset and base materials, we then used machine learning
predictions to iteratively select candidates from the 3000 possible
combinations to screen. For each atom, uncalculated descriptors were chosen to
reflect the fundamental chemical characteristics including: atomic number,
atomic mass, elemental melting temperature, charge, column and row on periodic
table, atomic radius 50, electronegativity51, and preferred valence. Primitive
features were collected by taking the average, standard deviation, minimum,
maximum, and range of the atomic descriptors for each material (a total of 50
primitives for each material). The key screening steps are: i) training a ML
model using the primitives of the materials in the database in a specific
iteration, ii) using the trained ML model to predict which of the unstudied
candidates will have the lowest $\frac{c}{a}$ ratio to screen next, and iii)
adding the newly screened materials to the database (the iterative ML
screening in Fig. 1). This process was repeated until the best 500 candidates
were screened.

During the protocol, it became clear that doped versions of certain base
materials are much more likely to have a lower lattice ratio than others
(specifically for BeO, AlN, ZnO, GaN). Therefore, in order to screen the best
candidates for each base material, once solid solutions of a given base
material are no longer predicted to give significant improvement to the proxy
of the host, the rest of the elemental combinations of that base material were
removed from the remaining active learning candidates. This ensures that we
suggest new variants for all nine different base materials, even if some have
a higher propensity toward high $\frac{c}{a}$ ratio than others. Additionally,
some elements are inherently unstable in the host material and the relaxation
calculations fail for a variety of reasons. In this circumstance, a high value
of $\frac{c}{a}$ ratio of 2.0 was assigned to these co-dopants so that the ML
predictions would avoid similar materials.

Figure 3: Validation of the predictive power of ML models. The low root mean
square error (RMSE) for multiple models indicates that all the viable
candidates are screened.

After screening, we performed five-fold cross validation for each ML model.
This is done to ensure that the ML models are accurately predicting the
lattice ratios from the primitive features so that we can be confident that
all viable candidates are screened correctly; the results are shown in Fig. 3.
We find that the random forest algorithm worked best to predict the lattice
ratios. However, with the exception of the LASSO regression, all of the
methods used are relatively accurate at predicting the proxy from primitives,
which is evident from the low RMSE given in Fig. 3. All of the methods have
particular difficulty with materials that are predicted to have $\frac{c}{a}$
ratios near the average yet are calculated by DFT to possess extremely low
lattice ratios; this is in part due to the fact that most of these extreme
values represent materials that become unstable and deviate from the original
wurtzite structure. Furthermore, because the only poor predictions are in the
low $\frac{c}{a}$ regime, we conclude that the unscreened materials predicted
to have high $\frac{c}{a}$ are unlikely to be good candidate piezoelectrics.
At the end of the screening, only about 1/5 of all the screened solid
solutions are found to be insulating and actually effective at reducing the
lattice ratio for the parent material; these materials are then verified by
subsequent $e_{33}$ calculations in accordance with the workflow in Fig. 1.

Figure 4: Importance of primary features to predicting lattice $\frac{c}{a}$
ratio. "At" stands for atomic, "std" for standard deviation, and the melting
temperature is of the material for each element in solid form. All features
involved no calculations are are summary statistics for the constituent
elements in each candidate material.

As shown in Fig. 3, the LASSO algorithm had the worst fitting for the ML
algorithms. By design, LASSO tends to lead to only a few features carrying
very high weight in the trained model; thus, the poorer fitting indicates that
many primitive features are needed in combination to accurately predict the
lattice $\frac{c}{a}$ ratio. However, determination of the most important
features still can provide insight to the structure-property relations,
guiding which elements are best to add to each wurtzite system. Since most of
the ML algorithms trained on the primitive material features provide reliable
prediction of the resulting $\frac{c}{a}$ ratio and insight into the possible
piezoelectric response, we performed feature selection to see which primaries
were most important (Fig. 4). Two major atomic characteristics stand out as
essential to a low $\frac{c}{a}$ ratio and consequently improved piezoelectric
response: electronegativity and mass. Materials containing extremely highly
electronegative elements as the anion, oxygen and nitrogen in particular, had
characteristically lower $\frac{c}{a}$ ratios. Additionally, materials with a
high standard deviation of electronegativity in constituent elements,
containing extremely electropositive elements as well, tended to have lower
$\frac{c}{a}$ ratios. With the high importance of mean row and atomic mass,
materials with a low average row and atomic mass also tended to have low
$\frac{c}{a}$ ratios. This means that the addition of small, electropositive
elements to AlN, ZnO, and BeO will reliably lead to heightened electronic
response in wurtzites. This finding is aligned with recent evidence of Sc and
B enhancing piezoelectricity in AlN, Mg in ZnO, and even the classic example
of adding Mg and Nb to lead titanate. However, since the LASSO algorithm had
relatively poor predictive power, we posit that these primitives alone are not
enough to predict the $\frac{c}{a}$ ratio, and that all (or almost all) of the
primitives provide enough information for accurate prediction of the lattice
ratio and piezoelectric properties from elemental descriptors.

During the verification of $e_{33}$ using Berry’s phase polarization
calculations (DFT response calculations in Fig. 1), we find that while all
good piezoelectrics have low $\frac{c}{a}$, not all wurtzite solid solutions
with low $\frac{c}{a}$ are good piezoelectrics. For example, some of the the
greatly reduced $\frac{c}{a}$ ratio systems unphysically distort the base
material out of the wurtzite phase, leading to an unstable system. We observe
that BeO in particular often becomes unstable with the addition of other
elements. Furthermore, we find that the trend of lower $\frac{c}{a}$ ratio
correlating with higher piezoelectric constant is not a single linear trend
for all different wurtzite materials, as suggested in previous work,8 but
instead we find a linear trend for dopants within each given base material, as
shown in Fig. 5. The resulting 30 materials from this final screening that
notably increased the piezoelectric coefficient of their respective base
material are listed in Table 2. All of these materials are predicted to
increase the $e_{33}$ of the parent material. However, not all are good
candidates for functional materials due to thermodynamic and practical
concerns (Fig. 1). For each material, the best candidate doping combinations
after considering such factors are highlighted in bold in Table 2.

Figure 5: Lattice ratios and piezoelectric responses for the final screened
materials. Each color corresponds to a base material and its respective co-
dopants.

While the proxy $\frac{c}{a}$ ratio helps to drastically reduce the
computational expense for the materials screening, it is important to further
validate ML-based predictions. The low $R^{2}$ values in Fig. 5 indicate that
there are complicating factors beyond the $\frac{c}{a}$ ratio that govern the
piezoelectric properties, particularly for zinc-containing materials (Fig. 5).
Additional factors, such as system Born effective charges, are discussed in
the supplementary material and will be investigated in the future. However, we
report that as a screening parameter, the $\frac{c}{a}$ proxy has been
effective to find valuable candidate wurtzite solid solutions which can
improve the piezoelectric response for all nine base materials.

Dopants | c/a | e33 | Improvement | Formation Energy | Avg. Dopant Cost
---|---|---|---|---|---
(Base) |  | (C/m2) |  | (eV) | (USD/kg)
(AlN) | 1.604 | 1.46 | - | - | -
B,Y | 1.572 | 1.64 | 1.12x | 4.299 | 1,200
Mg,Hf | 1.581 | 1.84 | 1.26x | 2.129 | 700
Sc | 1.575 | 1.86 | 1.27x | -0.069 | 15,000
Sc,B | 1.569 | 1.90 | 1.30x | 2.813 | 8,700
Be,Zr | 1.569 | 2.01 | 1.37x | 3.232 | 860
Be,Hf | 1.565 | 2.12 | 1.45x | 2.937 | 1,100
(BeO) | 1.612 | 0.356 | - | - | -
Li,Y | 1.458 | 0.791 | 2.22x | 2.338 | 75
Li,Sc | 1.376 | 0.954 | 2.68x | 2.955 | 7,560
Ga,Na | 1.488 | 1.01 | 2.84x | 9.267 | 140
Mg | 1.435 | 1.06 | 2.98x | 4.211 | 2
Al,Li | 1.563 | 1.06 | 2.98x | 3.593 | 60
(CdS) | 1.625 | 0.302 | - | - | -
Be,Sr | 1.577 | 0.561 | 1.86x | -1.161 | 420
B,K | 1.452 | 0.699 | 2.22x | 1.811 | 1,200
(CdSe) | 1.624 | 0.082 | - | - | -
Na,Sc | 1.618 | 0.264 | 3.22x | -2.627 | 7,500
Ca,Mg | 1.621 | 0.353 | 4.30x | -3.764 | 5
(GaN) | 1.633 | 0.629 | - | - | -
Mg,Ti | 1.614 | 0.889 | 1.41x | -2.628 | 3
Be,Zr | 1.601 | 0.946 | 1.50x | -1.750 | 850
B,Sc | 1.608 | 1.01 | 1.61x | -2.029 | 8,700
Sc | 1.613 | 1.08 | 1.72x | -4.935 | 15,000
B,Y | 1.598 | 1.15 | 1.83x | -0.886 | 1,200
(ZnO) | 1.610 | 1.27 | - | - | -
B,Na | 1.557 | 1.31 | 1.03x | -0.346 | 1,200
Mg | 1.598 | 1.40 | 1.10x | -4.369 | 3
Be,Mg | 1.585 | 1.43 | 1.13x | -3.750 | 420
Be,Ca | 1.560 | 1.64 | 1.29x | -3.555 | 420
(ZnS) | 1.642 | 0.243 | - | - | -
Be,Cd | 1.629 | 0.335 | 1.38x | 1.452 | 420
Be,Mg | 1.627 | 0.340 | 1.40x | -0.276 | 420
Be,Ca | 1.608 | 0.522 | 2.15x | -0.620 | 420
(ZnSe) | 1.647 | 0.001 | - | - | -
Na,Sc | 1.644 | 0.140 | 140x | -1.896 | 7500
Be,Mg | 1.632 | 0.198 | 198x | 0.154 | 420
(AgI) | 1.629 | 0.098 | - | - | -
Li | 1.628 | 0.138 | 1.41x | -4.948 | 120
Table 2: Notable piezoelectric response for co-doped systems. Base materials
are in parenthese and materials written in bold are of particular interest for
increasing the piezoelectric constant of the base system.

Overall, through a multi-step screening protocol and high-throughput study, we
identify previously unstudied solid solution candidates to improve the
piezoelectric response of nine chosen wurtzite base materials. During this
process, we emphasize the fundamental relation of $\frac{c}{a}$ lattice ratio
in wurtzites to its respective piezoelectric response and use it as a proxy to
develop a computationally inexpensive protocol for piezoelectric material
discovery. Furthermore, we are able to support the idea that primitives can be
effectively used in machine learning methods to predict basic material
properties such as lattice parameters, given a sizeable dataset. Finally, we
propose the best set of candidates for further experimental verification. We
hope that the present work serves as a practical example in the design of
materials with improved material properties through computationally efficient
multi-step high-throughput studies.

## Corresponding Author:

*E-mail<EMAIL_ADDRESS>
## ORCID:

Drew Behrendt: 0000-0003-4701-2722; Sayan Banerjee: 0000-0002-8586-9236;
Jiahao Zhang:0000-0002-8284-8122; Andrew M. Rappe: 0000-0003-4620-6496

## Acknowledgements:

D.B. and J.Z. thank the U. S. Department of Energy, Office of Science, Office
of Basic Energy Sciences Energy Frontier Research Centers program under Award
Number DE-SC00211118. S.B. acknowledges the Vagelos Institute for Energy
Science and Technology for the graduate fellowship. Computational support was
provided by the National Energy Research Scientific Computing Center (NERSC),
a U.S. Department of Energy, Office of Science User Facility located at
Lawrence Berkeley National Laboratory, operated under Contract No. DE-
AC02-05CH11231.

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