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Descriptor and scaling relations for ion mobility in crystalline solids

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Ion mobility is a critical performance parameter in electrochemical energy storage and conversion, but also in other electrochemical devices. Based on first-principles electronic structure calculations, we have derived a descriptor for the ion mobility in battery electrodes and solid electrolytes. This descriptor is entirely composed of observables that are easily accessible: ionic radii, oxidation states and the Pauling electronegativities of the involved species. Within a particular class of materials, the migration barriers are connected to this descriptor through linear scaling relations upon the variation of either the cation chemistry of the charge carriers or the anion chemistry of the host lattice. The validity of these scaling relations indicates that a purely ionic view falls short of capturing all factors influencing ion mobility in solids.The identification of these scaling relations has the potential to significantly accelerate the discovery of materials with desired mobility properties.
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Descriptor and scaling relations for ion mobility in
crystalline solids
Mohsen Sotoudeh
University of Ulm
Axel Groß ( axel.gross@uni-ulm.de )
University of Ulm https://orcid.org/0000-0003-4037-7331
Article
Keywords: ion mobility, descriptor, scaling relations, mono- and multivalent charge carriers, density
functional theory
DOI: https://doi.org/10.21203/rs.3.rs-309875/v2
License: This work is licensed under a Creative Commons Attribution 4.0 International License. 
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Descriptor and scaling relations for ion mobility
in crystalline solids
Mohsen Sotoudeh,and Axel Groß,,
Institute of Theoretical Chemistry, Ulm University, Albert-Einstein-Allee 11, 89081 Ulm,
Germany
Helmholtz Institute Ulm (HIU) for Electrochemical Energy Storage, Helmholtzstraße 11,
89069 Ulm, Germany
E-mail: mohsen.sotoudeh@uni-ulm.de; axel.gross@uni-ulm.de
1
Abstract
Ion mobility is a critical performance parameter in electrochemical energy storage
and conversion, but also in other electrochemical devices. Based on first-principles
electronic structure calculations, we have derived a descriptor for the ion mobility
in battery electrodes and solid electrolytes. This descriptor is entirely composed of
observables that are easily accessible: ionic radii, oxidation states and the Pauling
electronegativities of the involved species. Within a particular class of materials, the
migration barriers are connected to this descriptor through linear scaling relations
upon the variation of either the cation chemistry of the charge carriers or the anion
chemistry of the host lattice. The validity of these scaling relations indicates that a
purely ionic view falls short of capturing all factors influencing ion mobility in solids.
The identification of these scaling relations has the potential to significantly accelerate
the discovery of materials with desired mobility properties.
Introduction
Electrochemical energy storage devices play a central role in our attempts towards decar-
bonization through the storage of volatile renewable energy and the emission-free usage of
vehicles and mobile devices. Significant progress has been made in this respect due to the
development of advanced Li-ion battery technologies.1,2 In addition, recently so-called post
Li-ion technologies3,4 have drawn a lot of attention in order to address, among others, sus-
tainability issues associated with the materials typically used in Li-ion batteries.5,6 In post-Li
ion batteries, other charge carriers such as monovalent Na and K cations7,8 or divalent Mg
and Ca cations9–13 are used. These post-Li-ion batteries, in particular those based on multi-
valent ions, can compete with existing Li-ion batteries or even outperform them, as far as
energy density and safety are concerned,14,15 the latter in particular with respect to their
lower tendency for dendrite growth.16–20 Furthermore, as liquid electrolytes are prone to cor-
rosion processes and often represent fire hazards because of their flammability, all solid-state
2
batteries with higher safety and better electrochemical stability21 based on materials such
as inorganic oxides,22,23 hydrides,24–26 and chalcogenides27,28 have been intensively studied
for all possible charge carriers.
A critical parameter that significantly influences the performance of batteries is the ion
mobility both in the electrolyte and in the electrodes. 29–31 In particular batteries based on
multivalent ions such as Mg2+ are plagued with low ion mobility32–34 due to their stronger
interaction with the host structures compared to monovalent ions such as Li+. Hence the
identification and development of materials with improved ion mobility are essential for more
efficient electrochemical energy storage devices. However, ion conduction in solids is not only
important in battery materials but also in many other applications such as, e.g., solar cells.35
A very useful concept in order to accelerate materials discovery is based on so-called
descriptors.36,37 They represent fundamental materials properties or combinations thereof
that are correlated with a desired or undesired functionality of the material. This concept
has been very successfully used in heterogeneous catalysis,38 in particular in connection with
so-called scaling relations,39 but also already in battery research.20 The identification of
descriptors can significantly speed up the search for new materials with desired functional
properties because once they are identified only the particular descriptors need to be opti-
mized in a first step. Thus promising candidate materials can be proposed whose properties
can then be scrutinized in detail.
In fact, also with respect to ion mobility in solids a number of possible descriptors have
been proposed, based on, e.g., the lattice volume and ionic size,28,29 the choice of the anion
sublattice,29,40 the lattice dynamics,29,41,42 or the preferred crystal insertion site. 30 However,
many of the identified descriptors are restricted to some particular crystal structure. Fur-
thermore, some are based on materials properties that are not easily accessible. Hence it is
fair to say that so far no convenient descriptor has been established that is able to predict
ion mobility across a set of different structures.
Based on the results of first-principles density functional theory (DFT) calculations and
3
physico-chemical reasoning, here we propose such a convenient descriptor for the ion mobil-
ity, the so-called migration parameter or number, that is based on the product of Pauling’s
electronegativity, ionic radii and oxidation states of the involved compounds, all properties
that are easily accessible for any material. This particular descriptor, whose choice is also
supported by a statistical analysis of our first-principles results, goes beyond current propos-
als by considering also deviations from a purely ionic interaction between the migrating ion
and the host lattice. According to our calculations, the activation barrier for migration is
connected to this migration number via linear scaling relations within particular materials
classes. This allows to predict the activation barriers both for the variation of the cation
chemistry of the migrating ion as well as for the variation of the anion chemistry of the
host lattice. Thus this descriptor will most probably significantly accelerate the discovery
of materials with favorable mobility properties. As this migration number is based on ba-
sic physico-chemical quantities, it also enables a deeper fundamental understanding of the
principles underlying ion mobility.
Results and Discussion
From a microscopic viewpoint, migration or diffusion in solid crystalline materials occurs
by atomic hops in a lattice. Such jump processes are typically thermally activated, and the
corresponding tracer diffusion coefficient is given by
Dtr =Dtr
0exp Ea
kBT.(1)
Here Dtr
0is the pre-exponential factor, kBthe Boltzmann constant, and Tthe absolute
temperature. Eais the activation barrier corresponding to the energy barrier along the
minimum energy path connecting two equivalent intercalation sites, as illustrated in Fig. 1.
Such an minimum energy path can be determined by automatic search routines.43 In the
present work, we have used the nudged elastic band method (NEB) 44 in the DFT calcu-
4
0.0
0.5
1.0
1.5
2.0
2.5
0 10 20 30 40 50 60 70 80 90 100
CaO
Energy (eV)
Reaction path length %
Saddle-point
AA'
Ea
Figure 1: Illustration of a cation interstitial migration mechanism, using Ca diffusion in
CaO as an example. A diffusion event corresponds to the migration of the Ca cation from
the energetically most favorable octahedral site A to the nearest equivalent site Athrough
the transition state which corresponds to a saddle-point in the multi-dimensional potential
energy surface and which can be derived by first-principles electronic structure calculations.
The activation energy or diffusion barrier is denoted by Eawhich corresponds to the energy
difference between the saddle-point and the initial configuration.
lations to derive the activation barrier Ea. The electronic structure calculations were per-
formed using the Vienna Ab-initio Simulation Package (VASP) 45 employing the Projector
Augmented Wave (PAW)46 method with the exchange-correlation effects being described
with the Perdew-Burke-Ernzerhof (PBE) functional.47 Further details are provided in the
Supporting Information.
Motivated by the goal to identify the fundamental factors determining ion mobility in
solids, in a previous study28 we had derived the activation barriers for diffusion of a number
of ions of varying size and charge in the same host lattice, a chalcogenide spinel. We ob-
tained the expected results, namely that the size and the charge of the diffusing ion matter.
However, the ionic radius of the charge carrier alone could not explain the observed trends,
but rather the distance between the ion in the tetrahedral site and the nearest chalcogenide
atom. In order to further elucidate the mobility-determining factors, we decided to look at
structurally simpler compounds, namely binary AnXmmaterials with A being the migrating
5
ion. In total, we looked at 35 different compounds with Li, Mg, and Ca as the migrating
ion A.
For these binary materials, we again found that size and charge of the propagating ions
matter, but not in a very systematic way, as already observed by others.29 However, we
could recently show that the stability of ions in chalcogenide spinels can only be understood
if deviations from a purely ionic interaction are taken into account.48 It is essential to realize
that the considered binary materials span the whole range of interaction characteristics
between metallic and ionic bonding. Such bonding characteristics can in fact been classified
in so-called Van Arkel-Ketelaar triangles49 in which compounds are placed according to the
mean electronegativity χmean (x-axis) and the electronegativity difference ∆χ(y-axis) of the
constituting elements.
Fig. 2a shows the Van Arkel-Ketelaar triangle including the Mg binary compounds con-
sidered in this study. A large difference in electronegativity indicates ionic bonding charac-
teristics (shown in yellow), as present in MgO and MgF2. CsF (not shown) would lie at the
apex of the triangle. At the bottom of the triangle corresponding to a vanishing electroneg-
ativity difference, an increasing mean electronegativity is associated with more directional
bonding. Hence the lower right corner gathers covalent systems whereas the lower left corner
contains metallic systems.
The MgnXmbinaries considered in this study all fall along a line between metallic and
ionic bonding which is based on the fact that the cation in the binaries, Mg2+, has not
been varied. In detail, MgF2has the highest electronegativity difference ∆χindicating a
strong ionic bond. This is also true for MgO, whereas Mg2Si is associated with the lowest
value ∆χdemonstrating metallic bonding. The remaining compounds, Mg-halides, Mg-
chalcogenides, Mg-pnictides, and Mg-tetrels, are located between strong ionic and metallic
bonding indicated by the green area. They are divided into three groups. MgCl2, MgBr2, and
Mg3N2are characterized by a large electronegativity difference of about 1.7 demonstrating
a predominately ionic bonding (light yellow region). MgI2, MgS, and MgSe have ∆χ1.3,
6
1.0 1.5 2.0 2.5 3.0 3.5 4.0
Mean electronegativity (χmean)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Electronegativity difference (∆χ)
MgF2
MgO
MgCl2
Mg3N2
Mg2Si
Ionic
Metallic Covalent
MgS
MgSe
MgH2
Mg3P2
Mg3As2
MgTe
Mg2Ge
MgBr2
MgI2
a
0 5 10 15 20 25 30 35
(rA+rX)nAnXχ2(NA+NX)-1
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Migration barrier (eV)
b
Ca
Mg
Li
AnXm binary
Figure 2: AnXmbinaries considered in this study. (a) Van Arkel-Ketelaar triangle with
the considered MgnXmbinaries plotted as a function of the mean electronegativity and the
difference in the electronegativity of the two components. (b) Calculated activation energies
for the migration of A = Li, Mg, and Ca in AnXmbinaries as a function of the migration
number NAX
migr for various elements X according to Eq. (2). The solid lines correspond to
linear regressions of these results.
the other Mg binaries have electronegativity differences below 1.
The fact that also non-ionic components of the interaction contribute to the bonding
in nominally ionic crystals48 suggests that also the interaction characteristics within the
7
considered compounds represented by the electronegativity difference ∆χ2influences the ion
mobility. Together with the well-known dependence of the diffusion barriers on the ionic
radii (ri) and oxidation states (ni) of the involved compounds, this observation motivated
us to define the migration parameter or number Nmigr
NAX
migr = (rA+rX)nAnXχ2
AX/(NA+NX) (2)
as the product of these three quantities where the ionic radii rAand rXare given in ˚
A, and
nAand nXare the absolute values of the formal integer oxidation states or numbers. In
addition, also the number of atoms of the corresponding species in the unit cell of the crystal
NAand NXenters. In Fig. 2b, we plot the dependence of the migration barriers as a function
of the migration parameter for the three migrating ions Li, Mg and Ca in the low vacancy
limit. In spite of some outliers, overall the migration barriers nicely follow separate scaling
relations for each migrating ion
EA
a(X) = EA
0+CA(rA+rX)nAnXχ2
AX/(NA+NX) = EA
0+CANAX
migr .(3)
These presence of universal scaling relations strongly suggest that the same factors govern
the ion mobility in all considered binary compounds. It is no surprise that there are a
few outliers indicating that other critical contributions to the activation energies can play
a role, for example Coulomb interactions beyond those represented by the oxidation states,
quantum mechanical overlap effects and polarization.29
In order to verify that we identified the crucial parameters governing ion mobility in
these binary materials, we applied a statistical compressed-sensing approach using the sure-
independence screening and sparsifying operator SISSO,50 as described in detail in the Sup-
porting Information, to search for possible descriptors. We used the following input param-
eters or so-called primary features: number of atoms in the unit-cell (Natom) and the atomic
masses of the two elements in the binary compound (mA,mX), their formal oxidation
8
numbers (nA,nX) and ionic radii (rA,rX), the Pauling electronegativity (χA,χX) of both
elements, the A-X bond distances dAX, and the unit cell volume V. This approach allows
to vary the dimensionality Ω of the descriptor space, and the descriptor is expressed as a
linear combination of so-called features that are non-linear functions of the input parameter
or primary features. For Ω = 1, we obtained the descriptor
d= (((nX/nA)cos(nX))/((χX)6·sin(mX))) ,(4)
whereas for Ω = 2 we found a two-dimensional descriptor consisting of the two features d1
and d2:
d1= (nX)2×(rMg +rX),(5)
d2= (χX)3/(Natom).(6)
Indeed these findings confirm that the oxidation states reflecting the charge of the atoms, the
ion radii and the electronegativity differences are the determining factors for the migration
barriers. Interestingly, the unit cell volume Vwhich has been shown to substantially influence
the ionic mobility in some structural families28,29 does not show up in these statistically
derived descriptors. However, note that the functional dependencies found by the SISSO
operators do not allow for a straightforward interpretation of the physico-chemical factors
underlying the migration process.
Therefore we decided to look for a verification whether the observed scaling relations as a
function of the migration parameter (Eq. (2)) are also valid for other material types. As this
study was originally motivated by the results for migration barriers of An+in AB2X4spinel
structures, we reconsidered our previous results.28 For these structures, the NEB method
was again applied in the low vacancy limit. In Fig. 3, we have plotted the migration barriers
Ea(in eV) as a function of the migration parameter Eq. (2) for ASc2S4and MgSc2X4spinels
(panel a) and ACr2S4and MgCr2X4spinels (panel b), respectively. Note that the factor
9
4 8 12 16 20 24 28 32 36 40
(rA+rX)nAnXχ2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Migration barrier (eV)
a
ASc2S4
MgSc2X4
Li
Na
Mg
K
Ca
Sr
Te S
Se
O
4 8 12 16 20 24 28 32 36 40
(rA+rX)nAnXχ2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Migration barrier (eV)
ACr2S4
MgCr2X4
Te S
Se
O
Li Na
Mg
Ca
b
Figure 3: Panel a: migration barriers (in eV) in ASc2X4as a function of the migration
parameter (rA+rX)nAnXχ2
AX (Eq. (2)) for ASc2S4(black symbols) and MgSc2X4spinels
(blue symbols) for various mono- and multivalent cations An+and anions Xn. Panel b: the
same as in panel a, but with Sc replaced by Cr.
1/(NA+NX) has been omitted in the definition of the x-axis as this factor is constant
for all considered materials. Again, as in Fig. 2, we find a linear scaling of the migration
barriers upon variation of the anions Xn(blue symbols). Interestingly enough, we also find
additional scaling relations upon variation of the cations Li+, Na+, K+, Mg2+, Ca2+ , and
Sr2+ (black symbols) (note that the MgSc2S4and MgCr2S4spinels, respectively, are part of
10
0.4
0.8
1.2 ASc2S4
Li Na
Mg
Ca
0.4
0.8
1.2 ATi2S4
0.4
0.8
1.2 AV2S4
Migration barrier (eV)
0.4
0.8
1.2
12 16 20 24 28
ACr2S4
(rA+rS)nAnS ∆χ2
AMn2S4
Li Na Mg
Ca
AFe2S4
ACo2S4
12 16 20 24 28
ANi2S4
(rA+rS)nAnS ∆χ2
Figure 4: Migration barriers (in eV) in AB2S4spinels as a function of the migration number
NAS
migr for eight different transition metal cations B= Sc, Ti, V, Cr, Mn, Fe, Ci and Ni upon
variation of migrating cations A= Mg, Na, K, Mg, and Ca.
both corresponding subsets). These results demonstrate that the scaling relations Eq. (3)
are independently valid for the variation of either the cation chemistry of the migrating ions
An+or the variation of the anion chemistry of the host lattice ions Xn.
As Fig. 3 illustrates, upon variation of the host lattice cations Bn+present in the sulfide
spinels AB2X4, which are typically transition metal cations, the slope of the linear scaling
relations represented by the parameter CAin Eq. 2 changes. We have determined the height
of the migration barriers for the six additional transition metals B= Ti, V, Mn, Fe, Co and
Ni as a function of the migration number upon variation of the migrating cations An+and
collected the results in Fig. 4. We again find that the migration barriers follow linear scaling
Table 1: Difference EA
a(B) in eV between the lowest and the highest migration
barrier for the charge carriers A = Li, Na, K, Mg and Na in AB2X4spinels upon
variation of the eight considered transition metals B shown in Fig. 4.
Migrating ion Li+Na+K+Mg2+ Ca2+
EA
a(B) (eV) 0.08 0.19 0.42 0.44 0.61
11
relations, but with different slopes. It is interesting to note that the difference ∆EA
a(B)
between the lowest and the highest migration barrier upon variation of the eight considered
transition metals B increase with the size and the charge of the migrating cations An+, as
illustrated in Tab. 1. Apparently for increasing charge and size of the host lattice cations B,
the specific nature of the interaction between the cations A and B becomes more prominent,
as far as the migration barriers for A are concerned.
Note that in the migration number NAX
migr (Eq. 5), parameters of the migrating cations A
and of the anions X of the host lattice enter. However, in the spinels AB2X4there are also
further cations Bn+, typically transition metal cations, present that are not considered in
the migration number, but which should also be of significance in the A-ion transport. In
these materials, the B-X bond is dominantly covalent. In Fig. 5, we have plotted migration
barriers for MgB2X4spinels as a function of the squared electronegativity difference between
transition metal B and anion X (panel a) and the ionic radius of the transition metal B
(panel b) for a number of MgB2X4spinels. Note that there is some scatter in the data.
However, there is a clear minimum in the height of the migration barriers in panel 5a for
values of ∆χ22. Furthermore, the unit-cell volume of the spinel increases by substituting
a larger B cation into the structure. Again we find a clear minimum in the height of the
migration barriers in panel 5b, here for the ionic radius of the transition metal B at values of
rB1.1. These findings reflect that also the choice of the B cations play a role in minimizing
the ion migration barriers in the spinel compounds. However, we did not manage to identify
any linear scaling relations upon the variation of the cation B. Based on the identification
of these pronounced minima and the corresponding matching properties of Zr, we identified
MgZr2S4as a promising ion conductor with a high ion mobility, and indeed we found that
MgZr2S4has a rather low Mg migration barrier of only 0.3 eV.
We have applied the concept to yet another class of materials that are widely used as
battery materials, namely olivines. 51 Figure 6 shows the migration barriers in the olivine
AFeSiO4as a function of the migration parameter (rA+rO)nAnXχ2
AO (Eq. (2)) for varying
12
0 1 2 3 4
(χ)2in Pauling scale
0.3
0.4
0.5
0.6
0.7
Migration barrier (eV)
MgB2X4
a
0.6 0.7 0.8 0.9 1.0 1.1 1.2
Ionic radius (Å)
0.3
0.4
0.5
0.6
0.7
Migration barrier (eV)
MgB2X4
b
Figure 5: Mg migration barriers (in eV) as a function of the (a) squared electronegativity
difference between transition metal B and anion X, and (b) the ionic radius of the transition
metal B for a number of MgB2X4spinels.
charge carriers A. Again a convincing linear scaling relation has been obtained.
The fact that the migration parameter NAX
migr captures the essence of the migration barrier
height upon variation of the migrating cation A and the anion X of the host lattice calls for a
critical assessment of this parameter. There are some obvious factors influencing the height
of the migration barrier. For larger ions it will be harder to migrate through a given lattice,
13
24 28 32 36 40 44 48 52 56 60
(rA+rO)nAnOχ2
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Migration barrier (eV)
Li Na
Mg
Ca
AFeSiO4
Figure 6: Migration barriers (in eV) in the olivine AFeSiO4as a function of the migration
parameter (rA+rX)nAnXχ2
AX (Eq. (2)) for varying charge carriers A.
therefore it is no surprise that the ion radius rAenters the migration barrier. However, when
also varying the size of the anion of the host lattice, it becomes apparent that it is both
the size of the cation and of the anion represented by rA+rXthat is the critical length
parameter, as already stressed in a previous study.28 Furthermore, note that in many cases
the dependence of the mobility on the ionic radius is not monotonic,29 so any descriptor
of the ion mobility taking into account the ionic radius needs to reflect this non-monotonic
behavior.
It is also well-known that the charge of the migrating ion matters with respect to the ion
mobility. The higher the charge of an ion, the stronger its interaction with the environment
and thus the higher the migration barriers. This same argumentation of course also applies
to the charge of the ions constituting the host lattice as the ionic interaction scales with
the product of the charges of interacting ions. These charges enter the migration parameter
through the product of the oxidation numbers nAnX.
However, it is important to realize that in the migration of “ions” in a host lattice it is not
a priori clear that the “ions” keep their ionic charge. Any crystal containing migrating ions
has to be overall charge neutral because macroscopically charged matter is unstable. Hence
any charge on the migrating ions has to be compensated by the host lattice. Of course, the
assumption that strong ions remain charged in a host lattice makes a lot of sense and is the
14
basis of the concept of formal oxidation numbers. Still, formal atomic charges in a material
are no good observables because it can not be uniquely defined which electrons belong to the
migration ion and which to the host lattice as the electrons are shared between the bonding
partners. This is also the reason why there is a broad variety of different charge partition
schemes52–55 used in quantum chemical codes in order to derive atomic charge numbers which
can give quite different quantitative results. And furthermore, there are hardly any chemical
systems in which the interaction is either purely ionic or purely covalent or purely metallic.
Therefore it is not surprising that trends in the ion mobility cannot be fully understood on
the basis of formal oxidation states alone.
This deviation from the purely ionic interaction can be characterized by the difference in
the electronegativity ∆χ2of the interacting compounds which is also the basis for the Van
Arkel-Ketelaar triangle. In this context it should be noted that the Pauling electronegativity
in the form revised by Allred56 that has been used here is based on a quite accurate, semi-
empirical formula for dissociation energies, namely
(χAχB)2=Ed(AB) Ed(AA) + Ed(BB)
2.(7)
This illustrates that the square of the difference in the electronegativities takes the deviation
from a purely ionic interaction in a compound crystal into account. It is in fact true that
the stronger polarizability of “soft” anions has already been used to explain the higher ion
mobility in chalcogenides containing sulfur and selenide compared to oxides13 with their
softness reflected in the lower electronegativity of sulfur and selenide.57,58 Still this notion
had not been transferred into any descriptor concept before.
The fact that the migration parameter including ∆χ2yields such a good descriptor for
the height of the migration barriers reconfirms that a purely ionic consideration of ion mo-
bility in crystals does not capture all factors determining this mobility. It also means that
this deviation from ionicity is the reason for the observed non-monotonic behavior of the
15
migration barriers as a function of the ionic radii which is correctly taken into account by
including the factor ∆χ2in the migration parameter. It is also important to stress the fact
that the parameters entering the migration number are basically independent of the par-
ticular structure of the considered host lattice, as they correspond to general atomic and
ionic properties of the particular elements. The same parameters enter the scaling relations
for binaries, spinels and olivines, confirming the general fundamental nature of the scaling
relations.
Note that the linear scaling relations as a function of the migration parameter established
in our work do not allow the quantitative prediction of the height of migration barriers in
any particular system without any initially measured or calculated data. Thus they do not
correspond to a parameterization of the barrier height as a function of input parameter across
all families of possible structures. However, these scaling relations allow to make qualitative
predictions of the height of migration barriers, and once some migration barriers are known in
these structures, then even semi-quantitative predictions based on easily accessible materials
parameters can be made. This will be very beneficial for the identification of promising
candidate materials with improved mobility properties. Of course, this linear scaling is not
perfect, and we already identified some outliers. However, this descriptor is based on a
strict physico-chemical reasoning, so deviations from the scaling relations should point to
some interesting additional factors also influencing the ion mobility and thus to an enhanced
fundamental understanding of ion mobility.
Conclusions and Summary
In summary, we propose a descriptor called migration parameter for the ion mobility in
crystalline solids that is based on well-accessible materials parameters, namely ion sizes,
oxidation states and the Pauling electronegativity difference between anions and cations in
the compounds. Thus in contrast to previous attempts to derive descriptors for the ion
16
mobility we also take the deviation from ionic bonding in the compounds into account. For
a broad range of materials classes, we have shown that the height of the migration barrier
follows linear scaling relations as a function of this descriptor upon both the variation of the
cation chemistry of the migrating ion as well as upon variation of the anion chemistry of the
host lattice. This demonstrates the strong predictive power of the descriptor which should
accelerate the discovery of materials with improved migration properties in electrochemical
energy storage and conversion.
Supporting Information - Computational details
DFT calculations
All first-principles calculations were performed in the framework of density-functional the-
ory (DFT)59,60 employing the Projector Augmented Wave (PAW)46 method as implemented
in the Vienna Ab-initio Simulation Package.45,61,62 The exchange-correlation effects were de-
scribed by the generalized gradient approximation (GGA) using the Perdew-Burke-Ernzerhof
(PBE) functional.47 The calculations were optimized using 2×2×2 k-point mesh, with a plane
wave cutoff of 520 eV, and a convergency within 1 ×105eV per supercell.
The nudged elastic band (NEB)44,63 method is applied in the low vacancy limit to define
ion migration barriers. This means that one cation vacancy was created in a large supercell
to ensure the removal of defect-defect interactions across periodic boundaries in all cases, and
the minimum energy path for the propagation of a neighboring cation into this vacancy was
determined by the NEB calculations. All of the structures were fully relaxed until the forces
on the atoms were converged within 0.05 eV ˚
A1. The NEB calculations have been carried
out with seven distinct images for binary compounds, and four distinct images for ternary
spinels to evaluate the Mg-ion migration trajectory. Note that the diffusion in considered
compounds is referred to interstitial diffusion in the literature.
17
Machine-learning approach
The compressed-sensing approach, developed by Ouyang et al.,50 was performed based on
the DFT-calculated data for accurate physical descriptor findings in the classification of the
migration barriers. We have used the following input parameters to describe the migration
barrier energy of binary compounds and ternary spinels: the stoichiometric number of ele-
ments, the electronegativity of each element, the atomic masses of elements, the ionic radii of
elements, the valence electron numbers of each element, and the unit-cell volumes. Further-
more, the machine-learning process was operated within the scikit-learn package64 using the
Anaconda Distribution (Python 3.7.3, numpy 1.16.2).65,66 The random forest algorithm67
was employed for the data training and prediction and both regression and classification.
The output includes the migration barrier energies of compounds. The effect of B cation
in AB2X4spinel was then calculated after obtaining these results. The machine-learning
algorithm was verified by analyzing the mean absolute error (MAE).
Acknowledgments
Useful discussions with J¨urgen Janek, University of Gießen, and Holger Euchner, Helmholtz-
Institute Ulm, are gratefully acknowledged. This work contributes to the research performed
at CELEST (Center for Electrochemical Energy Storage Ulm-Karlsruhe) and was funded by
the German Research Foundation (DFG) under Project ID 390874152 (POLiS Cluster of
Excellence). Further support by the Dr. Barbara Mez-Starck Foundation and computer
time provided by the state of Baden-W¨urttemberg through bwHPC and the German Re-
search Foundation (DFG) through grant no INST 40/575-1 FUGG (JUSTUS 2 cluster) are
gratefully acknowledged.
18
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