Solid State Physics for the Structure of Uranium Oxide and Zinc Arsenide

Home , Cadmium arsenide, Uranium dioxide, Zinc arsenide

SOLID STATE PHYSICS FOR THE STRUCTURE OF

URANIUM OXIDE AND ZINC ARSENIDE

Lydia S Harris

A senior thesis submitted to the faculty of

Brigham Young University - Idaho in partial fulﬁllment of the requirements for the degree of

Bachelor of Science

Department of Physics

Brigham Young University - Idaho

December 2019

BRIGHAM YOUNG UNIVERSITY - IDAHO

DEPARTMENT APPROVAL

of a senior thesis submitted by

Lydia S Harris

This thesis has been reviewed by the research committee, senior thesis coordinator, and department chair and has been found to be satisfactory.

Date Lance J Nelson, Advisor

Date David Oliphant, Senior Thesis Coordinator

Date Evan Hansen, Committee Member

Date Todd Lines, Chair

ABSTRACT

SOLID STATE PHYSICS FOR THE STRUCTURE OF

URANIUM OXIDE AND ZINC ARSENIDE

Lydia S Harris

Department of Physics

Bachelor of Science

Material properties are based on the structure of the material. Two ways to determine the structure of a material are a computational search for the ground state structures, and an experimental look using X-ray diﬀraction. In this work, these two ways are described and utilized to ﬁnd the structure of two materials, uranium dioxide and zinc arsenide. Experimental techniques such as

X-ray diﬀraction are used in order to better understand existing materials, but computational searches can be used in materials discovery. The consequences of using computational techniques is that new alloys with desirable properties can be discovered using inexpensive computer resources, however, the existence of such alloys must be validated experimentally.

ACKNOWLEDGMENTS

I would like to thank my parents, family, and friends for their support in my life, a.k.a. getting me through all the things that aren’t schooling.

As far as schooling is concerned, I would like to thank Richard Hatt for converting me to the pure science of physics and for inviting me to repent of disliking coding. I have grown greatly as a physicist in my classes under the guidance of many professors in the physics department at BYU-Idaho. Special thanks to Evan Hansen, Lance Nelson, and Jon Johnson for helping me learn to do research in many different fields using many different techniques. These experiences have been helpful in my development as a physicist and for building my resume. This helped me to get the two internships I was able to complete: the first at the Idaho National Laboratory with Lance Nelson and the second at BYU with John Colton.

Another thanks to Lance Nelson for helping me out by reading and editing my thesis, as well as the other students that did the same.

For anyone reading this far, I may as well also oﬀer my unsolicited advice: start research as soon as possible in your physics career. It’s great for your resume and especially for your development. Don’t be afraid to ask for help from your peers and professors on homework and in life. And remember, physics is hard, but worth doing.

Contents

Table of Contents xi

List of Figures xiii

1 Introduction1 1.1 Finding the Structure...... 1 1.1.1 Experimentally...... 2 1.1.2 Computationally...... 3 1.1.3 Veriﬁcation...... 4 1.2 Three Problems...... 4 1.2.1 Ag-Au “Toy Problem”...... 4 1.2.2 Uranium Oxide Ground State Search...... 5 1.2.3 Crystalline Structure of Zinc Arsenide...... 5

2 Methodology7 2.1 Computational Methods...... 7 2.1.1 Density Functional Theory...... 7 2.1.2 Machine Learning for Exhaustive Searches...... 10 2.2 Experimental Methods...... 19 2.2.1 Symmetries in Crystals...... 19 2.2.2 X-ray Diﬀraction...... 24 2.2.3 Single Crystal XRD...... 26 2.2.4 Powder XRD...... 27

3 Experiments 33 3.1 Ag-Au...... 33 3.2 Uranium Oxide...... 34 3.2.1 DFT and Uranium...... 34 3.3 Zinc Arsenide...... 39 3.3.1 Collecting and Analyzing Data...... 40

4 Results and Discussion 43 4.1 Ag-Au...... 43 4.2 Uranium Oxide...... 46 xi xii CONTENTS

4.3 Zinc Arsenide...... 49 4.3.1 Initial P-XRD Results...... 49 4.3.2 Single Crystal XRD Results...... 49 4.3.3 Later Powder XRD Results...... 51 4.3.4 DFT results...... 52

5 Conclusion 55 5.1 Metals...... 55 5.2 Uranium Dioxide Potential...... 56 5.3 Zinc and Cadmium Arsenide...... 57

Bibliography 59

A Zinc Arsenide Structural Information 65

B VASP settings 75 List of Figures

1.1 A convex hull (in green) for an A-B system. Blue points indicate phases that are not on the hull and therefore unstable. Red points indicate stable phases. Figure from Ref [10]...... 2

2.1 Iterative workflow to solve a transcendental equation...... 9 2.2 Comparison of atomic wave functions of Mn using the PAW method (solid line) with the exact result (bullets) for a given energy and angular momentum. Shown also are their differences magnified by a factor of 10 (dash-dotted line), and their pseudo wave functions (dashed line). Figure from Ref [2]...... 11 2.3 The ith atom’s neighborhood is made up of each atom within some Rcut of itself. The total energy E is made up of the contributions from individual neighborhoods. The energy contribution, Vi, of neighborhood ni depends on the separation between atoms i and j, rij, and a discrete variable, zj, that represents the species of the atom in the neighborhood (I or II in this illustration). Figure from Ref [8].... 13 2.4 Simple 2-dimensional visualization of configuration space vs. energy with a best fit “line”. Each configuration is on the x-axis with it’s corresponding energy on the y-axis. In reality this graph would be N+1 dimensional, where N is the number of basis functions Bα (n).. 14 2.5 An illustration of the filling of configuration space. As more structures are added to the training set, finer details present themselves in the model...... 17 2.6 Active learning during relaxation. If MTP extrapolates too much (γ ≥ γtsh), the configuration is added to the pre-selected set, and MTP predicts the energies, fores, and stresses of the configuration. If a second threshold is reached (γ ≥ Γtsh), relaxation is terminated and no predictions are completed. Figure from Ref [8]...... 18 2.7 Anti-fluorite structure of oxides such as Li2O or Na2O...... 20

xiii xiv LIST OF FIGURES

2.8 Structure of Zn3As2 proposed by Dr. Campbell, isomorphic to structure of Cd3As2 given in Ref [1], a 25% cation deficient 2x2x4 anti- fluorite structure, referred to in this work as SymSG-A. Note that vacancies accounting for the zinc (grey) deficiencies can be seen in several places...... 20 2.9 A crystal from each of the 230 space groups. Figure from Ref [35]... 23 2.10 Low background sample holder for powder X-ray diffraction scans... 29 2.11 Cone of diffracted light that is recorded in P-XRD. Figure from Ref [23]. 30

3.1 Preferred structure of UO2, the fluorite structure with uranium (blue) atoms located on the face centered cubic sites, and oxygen (red) atoms located on the simple cubic sites...... 35 3.2 Two equivalent structures, with different directions for the layers. No- tice that on the right, every “1” atom lies in an uncircled layer, but on the left side, “1” atoms lie in both circled and uncircled layers. Choos- ing the correct direction preserves the layers and makes them able to be modeled as anti-ferromagnetic...... 37 3.3 An example of a super periodic structure (right) of the unit cell (left). The structures are equivalent, but are represented by different lattice and basis vectors...... 38

4.1 Convex hull of Ag-Au system as determined by 294 high-throughput ab initio calculations. The 50% concentration structure is the B10 structure shown in Fig 4.2. Figure from Ref [6]...... 44 4.2 The B10 crystal structure. It is similar to the fcc crystal structure. It is the structure of Ag-Au at 50% concentration. Figure from Ref [6]. 44 4.3 Convex hull of Ag-Au system as determined by the ML potential relaxation of 38,109 structures. This convex hull has shallow ground states that were not found in the high-throughput search of the same system shown in Fig 4.1. This may be due to fitting errors (this fitting has a mean absolute error of 0.8 meV/atom), as a few meV would place some of these structures above the convex hull. The training set for this model had 1556 configurations in it...... 45 4.4 One plane of a derivative fluorite structure with 2:1 oxygen (red) to uranium (blue) stoichiometry. Shown also are the forces on these atoms calculated by VASP (solid pink arrows) and predicted by the ML potential (dashed black arrows)...... 48 4.5 Powder XRD data for 99.9% (blue) and 99.999% (red) pure Zn3As2.. 50 4.6 Powder XRD data for 99.999% pure Zn3As2 (red) fit to the structure obtained from SC-XRD analysis (green peaks, blue profile)...... 53 Chapter 1

Introduction

The crystalline structures of materials give them their properties. Some of these properties include strength, hardness, heat and electrical conduction, magnetic properties, and even color. This means that there is great power in knowing the structure of a material because it determines their properties. Searching for materials with speciﬁc properties is easier if their structures are already known.

1.1 Finding the Structure

Materials science is the study of materials and their properties. With recent increases in computing power, a large part of this science has become theoretical and computational, but experimental validation is still necessary to verify the results. It is common to conduct exhaustive searches for material with desirable properties. Typically, the

first step is to identify the ground states of a system. The ground states of a system are those crystals that are thermodynamically stable and thus will form naturally. It should be noted that not all concentrations of an alloy have a ground state. To find the ground states of a material, the free energies of many different crystal structures

1 2 Chapter 1 Introduction

Figure 1.1 A convex hull (in green) for an A-B system. Blue points indicate phases that are not on the hull and therefore unstable. Red points indicate stable phases. Figure from Ref [10]. at diﬀerent concentrations for an alloy are calculated, and those that create a convex hull, or the set of points (concentration vs. energy) that enclose all of the energy solutions, are the ground state. There is an example of a convex hull in Fig 1.1.

These ground states can be conﬁrmed to exist experimentally by a metallurgist.

1.1.1 Experimentally

One common way to probe the structure of a material, and the way used in this work, is X-ray diffraction (XRD). X-rays are used because their wavelengths are close to the spacing between atoms (in the range of angstroms). This makes an atomic lattice a good diffraction grating for the light to pass through. Armed with a theoretical ground state structure for a material, the structure can be confirmed with XRD.

There are also methods for determining the structure without a previous guess for it, but these are more diﬃcult. 1.1 Finding the Structure 3

In this work, both of these kinds of experiments are attempted, ﬁrst by comparing the theoretical structure of a material to powder (P) XRD data and analysis, and then from no reference point by single crystal (SC) XRD.

1.1.2 Computationally

The ﬁeld of materials science often uses computational models to predict the properties of materials. Large materials databases have been generated using these methods, such as AFLOWLIB [6]. This is a database of quantum mechanical (QM) ab initio

first-principles calculations that can be mined for new, high-performing materials. Ab initio is defined as “from first principles” or “from the beginning,” meaning that the only inputs for the calculations are physical constants, in this case, the positions of the nuclei and the number of electrons. Other libraries use classical potentials, which use things like Coulomb’s law to describe the interactions between molecules, rather than using quantum mechanics, which is a more accurate description for the behavior of matter on an atomic scale. These types of problems can also be used to predict stable phases of alloys that lie on the convex hull, which allows for the discovery of materials. Quantum mechanical calculations are computationally costly, which places a restriction on this kind of materials prediction due to limits on availble computing power. The application of machine learning (ML) to materials predictions problems has the potential to make materials prediction much more efficient [8]. One such approach is called cluster expansion, and is effective in predicting stable structures that are derivatives of canonical lattice types but is not effective otherwise. The method used in this work [9, 27] is based on moment tensor potentials (MTPs) [28] that are described in section 2.1.2 and does not have the same limitations as other ML techniques, and is able to relax the structures off of their lattices, thus representing much more of configuration space. 4 Chapter 1 Introduction

1.1.3 Veriﬁcation

These two different methods (experimental/computational) are very different, but the purpose of them is the same: to determine the structures and properties of a material. It is not common for a scientist to do both, however, because of the very different nature of the two internships I completed, this work describes and utilizes both methods. When results from one method are verified by another method, it makes the results more believable. This is important to the scientific method, as truth cannot be proven, it can only be disproved. If two sources of information agree with each other, it is less likely to be false.

1.2 Three Problems

1.2.1 Ag-Au “Toy Problem”

The MTP machine learning algorithm method mentioned earlier is used in this work to ﬁnd the ground state structures, or convex hull, of the silver-gold metallic system.

Binary metallic systems have been studied quite thoroughly in the literature, and many of the interesting alloys have already been looked at. This problem is considered something of a “toy problem,” which is a problem where the answer is already known.

If the method used to study this kind of problem agrees with the literature, it helps a scientist to be conﬁdent that the method is a legitimate form of investigation. This problem has its own section here because of the interesting results relating to it that will be described later in this work. 1.2 Three Problems 5

1.2.2 Uranium Oxide Ground State Search

Once the ML algorithm was veriﬁed as discussed in the previous section, it was used to model Uranium dioxide (UO2). UO2 is used in nuclear fuel rods. Current models of the Uranium dioxide system used in molecular dynamics (MD) simulations are most often based on classical potentials. In order for such models to obtain desirable results many parameters must be tweaked. A better model would be one that is systematically improvable, which can be done with the MTP method described in this work. A better model of Uranium dioxide would lead to more accurate simulations of nuclear fuel under operating conditions, one of which is how and when the fuel will fail. When scientists know how and when a material will fail, they can design safer fuels and reactors.

1.2.3 Crystalline Structure of Zinc Arsenide

Recent work with semi-conductors in the lab of Dr. John Colton at Brigham Young

University (BYU) led to a curiosity about the structure of Zinc Arsenide (Zn3As2).

Review of the literature that has been published in peer-reviewed articles on Zn3As2 (see Table 2.1) shows that the bulk crystalline structure is unknown. In this work, 6 potential structures are investigated (see Table 1.1). There are signatures that can be collected in XRD that would be able to diﬀerentiate these structures. 6 Chapter 1 Introduction

Table 1.1 The proposed Zn3As2 structures investigated in this work. The sources for these structures are discussed in section 2.2.1.

Structure Grp (#) Source Note

SG 110 [37] -

SymSG-A 142 Campbell, [1] symmetric SG structures,

SymSG-B 142 Campbell 2 diﬀerent vacancy orderings

AltSymSG-A 142 Campbell alternate symmetric SG structures,

AltSymSG-B 142 Campbell 2 diﬀerent vacancy orderings

Calvert 133 [3] obtained from JCPDS Chapter 2

Methodology

2.1 Computational Methods

2.1.1 Density Functional Theory

Because of the size of many body QM problems for a solid (22g of copper is on the

order of N=1023 atoms) and the fact that the locations of the electrons must be known but are not in a many body system, it is impossible to solve the Schr¨odingerequation exactly for solids. This leads scientists to an approximation called density functional theory (DFT), which expressed the single electron wave functino using a basis set.

The Vienna Ab initio Simulation Package (VASP) [16] is used in this work to perform the quantum mechanical ab initio calculations. These are ﬁrst principles calculations, meaning the equation cannot be derived from anything more basic.

Schr¨odingerEquation

In quantum mechanics, the Schr¨odingerequation:

Hˆ |Ψi = E |Ψi (2.1)

7 8 Chapter 2 Methodology must be solved to ﬁnd the energy of a system. In the time-independent case, this is an ordinary diﬀerential equation: h¯2 d2 [− + V (x)]ψ(x) = Eψ(x) (2.2) 2m dx2 In the many body case with N particles, this becomes:

h¯2 X 1 ∂2 − N ψ(x , x ....x ) + V (x , x ....x )ψ(x , x ....x ) = Eψ(x , x ....x ) 2 m ∂x2 1 2 n 1 2 n 1 2 n 1 2 n n=1 n n (2.3)

Density functional theory is based on the assertion that the ground state energy of a system is a unique functional of the electron density [11]. In this energy functional, called the Kohn-Sham equation [14]: Z E[ρ] = Ts[ρ] + vext(r)ρ(r)dr + EH (ρ) + EXC (ρ) (2.4) where TS is the Kohn-Sham kinetic energy, vext is the external potential acting on the system, EH (ρ) is the Hartree or Coulomb energy, and EXC (ρ) is the exchange- correlation (XC) energy. Every term can be known exactly except the XC functional.

This functional is explained in the following section.

Because the Kohn-Sham equation depends on the electronic density and is also used to ﬁnd the density (known as a transcendental equation), an iterative approach is used to solve for the energy of the system. An iterative approach starts with a uniform electron density and uses the output from the right hand side of the equation from the previous iteration as the input for the left hand side of the equation. This continues until the input and output converge to the same value (within some and the equation is self consistent, as illustrated in Fig 2.1. DFT calculations are done in k-space, where the lattice describing the space has units of inverse length, the same units as the wave vector k. Sample points in the ﬁrst Brillouin zone (the region in reciprocal space that contains all of the unique lattice points) called k-points are chosen and appropriately weighted to replace the integrals in the Kohn-Sham equation. 2.1 Computational Methods 9

Figure 2.1 Iterative workﬂow to solve a transcendental equation.

Exchange Correlation Potentials

The XC functional can be approximated in many diﬀerent ways, the most basic of which is the Local Density Approximation (LDA):

Z LDA EXC (ρ) = ρ(r)xc(ρ(r))dr (2.5)

where xc is the exchange-correlation energy per particle of a homogeneous electron gas, assuming the electronic density behaves locally like a homogenous electron gas.

LDA has a tendency to underestimate the exchange energy and over-estimate the correlation energy.

Another approach is the Generalized Gradient Approximation (GGA):

Z GGA EXC (ρ) = ρ(r)xc(ρ(r), ∇ρ(r))dr (2.6) which is similar to LDA but includes the local gradient.

There are also other derivative methods of these two basic methods, and VASP requires the user to choose a method for approximating the XC functional. One 10 Chapter 2 Methodology derivative method, and the one used in this work, is the simpliﬁed rotationally invariant approach introduced by Dudarev et al. in Ref [7], which adds a penalty function to account for the double counting of the XC potentials.

Pseudo-potentials

The method of pseudo-potentials is often used to reduce the computational load of

DFT calculations. This method approximates the inner electrons (that typically have a higher energy, and as such a smaller wavelength, and get cut oﬀ in the plane wave basis) as a “frozen core” that place the electrons surrounding them in an eﬀective potential.

One common approximation is the projector augmented-wave (PAW) method.

The PAW method allows all-electrons orbitals to be reconstructed from the pseudo- orbitals, as shown in Fig 2.2.

VASP uses the PAW method, but a speciﬁc PAW potential must be chosen. VASP settings includes the speciﬁc settings, including the PAW potentials used, for this work.

2.1.2 Machine Learning for Exhaustive Searches

Traditional exhaustive searches (called high-throughput searches when they consist only of QM calculations) require large numbers of the DFT calculations discussed in the previous section. These calculations are computationally costly, so a method to reduce the number of these calculations would be beneﬁcial. Machine learning can be used to do this by using a small set of DFT data to train a model which can calculate the interesting values much quicker. The ML method used in this work is active learning and the MTP basis. 2.1 Computational Methods 11

Figure 2.2 Comparison of atomic wave functions of Mn using the PAW method (solid line) with the exact result (bullets) for a given energy and angular momentum. Shown also are their diﬀerences magniﬁed by a factor of 10 (dash-dotted line), and their pseudo wave functions (dashed line). Figure from Ref [2]. 12 Chapter 2 Methodology

MTP basis

The moment tensor potential (MTP) basis is a set of orthogonal basis functions given

by: X V (n) = ξαBα (n) (2.7) α Orthogonal basis functions are ones that obey the Dirac delta relationship: when

the indices are equal, the integral of their dot products over all space equals 1, and

when the indices are not equal, the integral of their dot products over all space equals

0, such that:

hBm|Bni = 1 when m = n (2.8)

hBm|Bni = 0 when m 6= n (2.9)

Any function of atomic conﬁguration can be expressed using the MTP basis. A crystalline structure can be evaluated on this basis to an arbitrary precision. A simple analogy to this kind of evaluation is the Fourier transform, where higher and higher frequencies can be added to make a better ﬁt to any function.

The basis functions Bα (n) of the MTP basis depend on the set of moment tensor descriptors: X Mµ,ν (ni) = fµ (|rij| , zi, zj) rij ⊗ ... ⊗ rij (2.10) j | {z } ν times

These descriptors are dependent on the immediate neighborhood, ni, of the ith atom,

within some Rcut, as shown in Fig 2.3. Each atom in the neighborhood of the ith

atom introduces four degrees of freedom to the energy contribution, Vi. The total

energy, E, depends on each energy contribution, Vi. The four degrees of freedom are the three coordinates in Euclidean space of the separation between atoms i and j,

rij, and a discrete variable, zj, that represents the species of the neighboring atom. These moment tensor descriptors are tensors of rank ν, where ν is the maximum 2.1 Computational Methods 13

Figure 2.3 The ith atom’s neighborhood is made up of each atom within some Rcut of itself. The total energy E is made up of the contributions from individual neighborhoods. The energy contribution, Vi, of neighborhood ni depends on the separation between atoms i and j, rij, and a discrete variable, zj, that represents the species of the atom in the neighborhood (I or II in this illustration). Figure from Ref [8]. depth of the calculations. ν is chosen based on the desired precision, with more levels providing more precision. This attribute makes the basis a systematically improvable functional form, similar to including more frequencies in a Fourier transformation.

The fµ (|rij| , zi, zj) term in Equation 2.10 is given by:

X f (ρ, z , z ) = c(k) Q(k) (ρ) (2.11) µ i j µ,zi,zj k where

(k) 2 Q (ρ) = Tk (ρ)(Rcut − ρ) (2.12) 14 Chapter 2 Methodology

Figure 2.4 Simple 2-dimensional visualization of configuration space vs. energy with a best fit “line”. Each configuration is on the x-axis with it’s corresponding energy on the y-axis. In reality this graph would be N+1 dimensional, where N is the number of basis functions Bα (n).

In Equation 2.12, Tk (ρ) are the Chebyshev polynomials on the interval [Rmin,Rcut].

The rij ⊗ ... ⊗ rij terms in Equation 2.10 contain angular information about the neighborhood ni and are tensors of rank ν. The basis functions Bα (n) are made up all of possible contractions of any number of Mµ,ν (ni) that result in a scalar.

(k) The ξα and cµ,zi,zj terms in Equations 2.7 and 2.11 are parameters that must be optimized. A Quasi-Newton optimization technique is used to fit these to the data provided by the training set. The configuration space created by these basis functions is N dimensional, where N is the number of basis functions the crystal was evaluated on. The optimization can be visualized in 2 dimensions, “configuration” vs energy, shown in Fig 2.4, where the appropriate terms are chosen to minimize the error of a best fit “line” to the training data. 2.1 Computational Methods 15

Algorithm and Automation

An algorithm has been created to build a training set and predict the observable

properties of many crystalline structures. The input for each step of the algorithm

is built or compiled by a separate software package [22]. The following steps are

repeated until the ML potential is able to interpolate (as opposed to extrapolate) the

energies, forces, and stresses of all the structures it is provided:

1. A list of structures that deﬁned our search space is created that includes all of the

crystal structures of interest for a system (e.g. in a metal system, most structures

of interest are derivative superstructures of fcc, bcc, and hcp lattice types).

• This can be the list of structures that did not relax on the previous iteration.

• Often the same list of structures is used for each iteration of the algorithm.

The extrapolation grade, γ, will decrease for structures with each successive

iteration, because the model has improved. This leads to more accurate energies,

forces, and stresses. In this work, this is the approach taken.

• Structures are only chosen that can be used to model the system of choice. For

example, not every derivative superstructure can be made anti-ferromagnetic

(AFM), so to model an AFM system, structures that cannot be AFM would be

excluded from the relaxation set in this step.

2. The structures speciﬁed in step1 are relaxed by the ML potential.

• Relaxation is the process by which the forces and stresses are minimized by

changing the positions of the atoms in the unit cell. These changes decrease the

energy of the conﬁguration.

• As the ML potential relaxes, it calculates an extrapolation grade, γ. If the

structure’s extrapolation grade is too high, the conﬁguration is added to a pre-

selected set and the relaxation is terminated. See Fig 2.6 and Section 2.1.2 for 16 Chapter 2 Methodology

more details.

• Note that on the ﬁrst iteration, the relaxation will terminate for every structure

because the training set is empty, and the ML potential must extrapolate.

3. The ML potential chooses structures from the pre-selected set to add to the training

set.

• Structures are chosen that most uniformly ﬁll the “missing” places in conﬁgu-

ration space.

4. VASP DFT calculations are run for the selected structures. The results are then

added to the training data set.

5. From the training data, the ML potential is trained. As more conﬁgurations are

added, ﬁner details become clear in the model, as illustrated in Fig 2.5.

(k) • A quasi-Newton method is used to optimize the ξα and cµ,zi,zj parameters in Equations 2.7 and 2.11.

When all of the structures have relaxed and a convex hull is created, it is good

practice to run the ground state structures relaxed by the ML potential against VASP

to verify that the energy predicted by the ML potential agrees with the energy cal-

culated by VASP. Fitting errors of even a few meV can change the ground state

conﬁgurations located on the convex hull, especially for very shallow ground states.

Active Learning

The ML potential, once the model has been trained according to the description given

in Section 2.1.2, must predict, using interpolation and extrapolation, the energies of

more crystalline structures. Extrapolation must be avoided, and the MTP algorithm

avoids it by using active learning. Active learning is a method where the algorithm 2.1 Computational Methods 17

Figure 2.5 An illustration of the filling of configuration space. As more structures are added to the training set, finer details present themselves in the model.

chooses additional data points to be added to the training data set. It does this by

calculating an extrapolation grade γ for each structure during relaxation, and if γ

breaks some threshold, γtsh, it is added to the pre-selected set of structures to add to

the training set. If it breaks a second threshold, Γtsh, the relaxation is terminated. From the pre-selected set, new structures to add to the training set are chosen. This workﬂow is illustrated in Fig 2.6. 18 Chapter 2 Methodology

Figure 2.6 Active learning during relaxation. If MTP extrapolates too much (γ ≥ γtsh), the conﬁguration is added to the pre-selected set, and MTP predicts the energies, fores, and stresses of the conﬁguration. If a second threshold is reached (γ ≥ Γtsh), relaxation is terminated and no predictions are completed. Figure from Ref [8]. 2.2 Experimental Methods 19

2.2 Experimental Methods

2.2.1 Symmetries in Crystals

It is likely that Zn3As2 has a structure isomorphic (having the same crystalline form) to Cadmium Arsenide (Cd3As2) due to their number of valence electrons. There are some proposals of structures in the literature (see Table 2.1), but it is not agreed upon. However, it is generally agreed upon that these isomorphic structures are a 2x2x4 super structure of anti-ﬂuorite (see Fig 2.7) with a 25% cation deﬁciency.

The space group proposed in the literature that is most referenced is group 110, and in this work, the corresponding structure is called the SG structure (short for

Steigmann-Goodyear, Ref [37]). An explanation of space groups is given in Section

2.2.1. Dr. Branton Campbell at BYU suggests that the structure of Zn3As2 has higher symmetry than the SG structure with an argument discussed in Section 2.2.1. He proposes that the structure belongs to the space group 142, which has an additional inversion symmetry relative to group 110. Ref [1] supports this claim, proposing that the structure of Cd3As2 is in space group 142, with the structure shown in Fig 2.8. 20 Chapter 2 Methodology

Figure 2.7 Anti-ﬂuorite structure of oxides such as Li2O or Na2O.

Figure 2.8 Structure of Zn3As2 proposed by Dr. Campbell, isomorphic to structure of Cd3As2 given in Ref [1], a 25% cation deficient 2x2x4 anti- fluorite structure, referred to in this work as SymSG-A. Note that vacancies accounting for the zinc (grey) deficiencies can be seen in several places. 2.2 Experimental Methods 21 ? 2 As 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 structure. As As As As As As As As As As As As As As 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 / Zn 2 As 3 As 3 9 - Zn . ˚ A) Determined By Cd 5 ∼ 9 . ˚ A) c ( 5 ∼ structure that is is proposed to be isomorphic to the Zn 2 As 3 Silvey (1958) [ 34 ] 142 c/a = 2.007 ref to [ 5 ] Zn Literature review of Zn Author (Year) [Ref] Grp (#) a ( Calvert (1978) [ 3 ] 133 11.789 23.635 P-XRD Zn Im et al. (2015) [ 12 ] 133 11.78 23.635 ref to3 ] [ Zn Ali et al. (2014) [ 1 ] 142 12.633 25.427 SC-XRD Cd Cole et al. (1956) [ 5 ] 142 11.78 23.65 SC & P-XRD Zn Lin-Chung (1969) [ 18 ] 110 11.783 23.652 ref to [ 37 , 43 ] Zn Izotov et al. (1977) [ 13 ] 110 11.778 23.643 ref to [ 26 ] Zn Kouklin et al. (2006) [ 15 ] 110 11.778 23.643 ref to [ 4 ] Zn Chelluri et al. (1986) [ 4 ] 110 11.778 23.643 ref to [ 13 ] Zn Natta & Passerini (1928) [ 21 ]- Stackelberg & Paulus (1935)] [ 36 137 8.95 12.65 - Cd Steigmann & Goodyear (1968) [ 37 ] 110 12.67 25.48 SC & P-XRD Cd Pietraszko & Lukaszewicz (1973) [ 26 ] 110 11.7786 23.6432 SC-XRD Zn Weglowski & Lukaszewicz (1968) [ 41 ] 110 11.775 23.646 SC-XRD Zn Table 2.1 22 Chapter 2 Methodology

Space Groups

The space group of a crystal is the set of all symmetry operators that are valid for the crystal. Symmetry operators include point operators, translations, and combinations of the two. There are 230 unique space groups in 3 dimensions, and they are numbered and referred to by their numbers. Fig 2.9 shows a real crystal that belongs to each of the space groups.

ISODISTORT

Branton Campbell at BYU and others developed a computer program called ISODIS-

TORT [38, 39] to see distortions of crystal structures. He used two approaches in conjunction with the software to come up with additional theoretical structures to investigate in this work, summarized in Table 1.1.

One approach was to distort the positions of the atoms in the parent SG structure belonging to space group 110 in order to change the space group to one with higher symmetry, space group 142. This yielded the SymSG A and B structures, which stands for “Symmetric Steigmann-Goodyear.”

The other approach was to input the information about the crystal class and the

2:1 stoichiometry of the crystal. The set of all point operators that appear as part of any symmetry operator in a crystal is called the point group of the crystal. Each space group with the same point group belongs to the same crystal class. This approach came up with the AltSymSG A and B structures, which stands for “Alternative

Symmetry Steigmann-Goodyear.” Turns out that the SymSG-A structure also showed up in the Cava paper [1] as the structure for Cd3As2, but the structure was determined experimentally. 2.2 Experimental Methods 23

Figure 2.9 A crystal from each of the 230 space groups. Figure from Ref [35]. 24 Chapter 2 Methodology

2.2.2 X-ray Diﬀraction

X-ray diﬀraction is a method where a coherent beam of X-rays is produced, and is

incident upon a crystalline sample to be studied. These X-rays are diﬀracted by the

crystal and an interference pattern can be observed in the scattered X-rays.

Bragg’s Law and the Laue Equation

X-rays scattered oﬀ of a surface of a crystal obey Bragg’s law for constructive inter-

ference:

nλ = 2 d sinθ (2.13) where n is an integer, λ is the wavelength of the X-rays, d is the spacing between planes, and θ is the angle of the incoming X-ray beam from the surface. This simple equation describes the constructive interference that happens when many parallel planes of atoms are reached by an incoming X-ray. There are many sets of planes to consider in a crystal, so analysis of XRD data is not as simple as ﬁnding a single d value for the crystalline structure.

Fourier Transform of Electronic Density

A more sophisticated approach is to think of a crystal as some repeated pattern of electronic density. This electronic density can be broken down to a three-dimensional

Fourier series on the reciprocal lattice that represents k-space. The incoming X-ray beam is diﬀracted by electromagnetic interaction, and the combination of vectors representing the incoming beam and the outgoing beam is one data point of this

Fourier transform, according to the Laue condition:

kfinal − kincident = G (2.14) 2.2 Experimental Methods 25

where kincident is the incoming k-vector of the X-ray, kfinal is the outgoing one, and G is a reciprocal lattice vector.

XRD can be used to study single grain crystals if the single crystal is larger than the incoming X-ray beam. To ﬁnd the electronic structure, an inverse Fourier transform of the reciprocal lattice vectors, G, is calculated. In practice, this direct calculation is not possible because only the intensity, not the phase, of the data points is collected, so an inverse Fourier transform cannot be performed. This is known as the “phase problem” of X-ray diﬀraction. In practice, the Fourier transform of a theoretical structure (guessed by the computer or inputted by a user) can be compared to the actual data (or Fourier transform without phase information). The structure is then solved iteratively in this dual-space until the two Fourier transforms agree.

HKL-Intensity

It is relatively easy to calculate the expected X-ray diffraction pattern from a known crystalline structure. This is done by finding all of the planes (identified by the Miller indices, that describe a vector normal to the plane, represented by an hkl index) that

X-rays could reflect off. Programs can easily find these planes and predict the angle using Bragg’s law where the diffracted X-ray will appear.

The information that a computer will generate based on a known crystalline structure will store the information in what is known as an HKL-Intensity ﬁle. This predicted data can be compared to the data that was actually collected to see if the crystalline structure is a good ﬁt to the data.

Using the inverse of the process just described, intensity values at a certain position collected by a detector are assigned to certain HKL values. An HKL-Intensity

ﬁle is generated with the calculated HKL value with their corresponding intensities collected. This is essentially what all reﬁnement techniques go back to. 26 Chapter 2 Methodology

2.2.3 Single Crystal XRD

Sample Size

By far the best XRD method is Single Crystal (SC) XRD. There is less multiplicity in the data points, because the data points are not collapsed down into one spatial dimension, 2θ. The detector is able to rotate around φ and θ directions. This leads to more accurate predictions for intensities, because they are not stacked.

The reason SC-XRD is not commonly used is because it is a highly sophisticated process, that requires a Ph.D. to complete, and the machine is very expensive to purchase and maintain.

The reason this project was a challenge for SC-XRD is because a single grain crystal larger than the incoming X-ray beam must be found. For low quality (and sometimes even high quality) samples, single crystal domains are often not very large, and even if there is one that is large enough, it is probably very diﬃcult to ﬁnd. The size of the single crystal domain that is desired is typically in the range of ∼ 75µm.

Dual Space Techniques

Dual space techniques are used to refine a crystalline structure in SC-XRD. The program performs an initial inverse Fourier transform of the Intensity data collected by the detector to find an initial structure. This initial structure then needs to be refined.

The structure is reﬁned using dual space techniques, meaning that the Fourier transform of the theoretical structure is compared to the actual data collected by the detector, and then the theoretical structure is tweaked until the ﬁtting errors are small enough to be acceptable.

Dual space techniques receives its name due to the fact that both real space 2.2 Experimental Methods 27

(theoretical structure atomic locations) and reciprocal k-space (inverse transform and raw data) are used to reﬁne the structure.

Atomic Displacement Parameters

Due to the Heisenberg Uncertainty relationship:

h¯ σ σ ≥ (2.15) x p 2 the position of a particle can never be exactly known while there is any information about the movement of the particle. This means that we describe particles in quantum mechanics in terms of probability densities.

We understand that there are things called orbitals where the electrons of an atom are likely to be found, but these are only probability densities. Because of this uncertainty in the position of an electron, the actual location changes over time, and as X-rays are incident on any electronic density, they interact slightly diﬀerently with the electrons as time passes.

This leads to something called atomic displacement parameters. The nucleus and electrons of an atom are likely to be located within some volume that can be described in real space. These are mathematically described either simply as spheres, or with more complexity as ellipsoids. If the refinement procedures make these parameters represent oddly shaped ellipses or fill unusually large or small spaces, it is likely that the hypothetical structure is not a good fit for the data.

2.2.4 Powder XRD

Another XRD technique is powder (P) diffraction. Much of the information that can be collected in single crystal (SC) diffraction is lost in this technique. In the absence of a large enough single grain crystal, powder diffraction is a agreeable alternative. 28 Chapter 2 Methodology

In P-XRD the sample is ground into a fine powder, and every possible orientation of the crystal is represented at once. Rather than collecting data by scanning a detector in multiple spatial dimensions, such as in SC-XRD, data can only be collected by scanning in one spatial dimension, because a cone of diffracted X-rays is produced rather than a single outgoing X-ray diffraction pattern. This can cause multiplicity of data points (i.e. multiple G vectors appearing in the same place).

Despite these challenges, this method can still be helpful in determining crystalline structure, if an initial guess for the structure of the crystal is supplied to the program.

These can be found in a database of standards, in the literature, or by symmetry arguments, such as those discussed in section 2.2.1.

Sample Size

The sample size for a P-XRD scan depends on the absorption of the material. A typical scan contains ∼ 2cm3 of powder. The powder can be ground up with a mortar and pestle to make a ﬁne powder.

The powder is prepared in sample holders that are appropriate for the absorption of the material. If the sample has low absorption, the X-rays will penetrate the material too far, and the signal will contain data from the holder. With a high enough absorption, less sample will be needed.

The holders are round, with an inlet that holds the powder, as shown in Fig 2.10.

They can be either back loaded or front loaded. To get scans of high angles, a kapton

ﬁlm (that interacts minimally with X-rays) can be placed on top to keep the powder from shifting during the scan. The peaks collected by the P-XRD (like the ones in Fig

4.5) have shapes that are convoluted by the smoothness of the powder in the sample holder. 2.2 Experimental Methods 29

Figure 2.10 Low background sample holder for powder X-ray diﬀraction scans.

Orientation of Samples

Because the sample is ground into a powder, every orientation of the crystal is present

at once (see Fig 2.11).

This is why only one spatial variable (2θ) can be collected. If these cones overlap each other, it is hard to determine their individual intensities. This is what makes

P-XRD not as advantagous as SC-XRD.

The programs used for P-XRD refinements know how to stack these HKL-Intensity values so that raw data can be compared to a hypothetical structure, but where there is multiplicity of peaks, the peak shapes can change significantly when the structure is refined. This can be the source of more uncertainty in the fit.

Preferred orientations are often present in P-XRD data due to the sample’s preparation, caused by irregular crystallite shapes (such as plate or needle like shapes) which makes it diﬃcult to prepare the sample with truly random orientations, as the analysis assumes is present. Because P-XRD assumes random orientations, if a preferred orientation is present, it can be modeled using one of two diﬀerent methods. 30 Chapter 2 Methodology

Figure 2.11 Cone of diﬀracted light that is recorded in P-XRD. Figure from Ref [23].

The March-Dollase factor can be used, which takes into account a single preferred orientation (an HKL index for the preferred orientation must be provided, determined manually from the peak with the largest discrepancy in intensity). The other method is spherical harmonics, which deﬁnes a shape that represents the amount of crystallites with each orientation.

Reitveld Reﬁnement

The process of fitting a hypothetical structure to P-XRD data is called Rietveld refinement. In Rietveld refinement, a hypothetical structure in the form of a cif

(crystallographic information ﬁle) is uploaded and a simulated P-XRD pattern is calculated (HKL-Intensity).

This pattern is then reﬁned by several parameters (e.g. scale factor, zero shift, unit cell parameters, atomic positions, peak shape parameters, etc..) to ﬁt the actual data more closely using a least squares optimization method. The program calculates 2.2 Experimental Methods 31

a best case R value, Rexp, from the signal to noise ratio of the data. The closer to 0 the Rexp value is, the better the signal to noise ratio is. A weighted R value, Rwp, is calculated that represents how close the simulated P-XRD proﬁle is to Rexp.

When the Rwp value is small (i.e. less than 10), the structure has been reﬁned completely, and the structure can be called a good match for the P-XRD data, although a thorough investigation of the graphical data representing the observed and calculated patterns is the best way to ensure a good ﬁt.

Atomic displacement parameters in P-XRD are always represented with spheres to prevent over fitting the data. These can either be modelled with an overall factor, applied to every atom in the configuration, or with individual parameters for each atom. One must always be careful not to over fit in Rietveld Refinement by refining too many parameters.

Peak shape parameters are used to describe asymmetries of peaks and other shapes with the instrumental broadening function Ω(θ), wavelength dispersion function Λ(θ), and the specimen function Ψ(θ), with the addition of a background function, b(θ), described mathematically by:

PSF (θ) = Ω(θ) ⊗ Λ(θ) ⊗ Ψ(θ) + b(θ) (2.16) that are convoluted as follows:

Z ∞ Z ∞ f(t) ⊗ g(t) = f(τ)g(t − τ)dτ = g(τ)f(t − τ)dτ (2.17) −∞ −∞

. 32 Chapter 2 Methodology Chapter 3

Experiments

3.1 Ag-Au

For the toy system of Ag-Au, we attempted to verify the ML potential. The convex hull of this system is well known (see Fig 4.1), and if it can be reproduced we will be able to conﬁdently move forward to more complex systems.

The main lattice types that are represented in metallic systems are face centered cubic, base centered cubic, and hexagonal close pack. These are the lattice types that were represented in the set of 38109 structures to relax.

On our first time through the algorithm, we attempted to create an initial training set of structures and calculations in VASP. In similar searches for materials that don’t use the MTP basis such as cluster expansion, this is the usual approach. We ended up with large fitting errors (on the order of a few meV/atom), and upon researching the literature further, we found that the model should be trained on an empty training set, and the algorithm should be allowed to choose the structures that represent configuration space the most uniformly. This prevents very similar structures from being chosen for the training set, and allows more flexibility in the model.

33 34 Chapter 3 Experiments

3.2 Uranium Oxide

Once the ML potential has been veriﬁed, it can potentially be used to create a quan-

tum mechanical model of UO2, a more complex system. Uranium dioxide is tradi- tionally treated classically.

The known structure of UO2 is ﬂuorite. Derivative ﬂuorite superstructures with the correct uranium to oxygen stoichiometry were searched in this work. Other structures were not searched because they are not known to converge well in DFT calculations because of a well known issue called the band-gap problem [25].

Uranium atoms are known to have a non-zero magnetic moment, and UO2 is known to be anti-ferromagnetic (AFM), with preferred orientation in the (100) plane.

The structure is shown in Fig 3.1. Because of these known attributes, to understand

the system more completely, an AFM system was constructed in this work.

3.2.1 DFT and Uranium

With modeling large atoms such as Uranium, there can be some diﬃculties with

getting the DFT calculations to converge to a computationally correct total energy.

This is because the traditional treatment of electrons in DFT calculations allows

the Coulomb repulsion to scatter the electrons, when in large atoms, the d and f electrons are strongly correlated and localized, which causes the basic XC potentials that approximate the electron density as uniform to break down. Because the energy of the system is dependent upon the electron density, an incorrect density will often predict an energy that is too high. 3.2 Uranium Oxide 35

Figure 3.1 Preferred structure of UO2, the ﬂuorite structure with uranium (blue) atoms located on the face centered cubic sites, and oxygen (red) atoms located on the simple cubic sites. 36 Chapter 3 Experiments

DFT+U

To remedy the traditional treatment of electrons in large atoms, DFT+U should be used. DFT+U is also known as LDAU, and the rotationally invariant method introduced by Liechtenstein et al. in Ref [17] is used in this work. Without the addition of the U parameter to DFT calculations, the calculation may converge, but it will likely converge to a non-physical solution.

The method used in this work to ensure DFT+U converges to the correct total energy follows a ramping scheme described in Ref [19] that begins with a U parameter of 0 and ramps it up to 4.5, using the charge density calculated in the previous iteration. This helps ensure convergence to a true total energy.

There are additional settings in VASP that must be used to employ the DFT+U method that are given in VASP settings.

Anti-ferromagnetic System

An AFM system is one where the magnetic moments, or spins, of individual atoms line up on alternating planes (e.g. every atom in one plane has a positive magnetic moment, every atom in the next plane has a negative magnetic moment, etc.), typically giving the crystal a net magnetic moment of zero. To accurately model a crystal as AFM, the layers in the desired plane are determined and spin is assigned, and then the neighboring cells are checked to make sure the periodicity is preserved. When the unit cell of the crystal is populated out into a repeating pattern, all of the atoms must still align in planes according to their original spin assignment(e.g. if an atom is

“up” spin in the primitive cell, the corresponding atom in all of the neighboring cells must also be “up” spin, and be in a layer of “up” spin (see Fig 3.2). If a structure does not meet these criterion in the (100) plane that is preferred by UO2, one of its derivative super periodic structures may meet them. 3.2 Uranium Oxide 37

Figure 3.2 Two equivalent structures, with diﬀerent directions for the layers. Notice that on the right, every “1” atom lies in an uncircled layer, but on the left side, “1” atoms lie in both circled and uncircled layers. Choosing the correct direction preserves the layers and makes them able to be modeled as anti-ferromagnetic.

Alternative unit cells

A super periodic structure is one that has the same atomic positions as the primitive unit cell, but is represented by a diﬀerent set of lattice and basis vectors. An example of this is given in Fig 3.3. A super periodic structure can also include more than just the primitive cell, but is always an integer multiple of the volume of the primitive cell. 38 Chapter 3 Experiments

Figure 3.3 An example of a super periodic structure (right) of the unit cell (left). The structures are equivalent, but are represented by diﬀerent lattice and basis vectors. 3.3 Zinc Arsenide 39

3.3 Zn3As2

The analysis completed in this work was initially based on some preliminary analysis of P-XRD data by a student in Dr. Colton’s group that indicated that the SymSG-A structure was a potentially good match for the data. Further examination of the analysis completed by this student showed that some peaks were manually inserted to ﬁt the data more closely. This means that not all of the peaks in the P-XRD data could be explained by the SymSG-A structure that had been used in the initial analysis. Further analysis showed that none of the 6 structures were a good match for the P-XRD data.

An alternative route had to be found for ﬁnding the structure of Zn3As2. If a large enough single crystal could be found the structure could be solved using SC-

XRD. This technique requires no previous guess for the structure. However, there is the issue of ﬁnding a large enough single crystal domain to study. A 99.9% purity sample obtained from Chemsavers was initially studied (there are three 9’s in the purity level, so in this work this samples is referred to as the 3N sample), and there were no single crystal domains found that were large enough to be studied. Upon studying a 99.999% sample obtained from Alfa Aesar (known in this work as the 5N sample), a piece was found that contained only two domains. This was manifested in the initial scans by twinning, where the data shows two bright diﬀraction spots where there would only be one if the sample were single crystal.

Twinning presents a challenge in analysis, as the two domains and their orientations have to be diﬀerentiated to solve the structure. To aid the CELL NOW

[30] program in diﬀerentiating the two domains, the proposed lattice parameters of a = 11.789 A˚ and c = 23.635 A˚ from the Calvert structure were inputted. The program was able to diﬀerentiate the two domains, that were related by a 90◦ rotation 40 Chapter 3 Experiments

about the h−1, 0, 0i vector. The components were present in roughly equal propor-

tions (52:48). The data was integrated and scaled using the two unit cell orientations.

All atoms were reﬁned anisotropically, meaning ellipsoids were used to represent the

atomic locations. Reﬁnements were generated using both the HKLF4 (containing

the averaged signal of the two domains’ corresponding reﬂections) and HKLF5 ﬁles

(containing only a single domain’s reflections). The final refinement presented here

is with the HKLF5 ﬁle, which produced slightly better statistics.

After reﬁnement, CheckCIF (a tool used to report on the consistency and integrity

of crystal structure determinations reported in CIF format) suggested two additional

twin laws (around rotation vectors h1, 0, 2i and h0, 1, 2i). We attempted refinements including each of these additional components, but the twin fractions refined to 2% or less with little or no improvement in the refinement statistics, so the additional twin laws were not included in the final refinement.

3.3.1 Collecting and Analyzing Data

Single Crystal XRD

The process for collecting and analyzing the 5N SC-XRD data for this work is as follows: a 40x40x25µm metallic gray/black crystal was harvested under oil in ambient conditions and placed at the tip of a polyimide loop. The crystal was mounted in a stream of cold N2 and centered in the X-ray beam using a video camera. Low- temperature (100 K) X-ray diffraction data comprising φ- and ω- scans were then collected using a MACH3 kappa goniometer coupled to a Bruker Apex II CCD detector with a Bruker-Nonius FR591 rotating anode X-ray source producing Cu Kα radiation (λ = 1.541 78 A).˚ The structure was solved from SC-XRD data using dual- space methods in SHELXT [32] and refined against F 2 on all data by full-matrix least 3.3 Zinc Arsenide 41 squares with SHELXL-2014 [33] using established refinement strategies [20]. Addi- tionally, the Bruker APEX3 suite was used to process the SC-XRD data; reflection intensities were integrated through the program SAINT [29], and appropriate absorption and extinction corrections were applied to the intensities via a multi-scan method using the program TWINABS [31].

Powder XRD

The process for collecting and analyzing the P-XRD data for this work is as follows: a metallic gray/black powder was prepared using a mortar and pestle and packed in a standard steel sample holder (back loaded for the 3N powder, front loaded with a

1.0mm aluminum insert for the 5N powder) with a Kapton ﬁlm placed on top. The powders were mounted in the PANalytical X-pert PRO diﬀractometer with an anode

X-ray source producing Cu Kα radiation (λ = 1.541 78 A).˚ Ambient temperature X- ray diﬀraction data comprising 2θ scans were collected with the X-Celerator detector.

The structures were reﬁned using a least squares method in X-Pert HighScore Plus

[24].

Additional Routes

We also wanted to get higher resolution data. This high resolution data can be obtained at the Advanced Photon Source at Argonne National Laboratory. The samples had to be prepared in Kapton capillaries. Because of the high absorption of

Zn3As2, the sample had to be prepared specially. Two sample preparation methods were used: one to dilute the sample, the other to make a thin layer. The dilution method is to mix the powder with powdered coﬀee creamer and pack the capillary with the diluted powder, sealing the ends with noncrystalline glue. The other method is to roll a small capillary in petroleum jelly and then in the powder, and place that coated 42 Chapter 3 Experiments capillary in a larger capillary, again sealing the ends with noncrystalline glue. We will also verify the low energy solution by doing DFT calculations on the crystalline structures presented in this work. I need to ﬁll in some details about that here still.... Chapter 4

Results and Discussion

4.1 Ag-Au

The convex hull of the Ag-Au system is well known (see Fig 4.1). On allowing the model to train on an empty training set, the model relaxed 38,109 structures with

1556 DFT calculations. The convex hull created in this work is shown in Fig 4.3.

The ﬁtting has a mean absolute error (MAE) of 0.8 meV/atom.

The search completed by the ML potential was much more thorough than the high-throughput ab initio search, and resulted in more ground state configurations, although they were quite shallow. Because we fit to so many diverse structures, a way to verify this model would be to remove from the relaxation set of 38,109 structures any structure with an energy greater than some value from the convex hull, and retrain the model from empty. This would ensure a better fit, and a more accurate convex hull, but would still include less DFT calculations than a high-throughput search over the same number of candidate structures.

43 44 Chapter 4 Results and Discussion

Figure 4.1 Convex hull of Ag-Au system as determined by 294 high- throughput ab initio calculations. The 50% concentration structure is the B10 structure shown in Fig 4.2. Figure from Ref [6].

Figure 4.2 The B10 crystal structure. It is similar to the fcc crystal structure. It is the structure of Ag-Au at 50% concentration. Figure from Ref [6]. 4.1 Ag-Au 45

Figure 4.3 Convex hull of Ag-Au system as determined by the ML potential relaxation of 38,109 structures. This convex hull has shallow ground states that were not found in the high-throughput search of the same system shown in Fig 4.1. This may be due to fitting errors (this fitting has a mean absolute error of 0.8 meV/atom), as a few meV would place some of these structures above the convex hull. The training set for this model had 1556 configurations in it. 46 Chapter 4 Results and Discussion

4.2 Uranium Oxide

UO2 is known to have the AFM ﬂuorite structure, shown in Fig 3.1, with spin aligning in the (100) direction. The ML potential found the correct ground state among the

other candidate derivative ﬂuorite structures that were investigated. The MAE of

the energy per atom is 6.5 ∗ 10−7 meV/atom. A summary of these results for several structures is given in Table 4.1.

Table 4.1 Predicted and calculated energies for several UO2 derivative ﬂu- orite structures. The mean abolute error (MAE) was 1.0 ∗ 10−4 meV for all structures, with a MAE per atom of 6.5 ∗ 10−7 meV/atom.

Total Energy (eV)

predicted calculated |diﬀ|

-117.250882889086 -117.25088298 9.1 ∗ 10−8

-140.126355786625 -140.12635566 1.3 ∗ 10−7

-150.328035440859 -150.32803526 1.8 ∗ 10−7

-181.549228746869 -181.54922873 1.7 ∗ 10−8

-168.618518091217 -168.61851806 3.1 ∗ 10−8

-137.376277272381 -137.37627747 2.0 ∗ 10−7

-124.495047858794 -124.49504792 6.1 ∗ 10−8

mean 1.0 ∗ 10−7

The MAE of the forces that the ML potential predicted is 0.09 eV/A.˚ The error

of the forces for one derivative ﬂuorite structure is given in Table 4.2. In Fig 4.4, one

plane of atoms in a derivative ﬂuorite structure with the calculated and predicted

forces is shown. 4.2 Uranium Oxide 47

Table 4.2 Predicted and calculated forces for one random UO2 structure. The mean absolute error (MAE) was 0.000034 eV/Afor˚ this structure. The MAE for all structures considered was 0.09 eV/A.˚ (note: The large forces are due to several uranium atoms being very close to eachother.)

Atom |Forces| (eV/A)˚

# predicted calculated |diﬀ|

1 24.56350 24.56350 0.000000

2 0.417038 0.416797 0.000241

3 24.56350 24.56350 0.000000

4 0.417038 0.416797 0.000241

5 11.33860 11.33860 0.000000

6 11.33860 11.33860 0.000000

7 11.33860 11.33860 0.000000

8 11.33860 11.33860 0.000000

9 2.790920 2.790840 0.000079

10 0.297829 0.297796 0.000033

11 2.790920 2.790840 0.000079

12 0.297829 0.297796 0.000033

13 1.597100 1.597090 0.000012

14 1.748360 1.748350 0.000012

15 2.544670 2.544670 0.000000

16 1.597090 1.597090 0.000000

17 1.748360 1.748350 0.000017

18 2.544680 2.544670 0.000011

19 1.597090 1.597090 0.000000

20 1.748360 1.748350 0.000017

Continued on next page 48 Chapter 4 Results and Discussion

Figure 4.4 One plane of a derivative ﬂuorite structure with 2:1 oxygen (red) to uranium (blue) stoichiometry. Shown also are the forces on these atoms calculated by VASP (solid pink arrows) and predicted by the ML potential (dashed black arrows).

Table 4.2 – Continued from previous page

Atom |Forces| (eV/A)˚

# predicted calculated |diﬀ|

21 2.544680 2.544670 0.000011

22 1.597100 1.597090 0.000012

23 1.748360 1.748350 0.000012

24 1.748360 1.748350 0.000012

mean 0.000034 4.3 Zinc Arsenide 49

4.3 Zinc Arsenide

4.3.1 Initial P-XRD Results

The initial P-XRD data that was collected on 3N purity powder had additional peaks to the 5N sample data. Their scans are shown in Fig 4.5. There are more peaks present in the 3N data than in the 5N data. Because the 0.1% impurity would not reﬂect strongly enough to show such high intensity peaks, it is likely that the extra peaks are due to at least one other phase of Zn3As2 being present. This mixed phase could be due to irregular cooling during synthesis that did not allow the crystal to relax into its preferred room temperature structure. Perhaps annealing the sample would remove these peaks. The absence of these peaks in the 5N data suggests that the higher purity sample had a higher quality synthesis process.

4.3.2 Single Crystal XRD Results

Because the patterns of the two twin components studied using SC-XRD mentioned previously overlap strongly, determining the symmetry of the crystals was not triv- ial. Space group 142 (I 41/acd) represented the best solution, though solutions were obtained in multiple other space groups (I 422 (97), I 4122 (98), I 4cm (108), I 41cd (110), I 4¯c2 (120), I 42¯ m (121), I 42¯ d (122)).

We reﬁned the structure solution in space group 110, but the presence of 50:50 inversion twinning (the inverted version of a structure also presenting a likely structural solution) strongly suggests that an inversion center is present and that space group 142 is a more accurate representation of the symmetry. In addition, the spe- ciﬁc structural solution in space group 142 had more physically reasonable atomic displacement parameters of the atoms than the structural solution in space group

110. 50 Chapter 4 Results and Discussion

Figure 4.5 Powder XRD data for 99.9% (blue) and 99.999% (red) pure Zn3As2. 4.3 Zinc Arsenide 51

Details of the crystal structure solved using SC-XRD are listed inA in Table A.1.

The Zn3As2 sample crystallized in the tetragonal space group 142 with the asymmetric unit containing three unique As and three unique Zn positions. Atomic coordinates and equivalent isotropic displacement parameters for all unique atoms are given in

Table A.2. Bond lengths and angles are listed in Table A.3. The symmetry operators required to produce the full unit cell are given in Table A.4. The full anisotropic displacement parameters are given in Table A.5.

4.3.3 Later Powder XRD Results

We used the structure determined from the single crystal analyses here as the initial structure for Rietveld reﬁnements on data collected using P-XRD. Upon allowing the atomic positions to relax during reﬁnement, the SymSG-A structure produced similar statistics to the structure determined by single crystal analysis, and upon inspection, the structures were determined to be the same.

The Rietveld reﬁnement of the structure found produces better statistics in the

5N powder than any of the other five models attempted from the literature (see Ta- bles 4.3, 4.4); peak positions were in large measure correctly predicted by both the structure indicated by the SC-XRD analysis and the SymSG-A structure, but the peak intensities were not always correctly predicted (see Fig 4.6). We hypothesize that this is due to the presence of both twinning and preferred orientation. As seen in the single crystal analyses, all crystals exhibit twinning. If one of the domains exhibits preferred orientation, it stands to reason that the other domain(s) also exhibit preferred orientation, but these would be different orientations. These preferred orientations were modeled using the method of spherical harmonics, with only two parameters refined to prevent overfitting the data.

I will also include ﬁts to Synchrotron data in these tables when that’s been ana- 52 Chapter 4 Results and Discussion

Table 4.3 Fits of 7 diﬀerent structures to 99.9% (3N) Zn3As2 P-XRD data.

Structure Grp (#) Rwp a (A)˚ c (A)˚

SG 110 34.85 11.80 23.57

SymSG-A 142 35.23 11.79 23.54

SymSG-B 142 35.15 11.80 23.59

AltSymSG-A 142 35.48 11.79 23.54

AltSymSG-B 142 36.52 11.81 23.57

Calvert 133 33.81 11.79 23.65

Solved via SC-XRD 142 33.57 11.79 23.65 lyzed....

4.3.4 DFT results

I don’t have these results yet, but I will have a data table and discussion here when

I have these results. 4.3 Zinc Arsenide 53

Figure 4.6 Powder XRD data for 99.999% pure Zn3As2 (red) ﬁt to the structure obtained from SC-XRD analysis (green peaks, blue proﬁle). 54 Chapter 4 Results and Discussion

Table 4.4 Fits of 7 diﬀerent structures to 99.999% (5N) Zn3As2 P-XRD data.

Structure Grp (#) Rwp a (A)˚ c (A)˚

SG 110 25.39 11.80 23.57

SymSG-A 142 17.93 11.78 23.65

SymSG-B 142 22.27 11.79 23.65

AltSymSG-A 142 27.16 11.80 23.57

AltSymSG-B 142 26.15 11.80 23.57

Calvert 133 26.08 11.79 23.65

Solved via SC-XRD 142 15.99 11.78 23.65 Chapter 5

Conclusion

5.1 Metals

In this work, it was shown that the MTP machine learning potential can create results on par with exhaustive QM calculations to ﬁnd the ground state conﬁgurations

(or convex hull) of a system. The technique offered an even more thorough search, representing more possible configurations with high accuracy. The next step is to run the structures closest to the convex hull through VASP to verify the low energy configurations and recreate the convex hull, as fitting errors of even a few meV could place a structure above or below the convex hull created by the ML potential.

Shallow structures on a convex hull are often hard to physically create, but deep structures can often be conﬁrmed to exist by experimental metallurgists. Applying this ML approach can give metallurgists a starting place for what concentration alloys to attempt to create. This collaboration can corroborate the accuracy of the ML potential.

An exciting area of application for this technique is to study High Entropy Alloys

(HEAs). There are some promising leads discussed in Ref [40] for the use of HEAs as

55 56 Chapter 5 Conclusion

super-alloys: alloys that can withstand high temperatures or that are stronger than

traditional alloys. A high-throughput search of such 4 or 5 body systems would be

far too expensive to ever complete, and as such these systems are a good candidate

for the machine learning approach that would lessen the computational load.

5.2 UO2 Potential

The ground state of UO2 was also conﬁrmed using this ML potential. The next step would be to train the ML potential on structures that have a random atomic displacement from the ideal ﬂuorite structure of UO2. This potential could then be used in classical MD simulations (such as those on LAMMPS) to determine thermody-

namic materials properties of UO2, such as heat capacity. This potential could also be used to compute grain boundary energies, an area of interest to other researchers

at BYU-Idaho.

An interesting research question that could follow is if the convex hull could be

created of the entire concentration range of the Uranium Oxide system. The challenge

here is that the U parameter used to ensure convergence in DFT calculations is based

on experimental data, and oﬀ of the ﬂuorite 2:1 stoichiometry, there is no way to

know if the calculation is well converged. Although the convex hull is well known and

could be used to verify such a work, it would be diﬃcult to comment on the accuracy

of the calculations.

The accuracy of the MTP machine learning model shows that this is a valid poten-

tial to model crystal systems with, and it is proposed that this could greatly accelerate

materials prediction by eﬃciently searching a large number of conﬁgurations with a

signiﬁcantly decreased amount of computational eﬀort. 5.3 Zinc and Cadmium Arsenide 57

5.3 Zn3As2 and Cd3As2

The structure found in this work for Zn3As2 belongs to space group 142, and is the same as the Steigmann-Goodyear structure that belongs to space group 110, except with an additional inversion symmetry. This solution was among the structures investigated in this work, but could not be veriﬁed with P-XRD alone. SC-XRD analysis was required to reach this conclusion.

The solution found is also isomorphic to the structure for Cd3As2 proposed in Ref [1]. This veriﬁes the claim in the literature that structures of these two materials are isomorphic. As these results disagree with the earlier literature regarding Zn3As2, we will publish a paper on our ﬁndings to set the record straight. This pattern will also be submitted to the XRD standards databases that the Calvert structure was obtained from.

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Zn3As2 Structural Information

Table A.1 Crystal data and structure reﬁnement for 5N Zn3As2.

Identiﬁcation code 19097 142 a 5

Empirical formula As2 Zn3

Formula weight 345.95

Temperature 100(2) K

Wavelength 1.54178 A˚

Crystal system Tetragonal

Space group I41/acd a = 11.7579(5) A˚ α= 90◦.

Unit cell dimensions b = 11.7579(5) A˚ β= 90◦.

c = 23.5938(12) A˚ γ = 90◦.

3 Volume 3261.8(3) A˚

Z 32

Density (calculated) 5.636 Mg/m3

Absorption coeﬃcient 36.740 mm−1

Continued on next page 65 66 Chapter A Zinc Arsenide Structural Information

Table A.1 – Continued from previous page

F(000) 4992

Crystal size 0.040 x 0.040 x 0.025 mm3

Theta range for data collection 6.513 to 68.082◦

Index ranges 0<=h<=10, 0<=k<=14, 0<=l<=28

Reﬂections collected 1756

Independent reﬂections 1756 [R(int) = ?]

Completeness to theta = 67.679◦ 99.3 %

Absorption correction Semi-empirical from equivalents

Reﬁnement method Full-matrix least-squares on F2

Data / restraints / parameters 1756 / 0 / 48

Goodness-of-ﬁt on F2 1.215

Final R indices [I¿2sigma(I)] R1 = 0.0304, wR2 = 0.0889

R indices (all data) R1 = 0.0361, wR2 = 0.0933

Extinction coeﬃcient 0.000024(3)

−3 Largest diﬀ. peak and hole 1.275 and -1.156 e.A˚ 67

Table A.2 Atomic coordinates ( x 104) and equivalent isotropic displacement 2 3 parameters (A˚ x 10 ) for 5N Zn3As2. U(eq) is deﬁned as one third of the trace of the orthogonalized Uij tensor.

x y z U(eq)

As(1) 7529(1) 2447(1) 6235(1) 7(1)

As(2) 7562(1) 5000 7500 7(1)

As(3) 5000 2500 7486(1) 7(1)

Zn(1) 6166(1) 6122(1) 6960(1) 10(1)

Zn(2) 6109(1) 1367(1) 6788(1) 11(1)

Zn(3) 8565(1) 3704(1) 6879(1) 10(1)

◦ Table A.3 Bond lengths [A]˚ and angles [ ] for 5N Zn3As2

As(1)-Zn(3) 2.4438(8)

As(1)-Zn(1)#1 2.4617(9)

As(1)-Zn(2)#2 2.4670(8)

As(1)-Zn(2) 2.4682(8)

As(1)-Zn(3)#3 2.7173(9)

As(1)-Zn(1)#4 2.7760(9)

As(2)-Zn(3) 2.4212(8)

As(2)-Zn(3)#5 2.4213(8)

As(2)-Zn(1) 2.4606(8)

As(2)-Zn(1)#5 2.4606(8)

As(2)-Zn(2)#6 2.8017(8)

As(2)-Zn(2)#7 2.8017(8)

Continued on next page 68 Chapter A Zinc Arsenide Structural Information

Table A.3 – Continued from previous page

As(3)-Zn(2)#8 2.4884(9)

As(3)-Zn(2) 2.4885(9)

As(3)-Zn(1)#5 2.4920(8)

As(3)-Zn(1)#9 2.4920(8)

As(3)-Zn(3)#6 2.6642(9)

As(3)-Zn(3)#10 2.6642(9)

Zn(1)-Zn(3)#7 3.0583(8)

Zn(1)-Zn(3)#11 3.0718(8)

Zn(3)-Zn(3)#3 2.9907(15)

Zn(3)-As(1)-Zn(1)#1 118.37(4)

Zn(3)-As(1)-Zn(2)#2 107.88(3)

Zn(1)#1-As(1)-Zn(2)#2 107.96(3)

Zn(3)-As(1)-Zn(2) 108.68(3)

Zn(1)#1-As(1)-Zn(2) 115.58(3)

Zn(2)#2-As(1)-Zn(2) 95.70(4)

Zn(3)-As(1)-Zn(3)#3 70.60(3)

Zn(1)#1-As(1)-Zn(3)#3 72.57(2)

Zn(2)#2-As(1)-Zn(3)#3 75.00(2)

Zn(2)-As(1)-Zn(3)#3 169.54(4)

Zn(3)-As(1)-Zn(1)#4 71.41(3)

Zn(1)#1-As(1)-Zn(1)#4 79.86(3)

Zn(2)#2-As(1)-Zn(1)#4 170.73(4)

Zn(2)-As(1)-Zn(1)#4 76.12(2)

Zn(3)#3-As(1)-Zn(1)#4 112.74(3)

Continued on next page 69

Table A.3 – Continued from previous page

Zn(3)-As(2)-Zn(3)#5 121.72(5)

Zn(3)-As(2)-Zn(1) 110.42(3)

Zn(3)#5-As(2)-Zn(1) 107.50(2)

Zn(3)-As(2)-Zn(1)#5 107.50(2)

Zn(3)#5-As(2)-Zn(1)#5 110.42(3)

Zn(1)-As(2)-Zn(1)#5 96.34(5)

Zn(3)-As(2)-Zn(2)#6 74.37(2)

Zn(3)#5-As(2)-Zn(2)#6 74.12(2)

Zn(1)-As(2)-Zn(2)#6 171.86(4)

Zn(1)#5-As(2)-Zn(2)#6 75.75(2)

Zn(3)-As(2)-Zn(2)#7 74.12(2)

Zn(3)#5-As(2)-Zn(2)#7 74.37(2)

Zn(1)-As(2)-Zn(2)#7 75.75(2)

Zn(1)#5-As(2)-Zn(2)#7 171.86(4)

Zn(2)#6-As(2)-Zn(2)#7 112.22(3)

Zn(2)#8-As(3)-Zn(2) 97.04(5)

Zn(2)#8-As(3)-Zn(1)#5 106.71(2)

Zn(2)-As(3)-Zn(1)#5 113.99(2)

Zn(2)#8-As(3)-Zn(1)#9 113.99(2)

Zn(2)-As(3)-Zn(1)#9 106.71(2)

Zn(1)#5-As(3)-Zn(1)#9 116.81(5)

Zn(2)#8-As(3)-Zn(3)#6 171.51(4)

Zn(2)-As(3)-Zn(3)#6 75.89(2)

Zn(1)#5-As(3)-Zn(3)#6 72.67(3)

Continued on next page 70 Chapter A Zinc Arsenide Structural Information

Table A.3 – Continued from previous page

Zn(1)#9-As(3)-Zn(3)#6 73.05(3)

Zn(2)#8-As(3)-Zn(3)#10 75.89(2)

Zn(2)-As(3)-Zn(3)#10 171.51(4)

Zn(1)#5-As(3)-Zn(3)#10 73.05(3)

Zn(1)#9-As(3)-Zn(3)#10 72.67(3)

Zn(3)#6-As(3)-Zn(3)#10 111.54(5)

As(2)-Zn(1)-As(1)#2 114.80(3)

As(2)-Zn(1)-As(3)#12 116.39(4)

As(1)#2-Zn(1)-As(3)#12 113.36(3)

As(2)-Zn(1)-As(1)#7 104.53(3)

As(1)#2-Zn(1)-As(1)#7 100.14(3)

As(3)#12-Zn(1)-As(1)#7 105.20(3)

As(2)-Zn(1)-Zn(3)#7 119.73(3)

As(1)#2-Zn(1)-Zn(3)#7 122.09(3)

As(3)#12-Zn(1)-Zn(3)#7 56.26(2)

As(1)#7-Zn(1)-Zn(3)#7 49.23(2)

As(2)-Zn(1)-Zn(3)#11 137.09(4)

As(1)#2-Zn(1)-Zn(3)#11 57.56(2)

As(3)#12-Zn(1)-Zn(3)#11 56.06(2)

As(1)#7-Zn(1)-Zn(3)#11 118.32(3)

Zn(3)#7-Zn(1)-Zn(3)#11 91.89(2)

As(1)#1-Zn(2)-As(1) 116.91(3)

As(1)#1-Zn(2)-As(3) 113.39(3)

As(1)-Zn(2)-As(3) 115.39(3)

Continued on next page 71

Table A.3 – Continued from previous page

As(1)#1-Zn(2)-As(2)#6 103.15(3)

As(1)-Zn(2)-As(2)#6 103.58(3)

As(3)-Zn(2)-As(2)#6 101.65(3)

As(2)-Zn(3)-As(1) 120.93(4)

As(2)-Zn(3)-As(3)#6 107.59(3)

As(1)-Zn(3)-As(3)#6 110.10(3)

As(2)-Zn(3)-As(1)#3 106.94(3)

As(1)-Zn(3)-As(1)#3 108.78(3)

As(3)#6-Zn(3)-As(1)#3 100.56(3)

As(2)-Zn(3)-Zn(3)#3 125.988(18)

As(1)-Zn(3)-Zn(3)#3 58.98(3)

As(3)#6-Zn(3)-Zn(3)#3 123.27(3)

As(1)#3-Zn(3)-Zn(3)#3 50.42(2)

As(2)-Zn(3)-Zn(1)#4 129.56(3)

As(1)-Zn(3)-Zn(1)#4 59.35(2)

As(3)#6-Zn(3)-Zn(1)#4 51.063(18)

As(1)#3-Zn(3)-Zn(1)#4 120.76(3)

Zn(3)#3-Zn(3)-Zn(1)#4 98.25(3)

As(2)-Zn(3)-Zn(1)#13 113.89(4)

As(1)-Zn(3)-Zn(1)#13 125.15(3)

As(3)#6-Zn(3)-Zn(1)#13 50.894(18)

As(1)#3-Zn(3)-Zn(1)#13 49.87(2)

Zn(3)#3-Zn(3)-Zn(1)#13 88.03(3)

Zn(1)#4-Zn(3)-Zn(1)#13 87.66(2) 72 Chapter A Zinc Arsenide Structural Information

Table A.4 Symmetry transformations used to generate equivalent atoms for 5N Zn3As2.

#1 y+1/4,-x+3/4,-z+5/4

#2 -y+3/4,x-1/4,-z+5/4

#3 -y+5/4,-x+5/4,-z+5/4

#4 -x+3/2,y-1/2,z

#5 x,-y+1,-z+3/2

#6 -x+3/2,-y+1/2,-z+3/2

#7 -x+3/2,y+1/2,z

#8 -x+1,-y+1/2,z+0

#9 -x+1,y-1/2,-z+3/2

#10 x-1/2,y,-z+3/2

#11 x-1/2,-y+1,z

#12 -x+1,y+1/2,-z+3/2

#13 x+1/2,-y+1,z 73

2 3 Table A.5 Anisotropic displacement parameters (A˚ x 10 ) for 5N Zn3As2. The anisotropic displacement factor exponent takes the form: −2π2[h2a ∗2 U 11 + ... + 2hka ∗ b ∗ U 12]

U11 U22 U33 U23 U13 U12

As(1) 6(1) 7(1) 8(1) 1(1) 0(1) 0(1)

As(2) 7(1) 7(1) 8(1) 0(1) 0 0

As(3) 7(1) 7(1) 8(1) 0 0 0(1)

Zn(1) 9(1) 9(1) 11(1) -1(1) -1(1) 1(1)

Zn(2) 10(1) 10(1) 11(1) -2(1) 2(1) -2(1)

Zn(3) 10(1) 10(1) 11(1) -2(1) 0(1) 0(1) 74 Chapter A Zinc Arsenide Structural Information Appendix B

VASP settings

Perdew-Burke-Ernzerhof PAW potentials used:

• Ag pv

• Au

• U

• O

Default settings used for all calculations:

• PREC = a

• LWAVE = False

• LREAL = auto

• ISMEAR = 1

• SIGMA = 0.1

DFT+U settings used:

• ENCUT = 550

• LWAVE = False

• LDAU = True

75 76 Chapter B VASP settings

• LDAUTYPE = 1

• LDAUL = 3 -1 -1

• LDAUU = # 0 0 (# changes with ramping scheme explained in Section 3.2.1)

• ICHARG = 1 (after the ﬁrst iteration of ramping scheme explained in Section

3.2.1)

• LDAUJ = 0.51 0 0

• ISMEAR = -5

• LMAXMIX = 6

• ISIF = 2

• NSW = 0

• NELM = 150

Settings used to assign magnetic moments:

• ISPIN = 2

• MAGMOM = +/- 2 for U, 0 for O

In the case of non-convergence, ionic steps were allowed at a setting of:

• NSW = 3 and this additional setting was added:

• IBRION = 2

In the case of the following error:

• VERYBAD NEWS! internal error in subroutine SGRCON The following setting was changed:

• INCLUDEGAMMA = False and KPOINTS was regenerated. If it still did not work, the following setting was changed from the default: 77

• SYMPREC = 10−4

In the case of the following errors:

• POSMAP internal error: symmetry equivalent atom not found, you might try de- creasing or increasing SYMPREC by an order of magnitude.

• VERY BAD NEWS! internal error in subroutine PRICEL (probably precision problem, try to change SYMPREC in INCAR ?): Sorry, number of cells and number

of vectors did not agree.

• RHOSYG internal error: stars are not distinct, try to increase SYMPREC to e.g. 1E-4.

The following setting was changed from the default:

• SYMPREC = 10−4

To run calculations in parallel, the following settings were added:

• LPLANE = True

• NCORE = (number of cores)

• LSCALU = False

• NSIM = 4

The k-points used are generalized Monkhorst-Pack grids [42]. 78 Chapter B VASP settings