Data analysis to understand coordination and topological environments in oxides

Data Analysis to Understand Coordination and

Topological Environments in Oxides

A ,

MASSACHUSETT 'rITI ITF

by

Tina Chen

JUN

0 8

2015

LIBRARIES

Submitted to the

Department of Materials Science and Engineering

in Partial Fulfillment of the Requirements for the Degree of

Bachelor of Science

at the

Massachusetts Institute of Technology

May 2015 (JV " 2' 53

2015 Chen

All rights reserved

The author

publicly paper

hereby grants to MIT permission to reproduce and to distribute

and electronic copies of this thesis document in whole or in part in

any medium now known or hereafter created.

-

Signature redacted

Signature of A uthor!.-... ... ...,.. ... ...

Department of Materials Science and Engineering

May 1, 2015

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Certified

by...

Gerbrand Ceder

R.P. Simmons Professors oMaterials Science and Engineering

Thesis Supervisor

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A ccepted by ... _{... , ... ... ... ... ...}

Geoffrey

Class of

'58

Associate Professor of Materials Science and Engineering

Data Analysis to Understand Coordination and

Toplogical Environments in Oxides

by

Tina Chen

Submitted to the Department of Materials Science and Engineering

on May 1, 2015 in Partial Fulfillment of the Requirements for the

Degree of Bachelor of Science in Materials Science and Engineering

Abstract

Local coordination and topology of ions determine several important properties of materials, including electronic structure, migration barrier, and diffusivity. In this thesis, we employ the Materials Project Database to investigate the coordination preferences of cations and topology of coordination polyhedra in oxides. We calculate the coordination environment preferences of several common cations in oxides, identifying lithium, sodium, calcium, and magnesium ion's preferred coordination numbers are 4-fold/6-fold, 6-fold, 6-fold, and 6-/4-fold coordination respectively. We also develop a method to quantify the connectivity between two polyhedra and determine whether they are point-sharing, edge-sharing, or face-sharing. We find that 4-fold coordinated lithium polyhedra mainly point-share while the 6-fold coordinated lithium polyhedra connectivites are face-sharing. We then build a tool to identify and insert "empty polyhedra" (i.e. coordination polyhedra which are bounded by ions but contain no central ion) which can help to provide a better descriptor of the structure topology. We also find that most connections with lithium polyhedra are with empty polyhedra and that in a connected set of two lithium polyhedra and one empty polyhedron, the coordination polyhedra tend to be either 6-4-6 or 4-6-4 with the empty

polyhedron in the center. Finally, we utilize the database to evaluate Pauling's first and second rules, which are guidelines for current understanding of coordination and topology, and observe that the rules

are generally accurate only within a 30% error margin.

Thesis advisor: Gerbrand Ceder

Table of Contents

1.3 Data-m ining and Data-based Analyses...9

2. Coordination Preference A nalysis ... 11

2.1 Coordination Num ber Definitions ... 11

2.2 Coordination Num ber Frequency Distributions... 13

2.3 Coordination Num ber as a Function of Energy Above Convex Hull ... 17

2.4 Form ation Energy...19

2.5 Discussion ... 21

3. Topology A nalysis and Tool Developm ent... 24

3.1 Topology Tool M ethodology ... 24

3.2 Lithium Topology as a Case Study ... 26

3.3 Discussion ... 28

4. Em pty Polyhedra A nalysis and Tool Developm ent... 30

4.1 Em pty Polyhedra Tool M ethodology... 30

4.2 Em pty Polyhedra in Li Oxides as a Case study... 32

4.3 Discussion ... 35

5. Pauling Rules...39

5.1 Pauling's First Rule... 39

5.2 Pauling's Second Rule ... 42

6. Conclusion...45

7. Bibliography...47

8. Appendix...47

List of Figures

Figure 1: Coordination numbers of univalent, divalent, and trivalent cations... 6

Figure 2: Li-O bond length frequency distribution...7

Figure 3: Pauling 's third rule...

8

F

igure 4: Voronoi polyhedra...1

1

Figure 5: Frequency distributions of Li, Na, Ca, and Mg coordination numbers...15

Figure 6: Comparison of O'Keeffe and effective coordination numbers...16

Figure 7: Energy above the convex hull...

17

Figure 8: Coordination number and energy above hull plots...18

Figure 9: Formation energy vs. coordination number plots...

20

Figure 10: Coordination numbers and formation energy differences between lithiated and

delithiated states. ...

2 1

Figure 11: Polyhedra connectivity types...25

Figure 12: Counting of shared anions...25

Figure 13: Overall Li to Li Polyhedra Connectivity Distribution...27

Figure 14: Distributions of Li to Li polyhedra connectivity by coordination number...

27

Figure 15: Processing steps for adding empty polyhedra to a structure...

31

Figure 16: Frequency distributions of Li to all polyhedra connectivities...32

Figure 17: Frequency distributions of the connectivities in Li-empty polyhedra-Li paths...34

Figure 18: Frequency distribution of the coordination numbers in Li-empty polyhedra-Li

p a th s...3

₄

Figure 19: Structures with common Li-empty polyhedra-Li coordination numbers...36

Figure 20: Pauling's second rule error distribution...

44

Figure 21: Exampled of 5-fold coordinated Li...

49

Figure 22: Example of triangular

prism Li-centered polyhedra...50

Figure 23: Examples of unstable 12-coordinated Li...

51

List of Tables

Table 1: Rules

for

topology method...24

Table 2: Pauling's Radius Ratio rule for coordination numbers. ...

39

1. Introduction

Local coordination and topology are two properties of atoms in materials that determine many other properties, such as electronic conduction and response to magnetic fields. Another property that local coordination and topology may help to predict is ion diffusion. Specifically, an ion's preference in coordination and topological environment may indicate the stability of that ion in different sites of a material. In this thesis, we aim to elucidate the local coordination and topological preferences of common cations with the end goal of developing tools for materials design.

1.1 Coordination Preferences

The current understanding of coordination in ionic compounds is primarily guided by the well-known Pauling's rules.' Pauling's first and second rules present a framework for understanding how ions coordinate. Pauling's first rule provides a guideline based on the ratio between the radii of the cation (re) and the anion (ra). In general, larger cations will have higher coordination numbers. Pauling's second rule ensures local electronegativity by considering the charge of coordinating ions.

Later studies looked more closely at

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coordination numbers of specific cations. Li -891 - - Be<D

B

Mg

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Brown analyzed the relative frequencies of Rb 6ee'ee Sr .eeeo| 00 in Q

Cs -ooooeeeeeo * ee.4eO _TBa ri 0 o

coordination numbers for different cations, _It

wobeee e Sn 0911-wo* AS

Pb *0oeese- 'o Sb

showing that cations in inorganic structures Cu Q-- " Bi O

have "preferences" toward certain Fe &a

o

cr

x

CO Odd * Mn 0 x

N Fe 0 x

coordination numbers.2

Using ICSD data Cu

Qeo-Zn .Ye y OO

available at the time, Brown conducted an LO 9 o

analysis of over

14,000 coordination

Figure 1: Coordination numbers of univalent, divalent, and

envionmntsof 00 iffeentcatonsand trivalent cations.

environments of 100 different cations and The area of the circle corresponds to the number of samples observed (normalized to the same total for each cation). The compared coordination numbers to those coordination number predicted by ionic radius ratio is

predicted by ionic radius ratios. One result of Brown's study was that some cations tended to have strong preferences for certain coordination environments. For example, Mg ions strongly prefer the

6-coordination environment, while Ca ions prefer 6 - 8 coordinations as seen in Figure 1. There were also,

however, ions which did not show preference for a single coordination environment but instead tended to appear in a range of coordination environments. He also discovered that the observed coordination numbers tended to be smaller than or equal to the coordination numbers predicted by Pauling's first rule. Smaller coordination numbers were associated with stronger bonds, and the Lewis-base strength of the anion strongly affected the coordination number.

While some studies attempted to find trends in coordination number, a few looked more closely at specific cations. In particular, Wenger studied Li's coordination and bonding to oxygen, comparing bond lengths and strengths for different coordinations of Li using a data set of 134 Li polyhedrons from 94 compounds, as shown in Figure 2.3 Wenger found that Li's coordinations ranged from 3-fold to 8-fold, and 4-fold coordination was the most preferred. Wenger also discovered that octahedral (6-fold)

coordination sometimes has uneven bonds, instead bonding in a 4+2 substructure rather than a symmetric

26

22-

U coordination:

E

6-2

111i&i'

nnnn,

Figure 2: Li-O bond length frequency distribution.

The frequency distribution of bond lengths between Li and 0, also grouped by Li coordination, shows a peak around 2.0 angstroms for 4-coordinated Li. 6-coordinated Li has bond lengths ranging from 2.1 angstroms to 2.3 angstroms. Based on this, we expect to find most 0 coordinated to Li within a 2.4 angstrom radius. Figure from Wenger, 1991.3

1.0 ~~, 0.38

Figure 3: Pauling's third rule.

An illustration of Pauling's Third Rule shows that two polyhedra with higher connectivity between them have closer cations. This results in higher connectivity being less stable due to the electrostatic repulsive forces between central cations. Figure from Nelson, 2014.4

octahedral substructure. Brown and Wenger demonstrate that cations often have preferred coordination numbers, and we look to verify these coordination number preferences using larger-scale data.

1.2 Topology in Materials

While Pauling's first and second rules have provided a framework for understanding how ions coordinate, his third and fourth rules describe features of topology that are generally preferred in ionic crystals. Pauling's third rule, demonstrated in Figure 3, uses electrostatic interactions to explain why point-sharing connections are more stable than edge- or face-sharing connections. Pauling's fourth rule states, again using an argument based on electrostatic interactions, that cations with higher charge and lower coordination number tend to form fewer connections with other polyhedra. Pauling's third and fourth rules thus provide an explanation for the trends observed in the topology of structures.

Other work in topology of structures has centered around understanding networks in structures. In one of the earlier works describing topology in terms of nets, Wells investigates the properties of periodic graphs using planar tessellations. Blatov et al analyzes nets through the enumeration of periodic tiling with application to zeolite nets.6 Carlucci et al analyzes the types of entanglement that appear in materials such as organic and metal-organic networks.7 More recently, Blatov et al have proposed the use of a computational software package that helps to visualize and analyze topology and networks. While these studies have focused on the overall connectivity and network of structures, here we focus not only on the network but also on the local topology and how polyhedra in the network are connected.

1.3 Data-mining and Data-based Analyses

In the studies of Brown and Wenger, only 100-200 Li samples were considered. Recent efforts to expand current materials databases, such as the Materials Project,9

have increased the potential to use data mining and data based analysis to make predictions about materials. One major application of data mining has been the prediction of crystal structures. Fischer combined data mining and quantum mechanics to predict binary crystal structures as well as the probabilities they will form.1 Hautier expanded on Fischer's work by predicting ternary crystal structures" as well as structures that could be formed from ionic substitutions.

Based on Hautier's work predicting ionic substitutions, Yang looked to expand on the ionic substitution similarity function Hautier et al developed by proposing methods for assessing the similarity of compositions, substructures, and crystal structures based on data-mined probabilities.3

' 14 One specific

application of her substructural similarity function was the prediction of Li intercalation sites in oxides. To predict Li intercalation sites, Li ions were removed from structures, and all possible Li sites were considered with respect to the Voronoi polyhedrons in the delithiated oxides. From the possible Li sites, the most likely Li sites were found by comparing the substructure similarity between the site and all known substructures in a training set. In our study, we develop tools that allow topology to be a feature added to Yang's structural similarity function.

This thesis aims to obtain a better understanding of coordination and topology with the goal of helping to develop tools for material design of oxides. A better understanding of coordination preferences could help us design a better descriptor for diffusion. Because diffusion involves the migration of a species with a changing local coordination environment, the local environment should affect the migration barrier.'5 If we consider that how strongly an ion prefers its coordination environments is

related to its energy in that site, then it is possible to trace out possible diffusion paths knowing the coordination environments of all potential cation sites within a structure.

A better understanding of topology would help current efforts to design a function for measuring

the similarity between structures. Specifically, a tool that could describe topology of a structure could be used as a descriptor by which structures are compared. By adding topology as a criterion for a structural

similarity function, we allow the function to identify materials with similar topology-dependent properties.

Preferred coordination environments for several common cations will be found using data analysis of the Materials Project Database. We then follow with the development of tools to obtain the connectivity of coordination polyhedra and the positions of empty polyhedra, which are subsequently used to find trends in the connectivity between cation polyhedra. Finally, we investigate the utility of Pauling's first and second rules using data from the Materials Project Database.

2. Coordination Preference Analysis

The environment of a site has a strong relationship with the stability of any cation in that site. Specifically, the interactions between the cation and its nearby neighbors determine how strongly the cation is tied to the site. We define coordination environment of a cation as the set of nearby anions bonded, or coordinated, directly to the cation. The coordination number of that cation is then the number of nearby anions coordinated to the cation. These anions form a polyhedron around the central site, with the shape of the polyhedra depending on the number of anions coordinated to the central site.

We employ the Materials Project Database to sample the coordination preferences of several cations in order to identify their preferred coordination. To do this, we first require a mathematical definition of coordination number to process the structutal data from the database. We then measure coordination preferences by the frequency of the appearance of coordination numbers based on our definition of coordination number. We also explore the use of other measures of coordination preference, including the energy above the convex hull and the formation energy. Finally, we discuss the potential materials design applications, specifically for diffusion, of being able to calculate coordination preferences as well as evaluate ours measures of coordination preference.

2.1 Coordination Number Definitions

While some structures have clear coordination polyhedra formed around cation sites, in others it is often unclear how atoms are coordinated to each other. We use two definitions of

coordination number to measure the coordination environment of any structure.

2.1. O'Keeffe Coordination Number

In his 1979 paper, O'Keeffe offers a coordination number

e

(b

Figure 4: Voronoi polyhedra.

a) The Voronoi polyhedron for the body-centered cubic array. b) The Voronoi polyhedron for the simple cubic array after a trigonal distortion. c) The Voronoi polyhedron for the face-centered cubic array after a small tetragonal distortion. Figure from O'Keeffe 1979.1s

based on weighting the faces of the Voronoi polyhedron, examples shown in Figure 4, around the central ion.'6 _{The planes that bisect lines drawn from the central ion to neighboring ions create the faces of the}

Voronoi polyhedron. Thus each coordinating anion contributes a face to the Voronoi polyhedron around the central cation. This method weighs the coordination number in proportion to the solid angle Qy subtended by that faces at the center. The neighbor subtending the largest angle is given a weight of 1, and other coordinating anions are given weights of Qx/Qmax. The coordination number is then the sum of these weights:

CNO'Keeffe = - ,(Equation 1)

DmaxI

2.1.2 Effective Coordination Number

Hoppe proposes the use of the effective coordination number (CNeffective).'7 The effective

coordination number is calculated by weighting the nearby anions based on the bond lengths (li) as compared to the bond length (min) of the closest anion. This method results in higher coordination numbers when anions are equidistant from the central ion and lower coordination numbers when anions have very different distances from the central ion.'8 Effective coordination number is calculated by summing the weights:

CNEffective = i exp

(

where lav is the weighted average bond length is equal to

= i l1exp (1-( )

lav =mi (Equation 3)

YEj exp (1-(min

However, to use this method, a radius must be given to determine the number of ions included in the calculation of the coordination number. In doing this, we make assumptions about the cation to anion bond lengths. To mitigate this difficulty, several radiuses are tested to approximate which radius gives the most accurate coordination number. Another possible way to address this difficulty is by varying the radius used based on a property like the distance to the nearest anion.

2.2 Coordination Number Frequency Distributions

In order to understand the preferences of cations, we look at the stability of the cations in different coordination environments. The main metric we use to measure stability is the frequency distribution of coordination numbers over all calculated coordination numbers for a single cation. Through this distribution, we find the coordination numbers that occur most frequently.

2.2.1 Data Collection and Analysis Methods

Data sets of 1,000 structures and their energy above the convex hull data were obtained from the Matgen version of the Materials Project Database. The Materials Project Database includes crystals from the ICSD database and from the Materials Project. This data was filtered using several criteria similar to the filtering methods described by Yang.'4

In Yang's thesis, only compounds satisfying the following criteria were included the data set:

(1) Compounds must be oxides; consider a compound as an oxide if it contains more than 20%

oxygen.

(2) Compounds must not be peroxides or superoxides; consider a compound peroxide and non-superoxide if 0-0 bond lengths are less than 1.5 angstroms.

(3) Compounds must not be marked high temperature or high pressure.

(4) Compounds must not have a mismatch between the reported composition and the ions given in the structure.

(6) Compounds must be charge balanced; consider a compound charge-balanced if the sum of the

charges of all species is less than 0.001 e.

From the data within the Materials Project Database, we filtered using several criteria based on those given in Yang's thesis.'4 Specifically, we used criteria (1), (2), (5), and (6). We exclude criteria (3) and (4) from our search because the Materials Project Database has already been cleaned of these data.

The coordination numbers were then calculated using both the O'Keeffe coordination number and effective coordination number formulae. Frequency distributions of O'Keeffe coordination number and effective coordination number were graphed as separate histograms with bin size of 0.2 coordination number units. The data set was further filtered by the energy above the convex hull value of the structure to obtain data sets containing more stable materials. We use 0.1 eV/atom as the cut-off for a subset of

stable compounds.

2.2.2 Coordination Number Distributions of Several Common Cations

We look at the frequency distributions of several common cations, namely Li, Na, Ca, and Mg. Based on the frequency distributions in Figure 5, we observe relative preferences of coordination numbers for each cation. For Li, the frequency distribution peaks at 4-fold coordination and is also relatively high in the 4-fold to 6-fold range using the O'Keeffe coordination number. With the effective coordination number, a peak at 6-fold coordination rather than a plateau from 4 to 6 coordination appears. This is in agreement with Wenger's results, which show that 4-fold coordination is the most preferred.3 For the Na distribution, we observe peaks at 4-fold and 6-fold coordination when using effective coordination number but a much broader peak at 6-fold coordination that decreases in gradually intensity farther from 6-fold coordination when using the O'Keeffe coordination number. The Ca distribution shows a greater preference for 6-fold coordination, with a strong peak in the effective coordination number distribution and a broad peak in the O'Keeffe coordination number distribution. The broad peak in the O'Keeffe

o Keefe Coordination Number Frequency Energy Above Hull Cutoff 0.1

I

M E above hull < 0. 1

b.

0-

-00 2 4 6 8 10 12 14 0 Keefe Coordination Number

0 Keefe Coordination Number Frequency

Enermw Above Hull CutOff 0.1 45s0 40D C. 33O

Na

zo

so

0 Ke

.

Ca

f

00

M E above hull < Infinite

M E above hull < 0.1

d.

1 6 8 10 12 14

o Keefe Coordination Number tfe Coordination Number Frequency

Energy Above Hull CutOff 0.1

M E above hull < Infinite

I E above hull<01

4 6 8 10 12 14 0 6eefe Coarditsion Number

o Keefe Coordination Number Frequency Energy Above Hull CutOff 0.1

M E above hull < infinite

400 E above hu < 0. 1 350 300 250 150 100 so 0 2 4 8 e 10 12 14

0 Cedfe Coordination Number

f.

h.

a.

Effective Coordination Number

Effective Coordination Number Frequency

ne'ov Cutoff - 0.-1

M E above hull < infinite

M E Above hull < 0.1

I

2 4 6 8 10 12 14

Effective Coordination Number Effective Coordination Number Frequency

Enerov Cutoff - 0.1

1200

E above hull < Infinite

1 E Above hull < 0.1 1000 800 600 400 200 2 4 6 8 10

Effective Coordination Number

Figure 5: Frequency distributions of Li, Na, Ca, and Mg coordination numbers.

Histograms showing frequency of appearance of coordination numbers, binned with bin size 0.2. Blue data includes all data in the data set. Green data includes only data having energy above hull less than 0.1. Graphs on the left (a, c, e, g) use O'Keeffe coordination number to calculate coordination number while graphs on the right (b, d, f, h) use effective coordination number. The first row (a, b) are frequency distributions for Li coordination numbers, the second row (c, d) are frequency distributions for Na coordination numbers, the third row (e, f) for Ca coordination numbers, and the fourth row (g, h) for Mg coordination numbers. Each set consists of 1,000 oxides containing the specified cation.

Effective Coordination Number Frequency

Energy Cutoff - 0.1

M E above hull < infinite 1200 M E Above hull < 0.1 1000 800 600 400 200 0 2 4 6 a 10 12 14 Effective Coordination Number

Effective Coordination Number Frequency

Eneray Cutoff -0.1

M E above hull < infinite

1200 M E Above hull < 0.1 1000 800 400 200

g.

Mg

distribution gradually decreasing in intensity away from 6-fold coordination with the ends of the peak ranging from 4-fold to 6-fold coordination, similar to the Na O'Keeffe distribution. The Mg distribution prefers both 4-fold and 6-fold coordination, showing strong peaks for those coordination numbers when using effective coordination number. It also shows these peaks when O'Keeffe coordination number is used, but these peaks decrease in intensity asymmetrically farther from top of the peaks at 4- and 6-fold coordination. O'Keeffe coordination numbers below 4-fold and above 6-fold coordination have very little intensity while the region between 4- and 6-fold coordination has a valley at 5-fold coordination. These results match those of Brown as well as common knowledge, which expects Li to have 4- and 6-fold coordination, Na to have fold coordination, Ca to have and 8-fold coordination, and Mg to have 6-fold coordination.2 A brief exploration of some of the substructures with uncommon coordination

numbers is given in Appendix A.

In the histograms in Figure 5, we note the

Comparison of O'Keefe and Effective

preference of more plateaus in O'Keeffe 14 . Coordination Number

coordination numbers-based frequency 12 ..

distributions and more distinct peaks in effective E 10

z O

C

coordination number-based frequency 8

E

distributions. Distributions using effective 0 6

561

coordination number also tend to extend to very u 0.

low coordination numbers (below 3-fold 2

coordination). This is likely because distances 0 _{S 2 4 6 8 10 12 14}

O'Keefe Coordination Number

between the cation and coordinating anions are not

Figure 6: Comparison of O'Keeffe and effective

evenly weighted, causing coordinating anions that are coordination numbers.

For each Li oxide in the sample of 1000 Li oxides, the

farther away to be very lightly weighted. Based on coordination numbers of each Li site were averaged

separately for both methods to obtain a single O'Keeffe coordination number and a single effective coordination this, the peaks around coordination 2 and 3 may be number for each oxide. The y = x line is given as a

reference for the case where O'Keeffe and effective

from distorted coordination 4 and coordination 6 coordination number calculate the same coordination

polyhedron (for example, in a coordination 4 polyhedron, 2 anions are closer to the central ion than the other 2 anions, resulting in a effective coordination number close to 2). This is supported by Figure 6, which plots the O'Keeffe coordination number versus the effective coordination number of each substructure. Most points fall below the y = x line, implying that the O'Keeffe coordination number of a substructure is almost always greater than or equal to the effective coordination number, and the effective coordination number is much more likely to underestimate the actual coordination number. Based on Figure 5, it appears that the O'Keeffe method is better at classifying coordination number outside of the 4-fold to 6-fold coordination number range. However, it does not differentiate between 4-fold and 6-fold coordination as well as the effective coordination number does.

2.3 Coordination Number as a Function of Energy Above Convex Hull

In addition to frequency, we also explore the use of energy above the convex hull as a measure of site stability. Specifically, when a cation is in its preferred site, it should be in a low energy state. This should be reflected in the energy above the convex hull, which is a measure of the stability of a structure relative to the most stable phase of the material.

2.3.1 Data Collection and Analysis Methods formation energy

Energy above convex hull is calculated as the _{A2B AB AB2 B}

formation energy difference between a compound and the fraction

convex hull trace.'9 In Figure 7, phase y is an unstable

ao

Phases A, B, a,, and P2 are all stable phases. The energy Figure 7: Energy _{Energy above hull measures how far a material}above the convex hull. is from its most stable form. The green line

above convex hull for phase y is calculated by the shows the stable phases of the material. The

arrow from the unstable phase, y, to the convex

difference between the formation energy of y and the hull is the eneir above hull. Figure from

Hautier, 2004.'

formation energy of stable phase on the convex hull with the same composition. Materials with zero energy above hull are in their most stable form. On the other hand, materials with positive energy above

hull are unstable because a more stable phase exists on the convex hull. Thus, energy above hull is a metric that measures the stability of compounds.

Again, we obtained data sets consisting of 1,000 structures and their energy above the convex hull data from the Materials Project Database as in Section 2.2.1. The coordination numbers were then

calculated using both the O'Keeffe coordination number and effective coordination number formulae. We then bin with a bin size of 0.5 coordination number units and average the lowest 1 0'hpercentile of data. This attempts to remove data from structures that are unstable due to factors other than the specific cation site.

2.3.2 Coordination Number and Energy Above Hull Results

We first look at the relationship between coordination number and energy above hull for Li-containing oxides. In Figure 8, the trends between coordination number and energy above hull are much less clear than those from the frequency distributions. Li is expected to have preferred coordination numbers of 4- and 6-fold coordination, and these specific coordination numbers indeed appear to have lower energy above convex hull. However, the data outside the 4- and

6-Evaluation of OKeefe Coordination Number Evaluation of Effective Coordination Number

0.6 Using 10 Percentile Values 0.6 Using 10 Percentile Values

* 10th percentile * 10th percentile . median - median 0.4 00.4 0.3 0.3 IV < 0.2 < 0.2 0.1 0. 4.0 2 4 6 8 10 2 14 10. 2 4 6 10 2 14

OKeefe Coordination Number Effective Coordination Number

Figure 8: Coordination number and energy above hull plots.

O'Keeffe coordination number (left) and effective coordination number (right) vs. the 10th percentile energy above hull value in eV/atom. The 10th percentile was chosen to remove unstable oxides since that instability

likely came from sources other than Li sites. The 50th percentile (median) is given in blue as a reference. Data was taken from a sample of 1,000 Li oxides.

fold coordination number region are statistically unreliable as the bins outside this area contain very few data points. In particular, bins above 8-fold coordination generally had less than 10 data points in their bins. Because of this and reasons mentioned further in the discussion, we believe that energy above hull is not an adequate measure of site stability for identifying coordination preferences.

2.4 Formation Energy

We also explored formation energy as another metric of site stability. Similar to the energy above convex hull, formation energy should be low when a cation is in a site with its preferred coordination number. One important distinction of the formation energy is that it is calculated relative to an unchanging reference, which is the energy of the elements in their elemental forms.

2.4.1 Data Collection and Analysis Methods

Formation energy, also called heat or enthalpy of formation, is also used to measure stability of a system. Formation energy is calculated as the change in energy when one mole of a substance is formed from its pure elements under standard state (standard temperature and pressure). The more negative formation energies indicate more stable compound because a system with lower formation energy requires more energy to be reverted to its pure elements. We then expect materials with more negative formation energy to be more stable and to be more likely to have preferred coordination numbers.

We obtained data sets consisting of 1,000 structures and their formation energy per atom data from the Materials Project Database as in Section 2.2.1. The coordination numbers were then calculated using both the O'Keeffe coordination number and effective coordination number formulae. We then bin with a bin size of 0.2 coordination number units and average the lowest 10th percentile of data. This

attempts to remove data from structures that are unstable due to factors other than the specific cation site. Additionally, we perform an analysis in which we isolate the formation energy of the cation by by subtracting the formation energy of the delithiated structure from that of the lithiated structure of a

material. For this, we obtained 146 stable materials and matched the space groups of the lithiated and delithiated states of the material. This decreases the sample size we are able to work with but allows us to isolate the energy from the site from energy due to other factors in the structure.

2.4.2 Coordination Number and Formation Energy Results

We start by looking at the plots for Li-containing oxides of coordination number and formation energy per atom. In Figures 9, showing the plots for Li-containing oxides, a trend is less evident than in the plots of energy above the convex hull. In particular, there appears to be no dip in formation energy correctly identifying Li's preferences as 4-fold coordination. In addition, the preferred coordination number (the coordination number with the lowest formation energy per atom) seems to be 2-fold coordination, which is incorrect One explanation is that many of the 4-fold coordinated materials are actually misclassified as 2-fold coordinated based on the method used to calculate effective coordination number. Another is that these 2-fold coordinated materials exist in 2-fold formation but are unstable. However, overall, the plots indicate that formation energy by itself is also not an adequate indicator of the coordination preferences of cations in oxides.

Evaluation of OKeefe Coordination Number Using

10 Formation Energy Percentile Values Evaluation of Effective Coordination Number Using

-10hpe l -1 10 Formation Energy Percentile Values

S-10th percentile 120h .ceti E *median *10th percentile median E-2.0 - _2. -3.0 -2.0-OC., 00 0I 0 0 2 4 6 a 10 12 04 0 2 4 6 8 10 12 14

OKeefe Coordination Number Effective Coordination Number

Figure 9: Formation energy vs. coordination number plots.

O'Keeffe coordination number (left) and the effective coordination number (right) vs. the average of the 10th percentile formation energy per atom value (in kJ/atom) for a sample of 1000 Li oxides. The 10th percentile was chosen to remove unstable oxides since that instability likely came from sources other than Li sites. The 50t percentile (median) is given in blue as a reference.

We then isolate formation energy _{Evaluation of Effective Coordination Number Using}

contributions from Li by taking the difference in 2 Formation Energy

E 0

formation energies between the lithiated and ~ delithiated states of a material and view these o -4

differences in relation to the effective . -5

E!

coordination numbers of Li in the material. In _ C

0

Figure 10, we observe the lowest differences in

E -7

0

formation energy around 4-fold and 6-fold

0 2 4 6 8 10 12 14

coordination. This agrees with the results from Effective Coordination Number

the Li frequency distributions. However due to Figure 10: Coordination numbers andformation energy differences

between lithiated and delithiated states.

The effective coordination number vs. the difference between our data set, which is all stable materials and formation energies of lithiated and delithiated states of 146

stable materials (in kJ/Li atom). For this graph, we used bins of only those with both lithiated and delithiated size 1 coordination number unit. Due to the nature of the

materials in our dataset, only coordination numbers between 3 states available, we are only able to obtain data and 6 were obtained.

for coordination numbers between 3 and 8. Therefore even though it may be the most accurate approach, the method of using lithiated and delithiated materials to measure site stability is limited.

2.5 Discussion

2.5.1 Applications of Coordination

Preferences

In our study of coordination preferences, we identified the coordination preferences of cations using the frequency distribution. For Li we identified 4-fold coordination as the most preferred

coordination number environment. 6-fold coordination was also a preferred coordination environment but was less preferred than 4-fold coordination. Originally we predicted that cations would have low energy when they were in their most preferred sites and higher energy when they were not in their preferred sites. The energy of a cation in a site is related to its migration energy barrier to other sites." We can then

describe a cation's likelihood of migrating from a site in terms of the cation's preference (or lack therefore) for the site.

Movement of a cation between two sites requires the cation to overcome the migration barrier between the sites. Part of this migration barrier is the energy difference between the target site and the current site of the cation. This means that when a cation is in its most preferred site, it is the least able to

migrate to another site. Conversely, when a cation attempts to move to a target site that does not have preferred coordination, it is unable to enter the target site due to the high energy state of the target site.15 Based on this reasoning, the best sites for cation migration are those that are moderately preferred - that is, they should be preferred enough that the cation is willing to enter but also not so preferred that the cation is unwilling to leave. In the case of Li, if we wanted to look for materials with better diffusion, based on Figure 5a and 5b, we might start by looking for materials with a large number of 5- or 6-fold coordinated Li sites because those coordination numbers seem relatively stable or low energy but still not as stable as 4-fold coordinated sites. In this way, coordination preferences can potentially help to identify materials with higher diffusion of different cations.

2.5.2 Coordination Preference Metrics

Our main metrics of stability in this thesis were frequency, energy above hull, and formation energy. While frequency is not a direct measure of stability (like energy above hull and formation energy are), frequency is a strong metric to use because it is able to isolate the effect of the coordination number. That is, a cation in a specific coordination environment will only appear frequently if it is relatively stable in that environment.

Energy above hull is a very good measure for the general stability of a compound but not for identifying the preference of coordination numbers. In particular, we are able to determine whether a compound is stable or not because materials with energy above hull of 0 are in their most stable form by definition. However, energy above hull is a characteristic of the entire material. It therefore takes into account instabilities from sources other than cation sites. This means that when we use energy above hull

as a metric for site stability, rather than compound stability, a site may be stable even though the energy above hull value for the material is non-zero.

Formation energy is similar to energy above hull in that it is a good measure of the stability of the compound but lacks as a measure of site stability for similar reasons. One advantage of formation energy

is that it has consistent energy references. While energy above hull has a different ground or lower boundary for each different compound, formation energy is always measured against the compound in its pure elemental form (that is when the compound is decomposed into its pure elements in standard state). However this also means that formation energy can only be used to measure relative stability between compounds since there is no clear binary way of determining whether a compound is more or less stable. Further, a more accurate floor to compare the lithiated and delithiated states to would be oxides of the same valence rather than the compound in pure elemental form. However, the formation energies with respect to oxides of the same valence are not readily available.

Part of our inability to use formation energy comes from a lack of data. Ideally, if we wanted to measure the stability of a site, say for Li in a compound, we could de-lithiate the compound, and then put a single Li ion in a specific site. Alternatively we could fully lithiate the compound and take out a single Li ion. In these cases, if we were able to calculate formation energy for the states before and after we add or remove a single Li ion, then this would be a much more direct measure of the stability of the cation in that specific site with that specific coordination number as most of the structure is unchanged.

3. Topology Analysis and Tool Development

The topology of a structure describes the next level of environment around a cation site. While coordination polyhedra describe how atoms are connected, topology describes how coordination polyhedra are connected. Understanding how polyhedra are connected has several applications, one of which is as a descriptor for structure. Specifically, if we want to be able to compare two structures, topology could be one descriptor with which we evaluate how similar those structures are. Another potential application is diffusion. Having a descriptor that measures how two polyhedra are connected can help us predict the migration paths between those polyhedra. We will focus mainly on Li oxides as they are abundant in the Materials Project Database due to their application as battery electrodes. In this section, we develop a tool to calculate how any two polyhedra are interconnected. We then use this tool to examine trends in topology by using data from the Materials Project Database.

3.1 Topology Tool Methodology

In order to understand topology, we first need a way to measure and calculate how two

polyhedral are connected. Specifically, we want to identify whether two polyhedra share a point, an edge, or a face. The types of connectivity are shown in Figure 11. We do this by counting the number of anions shared between the polyhedra, with the rules shown in Table 1.

Number of Shared Anions Connection Type

0 Unconnected

1 Point-sharing

2 Edge-sharing

> 3 Face-sharing

Table 1: Rules for topology method.

The rules used in the topology tool to define how polyhedra are connected. We count the number of shared anions between any two polyhedra to classify how those two polyhedra are connected.

Figure 11: Polyhedra connectivity types.

Structures demonstrating the types of connections between different polyhedra. The structure on the left shows a set of two point-sharing tetrahedra; the structure in the center shows a set of two edge-sharing tetrahedra; the structure on the right shows two sets of face-sharing polyhedra, one tetrahedron and one octahedron, on the top and bottom of the structure. The faces shared between the pairs of polyhedra are triangular faces.

If only one anion is shared between the two

polyhedra, then the two polyhedra are considered to be point-sharing. This is because the corners of the polyhedra are defined by the anions. If only one corner is shared between the polyhedra then by convexity that corner is the only point that is shared between the polyhedra. If the

Figure 12: Counting of shared anions.

polyhedra share two anions, then the polyhedra must be Schematic demonstrating how the topology

method counts the number of shared anions to determine the connectivity between two

edge-sharing, as in Figure 12. If the polyhedra share three polyhedra. The shared anions are circled in

red and the polyhedra are circled in blue.

or more anions, then they are most likely face-sharing.

Specifically, two polyhedra sharing three anions would be face-sharing via a triangular face, and two polyhedra sharing four anions would be face-sharing via a square face. We note that it is possible for three anions in a single line to result in edge-sharing between the polyhedra, but the occurrence of a polyhedra with this condition is very uncommon. If the polyhedra do not share any anions, then the polyhedra must be non-sharing and have no points of contact with each other.

The tool then uses this method of calculate connectivity to identify, for any cation-containing oxide, how polyhedra in the structure are connected. More specifically, it looks through the occurrences

of one type of cation in the structure and checks how it is connected to every other cation-centered polyhedron in the structure. It then counts the number of each type of connection (point-sharing, edge-sharing, or face-sharing). This tool then provides a description of the topology of the structure by quantifying how the polyhedra are connected within a single structure. In Figure 12, the structure on the left would give one point-sharing connection; the structure in the center would give one edge-sharing connection; and the structure on the right would give two face-sharing connection and two unconnected polyhedra.

3.2 Lithium Topology as a Case Study

We now investigate Li oxides using this tool in order to discern some trends in cation polyhedra topology. These trends give us a better general understanding of how Li polyhedral are connected in Li oxides.

3.2.1 Data Collection and Analysis Methods

We obtained a data set from the Materials Project Database with 10,000 Li-containing oxide using the method and filters described in Section 2.2.1. We then used the topology tool on each of these

10,000 structures and summed up the counts of each type of connection to see the overall trend of which

connectivities occurred most frequently between Li polyhedra. We then grouped the connections based on the coordination environment of the polyhedra involved in the connection and plotted the counts of each type of connection for common Li site coordination numbers.

For this second analysis, we remove the structures with energy above convex hull above

10-eV/atom so that we only account for the coordination numbers of stable structures. We also define coordination number using the effective coordination number, as described in Equation 2. While effective coordination number may sometimes underestimate coordination number, the effective coordination

Connectivity Frequency Distribution of

Li Polyhedra in Li Oxides

Non- Point

sharing Sharing Sharing Edge SharingFace Connectivity

Figure 13: Overall Li to Li polyhedra connectivity distribution.

Distribution of count of each type of Li polyhedra to Li polyhedra connection based on a data set of 10,000 Li-containing

oxides.

Connectivity Frequency Distribution of

120C 3.0 Coordination Number 100 800 U . 600 400 200 U Cr U-Non- Point sharing Sharing U Cr

U-Edge Face Connectivity Sharing Sharing

Connectivity Frequency Distribution of

1200 5.0 Coordination Number 1000 600 600 400 200

. mE

.

Edge Face Connectivity Sharing Sharing

Connectivity Frequency Distribution of

1200 4.0 Coordination Number 100 100 600 400

200

-srNon- Point Edge Face Connectivity sharing Sharing Sharing Sharing

Connectivity Frequency Distribution of 6.0 CoordInation Number 1200 2000 800 600 400 200 --6 0 Corinto Nube Non- Point sharing Sharing

Figure 14: Distributions of Li to Lipolyhedra connectivity by coordination number.

The top left figure shows the distribution for Li polyhedra that are (rounded) 3-fold coordinated; the top right figure shows the distribution with (rounded) coordination number 4; the bottom left figure shows the distribution with (rounded) coordination number 5; and the bottom right figure shows the distribution with (rounded) coordination number 6.

27

Edge Face Connectivity Sharing Sharing

number can be calculated much more quickly, and resulted in more precise peaks in the coordination number frequency distributions presented in Section 2.2.2.

3.2.2 Li Polyhedra Connection Distributions

We first look at the counts of each type of connection for all 10,000 Li-containing oxides. In Figure 12, we observe that the most common type of connection between Li polyhedral is point-sharing. Face-sharing connections between Li polyhedral is the least common, while edge-sharing is moderately common. There are also more non-sharing polyhedral than there are edge- or face-connections.

Then we look at the distributions of connections grouped by the coordination number of the Li polyhedra involved in the connection. These distributions account only for stable materials (defined as below 10- eV/atom). In Figure 13, we see that the Li polyhedra with coordination numbers 3 and 4 tend to connect to other Li polyhedra mainly through point-sharing. Li polyhedra with coordination numbers 5 and 6, on the other hand, tend to connect to other Li polyhedra mainly through edge-sharing. Although we only investigated Li in this analysis, other cations (Na, K, Ca, Mg) on which we used the topology tool showed similar trends with lower-coordinated polyhedra (particularly tetrahedra) more commonly connected to other polyhedra by point-sharing and higher-coordinated polyhedra (namely octahedra) more commonly connected to other polyhedra by edge-sharing.

3.3 Discussion

Topology helps us to understand how polyhedra are connected throughout a structure. From Figure 12, Li polyhedra seem to have relatively low connectivity to each other, with most connections between Li polyhedra being point-sharing and very few being face-sharing. In addition we observe more non-sharing polyhedra than there are edge- or face-sharing connections. Non-sharing and point-sharing connectivity are the dominant connectivity types in Li polyhedra. This agrees with Pauling's third rule, which states that point-sharing connections are more stable than edge-sharing or face-sharing

connections. Because edge-facing and face-sharing connections place the polyhedra closer to each other, it also brings the cations of the polyhedra closer together, increasing the electrostatic repulsion between

the two cations. The effect becomes stronger for lower coordination numbers because coordination polyhedral with lower coordination numbers are smaller. Thus Pauling's third rule also provides an argument for the higher proportion of edge-sharing in for 6-coordinated Li polyhedra compared to 4-coordinated polyhedra.

A possible application of the topology tool is as a descriptor for comparing structures. One

objective of comparing structures is to streamline the search for better materials. Because of the dependence between structure and many different properties, materials with similar structures should have similar properties. Thus by searching for materials with structures similar to materials with desirable properties, we are essentially searching for materials with those desirable properties. However, the ability to search for similar structures depends on the ability to compare those structures. Because some

properties depend on the topology of the structure, this information should also be a criterion to evaluate structural similarity.

Another application of the topology tool is toward understanding how cations migrate between cation sites. In the Li polyhedra case study, we discovered trends in how polyhedra connect to other polyhedra using the topology tool developed. In particular, we noted that there were in general fewer face-sharing connections between polyhedra than either point-sharing or edge-sharing connections. In terms of migration, face-sharing polyhedra can potentially correspond to the easiest path from one polyhedron to another. A cation in a polyhedron that is face-connected to another polyhedron can simply pass through the face to the other cation site. Edge-sharing and point-sharing polyhedra on the other hand lack such a direct path between the polyhedra because cations very rarely pass through an edge and cannot pass through a point. Instead, they must pass through one or more empty sites that could be occupied by a cation but are vacant, which we hereafter refer to as "empty polyhedra". Non-sharing polyhedra are required to pass through more than one empty polyhedron to migrate from one cation site to the other. Further investigation of empty polyhedra will be presented in the following section.

4. Empty Polyhedra Analysis and Tool Development

Structures can be thought of as networks of coordination polyhedra. Cations coordinate to surrounding anions or anions coordinate to surrounding cations, creating occupied polyhedra throughout the structure. In this thesis we consider polyhedra with cations in the center and anions as coordinating

ions because we focus on the coordination of cations. Between occupied polyhedra there is often empty space where there are no cation polyhedra. However, oftentimes this space still contains anions which together form polyhedron with no central cation. These polyhedra are empty but could be occupied by cations and therefore represent space through which cations could occupy or pass through during

migration. They also represent another way of measuring the topology of a structure and can be used as a criterion in evaluating similarity between structures. However, as mentioned earlier, cations prefer to exist

in only certain coordination polyhedra. These cations can only occupy empty space in which they would have these preferred coordination numbers. Thus, identifying empty polyhedra can help us predict which empty areas could feasibly be occupied by cations.

In order to better under connectivity between cation polyhedra, we build a tool to identify empty polyhedra in a structure. In this section, we build a tool to identify the empty polyhedra in oxides. We then investigate these empty polyhedra in Li oxides as a case study.

4.1 Empty Polyhedra Tool Methodology

The empty polyhedra tool developed in this thesis is able to section empty space within a structure into empty polyhedra. It first identifies all the potential empty polyhedron centers of the structure, following the methodology developed by Yang1 4, and then removes those that could not be possible cation sites.

Yang developed a methodology to identify possible Li sites in a structure based on a similarity criterion between the substructures of occupied Li sites and potential Li sites. The set of potential Li sites she uses is composed of all the corners of Voronoi polyhedra and the center of the faces of Voronoi polyhedra; this set is referred to as the Voronoi points of the structure.

This tool uses Yang's code to identify all the Voronoi points in the structure as the set of all possible empty polyhedron center sites. The tool then removes any repeated Voronoi points as well as any Voronoi points in any occupied cation polyhedra. Of those remaining points, the tool keeps only those Voronoi polyhedra that have the minimum number of anion neighbors in a given radius, with both the minimum neighbors and radius as designated by the user. This number should be based on the

coordination number preferences and the cation to anion bond lengths of the cation whose migration paths we are interested in. Additionally, for any Voronoi points that share four or more anion neighbors, we remove all Voronoi points in that set except for the one with the most neighbors. While this may remove square face-sharing polyhedra, it also avoids the scenario in which multiple Voronoi points represent the same empty polyhedra. Because the number of actual instances where polyhedra are face-sharing with square faces is very low, this criterion cleans data while likely not removing many actual empty

polyhedron sites. The remaining Voronoi points are the set of empty polyhedron center sites. The process

Figure 15: Processing steps for adding empty polyhedra to a structure.

The original structure (top left) is given Voronoi points in yellow, resulting in a structure with all Voronoi points (top right). Voronoi points that are not empty polyhedra structures are then taken out, resulting in a structure with empty polyhedra (bottom left without empty polyhedra visualized, bottom right with empty polyhedra visualized).

Connectivity Frequency Distribution of Connectivity Frequency Distribution of

4 Coordination Number 6 Coordination Number

5000

20000 M Li-L - U-U

M U-Other Cation 111 U-Other Cation

U L-Empty 4000 LI-Empty 15000 3000 0I U 10000 LaL- 2000 5000 ₁₀₀₀

0 Point Edge Face Connectivity ?I0 Point Edge Face Connectivity

Sharing Sharing Sharing Sharing Sharing Sharing Sharing Sharing

Figure 16: Frequency distributions of Li to all polyhedra connectivities.

Left: 4-coordinated Li-polyhedra, right: 6-coordinated Li-polyhedra. Plots are grouped by the type of polyhedra the Li polyhedra is connected to as well as by the type of connection between the polyhedra. Note that the scale in the two plots is different. Data was taken from a set of 500 Li-oxide structures.

for adding and removing Voronoi points is shown in Figure 14.

4.2 Empty Polyhedra in Li Oxides as a Case study

We now look more closely at Li oxides as a case study for the empty polyhedra tool. Through this case study, we show how we can further describe connectivity not just between a pair of cation polyhedra but also between a cation polyhedron and an empty polyhedron.

4.2.1 Data Collection and Analysis Methods

The structural data from a sample of 500 Li-oxides was collected from the Materials Project Database as in Section 2.2.1. Using the topology tool, we counted the connectivity between our cation of interest, Li, and all other polyhedra in the structure (other Li polyhedra, other types of cation polyhedra, and empty polyhedra). In order to investigate how empty polyhedra connect to cation polyhedra, we take a closer look at all connections between two Li polyhedra that share an empty polyhedron.

4.2.2 Li Polyhedra Connectivity to Other Polyhedra

We first look at how Li polyhedra are connected to all types of polyhedra. In Figure 15, we can observe the distributions of each type of connection for each type of polyhedra. Among the different

types of polyhedra, Li polyhedra appear to connect the most often to empty polyhedra and, in general, the least often to other Li polyhedra, except for 6-coordinated edge-sharing Li polyhedra, which coordinated

by edge-sharing to Li polyhedra slightly more than other types of polyhedra. We also notice that, while

very few Li or other cation polyhedra face-share with Li polyhedra, significantly more empty polyhedra face-share with Li polyhedra. It is noticeable that there is a relatively high proportion of face-sharing 6-coordinated Li polyhedra to empty polyhedra. We also note that there are no non-sharing Li polyhedra, which supports the idea that the empty polyhedra fill up the empty space in the structure. Any polyhedra should be connected to occupied polyhedra or empty polyhedra, with the total number of connected polyhedra defined by the shape of the polyhedra. For example, a tetrahedron should share faces with 4 polyhedra because it has 4 faces, edges with 6 polyhedra because it has 6 edges, and points with 4 polyhedra because it has 4 corners.

4.2.3 Li Polyhedra to Empty Polyhedra to Li Polyhedra Paths

We then look more closely at the connections between two Li polyhedra that also share an empty polyhedra to investigate how an empty polyhedron interacts with connected cation polyhedra. From Figure 16, we observe relatively high numbers of point-sharing and face-sharing between Li polyhedral and empty polyhedra for both point-sharing connected and edge-sharing connected polyhedra. This indicates that, when two Li polyhedra are connected by either points or edges and they also share an empty polyhedra, that empty polyhedra is usually point-connected or edge-connected to the two Li polyhedra. In particular, the relatively high number of Li polyhedra to empty polyhedra face-sharing suggests that the empty polyhedron is a polyhedron that fills the empty space between adjacent occupied polyhedra. We also note that a point-sharing and edge-sharing Li site seem to have around equal

probability of sharing a face with an empty polyhedron. This implies that a pair of point-shared Li polyhedra and a pair of edge-shared Li polyhedra are equally likely to have a face-shared empty polyhedron between the pair.