The Mechanism Behind the Calcium Aluminum Silicide Ternary Structural Preference and the Origin of Its Semimetal Behavior

THE MECHANISM BEHIND THE CALCIUM ALUMINUM SILICIDE TERNARY STRUCTURAL PREFERENCE AND THE ORIGIN OF ITS SEMIMETAL BEHAVIOR

by Torey Elizabeth Semi A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy (Applied Physics).

Golden, Colorado Date

Signed: Torey Elizabeth Semi

Signed: D.M. Wood Thesis Advisor

Golden, Colorado Date

Signed: Dr. Thomas Furtak Professor and Head Department of Engineering

ii ABSTRACT

CaAl2Si2 is the prototype of the CaAl2Si2 class of Zintl structures established to be useful as thermoelectrics. We propose that CaAl2Si2 be interpreted as an ordinary covalently bonded, tetrahedrally coordinated quasi-semiconductor consisting of a large distortion of the wurtzite structure with the almost fully ionized Ca inserted at an interstitial site. We support this interpretation via a structural mapping and calculations for both a charged binary primitive cell and a Si4 primitive cell.

Our intent is to explain the unusual structure of the CaAl2Si2 class of semiconductors, the origin of its semimetallic behavior, the basis for its stability and the eﬀect of substituting other column II atoms for Ca on these properties. To be clear, this work does not examine the nature of the true band gap, or the transport coeﬃcients of CaAl2Si2. GW corrections are not discussed, in view of the focus on the origins of stability of this peculiar structure.

iii TABLE OF CONTENTS

ABSTRACT ...... iii

LIST OF FIGURES ...... vi

LIST OF TABLES ...... xi

ACKNOWLEDGMENTS ...... xii

CHAPTER 1 INTRODUCTION ...... 1

1.1 Thermoelectric Materials ...... 2

1.2 Thermoelectric Eﬃciency ...... 5

1.3 Seebeck Coeﬃcient ...... 6

1.4 ‘Electron Crystal Phonon Glass’ ...... 7

1.5 Zintl Compounds ...... 9

1.6 Background and Motivation ...... 12

1.7 Other Members of the CaAl2Si2 Class ...... 16

CHAPTER 2 CALCULATIONAL METHODS ...... 18

2.1 Density Functional Theory ...... 18

2.2 Functionals ...... 22

2.3 The Scalar Relativistic Approximation ...... 24

2.4 Electronic Structure Codes and Pseudopotential Methods ...... 25

2.5 The PAW Method and its Implementation ...... 26

2.6 Convergence Studies ...... 30

CHAPTER 3 RESULTS AND DISCUSSION ...... 32

iv 3.1 Structural Properties of CaAl2Si2 ...... 32

3.2 Semimetallic Behavior of CaAl2Si2 ...... 38

3.3 Eﬀective Mass ...... 40

3.4 Structural and Electronic Calculations ...... 43

3.5 CaAl2Si2 and the Non-neutral Fictitious Binary Compound ...... 43

CHAPTER 4 MADELUNG ENERGY AND ELECTRONEGATIVITY ...... 52

4.1 Models for Stability ...... 52

4.2 Fixed Cell Volume and Its Impact on Total and Madelung Energies ...... 62

4.3 Electronegativity Arguments ...... 62

4.4 Bader Charge Analysis ...... 67

4.5 Summary ...... 69

CHAPTER 5 CONCLUSIONS AND FURTHER RESEARCH ...... 70

5.1 Association with Familiar Structure ...... 70

5.2 Utility of Fictitious Binary Reference Structure ...... 71

5.3 Arrangement of Atoms ...... 71

5.4 Stabilization Mechanisms ...... 71

5.5 Semimetal Behavior ...... 72

5.6 Future Work ...... 72

REFERENCES CITED ...... 74

APPENDIX A - THE CONNECTION BETWEEN DFT AND ELECTRONEGATIVITY ...... 79

v LIST OF FIGURES

Figure 1.1 Cartoon of thermoelectric device. The materials through which the electrons and holes travel must be diﬀerent from each other, to cause unequal responses to changes in temperature, and thus drive a current. Heavily-doped semiconductors fulﬁll this criterion...... 2

˚ Figure 1.2 Relaxed CaAl2Si2 unit cell. Vertical Si-Al bonds: 2.644 A. Toward horizontal Si-Al bonds: 2.569 A˚. a= 4.213 A˚, c = 6.941 A˚. c/a = 1.648 . . 3

Figure 1.3 A carrier concentration between 1019 and 1021 carriers/cm3, corresponding to that of heavily-doped semiconductors, is optimal for a good thermoelectric power factor S (S = α2σ) and a high zT ...... 4

Figure 1.4 Some CaAl2Si2 - class ternaries and quatenaries and their maximum zT values and corresponding temperatures...... 6

2− Figure 1.5 After subtracting the electron density of (Al2Si2) from that of CaAl2Si2, the remaining electron density reveals the presence of a weak covalent bond between Si and Ca. All calculations were run using ideal structural values to ensure coordinates are equivalent and the subtraction valid; no relaxation was introduced. The ﬁgure on the left shows a diagonal cut to emphasize the covalent bonding...... 11

Figure 1.6 Band structure of CaAl2Si2 suggesting that it is a semimetal...... 13

Figure 1.7 Illustration of La2O3 -like structure of CaAl2Si2. See Fig. Figure 1.8c for a single CaAl2Si2 primitive cell...... 14

Figure 1.8 Three different systems representing: (a) graphite, an sp2 covalent 3 configuration; (b), ZnO, an sp covalent configuration; (c), CaAl2Si2 primitive cell...... 14

Figure 1.9 Coordination environment for Si (bright blue) and Al (light blue). Al is flanked tetrahedrally by four Si. Si is four-coordinate in Al, but with an unusual umbrella configuration. Location of Ca ions (red) is shown to suggest their influence on the coordination environment...... 15

Figure 2.1 Here we compare PAW potential results with those of the ELK program. Despite the two programs using completely diﬀerent approaches to describe the basis of a system, their resultant band structures are nearly identical...... 28

vi Figure 2.2 Electron density shown from FHI (blue) and PAW (red) can be seen to match at interstitial sites in the CaAl2Si2 ternary. At ion core regions, the density reﬂects the manner in which it is modeled by each approximation; PAW is an all-electron potential and therefore records a high electron density near the nucleus, while the FHI potentials use only valence electrons (little electron density in core region)...... 28

Figure 2.3 CaAl2Si2 band structure comparison: Red=FHI CBs Blue=FHI VBs Yellow=PAW CBs Green=PAW VBs Then: orange means perfect overlap in CBs, cyan means perfect overlap in VBs. The PAW and FHI results match well...... 29

Figure 2.4 Convergence Study for CaAl2Si2, Si placed at origin. System is well-converged at an energy cutoﬀ of 90 Ha and 140 kpoints in the set. . . 31

Figure 3.1 Mapping between wurtzite structure and CaAl2Si2 structures. Top left: 2− Ideal wurtzite. Top right: Fictitious binary (Al2Si2) . Bottom left: 2− (Al2Si2) with green circles and black arrows indicating direction of Al2 plane motion to attain wurtzite structure. Bottom right: Ca atom 2− added to (Al2Si2) to complete ternary, demonstrating that distortion of the Al2 plane upwards and the insertion of a Ca atom into an interstitial site maps wurtzite to CaAl2Si2...... 35

Figure 3.2 Direct comparison between ZnO (wurtzite structure) and CaAl2Si2 in 3 5 wurtzite form. Moving the Al2 plane from the 8 to the 8 position in the primitive cell, and removing the Ca atom, returns the structure to a wurtzite conﬁguration...... 36

Figure 3.3 Left: ‘Upright’ CaAl2Si2. Center: Binary wurtzite with interstitial sites denoted by V 0s. A0s signify anion sites, C0s cation sites. The red circle indicates the location of the Ca ion in its wurtzite interstitial site, were Al atoms shifted up to match wurtzite plane, as in right image. Right: ‘Inverted’ CaAl2Si2 clearly a distorted wurtzite structure...... 36

Figure 3.4 Extended cell of CaAl2Si2 showing isosurfaces of low valence electron density, and wurtzite-structure interstitial sites. The unit cell is outlined in black. The red atoms are Ca and occupy one interstitial region, the yellow footballs are low electron density volumes centered on the second interstitial sites...... 37

Figure 3.5 Illustration of z-axis scan of Si atom in Al2 site showing local and global minima for fictitious binary, planar and wurtzite configurations of Si4. Total energy calculations; volume held fixed...... 39

vii Figure 3.6 Si4: Positions of Al2 plane corresponding to the metastable state, the barrier, or planar conﬁguration, and the global minimum...... 39

Figure 3.7 Upper left: Si4 in wurtzite and umbrella phases. Upper right: Band structure of CaAl2Si2. Bottom row: Si4 band structures in wurtzite and umbrella conﬁgurations, left to right. When forced into an umbrella phase, Si4 exhibits semimetal characteristics, very similar to CaAl2Si2. . . 41

Figure 3.8 Abinit vs ELK PBE GGA calculations (same exchange-correlation functional used): CaAl2Si2 band structure and density of states comparison. Upper frame shows the band structure generated by Abinit with blue and red, that by ELK with green. Lower frame, DOS in blue is via Abinit, red via ELK. Excellent agreement between two methods in both cases (discrepancy above 0.1 Hartree is simply due to an input parameter inconsistency). Important in DOS that electron densities around Fermi level are equivalent...... 42

Figure 3.9 Eﬀective masses at M point of CaAl2Si2 ternary...... 44

2− Figure 3.10 Electron densities of (Al2Si2) and CaAl2Si2 are very similar. Cut through 111 plane to include Ca interstitial site. Top row is CaAl2Si2, with and without isosurfaces and lattice planes. Likewise for the bottom 2− row, but for (Al2Si2) , our ﬁctitious binary that has no Ca atom. . . . . 46

Figure 3.11 Illustration of square modulus analysis of M9 band (conduction band minimum) of CaAl2Si2. The Ca atom in the cell on the right is removed in order to see the isosurface beneath it...... 47

Figure 3.12 Band structure comparison between CaAl2Si2 (left) and the non-neutral 2− ﬁctitious binary (Al2Si2) (right), displaying a close similarity and inferring that the Ca ion has a greater inﬂuence structurally than electronically. The band structure retains its semimetal property...... 47

Figure 3.13 The diﬀerence in electronic density between CaAl2Si2 and the non-neutral ﬁctitious binary (AlSi)2−, revealing a tiny covalent bond between Ca and Si...... 48

Figure 3.14 Left to right: Unit cells of BeAl2Si2 (green Zintl ion), MgAl2Si2 (orange Zintl ion), CaAl2Si2 (red Zintl ion), SrAl2Si2 (blue Zintl ion), BaAl2Si2 (purple Zintl ion). Not to scale; shows sketch of diﬀerences in unit cells. . 49

viii Figure 3.15 CaAl2Si2 band structures for the ternary’s preferred, planar and wurtzite geometries, respectively, from top to bottom. The ﬁgures clearly illustrate that for a CaAl2Si2 - class ternary with a preference for the umbrella conﬁguration, the structure is a semimetal. In the wurtzite arrangement, it is a semiconductor...... 50

Figure 3.16 Total energy of unmodified CaAl2Si2 structure (top left) lower than that of CaAl2Si2 structure with positions of Al and Si atoms switched. Energy difference is about 0.0374 Ha (approximately 1.02 eV). Supports stability of umbrella bonds in CaAl2Si2, as well as connection between semimetal behavior and umbrella configuration...... 51

Figure 4.1 Wurtzite ternary suggested by Nilthong ...... 53

Figure 4.2 Contributions of conventional components of total energy to the total energy of the system. The Ewald (EW) portion is equivalent to the electrostatic energy of the system, and is the dominant term. Abbreviations are explained in the text...... 55

Figure 4.3 Madelung energy: CaAl2Si2. The z values on the vertical axis reflect a continuum of placements of the Al2 atom as a fraction of lattice 5 1 constant c, including the wurtzite (z= 8 ), planar (z= 2 ) and umbrella 3 (z= 8 ) configurations. The charge state of the Ca ion, from 0 to +2, defines the horizontal axis. The contour lines reveal the stability of a given combination; here it is clear that with Ca fully ionized, the CaAl2Si2 umbrella arrangement is lowest in energy. To observe this, follow a line of isosurface minima from Ca with no charge to Ca with +2 charge (left to right). The line meets Ca with +2 charge just below the planar configuration (0.5), thus putting it in the region of umbrella stability. The upper and lower halves of the diagram are shaded; the lighter indicates a preference for wurtzite, the darker for the umbrella. The figure on the right is enlarged to show unequivocally that CaAl2Si2 prefers the umbrella structure preference...... 56

Figure 4.4 Si4: Red line: Ewald energies extracted from Abinit, equivalent to Madelung energy. Open circles: Madelung energy data points from D.M. Wood program; agree perfectly with Ewald energies, as they should. Blue line and points: Total energies produced by scan of z coord of Al2 (fraction of lattice constant c) through wurtzite, planar and umbrella conﬁgurations, produced by Abinit. Clear preference for wurtzite conﬁguration demonstrated by all methods. Inclusion of detailed electronic structure information is not necessary to determine structural preference of system...... 57

ix Figure 4.5 CaAl2Si2: scan of z coord of Al2 (fraction of lattice constant c). Clear preference for umbrella structure...... 58

Figure 4.6 Positions of z(Al2) as a fraction of lattice constant c vs. energy in Hartrees. Ewald energies calculated with ABINIT agree with Madelung energies output from D.M.Wood’s Madelung energy code (red line and red squares), and show a minimum at the umbrella conﬁguration 3 (z(Al2/c)= 8 . The blue circles show the total energy of the CaAl2Si2 system for the same positions and also veriﬁes that CaAl2Si2 stabilizes in the umbrella structure...... 60

2− Figure 4.7 Fictitious binary (Al2Si2) : Scan of z coord of Al2 (fraction of lattice 2− constant c). Note absence of Ca ion. (Al2Si2) shows a preference for wurtzite structure...... 61

Figure 4.8 Illustration of eﬀect of ﬁxed volume on total energy and Madelung energy comparisons, using pressure and stress...... 63

Figure 4.9 Electron densities of CaAl2Si2 and BeAl2Si2. We see that the Ca atom loses its charge to allow the formation of Si-Al bonds (top), and that Be actually steals charge from the Si-Al bonds, retaining most of its own charge. The deciding factor is the electronegativity of the interstitial atom...... 65

Figure 4.10 Band structure for BeAl2Si2; interstitial Ca atom replaced by Be. Band structure is clearly metallic. BeAl2Si2 is not stable in the umbrella structure (prefers planar or wurtzite) and therefore will not be a semimetal...... 66

Figure 4.11 Scan of z coord of Al2 (fraction of lattice constant c), with fixed volume, through umbrella, planar, wurtzite and beyond. Data with blue squares is for Si4 (SSSS), red circles for the fictitious binary with Al atoms replaced by Ga atoms (SGGS) and green triangles for the fictitious binary (Si and Al; SAAS). Clear preference is seen for the wurtzite structure for all combinations, as is expected for any tetrahedrally coordinated, 16-electron-per-primitive-cell system. The correspondence between electronegativity difference and depth of preference for wurtzite over umbrella is discussed in the text...... 67

x LIST OF TABLES

Table 1.1 Pauling Electronegativities (χ) of Certain Elements ...... 1

Table 1.2 Pauling Electronegativity Diﬀerences Between Column III and IV Elements . 1

Table 1.3 Lattice Parameters of CaAl2Si2 (in Angstrom) ...... 12

Table 3.1 Atom Positions for Umbrella, Planar and Wurtzite Ternaries ...... 33

Table 3.2 Structural parameters for ideal wurtzite and as calculated by T. Semi for CaAl2Si2 ...... 43

Table 3.3 Comparison of XAl2Si2 Structural Characteristics; X = Column II atoms; Be, Mg, Ca, Sr, Ba ...... 49

Table 4.1 Bader charges computed via the ABINIT AIM utility. int indicates interstitial site (Ca or Be atom); WZ, umbr, and plan indicate, respectively, ideal wurtzite, ‘umbrella’, and planar structures. Negative (positive) values indicate electron charge has ﬂowed into (out of) Bader basin about speciﬁed atom...... 68

xi ACKNOWLEDGMENTS

There are so many people to whom I am profoundly grateful. Without their support, encouragement, mentorship and eagerness to share their time, skills, knowledge and experi- ence with me, this journey I chose would not have been nearly as colorful and rewarding as it has been! A sincere and heartfelt thank you to my advisor of the past two years, David M. Wood. He picked me up, brushed me off and set me back on my way, never once losing faith in my ability to get here. And I wish to thank Uwe Greife for giving me a chance in the first place. Thank you to Tim Kaiser, who has seen me through a good portion of my challenges at Mines! and has provided me with many a wonderful opportunity. To Jim Bernard, my mentor from way back when all of this started, without whose sanity and calm I would be much the worse. Thank you to my committee members, Cristian Ciobanu, Craig Taylor, Eric Toberer and Zhigang Wu for their time and efforts on my behalf. I would very much like to thank Abram van der Geest and Liangzhe Zhang, with whom I spent my formative years at Mines (!). Thank you to my mom, and my dad, who have been there always for me. To my pals, Babette and Lolita. Last and most importantly, my husband, I don’t know what I would have done without you. You’ve been my rock and my light through all of these years. Thank you.

xii CHAPTER 1 INTRODUCTION

CaAl2Si2 is a ternary semiconductor, the prototype of the CaAl2Si2 class of Zintl structures known to be useful as thermoelectrics[1]. One aspect of condensed matter physics is to consider the atomic and electronic structure of a system, and this dissertation looks at the

CaAl2Si2 class of structures from this approach, with the intent to broaden options for thermoelectric materials. To establish the value of a rigorous investigation and reformulation of

the structural properties and preferences of CaAl2Si2, a brief introduction to thermoelectrics

ensues. This is followed by a thorough discussion of the known properties of CaAl2Si2, before plunging in to the heart of this research.

Table 1.1: Pauling Electronegativities (χ) of Certain Elements

Element F Fr H Be Mg Ca χ 3.98 0.7 2.2 1.57 1.31 1.00 Signiﬁcance Highest Lowest Ref Col II Col II Col II Element Sr Ba Ra Si Al Ga χ 0.95 0.89 0.90 1.9 1.61 1.81 Signiﬁcance Col II Col II Col II Col IV Col III Col III

Table 1.2: Pauling Electronegativity Diﬀerences Between Column III and IV Elements

Col IV → B Al Ga In Col III ↓ 2.04 1.61 1.81 1.78 C 2.55 0.51 0.94 0.74 0.77 Si 1.9 0.14 0.40 0.09 0.12 Ge 2.01 0.03 0.40 0.20 0.23 Sn 1.96 0.08 0.35 0.15 0.18

To successfully address the energy predicament in which we presently ﬁnd ourselves, locally and globally, it is essential to both develop a wide range of resources and to use

1 current supplies eﬃciently. According to research done at Lawrence Livermore National Laboratory, over 50% of energy used domestically is waste heat, a by-product of energy generated by the heat engine cycle[2, 3]. A more recent publication[4] suggests that two- thirds of energy consumed is squandered. Clearly, the ability to recover even a small portion of this as usable energy would be extremely advantageous. Thermoelectric materials are capable of facilitating this conversion, and are therefore of signiﬁcant interest in the new energy vista.

Heat Source

Cool Side

Figure 1.1: Cartoon of thermoelectric device. The materials through which the electrons and holes travel must be diﬀerent from each other, to cause unequal responses to changes in temperature, and thus drive a current. Heavily-doped semiconductors fulﬁll this criterion.[5].

1.1 Thermoelectric Materials

Thermoelectric devices (see Figure 1.1[5]) exploit the properties of pertinent materials in order to generate voltage from a temperature gradient and current from heat ﬂow, thus allowing useful recycling of waste heat[6]. Examples of thermoelectric applications include

2 (1,0,1)

(2/3, 1/3, 3/4) Ca

(1/3, 2/3, 1/2) Si2

Al2 (0, 0, 3/8) (1/3, 2/3, 1/8) Al1

Si1 (0,0,0) (1,0,0)

˚ Figure 1.2: Relaxed CaAl2Si2 unit cell. Vertical Si-Al bonds: 2.644 A. Toward horizontal Si-Al bonds: 2.569 A˚. a= 4.213 A˚, c = 6.941 A˚. c/a = 1.648 radioisotope thermoelectric generators, which convert the heat released by radioactive de- cay to power satellites and space probes, and automotive thermoelectric generators, which salvage usable energy from the exhaust (waste heat) to run components like headlights, and ultimately reduce fuel consumption. Another promising use combines solar energy systems and thermoelectric systems to take advantage of both high-frequency (solar cells) and low-frequency solar radiation[7, 8]. A thermoelectric material hosts free electrons and holes (a heavily doped semiconductor whose carrier concentration falls between 1019 and 1021 carriers/cm3 is a criterion for good performance; see Figure 1.3[6]). By connecting an electron conducting (n-type) and hole conducting (p-type) material in series, a net voltage is produced that can be driven through a load[7]. These free carriers move charge and heat, and can be described to a first approximation as a gas of charged particles. Briefly, a neutral gas at fixed volume subject to a temperature gradient will first accumulate a higher density of molecules at its cold end, since particles on the hot end will diffuse faster than those

3 at the cold end. The resulting density gradient will force molecules back toward the hot end. When equilibrium is reached, no net flow remains. If the particles are charged, the high density of particles at the cold end will cause a repulsive electrostatic force, creating a voltage, or electric potential (this is the Seebeck effect). Connecting the hot ends of the n-type and p-type components and putting a load across the cold ends results in a current flowing through the load, producing electrical power (P = I2R, where P is power, I is current and R is resistance, or load). The voltage is a consequence of the temperature gradient, the current arises from thermal motion (heat flow)[6, 9, 10].

Figure 1.3: A carrier concentration between 1019 and 1021 carriers/cm3, corresponding to that of heavily-doped semiconductors, is optimal for a good thermoelectric power factor S (S = α2σ) and a high zT [6].

In addition to turning waste heat into usable energy, thermoelectric devices generally have solid state components and require no circulating ﬂuids or moving parts (except electrons!), for quiet and reliable operation. Further attractive features include their ﬂexibility of form

4 and scalability[6, 10]; their design can be tailored to suit many situational requirements; large, small, irregular shapes, environment.

1.2 Thermoelectric Eﬃciency

Identifying good thermoelectric materials presents some captivating and complex challenges. Performance is described by the dimensionless thermoelectric ﬁgure of merit, expressed as[11]:

zT = σα2T/κ, (1.1) where σ represents electrical conductivity, α is the Seebeck coefficient and T indicates absolute temperature. κ denotes the thermal conductivity and includes both the electronic and lattice (vibrational) contributions: κ = κe + κl. A high zT indicates a useful material; to achieve this, it is evidently desirable to maximize the electrical conductivity and minimize the (phonon) thermal conductivity of a given system. Examples of maximum zT s and associated temperatures attainable by various CaAl2Si2 - class ternaries and quartenaries are shown in Figure 1.4[12]. From a transport property perspective, finding an arrangement with a high zT is decidedly an intriguing task, as these two requirements tend to be conflicting properties in most materials[11]. A modified form of zT , denoted ZT¯, where T¯ is the average temperature of the hot and cold ends of the thermoelectric device and Z distinguishes the device from the material, enters into the definition of the thermoelectric efficiency. Limited like all heat engines by the Carnot efficiency (ruled by the second law of thermodynamics)[13], the thermoelectric device efficiency η is expressed as: √ ∆T 1 + ZT¯ − 1 η = √ (1.2) T Th 1 + ZT¯ + c Th where

(S − S )2 T¯ ZT¯ = p n (1.3) h 1 1 i2 (ρnκn) 2 + (ρpκp) 2

5 zT

Figure 1.4: Some CaAl2Si2 - class ternaries and quatenaries and their maximum zT values and corresponding temperatures[12].

Here, ρn and ρp are the electrical resistivities of the n- and p-doped materials, respectively,

κn and κp their thermal conductivities and αn and αp their Seebeck coefficients. For small temperature differences and in the event that thermoelectric properties of the n- and p-type components are equivalent, ZT¯ can be replaced with zT . Otherwise, material differences are of great consequence and ZT¯ is the true measure of efficiency[6, 14]. For reference, a zT of one corresponds to an efficiency of about 17% at room temperature.

1.3 Seebeck Coeﬃcient

zT is also directly proportional to the square of α, the Seebeck coefficient; clearly a high absolute value of this quantity is beneficial in a thermoelectric material. The Seebeck coefficient emerges from the previously described Seebeck effect. Also called the thermopower, α is, to a very good approximation[15], the magnitude of the ratio of the thermoelectric voltage across a material in response to a corresponding change in temperature:

6 ∆V α = − . (1.4) ∆T The above representation of α derives from the deﬁnition of electrochemical force in the context of the Boltzmann equation describing transport phenomena (G~ = E~ + ∇~ µ/e) and in terms of the Seebeck coeﬃcient (G~ = α∇~ T )[16, 17]. As the emphasis of this work is mainly

on the structural properties of the CaAl2Si2 ternaries, and not so heavily on thermoelectrics, further detail will not be explored in this work. For a degenerate semiconductor, with parabolic bands and the approximation of energy- independent scattering, the Seebeck coeﬃcient can be written as[6]:

8π2k2 π 2/3 α = B m∗T , (1.5) 3eh2 3n

where kB is the Boltzmann constant, h Planck’s constant, T temperature, e the electron charge and n represents the carrier concentration. m∗ is the effective mass of the charge carrier energy density of states. This rendition shows how thermoelectric efficiency depends on various transport properties; effective mass and carrier concentration, electrical and thermal conductivity, and presents a starting point at which their individual and combined contributions can be optimized. The connection to σ is implicit in the carrier concentration; for a semiconductor, σ = neµ, with µ the carrier mobility.

1.4 ‘Electron Crystal Phonon Glass’

Examples of inherent conﬂicts can be viewed as follows. High Seebeck coeﬃcients are characteristic of doped crystalline semiconductors, electrical conductivities (σ = neµ) are high in crystalline metals and thermal conductivities are low in glasses (the electronic portion of κ = κl + κe is a straightforward consequence of the Wiedemann-Franz Law; κe = LσeT

2 2 π kB with L the Lorenz number = 3e2 , leaving κl as the variable to minimize; in fact there are

a variety of ways to ﬁne-tune κl). A large eﬀective mass suggests a high DOS at the Fermi level but also low electrical conductivity; the low group velocity associated with these heavy carriers implies low mobility[2]. The relationship:

7 τe µ = ∗ (1.6) mi supports this claim, with µ the mobility, τ the time between scattering events, e the carrier

∗ charge and mi the transport effective mass[16]. Phonons must undergo strong scattering in order to minimize the lattice thermal conductivity. In contrast, electrons need to encounter little scattering in order to sustain high conductivity[10]. These observations suggest that an ‘electron crystal-phonon glass’ (ECPG) structure would constitute an optimal thermoelectric material[6]. Strategies to attain one or more ECPG systems and therefore a plausible zT , which is considered to be at least 1.0 and has reached as high as 2.0 [1, 11] have most recently focused on ways to minimize thermal conductivity. Approaches involve disordered unit cell, nanostructuring and complex unit cell consideration. Disordered unit cell methods make use of skutterudites, clathrates; structures with random vacancies and void spaces that disrupt phonon transport[18]. Examples include CoSb3 with multiple cofillers; Ba, Yb and La; a zT of 1.7 has been achieved at 850K[19]. Nanostructuring takes advantage of grain boundaries to scatter phonons and thin-film superlattices tantalize with increased Seebeck coefficients and decreased thermal conductivity. Nanocrystalline powders such as p-type BixSb2−xTe3 have increased zT in bulk materials to 1.4 at 373K[20]. An instance of high zT in a system with a complex unit cell is In4Se3−δ, which realizes a zT of 1.48 at 705K. Complex unit cell concepts are also well-represented by Zintl compounds, which exhibit valence balance, an important characteristic common to all thermoelectric materials, and a substructure arrangement, in which ideally, the mechanisms for phonon glass and electron crystal behaviors are kept separate, or minimally interfere with each other. At present, the general consensus is that (heavily doped) semiconductors provide the most effective systems in commercial thermoelectric applications[6, 21, 22]. Their band structures evince more desirable thermoelectric qualities than those of metals, in the guise of electrical conductivity. The Fermi level of a good thermoelectric material resides close to one edge of

8 the band gap[21]; in order for electron transport to occur, a state must be partially ﬁlled, and states with that characteristic have energies that place them near either the conduction or valence band. In a semiconductor, the Fermi level lies in the band gap below the conduction

band, the density of states is asymmetric about EF and the average valence electron energy is

greater than EF . This setting is favorable to a shifting of charged particles to a lower energy state. In a metal, the Fermi level exists in the conduction band, offering a symmetric density of states about EF and an average conduction electron energy comparable to EF [23]. Since charge transport is a main contributor to a high zT , semiconductors are far more likely to be a successful choice than metals. Heavily-doped, or degenerate, semiconductors, with carrier concentrations on the order of 1019 to 1021 carriers/cm3, are generally the best-performing semiconductors[2]. Adding dopants (impurities that allow tuning of a material’s electrical properties) to a semiconductor creates a population of bound states; p-type doping adds holes near the top of the valence band and n-type doping results in states being formed just below the bottom of the conduction band. Doping decreases the separation in energy between the Fermi level and the energy level of the bound states introduced by the corresponding dopant type. The resulting increase in carrier concentration augments the mobility, and hence the electrical conductivity, without adversely affecting the thermal conductivity. With a large enough band gap, single-carrier dominance can be achieved, ensuring a maximum net Seebeck coefficient for the system (it is considered optimal that one carrier-type dominate in order to avoid bipolarization[24]). Despite advancements and auspicious outcomes of present research efforts; Tl-doped PbTe was recently (and only preliminarily) shown to have a zT of 1.5 at 773K[25] and a zT of 2.2 at 915K was achieved last year with Na-doped PbTe-SrTe[4],

binary semiconductor compounds such as Bi2Te3 and PbTe serve presently as the industry standard thermoelectric materials, with a zT of about one[1].

1.5 Zintl Compounds

CaAl2Si2 systems are naturally described as Zintl phases. The Zintl postulate, formulated by E. Zintl[26–28] and subsequently developed by others [29, 30], describes the correlation

9 between number of valence electrons and crystal structure [31]. Structural components are linked to chemical stability via closed electron shell conﬁgurations (Zintl phases are valence balanced, or have stoichiometrically balanced valences[32]. A Zintl phase is also semiconducting in nature; as a valence compound its constituents have disparate electronegativities, suggestive of anion and cation contributions[22]. Zintl electron counting is central to the Zintl concept, and is anchored by two main ideas: an (ideally) complete charge transfer occurs between the electropositive (electron-losing) element and the electronegative (electron- attracting) bonding members, and with the use of the donated electrons, the latter adhere to the octet rule. For example, anion constituents of a given system require electrons; the cation associated with this system is viewed primarily as an electron donor that exists to fulﬁll the

2+ 2− anions’ valence needs; in CaAl2Si2, Ca is the electron donor, (Al2Si2) the acceptor[33].

CaAl2Si2 is somewhat unusual, however, in that the Ca atom does exhibit weak Ca-Si covalent bonding, as is veriﬁed by Figure 1.5, which ampliﬁes the charge left after the electron

density of the fictitious binary has been subtracted from that of the CaAl2Si2 structure. Ac- cording to Alemany, et.al.[31], however, the presence of this covalent bond does not detract from the ternary’s eligibility to be treated as a Zintl phase. A Zintl phase neatly partitions the ionic region from the covalently bonded one (reflecting the separation of the ‘phonon- glass’ from the ‘electron-crystal’ attributes, as desired). The cation/anion interaction is mainly ionic; the electronegativities of Si and Al are enough alike to encourage covalent bonding, while the electron density can be fine-tuned by replacing Ca2+ with other donor elements[34]; we explore column II element substitution in another section. Compounds for which these notions are valid are termed Zintl phases, and are prevalent among structures with useful zT [31]. Another way to expand the number of options for thermoelectric materials is to exploit the structural complexity and tunability of Zintl phase, ternary semiconductors. Here we focus on the group of ternaries of CaAl2Si2 -type. By constructing a comprehensive structural basis for this class of systems we will gain substantial, significant insight into into its latent qualities

10 Si

2− Figure 1.5: After subtracting the electron density of (Al2Si2) from that of CaAl2Si2, the remaining electron density reveals the presence of a weak covalent bond between Si and Ca. All calculations were run using ideal structural values to ensure coordinates are equivalent and the subtraction valid; no relaxation was introduced. The ﬁgure on the left shows a diagonal cut to emphasize the covalent bonding.

11 and, most importantly, apply predictive capabilities to a very wide range of potentially new thermoelectric materials with enhanced zT .

1.6 Background and Motivation

CaAl2Si2 is a double octet ternary of space group P3m1[35,¯ 36], with 5 atoms and 16 valence electrons per unit cell. Its primitive cell is hexagonal, and its experimental lattice parameters are a = 4.130A˚ and c = 7.145A˚[35], with Si and Al atoms site-specific[34, 37] (see Table 1.3). It was initially thought to be metallic [35, 38] but more recently has been recognized as a semimetal[31, 39, 40]; Figure 1.6 demonstrates this quality, among others. Its Fermi surface is characterized by two hole pockets at the Γ point and one electron pocket at the M point and the Fermi level is found at the minimum of the total DOS curve. It is superconducting below 1.4K[39]. Its transport properties undergo an abrupt change at 150K, at which point the Hall coefficient RH changes sign (RH is 1300 at 5K and -51 at 300K[35, 36]) and the dominant carrier type goes from holes below 150K to electrons above 150K[39]. Electrical resistivity ρ displays a weak dependence on temperature; after increasing with temperature it saturates at 200K (inconsistent with Bloch-Grunëisen theory)[41]. The electron-phonon coupling constant λ is 0.17, suggesting very weak electron- phonon interaction (and justifying its superconductivity limit of 1.4K)[41]. Above features most relevant to this work will be expanded upon later in the text, including band attributes at the band gap edges.

Table 1.3: Lattice Parameters of CaAl2Si2 (in Angstrom) Experiment[35] Calculation Semi: Calculation Semi: Comparison Ideal X-ray Diﬀraction FP-LMTO[39] Planewave To Experiment a 4.414 4.142 4.213 0.95 c 7.133 7.137 6.941 0.97 c/a 1.616 1.723 1.648 0.99 1.633

Prevailing literature regards the CaAl2Si2 class of ternaries as La2O3 -type layered trigonal structures, with Ca placed at the origin[37, 40] (see Figure 1.7). It also describes CaAl2Si2

12 Band Structure from Bnds-OtherAtOptGeom.out

0 εF

-5 Energy (eV)

-10

Γ M K Γ A L H A M

(a) Bril- (b) CaAl2Si2 band structure. Note the dip by the lowest con- louin Zone CaAl2Si2 duction band below the Fermi level at the M point.

Figure 1.6: Band structure of CaAl2Si2 suggesting that it is a semimetal.

2− +2 as having double-corrugated hexagonal layers of tightly-bound (Al2Si2) with Ca ions

2− intercalated between them[33, 39]. The (Al2Si2) layers are said to form two six-membered

2− (Al2Si2) sheets, stacked in a corrugated (or puckered[37]), chair-like fashion[33, 37]. These descriptions, while accurate, do not effectively describe the CaAl2Si2 structure, nor offer any germane insight; for example, the La2O3 ideal is a binary system, while CaAl2Si2 is a ternary. Most glaring, and with the most consequences, a fundamental reason for the mysterious corrugation is lacking; the La2O3 characterization does not give a convincing justification for the evident stability of the ‘puckered’ configuration. This buckling terminology is only useful if one expects planar behavior; it would seem more efficacious to keep close to familiar wurtzite and zincblende compounds; III-V and II-VI binaries tweaked to create wurtzite and zincblende ternaries[42]. Figure 1.8 demonstrates these sp2 and sp3 configurations, with a

CaAl2Si2 unit cell for comparison.

We will show that CaAl2Si2, exemplifying others in its class, is an ordinary covalently bonded solid, with tetrahedrally coordinated Si and Al atoms. We will establish that it is similar to wurtzite, with a Si atom at the origin and with each Ca ion at a particular interstitial site. As with a typical Zintl compound, Ca donates its electrons, allowing Si and

13 Figure 1.7: Illustration of La2O3 -like structure of CaAl2Si2. See Fig. Figure 1.8c for a single CaAl2Si2 primitive cell.

Si Zn Ca

O C Al

Figure 1.8: Three different systems representing: (a) graphite, an sp2 covalent configuration; 3 (b), ZnO, an sp covalent configuration; (c), CaAl2Si2 primitive cell.

14 Al to bond tetrahedrally; the Si atoms, however, form a curious umbrella-like conﬁguration with the Al atoms (see Figure 1.9). This is unusual in that Si generally forms bonds with alignment similar to that shown by Al, not with all four bonds on the same side on the Si atom. For reasons related to electrostatic energy that will be described, the Si atoms prefer

to be near the Ca. Our calculations have veriﬁed that when atoms in CaAl2Si2 are fully

relaxed, the CaAl2Si2 structure is adopted.

Figure 1.9: Coordination environment for Si (bright blue) and Al (light blue). Al is flanked tetrahedrally by four Si. Si is four-coordinate in Al, but with an unusual umbrella configuration. Location of Ca ions (red) is shown to suggest their influence on the coordination environment.

In order to explain the structural preferences of the CaAl2Si2 class of compounds, it is crucial to choose an electronic structure code capable of providing an integrated computational framework. We have selected ABINIT [43–45], an open-source electronic structure package based upon density functional theory (DFT)[46], for its ﬂexibility and capabilities. It allows for computation involving non-neutral cells, which is useful for this problem. An- other attractive feature is the ability to generate one’s own PAW (projector augmented wave) potentials [47]. Standard PAWs could be used for Al and Si as they are simple materials, but

15 the 3d electron configuration in Ca can have a significant effect on the pseudopotential. This is a fairly intricate process, and significant effort was put forth to create and test a PAW potential for Ca. Further discussion of PAW potentials, including support for the ultimate choice of pseudopotential, ensues in Chapter 2 and a detailed treatment can be found in Appendix B.

CaAl2Si2 exhibits some rather unusual electronic and structural properties[36, 39], providing a challenge in their full characterization. The chief focus of this project is the structural

preferences of CaAl2Si2; we wish to discover and explore the mechanism by which CaAl2Si2 stabilizes in the peculiar form that it does. Along the way, we discover contributing aspects,

2− such as information we from the fictitious binary (AlSi) , whether or not CaAl2Si2 is related to a familiar structure, whence the semimetal behavior, reasons for the adopted arrangement and the stabilizing influences on this energetically preferred configuration. We explore three related compounds; the ternary semiconductor CaAl2Si2, a non-neutral fictitious binary con-

sisting only of the Si2 and Al2 portions of the ternary compound and a wurtzite structure, a II-VI semiconductor with 16 valence electrons per primitive cell. Calculations we have

2+ performed conﬁrm that the CaAl2Si2 structure is indeed distorted wurtzite, with the Ca ion at one interstitial site. Close examination of the non-neutral binary, in which there is no atom at the Ca site but the two electrons the Ca atom would have donated are present, has allowed us to assess the extent to which any column II element can be substituted for Ca; an analysis with some intriguing results set forth in Chapter 3.

1.7 Other Members of the CaAl2Si2 Class

The thermoelectric properties of diﬀerent groups of systems belonging to the CaAl2Si2 class have recently been studied, and it is worth including examples to appreciate the breadth

of the CaAl2Si2 class. Ternary antimonides (AX2Sb2; X = Cd, Zn, A= Ca, Yb, Eu) are known to have zT s between 0.5 and 1.1 and their structural complexities and tunabilities have been

found to be advantageous in maximizing zT [48]. Ternary bismuthides (AMg2Bi2; A = Ca, Eu, Yb) have been shown to possess similar characteristics, with low carrier concentrations

16 and high mobilities. It is also noted that rare-earth compounds have greater mobilities than alkaline-earths[49]. SrAl2Si2, isostructural to CaAl2Si2, has received considerable attention as well, and compared to the binary La2O3. For SrAl2Si2, an alleged pseudogap at Fermi level DOS gives rise to exotic transport properties[50]. It is determined that by substituting Y for Sr, a notable decrease in the electrical resistivity, and therefore the Seebeck coeﬃcient, occurs[51]. It would be instructive to investigate these variations in further detail, and to compare them with CaAl2Si2.

17 CHAPTER 2 CALCULATIONAL METHODS

There are many components that ﬁt together theoretically and computationally to facilitate a successful calculation. This chapter details those requirements.

2.1 Density Functional Theory

Density Functional Theory (DFT) is the method chosen to solve the Hamiltonian describing the electronic structure and consequently predicting the behavior of the Al, Si and

Ca atoms in CaAl2Si2 at the atomistic level. DFT is considered an ab-initio technique; strictly speaking, however, ab-initio refers to the ability to describe all properties of a system based on knowledge of atomic number only, and the immediate implementation requires more parameters than Z. This is rationalized in that the DFT approach is evolving, and parameterization is minimized based upon current wisdom. The Hamiltonian that describes a many-body system of electrons and nuclei can be written as follows:

The ﬁrst term denotes the kinetic energy of the electrons (Tˆ), the second represents ˆ the potential acting on the electrons due to the nuclei (Vext) and the third term is the ˆ electron-electron interaction (Vint). The kinetic energy of the nuclei/ions follows; this term is neglected in the adiabatic approximation, in which the time scale of the motion of the nuclei is so much greater than that of the electrons that the electrons can be presumed to be in an instantaneous state. The ﬁnal term describes the classical interaction of the nuclei, and any other contributions to total energy that are not concerned with electronic structure

(EII ).

18 The fundamental Hamiltonian for the theory of electronic structure can be summarized as follows:

ˆ ˆ ˆ ˆ H = T + Vext + Vint + EII . (2.2)

The task is to solve the many-body Schr¨odinger (Dirac) equation with this Hamiltonian, which shall be addressed using density functional theory. The primary principal behind DFT is that any property of a system of interacting particles can be represented as a functional of the ground state density. In other words, a single scalar function of the ground state density determines all information in the many-body wave functions for the ground state (and in principle, for all excited states)[52]. This methodology allows replacement of a system of 3N variables with N systems of 3 variables. The statement of DFT does not specify the formulation of these functionals. Hohenberg and Kohn suggested that DFT be expressed as an exact theory of many-body systems, whose Hamiltonian can be written in the form:

2 2 ˆ −~ X 2 X 1 X e H = ∇i + Vext(~ri) + (2.3) 2me 2 |~ri − ~rj| i i i6=j where we use Hartree atomic units: 4π ~ = me = = 1. (2.4) 0 This is in fact the fundamental electronic structure Hamiltonian, without the classical interaction of ions. Accompanying this ansatz, Hohenberg and Kohn proposed two theorems, which form the basis for the practical implementation of DFT. Theorem 1 allows the system to be described with electron density as a basic variable. Theorem 2 enables the construction of functionals of the electron density; it states that there exists a universal (for all electrons) functional for the energy in terms of the electron density that is valid for any external potential (for example, that of the nuclei acting on the electrons). The focus here is on the external potential; the global minimum value of the functional for a given external potential is the exact ground state energy of the system, and

19 the electron density that minimizes it is the exact ground state density[52]. Another way to interpret Theorem 2, as is pointed out by Liechtenstein[53], is that the ground state of an electronic system should be uniquely determined by any quantity that has a nonzero macroscopic expectation value. Since the energy of the system is nonzero, and is deﬁned as the expectation value of the Hamiltonian, a total energy functional can be formulated: Z 3 EHK [n] = T [n] + Eint[n] + d rVext(~r)n(~r) + EII (2.5)

Here the expectation value of the external potential is expressed in integral form, over the density function, and the adiabatic approximation invoked. Obstacles are nonetheless inherent in the Hohenberg-Kohn energy formulation. It still describes a many-body interacting system. To combat this less-than-favorable situation, Kohn and Sham proposed to choose a non-interacting system that possesses the same ground state density of the interacting system; this new system is dubbed the ‘auxiliary system’. The idea is that now the system is exactly solvable in terms of independent-particle equations for the non-interacting system, with all of the remaining ‘many-body’ terms incorporated into what is termed the exchange-correlation functional of the electron density[52]. The Kohn-Sham rewrite of the Hohenberg-Kohn energy functional is then as follows; Z EKS = Ts[n] + d~rVext(~r)n(~r) + EHartree[n] + EII + EXC [n], (2.6) where

N σ X X X σ 2 n(~r) = n(~r, σ) = |ψi (~r)| (2.7) σ σ i=1

N σ N σ 1 X X 1 X X Z T = − hψσ|∇2|ψσi = d3r|∇ψσ(~r)|2 (2.8) s 2 i i 2 i σ i=1 σ i=1 and

Z ~0 1 3 3 0 n(~r)n(r ) EHartree[n] = d rd r . (2.9) 2 |~r − r~0|

20 It is instructive to note that n(~r), the electron density, is given by the sums of squares of the orbitals of each spin. This relationship forms the basis for the self-consistent calculation of the electron density. An initial guess is made, the Schroedinger equation solved and a wave function obtained. The next iteration begins with the electron density based on this new wave function; the process is repeated until the speciﬁed tolerance is reached.

Ts[n] represents the independent-particle kinetic energy. EHartree is the self-interaction of the electron density, treated as a classical charge density. This self-interaction energy will cause modiﬁcations to be made, in the form of double counting schemes, for LSDA+U implementations. The purpose of the Hohenberg-Kohn and Kohn-Sham formulations is to ﬁnd the ground state of the electronic system. To that end, the Kohn-Sham energy must be minimized with respect to the electron density. An examination of the equation for EKS reveals that all terms except for the kinetic energy are functionals of the electron density, and the kinetic energy term a functional of the orbitals. It is, therefore, admissible to apply the variational principle, giving the Kohn-Sham variational equations: δEKS δTs δEext δEHartree δExc δn(~r, σ) σ∗ = σ∗ + + + σ∗ = 0. (2.10) δψi (~r) δψi (~r) δn(~r, σ) δn(~r, σ) δn(~r, σ) δψi (~r)

These equations are subject to the orthonormalization conditions:

σ σ0 hψi |ψ i = δi,jδσ,σ0 , (2.11)

σ and, using the previous expressions for n (~r) and Ts the following substitutions can be made:

δTs 1 2 σ σ∗ = − ∇ ψi (~r) (2.12) δψi (~r) 2 σ δn (~r) σ σ∗ = ψi (~r). (2.13) δψi (~r)

The method of Lagrange multipliers can now be applied; h i δ hΨ|Hˆ |Ψi − E(hΨ|Ψi − 1) = 0, (2.14)

21 which is equivalent to the Raleigh-Ritz principle that the functional

ˆ ΩRR = hΨ|H − E|Ψi (2.15)

be stationary at any eigensolution |Ψmi. Variation of hΨ| gives

hδΨ|Hˆ − E|Ψi = 0, (2.16) which must hold for all hδΨ|.

This condition is satisﬁed only if

Hˆ |Ψi = E|Ψi, (2.17) which leads to the Kohn-Sham Schroedinger-like equations:

σ σ σ (HKS − εi )ψi (~r) = 0 (2.18)

1 Hσ (~r) = − ∇2 + V σ (~r) (2.19) KS 2 KS δE δE V σ (~r) = V (~r) + Hartree + XC (2.20) KS ext δn(~r, σ) δn(~r, σ)

σ = Vext(~r) + VHartree(~r) + VXC (~r). (2.21)

2.2 Functionals

As described in Section 2.1, the number of electrons in CaAl2Si2 requires careful treatment. The local (spin) density approximation[54] (LSDA, or LDA commonly), which maps the many-body electron problem into one of single electron orbitals, has proven remarkably accurate for many elements of the periodic table. It fails, however, for the transition metals, rare-earth metals and actinides. For example, for UO2, LDA predicts a metallic instead of insulating ground state, and a collapse of the ﬂuorite structure[55, 56]. This breakdown has been attributed to LDA’s inability to properly describe the correlations between the two 5f electrons. The strong on-site Coulomb repulsion of these electrons is responsible for the

22 emergence of the band gap, and therefore must be taken into account by the Kohn-Sham functional in order to simulate the electronic, magnetic and crystal structure of UO2. The addition of the +U functional to the L(S)DA functional corrects for the failure of L(S)DA to properly describe the electronic and magnetic structure of the class of Mott

Hubbard insulators, to which UO2 belongs. The success of this model is attributed to the incorporation of orbital ordering, as well as to the recognition that the screened on-site Coulomb interactions are responsible for spin and orbital polarization; not the exchange interactions of the homogeneous electron gas[53]. Since LDA is based on mean-field theory, in which large S is the controlling factor as opposed to coupling constants that arise from interaction strength, it does a fairly decent job of describing MHIs when applied to Hubbard models. But what’s missing is the apparent necessity to treat atomic orbital degeneracies at the same level as spin degeneracies[53]. Claims have been made[57, 58] that the implementation of a DFT code (specifically VASP) with an exchange-correlation GGA functional, instead of the more complex L(S)DA +U functional, is sufficient to generate reliable information with respect to the energetics of UO2. Use of the GGA without the +U alleviates the complexities (and uncertainties) associated with the +U parameters. Although GGA is known to give an incorrect ground state band structure for UO2 (metallic instead of insulating), if this can be shown to be of negligible impact on energetics, then the problem of unknown electron density around a defect, the issue of metastable vs ground states and the use of experimental exchange and Coulomb energies can be avoided. Since the ultimate goal of this research is to predict trends of activation energies and diffusion coefficients in concert with lattice strain, quite possibly the GGA implementation will adequately address this purpose.

But for CaAl2Si2, the LSD approximation is adequate. Some improvement can be seen with the generalized gradient approximation (GGA). Developed as an improvement to LSDA, GGA incorporates the gradient of the electron density of a uniform electron gas to represent the exchange-correlation energy of a given spatial point in an electronic system. It has been

23 shown to enhance total energies, atomization energies, energy barriers and structural energy diﬀerences with respect to LSDA[59]. GGA tends to expand and soften bonds, resulting in corrections or over-estimations to LSDA calculations. Generally, for systems with density inhomogeneity[59], GGA seems to produce more accurate output; for this reason we select GGA as the functional for this work.

2.3 The Scalar Relativistic Approximation

A brief mention should be made regarding relativistic density functional theory, since it embraces the equation that fully describes the systems for which we are solving, the Dirac equation. Arguments analogous to the non-relativistic many-body case, in which it is shown that the external potential is uniquely determined by the ground state electron density, can be made for the relativistic system. Without going into details, suffice to say that the full Dirac equation is difficult to solve for the same reasons as the Schrödinger equation, with the added complexity of the relativistic terms (i.e. spin-orbit coupling). Neglecting the spin-orbit coupling term defines the scalar-relativistic approximation (SRA), whose implementation for DFT is described by Koelling and Harmon[60]. The significance of the scalar-relativistic ‘Dirac’ equation is that it captures a large portion of the relativistic effects, which is the reason it is used by many electronic structure codes. In VASP, for example, the projector- augmented wave (PAW) potentials for the actinide elements are built according to the scalar relativistic approximation. Likewise, RSPt (an LMTO, or linear muffin-tin orbital code) in its current form uses the same approximation. To obtain fully relativistic results, spin-orbit coupling must also be included; in VASP, this is calculated non-self-consistently once the charge density for the scalar-relativistic Hamiltonian is achieved. For ABINIT, the code employed in this dissertation, PAW potentials can be constructed by the user, who has the choice of non-relativistic or SRA for the atomic problem (there is no option for fully relativistic). All atomic PAW potentials used in this research specify the SRA, for accuracy, versatility and transferability among different systems.

24 2.4 Electronic Structure Codes and Pseudopotential Methods

An imposing hurdle to accurate electronic structure calculations is the ability to represent the oscillations of the wavefunctions near the nucleus and their relative smoothness in the bonding regions in a computationally practical manner. There are two classes of calculational methods that dominate electronic structure codes; linear all-electron methods and pseudopotential methods. Linear methods (linear in energy) are based on the augmented plane wave (APW)[61, 62] and Korringa, Kohn and Rostoker (KKR) [63, 64] approaches. The pseudopotential method draws from norm-conserving ab-initio pseudopotentials as developed by Hamann, Schlüter and Chiang[65]. In both methods, wavefunctions are radial solutions to the Schrödingerequation times the spherical harmonics inside a sphere radius, represented by partial wave expansions, and an envelope function expanded in Fourier series or plane waves or some other appropriate basis outside the sphere radius. Norm-conserving pseudopotentials substitute a pseudo wavefunction (PWF) for the all- electron valence wavefunction (AE) inside some core radius; the AE is a full one-electron wavefunction, obtained by solving the Kohn-Sham equations. The PWF is soft and nodeless, and within the core radius, its norm must be equivalent to that of the AE. Ultrasoft pseudopotentials[66] eliminated the norm-conserving constraint, incorporating the resulting charge density difference between the AE and PS constructions as an augmentation in the core region. These augmentations in turn needed to be pseudized for the sake of computational efficiency. In effect, improvements to the technique added to the number of parameters required to specify the pseudopotential and increased the complexity of their construction[67]. The PAW method avoids the charge augmentation pseudization, by involving only the full AE wave functions and potentials[68]. PAWs can include a frozen core approximation for computational efficiency, in which the properties of the core electrons are fixed. In contrast, the APW methods, which are based on the linear methods, match value and energy derivative of the envelope functions at the sphere boundary to those of the

25 partial waves inside the sphere. Because of the energy dependence, the matching results in nonlinear equations, which are computationally demanding since each eigenstate must be solved for individually. Linearization of this method led to more practical implementations, including the LMTO. The hallmarks of the LMTO method are that the muﬃn-tin orbitals are generated from the Kohn-Sham hamiltonian and that calculations can satisfactorily be performed with a minimal basis set[52]. As previously stated, the main thrust of this research is carried out using the ABINIT code. ABINIT is a plane-wave code in which a range of pseudopotential formulations is available. We choose the PAW method for calculations involving CaAl2Si2 itself, and norm- conserving Troullier-Martins[69] pseudopotentials for cases in which we are searching for trends. The code incorporates the frozen-core approximation, and solves the Kohn-Sham equations self-consistently. Core and valence electrons can be treated non-relativistically or via the scalar-relativistic approximation. To improve upon this approximation, spin- orbit coupling can be added as a perturbation (a non-self-consistent direction of the orbital moment).

2.5 The PAW Method and its Implementation

The main virtue of the PAW method is its capacity to integrate the most desirable aspects of the linear methods and the pseudopotential methods and produce a unified approach to electronic structure codes. It is an all-electron, not a pseudopotential, method. Its explicit inclusion of one-center terms (on a radial grid) requires more computational effort, but scales linearly with the number of atoms and is thus negligibly small, an improvement over the plane-wave rendition. It allows for an added flexibility that both increases computational efficiency and accuracy. Larger core radii, and thus softer pseudopotentials can be implemented without greatly affecting transferability, and the norm-conserving condition is relaxed, resulting in the incorporation of smoother partial waves and a smaller basis set[67]. The essence of the PAW approach is as follows. Blöchl[67] introduced a linear transformation from the PWF wavefunctions to the AE wavefunctions, thus allowing physical quantities

26 expressed as expectation values of some operator on PWF wavefunctions to be interpreted as physical AE quantities, including the total energy functional. The transformation maps the physically relevant wavefunctions in the Hilbert space of all wavefunctions orthogonal to the core states to pseudo wavefunctions in a new pseudo Hilbert space. The transformation is chosen such that it reflects specific atomic traits, and differs from the identity by the sum of local, atom-centered contributions. A much more thorough discussion of the PAW method is offered in B, however, considerable effort has been spent addressing the issue of 3d electron orbitals in the Ca PAW potential. We have since elected to use norm-conserving (FHI) potentials for most calculations, since although this is a technical matter, the contribution of the Ca 3d orbitals to the structural (cell constants and bond lengths)[31] and electronic attributes[39] could be significant. Ca in its ground state has electron configuration 1s22s22p63s23p64s23d0; the 3s2 and 3p6 states are semicore and can be treated as either core electrons or valence electrons. The choice has notable consequences on accuracy from an atomic perspective, and more importantly, for the existence of ghost states[68, 70]. Using the PAW generation code atompaw[71], we have created and thoroughly tested our own Ca PAW dataset, which we use in all calculations in which we PAW potentials are implemented. In order to validate some of the calculation methods, the ELK software program was invoked. ELK is an all-electron electronic structure code, meaning that it does not use pseudopotentials and operates from a completely different basis from that of ABINIT. ABINIT uses plane-waves, ELK atomic orbitals; this is a good way to discern the reliability of one’s calculations. Figure 2.1 and Figure 2.2 place FHI and PAW results on the same graph; clearly the bands and the electron densities behave as expected and are in very good agreement. Similarly, Figure 2.3 favorably compares the FHI norm-conserving pseudopotentials with the PAW method results. Although observing some difference might have provided some rationale for the necessity of PAWs in CaAl2Si2 calculations, the corroboration instead serves to instill confidence in the PAW results.

27 Band Structure from Bnds-OtherAtOptGeom.out

5 Brillouin zones and rotational

0 b3 εF

R S L S H Energy (eV) U P T

-5 M T K b

In units of 4 a 3, 4 a 3, and 2 c, along the three primi 0 0 0 ,A 0 0 1 2 ,M 1 2 0 0 ,K 1 3 1 3 0 , -10 L 1 2 0 1 2 . Γ MK ΓA LH A M

Figure 2.1: Here we compare PAW potential results with those of the ELK program. Despite the two programs using completely diﬀerent approaches to describe the basis of a system, their resultant band structures are nearly identical.

PAW FHI/TM

0.1 Si Al Si

0.01 Electron density (elecs/cubic bohr) (elecs/cubic density Electron

0.001 0 1 2 3 4 5 6 Distance along c axis through (000), ang

Figure 2.2: Electron density shown from FHI (blue) and PAW (red) can be seen to match at interstitial sites in the CaAl2Si2 ternary. At ion core regions, the density reﬂects the manner in which it is modeled by each approximation; PAW is an all-electron potential and therefore records a high electron density near the nucleus, while the FHI potentials use only valence electrons (little electron density in core region).

28 5

0 ¡F

-5 Energy (eV)

-10

-15 A B C A D E F D B

Figure 2.3: CaAl2Si2 band structure comparison: Red=FHI CBs Blue=FHI VBs Yel- low=PAW CBs Green=PAW VBs Then: orange means perfect overlap in CBs, cyan means perfect overlap in VBs. The PAW and FHI results match well.

29 2.6 Convergence Studies

It is crucial in any computational electronic structure study that convergence be established for several parameters; this is one of the first actions taken when calculations are performed. The idea is to find an acceptable balance between accuracy and computational efficiency. The number of k-points required to adequately represent the Brillouin Zone, and the energy cutoff, which sets the number of plane waves used to describe the system, are the two most critical. The convergence study begins with determining the optimal set of k-points for the given system and performing a self-consistent (SCF) energy minimization calculation. Since they cannot be determined independently, a guess is made for a reasonable energy cutoff as initial input. Specifying the unit cell, type and positions of atoms and a stopping criterion for the SCF loop, and allowing ABINIT to choose a favorable k point set, produces a list of possible k point sets, each including number of k points, a grid number for reference and a merit factor. Selecting the most promising and running total energy calculations on each results in an optimal k point set; in this case two sets are well-converged, one more tightly (140 k points) than the other (66 k points), with the second useful should compute time be more critical than great accuracy. With the k point grid established, a range of energy cutoffs now comprises the datasets for the next calculation. Similar reasoning is applied; balancing accuracy and compute time again influencing the cutoff value. 90 Ha is well-converged; 70 Ha is adequate. Fig- ure 2.4 demonstrates excellent convergence for both k points and cutoff. As a general rule, calculations involving CaAl2Si2 or that require high accuracy will be performed using the well-converged k point set and energy cutoff, while those whose focus is more on trends than precise values will rely on the energy cutoff of 70 Ha and the lesser number of k points.

30 # Kpts in Special k Set 20 40 60 80 100 120 140 160 -13.6392

-13.63925

-13.6393

NumKpts = 140 (ECUT) -13.63935 Ecut = 90 Ha (KPTS)

Total En(Ha) Total -13.6394 CaAl2Si2 Si1 at origin Metallic GGA -13.63945 c/a ideal

-13.6395 0 20 40 60 80 100 120 140 160

Ecut, Ha 120403

Figure 2.4: Convergence Study for CaAl2Si2, Si placed at origin. System is well-converged at an energy cutoﬀ of 90 Ha and 140 kpoints in the set.

31 CHAPTER 3 RESULTS AND DISCUSSION

The CaAl2Si2 class of ternaries is a subset of the AM2X2 group of ternary silicides, with A a rare earth or alkaline-earth element, M among main group or transition metal elements and X usually either Si or Ge, but more generally from group 4 or 5, in some cases 3. The

AM2X2 group is comprised of several subsets, including ThCr2Si2 (space group I4/mmm),

CaAl2Si2 (space group P 3¯m1), α-BaCu2S2 (space group P nma) and EuIn2P2 (space group

P 63/mmc)[72]. CaAl2Si2 is the second most populous subset to ThCr2Si2; at one point there were approximately 400 members in the latter subgroup to merely dozens in the former[73].

The relative scarcity of CaAl2Si2 compounds is a result of restrictions inherent to that class, including d shell levels that must be in a d0, d5 or d10 conﬁguration to comply with the 16- electron electrovalency requirement[72]. Here the small electronegativity diﬀerence between X and Al components closes the bandgap and stabilizes the system[22, 72, 74]). Recent

research indicates there are many more as yet undiscovered AM2X2, and speciﬁcally CaAl2Si2 ternaries, that may be useful thermoelectric materials[22]. The atypical yet compelling

properties emerging from current work (EuZn2Sb2 at 700 K has a zT of 0.92 and EuCd2Sb2

a zT of 0.66 at 616 K [75]) illustrate the importance of continuing to investigate the CaAl2Si2 class of materials[72].

3.1 Structural Properties of CaAl2Si2

2− For a thorough analysis, a fundamental physical understanding of the (Al2Si2) layers in

CaAl2Si2 must be established. With emphasis on the message that CaAl2Si2 is relevant only in that it is prototypical of its class, a discussion of its structural and electronic properties

ensues. CaAl2Si2 is of space group P 3m1, and, in terms of the Zintl concept, consists of

2− 2+ covalently bound blocks of (Al2Si2) intercalated by layers of Ca arranged trigonally in

32 the a-b plane (see Figure 1.7c). CaAl2Si2 sports a hexagonal unit cell, whose primitive cell vectors we conveniently choose to be[76], as manifest in Figure 1.2: √ 1 3 ~a = axˆ − ayˆ (3.1) 1 2 2 √ 1 3 ~a = axˆ + ayˆ (3.2) 2 2 2

~a3 = cz,ˆ (3.3)

with a ﬁve-atom basis. Ordered according to positions along the c axis and in terms of ~a1, ~a2

3 c q 8 and ~a3, with ideal values u = 8 and a = 3 [77], the basis is:

Si1: (0, 0, 0)

1 Al2: (0, 0, u + 2 )

1 2 Al1: ( 3 , 3 , u)

1 2 1 Si2: ( 3 , 3 , 2 )

2 1 3 Ca: ( 3 , 3 , 4 )

Table 3.1: Atom Positions for Umbrella, Planar and Wurtzite Ternaries

Atom → Si1 Al1 Al2 Si2 Ca Structure 1 2 1 3 1 2 1 2 1 3 Umbrella 0 0 0 3 3 8 0 0 8 3 3 2 3 3 4 1 2 1 1 2 1 2 1 3 Planar 0 0 0 3 3 0 0 0 2 3 3 2 3 3 4 1 2 1 5 1 2 1 2 1 3 Wurtzite 0 0 0 3 3 8 0 0 8 3 3 2 3 3 4

3 This is a distorted wurtzite structure (with ideal value of u = 8 ), with a Ca atom at a speciﬁc interstitial site. (See Table 3.1 for atom positions for wurtzite, planar and umbrella

conﬁgurations.) Figure 3.1 follows the mapping of the wurtzite structure to the CaAl2Si2 class of ternaries. The top left ﬁgure is the ideal wurtzite, meaning that the primitive

33 c q 8 vectors are hexagonal, the a ratio is 3 and all bonds are tetrahedral and of the same 2− length. Moving clockwise, the ﬁctitious binary (Al2Si2) (the ideal CaAl2Si2 ternary with the Ca atom removed) is shown. The direction of the arrows in the bottom left panel indicate that the Al2 atom must move upward parallel to the c axis to match the wurtzite binary (top left). The Ca atom is inserted in an interstitial site (bottom right) and demonstrates that

the CaAl2Si2 ternary is structurally equivalent to an inverted wurtzite binary. Figure 3.2 emphasizes this structural equivalency from a slightly diﬀerent perspective, adding more weight to the importance of the distorted wurtzite description. The role of the Ca atom

2− is primarily to donate its electrons to the anion (Al2Si2) layer, in true Zintl compound fashion, stabilizing the hexagonal crystal structure that is a distorted wurtzite system. The Ca atom also forms a slight covalent bond with the Si atom, as will be seen presently. Work done by Rungrote Nilthong[42] establishes structural characteristics of a wurtzite- based ternary (see Figure 3.3). He describes the interstitial sites of such a compound; they are located at the octahedral sites nearest the cations and anions. We ﬁnd that the Ca atom

in the CaAl2Si2 primitive cell is intercalated at one of these identiﬁed octahedral locations,

2 1 3 specifically at ( 3 , 3 , 4 ), in terms of the chosen primitive cell vectors. Although this does not match one of the interstitial sites explicitly specified in Nilthong’s thesis, the sites are equivalent when the two structures are aligned with the same primitive cell vectors[42]. Pursuant to the distorted wurtzite hypothesis, we have chosen one of the Si atoms as the origin. That this is a natural choice is supported by the observation that the two Al atoms are recipients of the electrons donated by Ca2+, turning them into Si look-alikes, thus fulfilling requirements for the octet; it is also supported by electronegativity arguments, namely that two atoms in a system with the highest difference in electronegativity prefer to locate closest to each other[33] (see Table 1.1 and Table 1.2).

Figure 3.4 reveals pockets of low electron density in interstitial regions in CaAl2Si2 that correspond to those of the binary wurtzite (center image, Figure 3.3). The Ca atom sits in the location marked by subscript 1 on the V 0s, and an egg-shaped patch of low density exists

34 Figure 3.1: Mapping between wurtzite structure and CaAl2Si2 structures. Top left: Ideal 2− 2− wurtzite. Top right: Fictitious binary (Al2Si2) . Bottom left: (Al2Si2) with green circles and black arrows indicating direction of Al2 plane motion to attain wurtzite structure. 2− Bottom right: Ca atom added to (Al2Si2) to complete ternary, demonstrating that distortion of the Al2 plane upwards and the insertion of a Ca atom into an interstitial site maps wurtzite to CaAl2Si2.

35 Figure 3.2: Direct comparison between ZnO (wurtzite structure) and CaAl2Si2 in wurtzite 3 5 form. Moving the Al2 plane from the 8 to the 8 position in the primitive cell, and removing the Ca atom, returns the structure to a wurtzite conﬁguration.

Figure 3.3: Left: ‘Upright’ CaAl2Si2. Center: Binary wurtzite with interstitial sites denoted by V 0s. A0s signify anion sites, C0s cation sites. The red circle indicates the location of the Ca ion in its wurtzite interstitial site, were Al atoms shifted up to match wurtzite plane, as in right image. Right: ‘Inverted’ CaAl2Si2 clearly a distorted wurtzite structure.

36 2− where the second pair of V’s is identiﬁed. These latter volumes are within the (Al2Si2) anion layer and indicate the attraction of the Ca electrons to the Al ions, needed to fulﬁll valency requirements.

Figure 3.4: Extended cell of CaAl2Si2 showing isosurfaces of low valence electron density, and wurtzite-structure interstitial sites. The unit cell is outlined in black. The red atoms are Ca and occupy one interstitial region, the yellow footballs are low electron density volumes centered on the second interstitial sites.

2− Close examination of the (Al2Si2) layers oﬀers further insight into structural features of

1− 2− CaAl2Si2. Stacked AlSi layers comprise (AlSi) strata which in turn make up the (Al2Si2) slabs. As supported by the work of Zheng and Hoﬀman[33], within these slabs, Al and Si are both four- coordinated, albeit in very diﬀerent forms. Al expresses tetrahedrally, a fairly common coordination, while Si exhibits an umbrella-shaped coordination (see Figure 1.9).

The umbrella conﬁguration is unusual for Si4; ordinarily Si prefers the tetrahedral arrange-

2− ment. Zheng et. al. imply that the (Al2Si2) layer, being isoelectronic to AlN, conjures up similarities to a wurtzite lattice intercalated with Ca ions, and when the structure transforms

2− from wurtzite slabs to (Al2Si2) layers, Al maintains its tetrahedral coordination while Si

acquires the umbrella shape[33]. The explanation for this peculiar penchant of CaAl2Si2 is addressed in Chapter 4. Zheng, et.al. also conclude that for a material with two non- equivalent lattice sites, the more electropositive of the two types should sit closest to each

37 other. In our case, the Al-Al distance is shortest, with Al more electropositive than Si, in agreement with their findings. It is our contention that the Madelung energy plays a large role in this site preference dance, which is mentioned only in passing by Zheng et. al.. More detail follows in Chapter 4; here we have verified that in pure Si, the umbrella configuration is a locally stable state;

1 if the internal parameter u (and therefore the u + 2 parameter) were to be slightly adjusted back and forth, the Si structure would prefer to return to the umbrella state. Total energy

calculations were run on Si4 at constant volume, meaning c and a were held ﬁxed, as the Si atom that corresponds to Al2 in our ﬁctitious binary was moved along the z axis. From

3 its starting location (its position in the ground state ternary) at 8 , it passed through a 1 1 2 5 planar conﬁguration (z= 2 ) to ﬁnish at wurtzite coordinates ( 3 , 3 , 8 ). The results are shown in Figure 3.5; with the umbrella state a local minimum and the wurtzite arrangement the

most stable. Figure 3.6 depicts the structure of Si4 in each configuration. Globally, however, the Si solid would settle into the widely recognized diamond structure; in fact it has been shown that Si prefers a diamond configuration to wurtzite; total energy calculations reveal a difference of 0.024eV in diamond’s favor[78].

3.2 Semimetallic Behavior of CaAl2Si2

Based on simple electron counting arguments (16 electrons per primitive cell, eight bands

ﬁlled), we expect CaAl2Si2 to be an insulator or semiconductor. Given that Al is a column

III element and Si column IV it should behave as a III-V or IV-IV semiconductor. CaAl2Si2, however, is a semimetal[35, 41, 79]. In fact, without precise DFT band structure calculations and deﬁnitive experimental evidence, the possibility exists that it could be a semiconductor.

CaAl2Si2 does exhibit characteristics typical of a semimetal, including a band structure with no band gap, a small overlap between the bottom of the conduction band and the top of the valence band, and a small density of states at the Fermi level. Figure 1.6a conﬁrms the ﬁrst two identifying factors, and the band structure in Figure 3.8 illustrates the third.

Another interesting qualiﬁer involves charge carrier type; the Hall coeﬃcient in CaAl2Si2

38 -15.5

Si4 -15.55 Scan of Al2 Position and Corresponding Total Energy -15.6

-15.65 Ideal CaAl2Si2 Planar Total Energy, Ha/primitive cell -15.7 Wurtzite

-15.75 0.2 0.3 0.4 0.5 0.6 0.7 0.8 z "Al2", fraction of c 011326

Figure 3.5: Illustration of z-axis scan of Si atom in Al2 site showing local and global minima for fictitious binary, planar and wurtzite configurations of Si4. Total energy calculations; volume held fixed.

Figure 3.6: Si4: Positions of Al2 plane corresponding to the metastable state, the barrier, or planar conﬁguration, and the global minimum.

39 undergoes a sign reversal at 150K, at which temperature the dominant carrier type switches from holes (below 150K) to electrons (above 150K)[35]. Dependence of electron and hole mobilities on temperature is what drives this sign change, not the more prevalent mechanism of unequal carrier densities[39]. It is established that in a semimetal, both holes and electrons contribute to the electrical conductivity[15], a phenomenon observed by several researchers

in CaAl2Si2[35, 39]. In the absence of GW calculations, which would go a long way toward conﬁrming that

CaAl2Si2 is a semimetal, and also as support for future GW calculation results (assuming

they indicate a semimetal!), we again examine our familiar Si4 structure. Figure 3.7 shows the positions of the Al2 plane in Si4, in wurtzite and umbrella phases, left to right, and the ground state CaAl2Si2 band structure in the top row. The bottom row displays band structures of

Si4 in the wurtzite and umbrella conﬁgurations, again left to right. The signiﬁcance brought to these calculations by Si4 is that, because Si4 is monoatomic and has no interstitial atoms, any conclusions made about its electronic structure are free of chemistry considerations.

Forcing Si4 into the umbrella structure and calculating its band structure make clear that the resultant overlap of the Fermi level by the conduction band at the M point is a consequence solely of this strain. By direct analogy, we can then conclude that the stability of the umbrella conﬁguration in CaAl2Si2 is due entirely to strain. More generally, we have shown that any

CaAl2Si2 - type structure that prefers the umbrella arrangement will be a semimetal.

3.3 Eﬀective Mass

Having broached the band structure of CaAl2Si2, we now calculate the effective mass, whose relevance, as described in ??, is that it is directly proportional to the Seebeck coefficient, whose square, in turn, is directly proportional to the thermoelectric figure of merit. The ELK LAPW suite of programs permits easy computation of both the effective mass tensor at specified k points and of the Fermi surface. The close agreement between the band structures as computed with ELK and with ABINIT (see Figure 2.1 and Figure 2.3 suggests that effective masses will also agree. In principle the effective masses should be computed at

40 wurtzite umbrella

Figure 3.7: Upper left: Si4 in wurtzite and umbrella phases. Upper right: Band structure of CaAl2Si2. Bottom row: Si4 band structures in wurtzite and umbrella conﬁgurations, left to right. When forced into an umbrella phase, Si4 exhibits semimetal characteristics, very similar to CaAl2Si2.

41 Band Structure from Bnds-OtherAtOptGeom.out

5 R S L S H

U P T

M T K

0 εF In units of 4 a 3, 4 a 3, and 2 c, along the three pri 0 0 0 ,A 0 0 1 2 ,M 1 2 0 0 ,K 1 3 1 3 0 L 1 2 0 1 2 . Energy (eV)

-5

-10 Γ MK ΓA LH A M

400 Ca Al2 Si2 ideal structure cell volume = AB value GGA/PBE revised 300

FLAPW (elk) PW (ABINIT) 200

100 DOS (states/Ha per prim cell)

0 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 Band energy (Ha w/r to Fermi level at 0)

Figure 3.8: Abinit vs ELK PBE GGA calculations (same exchange-correlation functional used): CaAl2Si2 band structure and density of states comparison. Upper frame shows the band structure generated by Abinit with blue and red, that by ELK with green. Lower frame, DOS in blue is via Abinit, red via ELK. Excellent agreement between two methods in both cases (discrepancy above 0.1 Hartree is simply due to an input parameter inconsistency). Important in DOS that electron densities around Fermi level are equivalent.

42 the Fermi level, but for simplicity we evaluated them right at M (for the lowest conduction band) and at Γ (for the valence band maximum). The eigenvalues of the (band) eﬀective

∗ mass tensor, corresponding to the principal axes, are me/me = {0.88, 0.77, 0.090} for the conduction band minimum at M. The Fermi level also cuts two distinct valence bands (see Figure 3.9) near the Γ point.

∗ We ﬁnd hole eﬀective masses along principal axes mh/me = {1.06, 0.33, 0.20} for the top of the valence band and {0.41, 0.097, 0.094} for the next lowest valence band (degenerate with the top at Γ).

3.4 Structural and Electronic Calculations

In support of the interpretation of CaAl2Si2 as a distorted wurtzite structure, we begin structural relaxation calculations with the ideal values for wurtzite, including internal coor-

1 3 c q 8 dinates of u and u + 2 , with u = 8 and the ideal a ratio of 3 [77]. In ABINIT, one ﬁrst zeroes the forces on the atoms, then relaxes cell dimensions and internal cell parameters to

c obtain the ground state properties of the system in question. The relaxations indicate a a of 1.647 and a cell internal parameter u of 0.381, both slightly larger than the ideal values (displayed in Table 3.2). The optimized parameters will be used unless otherwise stated.

Table 3.2: Structural parameters for ideal wurtzite and as calculated by T. Semi for CaAl2Si2 c/a value u Ideal Wurtzite 1.633 0.375 Semi Calculation CaAl2Si2 1.647 0.381

3.5 CaAl2Si2 and the Non-neutral Fictitious Binary Compound

In order to study the role of Ca in our distorted wurtzite structure, we compare the elec-

tronic structure of CaAl2Si2 and the non-neutral ﬁctitious binary, which consists of CaAl2Si2 with the Ca atom removed and two electrons added to the unit cell. In this case we adopt

43 ELK, using structural params of AB

A M pt Fermi surface

L H

Γ TOP VIEW M M K b2

m*/mel = {0.77,0.88,0.090}

Figure 3.9: Eﬀective masses at M point of CaAl2Si2 ternary.

44 c ideal structural parameters a and u. Preliminary calculations reveal that there is very little diﬀerence between their electron densities (see Figure 3.10) and their band structures (see

2− ??); bands of (AlSi) are very close to those of CaAl2Si2, as are the ground state charge densities. When examined in more detail (Figure 3.13), the electron density differences expose a tiny bond between Ca and Si. These observations attest to the role of Ca as a structural one, without much effect on the electronic properties. Previous research supports this finding; that the 3d Ca orbital plays a structural role in CaAl2Si2 but leaves the band structure mostly unaffected[31, 39].

Huang, et. al. [39] assert that the bottom of the conduction band of CaAl2Si2 is of Ca 3d character. Square modulus calculations of wavefunctions and decomposition of conduction band states by angular momentum character about atom sites that we have performed refute this idea. Figure 3.11 shows the results of these calculations for the band and at the high symmetry point referred to by Huang (band 9, location M; seeFigure 1.6). Figure 3.11a suggests that there is Ca character; this is conﬁrmed when we remove the model Ca atom to reveal the electron density at this site (Figure 3.11b). But in no way do these results imply that the band is of Ca 3d character. Again invoking Figure 3.12, bands with no Ca present very closely resemble those with Ca present. This is signiﬁcant in determining whether or not the Ca 3d orbitals need to be included in the pseudopotential for accurate representation of CaAl2Si2.

Another important mechanism for stability in CaAl2Si2 can be observed in Figure 3.11. The ﬁgures are square moduli of the conduction band minimum, and establish that this overlapping band is very sensitive to the z(Al2) position. The large amplitude of the Al2

2− plane and the big square modulus near the (Al2Si2) region (shown in both (a) and (b)) conﬁrm this, and also convey that the bonds are strained by the repulsion of Al2 by the Ca atom.

We believed, based on the great likeness between the band structures of the CaAl2Si2 ternary and the ﬁctitious binary (the former without the Ca atom), that replacement of the

45 2− Figure 3.10: Electron densities of (Al2Si2) and CaAl2Si2 are very similar. Cut through 111 plane to include Ca interstitial site. Top row is CaAl2Si2, with and without isosurfaces and 2− lattice planes. Likewise for the bottom row, but for (Al2Si2) , our ﬁctitious binary that has no Ca atom.

46 Ca

Figure 3.11: Illustration of square modulus analysis of M9 band (conduction band minimum) of CaAl2Si2. The Ca atom in the cell on the right is removed in order to see the isosurface beneath it.

Figure 3.12: Band structure comparison between CaAl2Si2 (left) and the non-neutral ﬁcti- 2− tious binary (Al2Si2) (right), displaying a close similarity and inferring that the Ca ion has a greater inﬂuence structurally than electronically. The band structure retains its semimetal property.

47 Figure 3.13: The diﬀerence in electronic density between CaAl2Si2 and the non-neutral ﬁctitious binary (AlSi)2−, revealing a tiny covalent bond between Ca and Si.

Column II atom (Ca) by any other Column II atom would have no effect on structural properties. Follow-up calculations, with the cations replaced, suggested otherwise, and revealed some interesting behavior. Table 3.3 and Figure 3.14 display some properties of each as calculated with ABINIT, and some images mainly meant for illustrative purposes. For example, the compound with Be has a shape closer to square than the rectangular form of CaAl2Si2; the atoms at the Si2 and Al2 positions are much closer to planar than any of the others. The electronegativity difference between Be and Si is much smaller than that between Ca and Si, and in fact, Be is metallic as would be predicted (electronegativity and BeAl2Si2 is further analyzed in Chapter 4). The Ba ternary has at least three separate space groups defining its subclasses. One is a Zintl compound, and although a simple analysis indicates that it is a semiconductor, the temperature dependence of the electrical resistivity reveals

Pauli paramagnetic behavior, thus implying it is a metal[72]. MgAl2Si2[80] and SrAl2Si2[34]

(isostructural to CaAl2Si2) are closest in behavior to CaAl2Si2.

The band structures of CaAl2Si2 in each of the three arrangements, umbrella, planar and wurtzite, are shown in Figure 3.15. At planar and umbrella positions of the Al2 plane, the ternary is semimetallic. In wurtzite form, it becomes semiconducting. This is consistent with

48 Table 3.3: Comparison of XAl2Si2 Structural Characteristics; X = Column II atoms; Be, Mg, Ca, Sr, Ba

Ternary Zintl Zintl Ion c/a Value Si1-Al2 Ion Eneg Distance (A)˚

BeAl2Si2 Be 1.57 1.241 2.623 MgAl2Si2 Mg 1.31 1.621 2.587 CaAl2Si2 Ca 1.00 1.647 2.644 SrAl2Si2 Sr 0.95 1.648 2.667 BaAl2Si2 Ba 0.89 1.673 2.74

Figure 3.14: Left to right: Unit cells of BeAl2Si2 (green Zintl ion), MgAl2Si2 (orange Zintl ion), CaAl2Si2 (red Zintl ion), SrAl2Si2 (blue Zintl ion), BaAl2Si2 (purple Zintl ion). Not to scale; shows sketch of differences in unit cells. our claim that as the Al2 plane moves up, the electronegativity difference between Si and Al begins to be of account, the band gap widens and the system becomes less metallic and more insulating. It also suggests that strain on the Al-Si bonds is the root of the semimetallic behavior of CaAl2Si2. Verification that the arrangement of Ca, Si and Al atom sites in the ternary with which we have been working is the one that gives the lowest energy configuration, and hence is the preferred structural order, was accomplished by performing relaxations on all combinations of Al and Si atoms. For example, calling our structure SAASC to reflect the order in the z-axis direction, we relaxed ASASC, AASSC, ASSAC and SASAC. Results confirmed that SAASC is most stable. Figure 3.16 is a particularly clear example; ASSAC is compared with SAASC. The ASSAC configuration is still umbrella, however it is evident that the Al2 plane

49 Band Structure CaAl2Si2: Preferred Structure

10 A

R S 5 L S H

U P T 0 εF Energy (eV) M T K

-5 b1

-10 In units of 4 a 3, 4 a 3, and 2 c, along the three pri Γ MK ΓA LH A M 130204 0 0 0 ,A SAASCbs0 0 1 2 ,M 1 2 0 0 ,K 1 3 1 3 0 L 1 2 0 1 2 . .fhi

Band Structure: SAASC in Planar Ideal Structure

0 εF Energy (eV)

-5

-10

Γ MK ΓA LH A M 130213 SAASCplanarBS .fhi

Band Structure: SAASC Wurtzite Configuration

0 εF Energy (eV)

-5

-10

Γ MK ΓA LH A M E = 0.12561 eV G (indirect) 130213 SAASCwurtziteBS .fhi

Figure 3.15: CaAl2Si2 band structures for the ternary’s preferred, planar and wurtzite geometries, respectively, from top to bottom. The ﬁgures clearly illustrate that for a CaAl2Si2 - class ternary with a preference for the umbrella conﬁguration, the structure is a semimetal. In the wurtzite arrangement, it is a semiconductor.

50 is nearly horizontal (planar); ground state energy is higher than SAASC by about 1.02 eV. The ASSAC band structure is distinctly semimetal, which corroborates our ﬁnding that the

two (umbrella and semimetal attributes) go hand-in-hand for CaAl2Si2.

Brillouin zones and rotational symmetry 28

R S L S H

U P Band Structure CaAl2Si2: Preferred StructureT CaAl2Si2 Band Structure: Si and Al Positions Switched M T K b2

b 5 10 1

In units of 4 a 3, 4 a 3, and 2 c, along the three primitive vectors b1, b2, and b3; 5 0 0 0 ,A 0 0 1 2 ,M 1 2 0 0 ,K 1 3 1 3 0 ,H 1 3 1 3 1 2 , and 0 εF L 1 2 0 1 2 .

0 εF Energy (eV) Energy (eV) -5

-5 9th February 2003 c 2003, Michael Marder

-10 -10

Γ MK ΓA LH A M Γ MK ΓA LH A M 130204 nkpts 140 SAASCbs ecut 90 Ha 120426 .fhi

Figure 3.16: Total energy of unmodified CaAl2Si2 structure (top left) lower than that of CaAl2Si2 structure with positions of Al and Si atoms switched. Energy difference is about 0.0374 Ha (approximately 1.02 eV). Supports stability of umbrella bonds in CaAl2Si2, as well as connection between semimetal behavior and umbrella configuration.

51 CHAPTER 4 MADELUNG ENERGY AND ELECTRONEGATIVITY

To probe further into the stability of the unusual structure of CaAl2Si2, we look at some alternate approaches (less complex but coarser) to DFT to estimate energies, hence structural stability, including the Madelung energy and electronegativity arguments. The Madelung energy, also called a system’s electrostatic energy, takes into account no chemistry apart from nominal valence charge, making it the most elementary method, while electronegativity is a property of an atom which determines its actual charge in the presence of other, diﬀer- ent atoms, and therefore more chemically (but not structurally) speciﬁc. DFT, of course, involves many more attributes of the particular system, its main approximation encoded in the exchange-correlation energy. Focusing on the Madelung energy and electronegativity of

CaAl2Si2 enables the prediction and/or corroboration of DFT results, and the rapid scanning of the dependence of energy on structural details.

4.1 Models for Stability

We emphasize that we wish to ascertain whether this minimal approach corroborates

DFT results that show the lowest energy configuration for CaAl2Si2 to be the geometry in which the Ca atom is at a specific interstitial; one closest to a Si atom, and that, for a fixed Ca location, the overall stability puts Si near the Ca atom. Cyclic permutations of the positions of the Al and Si atoms for each possible interstitial site that could house Ca were

performed; there are four sites suggested by Nilthong [42] (see Figure 4.1), labeled V2C ,V2A,

V1C and V1A. Site V1C did indeed emerge as the energetically preferred interstitial for Ca, in the corresponding ‘inverted’ wurtzite structure. (See also Figure 3.3.) Here our objective was simply to verify that our choice of underlying structure based upon electrostatic arguments is cogent, and in accord with our previous statement.

52 Figure 4.1: Wurtzite ternary suggested by Nilthong [42]

53 Formally, the Madelung energy is defined as the total Coulomb energy of an infinite array of N ion points of charge Ze in a uniform compensating negative background. In our case, the array is composed of different types of point ions, each with its relevant charge. It is the dominant constituent of a system’s total energy. Figure 4.2 displays the various energies

comprising the total energy of, in this case, a ground state CaAl2Si2 system relaxed using FHI potentials. Top to bottom, the key shows the kinetic energy (KE), Hartree energy (HE), exchange-correlation energy (XC), Ewald energy (EW), energies due to pseudopotential chosen (PSCORE, PSLOC and PSPNL), and the entropy term (which accounts for deliberate thermal smearing at a ﬁnite temperature, selected to facilitate ﬁnding the Fermi energy). With a strictly Madelung energy code (D.M. Wood), we have examined several aspects of

the CaAl2Si2 structural preferences. While the selection of the eﬀective charges on atom sites can be subtle, we select charges as they appear within a total-energy DFT calculation. That

is, point ion charges have their nominal valence charge. For the calculation, we designate QAl

to be +3, QSi to be +4 and allow QCa to range from 0 (neutral) to +2 (fully ionized). We

wish to analyze the eﬀect of the charge state of the Ca ion on the stability of the CaAl2Si2 structure. Figure 4.3 shows contours of constant energy for various locations of Al2 along the z plane, as a fraction of the c value of the lattice constant, as the ionization state of the Ca ion is taken from 0 to +2. The outcome supports our contention that fully ionized Ca2+ repels the Al2 atom, forcing the Al2 plane away from the Ca ion and favoring the CaAl2Si2 (umbrella) conﬁguration over that of wurtzite (tetrahedral). The line through the isosurface

5 minima starts at the wurtzite arrangement, where z(Al2) = 8 and the Ca atom is not ionized, passes through the planar state with a partially ionized Ca and extends to its lowest point,

3 the umbrella structure at z(Al2) = 8 ; Ca is now fully ionized. The regions of the isosurfaces indicating a preference for wurtzite (upper half of figure) and for the umbrella (lower half of figure) are made clear and it becomes obvious that the lowest minimum is well below the planar point and into the umbrella area. The requirement that Ca be almost fully ionized (a consequence of its low electronegativity; see below) for this to occur is also confirmed,

54 emphasizing once more that a charged Ca ion repels the Al2 ion, and stabilizes the ternary

in the umbrella conﬁguration. Energy in Hartree in Energy

Figure 4.2: Contributions of conventional components of total energy to the total energy of the system. The Ewald (EW) portion is equivalent to the electrostatic energy of the system, and is the dominant term. Abbreviations are explained in the text.

The beauty of electrostatic energy arguments is the simplicity; it is very often possible to predict stabilities and structural preferences for a given system based only on ionic charge.

Using Madelung energy arguments, we have just established that the CaAl2Si2 - class structure depends sensitively on the interstitial ion. In this case, it is Ca, with a fairly high electronegativity difference with Si that enables the addition of charge and thus repulses the Al2 plane ion (Al), pushing it into its umbrella formation. As we have demonstrated in Chapter 3, the stability of this configuration then guarantees the material to be a semimetal. Stated another way, we can make the significant observation that the choice of interstitial atom will determine the electronic structure of the system. This assertion, supported by the charge state reasoning, can be corroborated with calculations involving the position of the Al2 plane in the CaAl2Si2 structure. Starting with

Si4, we establish ﬁrst the utility of the approach. We choose the pure Si structure since it is a simpler system than CaAl2Si2, and one whose electron density is closely related; it is a good reference material for CaAl2Si2. Use of the Si4 structure also avoids any impact

55 1.0

wurtzite

0.8 0.60 -15

-16 0.55 0.6 -17 0.50

0.4 z(Al2) 0.45 ) as fraction of c 2

0.2 0.40

z(Al 1.80 1.85 1.90 1.95 2.00 umbrella Q on Ca site 0.0 0.0 0.5 1.0 1.5 2.0 Q on Ca site

Figure 4.3: Madelung energy: CaAl2Si2. The z values on the vertical axis reflect a continuum 5 of placements of the Al2 atom as a fraction of lattice constant c, including the wurtzite (z= 8 ), 1 3 planar (z= 2 ) and umbrella (z= 8 ) configurations. The charge state of the Ca ion, from 0 to +2, defines the horizontal axis. The contour lines reveal the stability of a given combination; here it is clear that with Ca fully ionized, the CaAl2Si2 umbrella arrangement is lowest in energy. To observe this, follow a line of isosurface minima from Ca with no charge to Ca with +2 charge (left to right). The line meets Ca with +2 charge just below the planar configuration (0.5), thus putting it in the region of umbrella stability. The upper and lower halves of the diagram are shaded; the lighter indicates a preference for wurtzite, the darker for the umbrella. The figure on the right is enlarged to show unequivocally that CaAl2Si2 prefers the umbrella structure preference.

56 all bonds same length

all bonds same length

isosurface 0.55 elecs/bohr3

-15.50 -12 Ideal c/a; a from fully relaxed Si in Ca Al2 Si2 structure Ca Al2 Si2, April 2012 December 2012 -12.5

-15.55 Madelung energy, Ha/prim cell

2 ABINIT -13 DMWprog Madelung interpolated1 ﬁrst deriv -15.60 'Ewald' from AB -13.5

0.4 0.5 0.6 0.7 100

planar -14 -1 -15.65 idl tern 80 WZ 60 -2 -14.5

Total energy, Ha/primitive cell Ha/primitive energy, Total 40 -15.70 interpolated second deriv -15 20

0.4 0.5 0.6 0.7 -15.75 -15.5 0 0.2 0.4 0.6 0.8 1 z "Al2", fraction of c

Figure 4.4: Si4: Red line: Ewald energies extracted from Abinit, equivalent to Madelung energy. Open circles: Madelung energy data points from D.M. Wood program; agree perfectly with Ewald energies, as they should. Blue line and points: Total energies produced by scan of z coord of Al2 (fraction of lattice constant c) through wurtzite, planar and umbrella con- ﬁgurations, produced by Abinit. Clear preference for wurtzite conﬁguration demonstrated by all methods. Inclusion of detailed electronic structure information is not necessary to determine structural preference of system.

57 CaAl2Si2: Scan of Al2 Position -10.6 and Corresponding Total Energy

Planar

Wurtzite Umbrella

-10.8 Total Energy, Ha/primitive cell

0.2 0.3 0.4 0.5 0.6 0.7 0.8 z "Al2", fraction of c

Figure 4.5: CaAl2Si2: scan of z coord of Al2 (fraction of lattice constant c). Clear preference for umbrella structure.

58 electronegativity differences may have on the stable form, since the system is monoatomic. Figure 4.4 shows total energies generated by an ABINIT DFT calculation, in which the cell volume is frozen (blue line and data). We see clearly the local minimum in the umbrella position, a slight barrier at the planar configuration and a global minimum for the wurtzite structure. As we see in Figure 3.6, two configurations corresponding to two different c values for the Al2 plane permit equal bond lengths. Only one of these, however, that of the wurtzite arrangement, permits perfect tetrahedral coordination. The reason for the higher energy at the umbrella structure is that bond bending is required to adopt this arrangement, leaving the tetrahedrally coordinated structure (no bond bending or stretching) most stable. We add the Madelung energy calculation from D.M. Wood’s code (open circles on red line); the Ewald energy as found by ABINIT (red line) matches the data perfectly. (The latter two energies are equivalent, providing excellent confirmation of the reliability of both programs and lending confidence to results.) The Madelung energy calculation misses the local (umbrella) minimum, but most importantly, finds the true (wurtzite) minimum, and we see that for Si4, the uncomplicated electrostatic energy calculation provides an accurate prediction of its structure.

We apply the same strategy with CaAl2Si2. Further support for our distorted wurtzite claim comes from the scan over the continuum of the three configurations for CaAl2Si2; see Figure 4.5. The preference is clearly for the umbrella/distorted wurtzite structure! Again comparing the ABINIT Ewald energy with the D.M.Wood Madelung energy, we have solid agreement, and all three energy measurements confirm that CaAl2Si2 is most stable in the umbrella arrangement. In Figure 4.6, as with the Madelung energy calculations (Figure 4.3), anything minimum below planar is unequivocally of umbrella configuration. Looking at the fictitious binary, the preference for wurtzite reasserts itself, as predicted and expected (shown in Figure 4.7).

59 -12.4 Ca Al2 Si2, equilib cell volume -12.6

-12.8 Ewald (AB) -13.0 total Madelung pgm -13.2

-13.4 Energy (Ha/cell) Energy -13.6

-13.8

-14.0 0 0.2 0.4 0.6 0.8 1 z(Al2)/c

Figure 4.6: Positions of z(Al2) as a fraction of lattice constant c vs. energy in Hartrees. Ewald energies calculated with ABINIT agree with Madelung energies output from D.M.Wood’s Madelung energy code (red line and red squares), and show a minimum at the umbrella 3 conﬁguration (z(Al2/c)= 8 . The blue circles show the total energy of the CaAl2Si2 system for the same positions and also veriﬁes that CaAl2Si2 stabilizes in the umbrella structure.

60 SAAS in CaAl2Si2 Structure

-11.5

Scan of Al2 Position -11.55 and Corresponding Total Energy

-11.6

-11.65 Ideal CaAl2Si2 Planar Wurtzite Total Energy, Ha/primitive cell -11.7

-11.75 0.2 0.3 0.4 0.5 0.6 0.7 0.8 z "Al2", fraction of c 130127

2− Figure 4.7: Fictitious binary (Al2Si2) : Scan of z coord of Al2 (fraction of lattice constant 2− c). Note absence of Ca ion. (Al2Si2) shows a preference for wurtzite structure.

61 4.2 Fixed Cell Volume and Its Impact on Total and Madelung Energies

Because the Madelung energy must be computed for fixed cell volume (the cell would collapse to zero dimensions to minimize the Coulomb energy), we compared the Madelung energy to the total DFT energy for FIXED cell volume. We carried out DFT scans of z(Al2) for fixed cell volume, since ABINIT cannot readily relax c and a while atom locations are fixed. However, it does report stresses during such a scan.

We use Si4 as an example to estimate the error we make. Figure 4.8 shows the energy (panel(a0)) and the stress (panel (b)) as z(Al2) is scanned. For ﬁxed cell volume we expect the pressure (related to the energy density) to track the total energy, as we see in panel (b).

c We note that σ33 dominates, and ask what change in a is needed to make the error in energy (due to not relaxing the cell volume) equal in magnitude to the structural preference energy in panel (a), ≈ 0.03 Ha. √ ! 3 c ∆E ≈ σ a3 ∆ (4.1) pref 33 2 a

c c where ∆ a is the error in the a ratio we make by failing to relax the cell volume. a c Using ≈ 8, σ33 ≈ 0.003 Ha/cubic bohr, we ﬁnd that a 15% shift in the value would a0 a be needed to shift the order of stability in panel (a). This amount of shift is unlikely to occur even should we let the cell relax, thus we have conﬁdence in our approximation.

4.3 Electronegativity Arguments

References to electronegativity have permeated this study of the structural preferences of CaAl2Si2, enough to warrant a more thorough investigation of its usefulness. We claim that CaAl2Si2 is a distorted wurtzite structure, and that several factors cause this distorted arrangement to be energetically favorable. Electronegativity is one that plays a vital role in the stability of its unusual features. A brief discussion of its connection with density functional theory (DFT) can be found in Appendix A. A more in-depth account of DFT is given in Chapter 2.

62 all bonds same length

all bonds same length

isosurface 0.55 elecs/bohr3

-15.50 -12 Ideal c/a; a from fully relaxed (a) Si in Ca Al2 Si2 structure Ca Al2 Si2, April 2012 December 2012 -12.5

-15.55 Madelung energy, Ha/prim cell

2 ABINIT -13 DMWprog Madelung interpolated1 ﬁrst deriv -15.60 'Ewald' from AB -13.5

0.4 0.5 0.6 0.7 100

planar -14 -1 -15.65 idl tern 80 WZ 60 -2 -14.5

Total energy, Ha/primitive cell Ha/primitive energy, Total 40 -15.70 interpolated second deriv -15 20

0.4 0.5 0.6 0.7 -15.75 -15.5 0 0.2 0.4 0.6 0.8 1 z "Al2", fraction of c 0.002 Fixed prim cell scan of z(Al2), SI4 in WZ structure (b) 20 0.001 Pressure (GPa) Pressure

0.000 10

sigma(11) sigma(33) -0.001 pressure 0

Stress (Ha/cubic bohr) Stress (Ha/cubic -0.002 -10

-0.003 0.2 0.3 0.4 0.5 0.6 0.7 0.8 z(Al2), fraction of c

Figure 4.8: Illustration of eﬀect of ﬁxed volume on total energy and Madelung energy comparisons, using pressure and stress.

63 To tease out the underlying principles for the stable geometry in terms of electronegativity, we expand our investigation to include BeAl2Si2, where the interstitial atom Ca has been replaced with Be, another column II atom. Besides refuting once and for all the idea that any column II atom can replace Ca without aﬀecting structural properties, we reinforce our previous discovery that this interstitial is responsible for the electronic characteristics of the material.

− 1 The (Pauling) electronegativity diﬀerence between Si and Ca (0.9 eV 2 ) is much greater

− 1 than that between Si and Be (0.33 eV 2 ). Ground state electronic structure calculations reveal that BeAl2Si2 is metallic (see Figure 4.10), and takes on a (barely) planar configuration (the angle between Si2 and Al2 is 91.0677◦, making it still an umbrella structure, but very nearly planar and clearly moving toward wurtzite). Figure 4.9 details the electron density of both CaAl2Si2 and BeAl2Si2. Again, electronegativity differences between the interstitials and the Si2 atom are responsible for structural aspects of the ternaries, in this case bonding behaviors. As a side note of interest, the impact of electronegativity differences between the two

2− atoms that comprise the anion double layer (Al2Si2) was investigated. Figure 4.11 plots the Al2 position against total energy differences for each system. We replaced Al with Ga and also used Si4 in this analysis; the electronegativity difference between Si and Al is 0.29, that between Si and Ga is 0.09 and 0.0 for Si only (see Table 1.2). This electronegativity difference has been reported to determine stability in CaAl2Si2 - class compounds that are not electrovalent (have 17 electrons rather than 16 in the primitive cell) such as LnAl2Si2[31, 72]). In our calculations with these three fictitious binaries, however, there does not appear to be a clear correspondence between electronegativity difference and depth of preference for wurtzite over umbrella. The difference between Si and Si is obviously 0, yet that structure is neither the most nor the least stable of the three. For the two column III atoms, Ga and Al, the difference in electronegativity between Si and Ga is less than that between Si and Al and the preference for the wurtzite configuration is more pronounced, although more data

64 Ideal geometries

CaAl Si 2 2

BeAl Si 2 2

Figure 4.9: Electron densities of CaAl2Si2 and BeAl2Si2. We see that the Ca atom loses its charge to allow the formation of Si-Al bonds (top), and that Be actually steals charge from the Si-Al bonds, retaining most of its own charge. The deciding factor is the electronegativity of the interstitial atom.

65 Band Structure from SAASBeBSgo.out

0 εF Energy (eV)

-5

-10

Γ MK ΓA LH A M 130401 SAASBeBSgo .fhi

Figure 4.10: Band structure for BeAl2Si2; interstitial Ca atom replaced by Be. Band structure is clearly metallic. BeAl2Si2 is not stable in the umbrella structure (prefers planar or wurtzite) and therefore will not be a semimetal.

66 points are needed to make a meaningful statement concerning column III constituents.

-0.66 Scan of Al2 Plane Position and Corresponding Total Energy

planar -0.68 umbrella

wurtzite -0.7

-0.72 SSSS SGGS SAAS

Total Energy Difference: Ha/primitive cell Energy Difference: Total -0.74

0.3 0.4 0.5 0.6 0.7 0.8 z(AL2): fraction of c

Figure 4.11: Scan of z coord of Al2 (fraction of lattice constant c), with fixed volume, through umbrella, planar, wurtzite and beyond. Data with blue squares is for Si4 (SSSS), red circles for the fictitious binary with Al atoms replaced by Ga atoms (SGGS) and green triangles for the fictitious binary (Si and Al; SAAS). Clear preference is seen for the wurtzite structure for all combinations, as is expected for any tetrahedrally coordinated, 16-electron-per-primitive- cell system. The correspondence between electronegativity difference and depth of preference for wurtzite over umbrella is discussed in the text.

4.4 Bader Charge Analysis

A Bader charge analysis lends additional validity to our electronegativity arguments. This method divides a system to seek a good approximation of the total electronic charge associated with each atom in a particular environment. Typically, the charge density will be at a minimum between atoms, and from these minima, a Bader volume (basin of attraction)

67 can be constructed around each atom. The charge enclosed gives a good estimate of the atomic charge. Table 4.1 displays Bader charges computed (D.M. Wood and Seth Ritland) using the AIM utility of ABINIT. The first two rows give an idea of the error in the Bader charges (the accuracy), the final column, the precision, or the reproducibility of the sum quantities. The analyses are for a Si structure in both wurtzite and umbrella configurations, the fictitious binary in its planar and umbrella forms, CaAl2Si2 in its preferred structure and BeAl2Si2 in both planar and umbrella organizations. In the latter ternary, the Ca atom

2− has been replaced with another column II atom, Be (Z=4). Except for (Al2Si2) , for which a charge of negative two is expected, all other (neutral) compounds have Bader sums of zero. This is because the net flow of electron charge through each Bader basin of attraction must be (very nearly) zero. For the Si structures, no charge needs to flow, since all atoms are identical and precisely tetrahedrally coordinated. In the ternaries, electron charge flows out of the basins whose atoms have lower electronegativities (Al and Ca or Al and Be) and into the Si basins (highest electronegativity). Here we see that, since Be has a higher electronegativity than Ca, its charge flux is lower than that of Ca, and both Be and Ca have lower electronegativities than Si. Electric charge flows out of the Al basin and into the Si basin for the fictitious binary. All of this behavior is consistent with electronegativity arguments, even though the electronegativities did not participate in the analysis. Table 4.1: Bader charges computed via the ABINIT AIM utility. int indicates interstitial site (Ca or Be atom); WZ, umbr, and plan indicate, respectively, ideal wurtzite, ‘umbrella’, and planar structures. Negative (positive) values indicate electron charge has flowed into (out of) Bader basin about specified atom.

↓structure/site→ Si1 Al1 Al2 Si2 int sum Si (WZ) -0.01 +0.01 -0.01 +0.01 – +0.001 Si (umbr) -0.02 +0.05 -0.02 -0.00 – -0.001 −2 (Al2Si2) (umbr) -2.12 +1.34 +1.09 -2.31 – -2.000 CaAl2Si2 (umbr) -1.99 +1.57 +1.37 -2.27 +1.32 -0.002 BeAl2Si2 (umbr) -1.63 +1.53 +1.35 -1.82 +0.57 -0.001 BeAl2Si2 (plan) -1.67 +1.48 +1.25 -1.69 +0.63 -0.000 −2 (Al2Si2) (plan) -2.20 +1.36 +1.13 -2.29 – -2.001

68 4.5 Summary

The Madelung energy and electronegativity arguments presented in this section make clear a central theme of this dissertation; that for compounds in the CaAl2Si2 class, the tendency of a structure to stabilize in the wurtzite or umbrella conﬁguration is determined by the electronegativity of the interstitial atom.

69 CHAPTER 5 CONCLUSIONS AND FURTHER RESEARCH

CaAl2Si2 is the prototype of a subset of the AM2X2 group of ternaries, which is known to harbor many materials useful in thermoelectric devices. This dissertation has touched upon diﬀerent members of this group, but has mainly focused on the peculiar nature of the

structure of CaAl2Si2 itself. Calculations and most notably, the analysis and interpretation of

their results, have enabled a thorough examination and a careful reconsideration of CaAl2Si2.

A new insightful, comprehensive picture of the unusual structural properties of CaAl2Si2 has emerged, allowing classification of CaAl2Si2 as a distorted wurtzite structure. From a condensed matter point of view, determining the physics behind these structural preferences has proven just as significant and compelling as its thermoelectric capabilities. As mentioned in the introduction, several factors that contribute to the structural analysis of CaAl2Si2 were investigated along this path. To summarize, important points are highlighted, all contributing to a broader selection of thermoelectric materials that can be fine-tuned for specific energy-saving applications.

5.1 Association with Familiar Structure

The mainstream approach to the primitive cell for CaAl2Si2 has been to select the Ca atom as the origin; an intuitive choice based on symmetries. A wurtzite binary, however, is a plausible reference for CaAl2Si2 and similar compounds. The simple action of reassigning the origin to a Si atom (which we label Si1) reveals another choice of primitive cell; one that connects our structure with the familiar wurtzite class of semiconductors and increases the knowledge base for developing and ﬁne-tuning wurtzite ternaries. We have successfully categorized CaAl2Si2 as a distorted wurtzite structure in which the wurtzite is inverted and the Al2 plane forms an umbrella-like structure with respect to the Si2 atom (see Figure 1.2), instead of the usual tetrahedral arrangement found in the wurtzite binary. This realignment

70 is stabilized by the low electronegativity of Ca (which loses most of its charge to Al-Si covalent bonds) and the resultant Coulomb repulsion between Si and Ca ion cores.

5.2 Utility of Fictitious Binary Reference Structure

2+ At the outset of this project, we surmised that the ﬁctitious binary (CaAl2Si2) could be used to demonstrate that since removal of the Ca atom from the ternary barely aﬀects its electronic structure, Ca ought to be replaceable with any column II atom, without altering

other structural characteristics. We have seen, however, that in order for other AM2X2 compounds, where A represents a column II atom, to stabilize in the umbrella structure, the A atoms must be fully ionized as are the Ca atoms, which is not generally the case. The value of the ﬁctitious binary, then, was to expand our knowledge of and predictive capabilities for the CaAl2Si2 class of ternaries.

5.3 Arrangement of Atoms

Analogies with existing cubic and hexagonal semiconducting Zintl compounds clarify the reasons that the Ca atom sits in one of the previously identiﬁed ternary wurtzite interstitial sites[42]. Electronegativity arguments again provided other clues. Given a choice, an atom tends to be more attracted to another sharing the largest electronegativity diﬀerence; in our case that would be Ca and Si. In its particular interstitial site, Ca is closest to Si. Identify-

2− ing CaAl2Si2 as a Zintl compound further supports its atomic arrangement; the (Al2Si2) constituent serves as the anion to the Ca2+ cation in the fairly common Zintl interpretation.

This property especially factors into the usefulness of CaAl2Si2 as a thermoelectric material.

5.4 Stabilization Mechanisms

Our work established several mechanisms crucial to the stability of CaAl2Si2. Symmetry, as usual, plays its part. Probably most inﬂuential is the repulsion between the Al2 and Ca atoms; this interaction forces the entire plane of Al atoms to relocate from its wurtzite ori- entation to the CaAl2Si2 characteristic umbrella conﬁguration. Madelung energy arguments,

71 Bader charge analyses and electronegativity also oﬀer convincing evidence for the stability of

CaAl2Si2 in its umbrella conﬁguration, as was made clear in Figure 4.6. The insight that the umbrella vs. wurtzite conﬁguration is determined by the electronegativity of the interstitial atom is of great consequence.

5.5 Semimetal Behavior

Although some literature oﬀers some reasonable claims to metallic behavior of CaAl2Si2, it is now fairly well substantiated as a semimetal. By virtue of its electronic structure; its lack of bandgap, the overlap of its conduction band into the valence band and a negligible density of states, CaAl2Si2 qualiﬁes as a semimetal. An important discovery of this dissertation is that compounds in the CaAl2Si2 class that prefer the umbrella structure will be semimetallic.

5.6 Future Work

As this investigation of the mechanisms behind the structural preferences of CaAl2Si2 has taught us, this topic is full of avenues of exploration and far from being drained of vibrant research opportunities. Some possibilities follow. The work in this thesis focused on the stability of an unusual structure, not on its band gap. It is crucial to perform GW or similar quasiparticle calculations to determine the formal status of the pure material as a semiconductor or semimetal. CaAl2Si2 itself has not been exhaustively studied; possibly more insights can be drawn from replacing the Ca atom with other column II atoms and performing in-depth investigations. Likewise, quaternary materials of this same class may oﬀer additional ﬁne-tuning to thermoelectric advantage.

As described in the Introduction, there are many compounds of formula AM2X2, the over- arching group that includes CaAl2Si2 - type structures. Ternary antimonides, bismuthides, compounds with A = rare-earths are a few. A wide variety of tunable parameters presents itself with these materials, oﬀering many possibilities for good thermoelectrics. Finally, investigation for its own sake, as seen in this dissertation, can result in rewarding revelations with as-yet-unknown insights and applications. Continuing to take fresh looks at presently

72 accepted norms may hold intriguing promise.

73 REFERENCES CITED

[1] E. Toberer, A. May, B. Melot, E. Flage-Larsen, and G. Snyder. Dalton Trans., 39:1046, 2010.

[2] E. Toberer, A. May, and G. Snyder. Chem. Mater., 22:624, 2010.

[3] L.L.N.L. Energy, Environment Directorate. https://publicaﬀairs.llnl.gov/news/energy/energy.html, 2010.

[4] K. Biswas, J. He, I.Blum, T. Hogan, D. Seidman, V. Dravid, and M. Kanatzidis. Nature, 489:414, 2012.

[5] A. Cuenat. http://www.npl.co.uk/upload/img/nanostructuredthermoelectrics- diagram1.jpg, 2010.

[6] G. J. Snyder and E. S. Toberer. Nat. Mater., 7:105, 2008.

[7] G. J. Snyder, Fall 2008.

[8] M. Kanellos. http://www.greentechmedia.com/articles/read/tapping-americas-secret- power-source-5259/, 2008.

[9] G. J. Snyder and T. Ursell. Phys. Rev. Lett., 91:148301, 2003.

[10] D. M. Rowe, editor. CRC Handbook of Thermoelectrics. CRC Press, Boca Raton, 1995.

[11] D. Parker and D. Singh. Phys. Rev. B, 85:125209, 2012.

[12] C.Candolﬁ, J. Zhao, F. Steglich, and Y. Grin. Research Reports: Intermetallic Phases as Thermoelectrics, 2011.

[13] J.P. Sethna. Statistical Mechanics: Entropy, Order Parameters and Complexity. Oxford University Press, Oxford, G.B., 2006.

[14] A.F Ioﬀe. Physics of Semiconductors. Academic Press, Inc., New York, NY, 1960.

[15] N. W. Ashcroft and N. D. Mermin. Solid State Physics. Brooks Cole, New York, 1976.

[16] M. P. Marder. Condensed Matter Physics. JW Sons, Hoboken, NJ, 2010.

74 [17] S. Maekawa, T. Tohyama, S. E. Barnes, S. Ishihara, W. Koshibae, and G. Khaliullin. Physics of Transition Metal Oxides. Springer-Verlag, Berlin Heidelberg New York, 2004.

[18] J.-F. Li, W-S. Liu, L.-D. Zhao, and M. Zhou. NPG Asia Mater., 2:152, 2010.

[19] X. Shi, J. Yang, J.R. Salvadore, M. Chi, J.Y. Cho, H. Wang, S. Bai, J. Yang, W. Zhang, and L. Chen. J. Am. Chem. Soc., 133:7837, 2011.

[20] B. Poudel, Q. Hao, Y. Ma, M. S. Dresselhaus, G. Chen, and Z. Ren. Science, 320:634, 2008.

[21] E. Toberer, A. Zevalkink, N. Cristoso, and G. Snyder. Adv. Funct. Mater., 20:4375, 2010.

[22] H. Zhang, J. Zhao, Y. Grin, X. Wang, M. Tang, Z. Man, H. Chen, and X. Yang. J. Chem. Phys., 129:164713, 2008.

[23] D. M. Rowe. Thermoelectrics handbook: macro to nano. CRC/Taylor & Francis, Boca Raton, 2006.

[24] J-S. Rhyee, K. Lee, S. Lee, E. Cho, S. Kim, E. Lee, Y. Kwon, J. Shim, and G. Kotliar. Nature, 459:965, 2009.

[25] J. Heremans, V. Vovovic, E.S. Toberer, A. Saramat, K. Kurosaki, A. Charoenphakdee, S. Yamanaka, and G.J. Syner. Science, 321:1554, 2008.

[26] E. Zintl and W. Dullenkopf. Z.phys.Chem., B16:183, 1932.

[27] E. Zintl and G. Brauer. Z.phys.Chem., B20:245, 1933.

[28] E. Zintl. Angew.Chem., 52:1, 1939.

[29] R. Nesper. Prog. Solid State Chem., 20:1, 1990.

[30] J. Corbett. Inorg. Chem., 39:5178, 2000.

[31] P. Alemany, Miquel Llunell, and E. Canadell. J. Comput. Chem., 29:2144, 2008.

[32] G.J. Snyder. http://www.hri.res.in/ cmschool/ISCFM/school/JeﬀreySnyder2.pdf, 2012.

[33] C. Zheng, R. Hoﬀman, R. Nesper, and H-G von Schnering. J. Amer. Chem. Soc., 108: 1876, 1986.

75 [34] S.M. Kauzlarich, S.R. Brown, and G.J. Snyder. Dalton Trans., 2007:2099, 2007.

[35] M. Imai, H. Abe, and K. Yamada. Inorg. Chem., 43:5186, 2004.

[36] Y.K. Kuo, K.M. Savakumar, J.I. Tasi, C.S. Lue, J.W. Huang, S.Y. Yang, D. Varshney, N. Kaurav, and R.K. Singh. J. Phys. Condens. Matter, 19:176206, 2007.

[37] J. Burdette and G. Miller. Chem. of Mat., 2:12, 1990.

[38] C. Kranenberg, D. Johrendt, and A. Z. Mewis. Anorg. Allg. Chem., 625:1787, 1999.

[39] G. Huang. J. Physics: Conf Series, 29:73, 2006.

[40] C. S. Lue, S. Y. Wang, and C. P. Fang. Phys. Rev. B, 75:235111, 2007.

[41] G. Huang, M. Liu, L.F.Chen, and D.Y. Xing. J. Physics Condens. Matter, 17:7151, 2005.

[42] Rungrote Nilthong. Cubic Ternary Alternatives to Wurtzite Semiconductors. PhD thesis, Colorado School of Mines, 1999.

[43] X. Gonze et. al. Computer Phys. Commun., 180:2582, 2009.

[44] X. Gonze et. al. Zeit. Kristallogr., 220:558, 2005.

[45] X. Gonze et. al. Comput. Mat. Sci., 25:478, 2002.

[46] P. Hohenberg and W. Kohn. Phys. Rev., 123:B864, 1964.

[47] M. Torrent, F. Jollet, F. Bottin, G. Z`erah, and X. Gonze. Comput. Mat. Sci., 42:337, 2008.

[48] X.-J. Wang, M.-B. Tang, H.-H. Chen, X.-X. Yang, J.-T. Zhao, U. Burkhardt, and Y. Grin. Appl. Phys. Lett., 94:092106, 2009.

[49] A. May, M. McGuire, and D. Singh. Phys. Rev. B, 85:035204, 2010.

[50] M. Weinert and R.E. Watson. Phys. Rev. B, 58:9738, 1998.

[51] C. S. Lue, C. P. Fang, A. C. Abhyankar, J. W. Lin, H. W. Lee, C. M. Chang, and Y. K. Kuo. Intermetallics, 19:1448, 2011.

[52] R.M. Martin. Electronic Structure. Cambridge University Press, New York, 2004.

76 [53] A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen. Phys. Rev. B, 52:R5467, 1995.

[54] J. P. Perdew and A. Zunger. Phys. Rev. B, 23:5048, 1981.

[55] S. L. Dudarev, D. N. Manh, and A. P. Sutton. Phil. Mag. B, 75:613, 1997.

[56] P. J. Kelly and M. S. S. Brooks. J. Chem. Soc., Faraday Trans. II, 83:1189, 1987.

[57] Y. Yun, H. Kim, H. Kim, and K. Park. J. Nucl. Mat., 378:40, 2008.

[58] M. Freyss, N. Vergnet, and T. Petit. J. Nucl. Mat, 352:144, 2006.

[59] J. P. Perdew, K. Burke, and M. Ernzerhof. Phys. Rev. Lett., 77:3865, 1996.

[60] D. D. Koelling and B. N. Harmon. J. Solid State Phys., 10:3107, 1977.

[61] J.C. Slater. Phys Rev., 51:846, 1937.

[62] P.M. Marcus. Int. J. Quantum. Chem., IS:567, 1967.

[63] J. Korringa. Physica, 13:392, 1947.

[64] W. Kohn and N. Rostoker. Phys. Rev, 94:111, 1954.

[65] D.R. Hamann, M. Schl´’uter, and C. Chiang. Phys. Rev. Lett., 43:1494, 1979.

[66] D. Vanderbilt. Phys. Rev. B, 41:7892, 1990.

[67] P. E. Bl¨ochl. Phys. Rev. B, 50:11953, 1994.

[68] D. Joubert G. Kresse. Phys. Rev. B, 59:1758, 1999.

[69] N. Troullier and J.L. Martins. Phys. Rev. B, 43:8861, 1991.

[70] N.A.W. Holzwarth, G.E. Matthews, R.B. Dunning, , A.R. Tackett, and Y.Zeng. Phys. Rev. B, 55:2005, 1997.

[71] N.A.W. Holzwarth, A.R. Tackett, and G.E. Matthews. Comput. Phys. Comm., 135: 329, 2001.

[72] C. Condron, H. Hope, P.M.B. Piccoli, A. Schultz, and S. Kauzlarich. Inorg. Chem., 46: 4523, 2007.

[73] C. Zheng and R. Hoﬀman. J. Solid State Chem., 72:58, 1988.

77 [74] C. Kranenberg, D. Johrendt, and A. Z. Mewis. Solid State Sci., 4:261, 2002.

[75] H. Zhang, M. Baitinger, M. Tang, Z. Man, H. Chen, X. Yang, Y. Liu, L. Chen, Y. Grin, and J. Zhao. Dalton Trans., 39:1101, 2010.

[76] Naval Research Laboratory. http://cst-www.nrl.navy.mil/lattice/, 2012.

[77] H. Morkoc and U. Ozgur. Zinc Oxide: Fundamentals, Materials and Device Technology. WILEY, Weinheim, 2009.

[78] J. S˘aanchez-Dehesa, J.A. Verg˘es, and C. Tejedor. Solid State Comm., 38:871, 1981.

[79] M. ElMassalami, L. Soares de Oliveira Paixao, and F.A.B. Chavez. J. Phys: Conden. Mat., 23:245701, 2011.

[80] S. J. Anderson, C.D. Marioara, R. Vissers, A. Frøther, and H.W. Zandenberg, 2006.

[81] C.-G. Zhan, J.A. Nichols, and D.A. Dixon. J. Phys. Chem. A, 107:4104, 2003.

[82] R.G. Parr, R.A. Donnelly, M. Levi, and W.E. Palke. J. Chem. Phys., 68:3801, 1978.

[83] P. Geerlings, F. de Proft, and W. Langenaeker. Chem. Rev., 103:1793, 2003.

[84] J. Korringa. Physica, 13:392, 1947.

[85] O.K. Andersen. Phys Rev. B, 12:3060, 1975.

[86] D. Vanderbilt. Phys. Rev. B, 32:8412, 1985.

[87] Peter David Haynes. Linear scaling methods in ab-initio quantum-mechanical calculations. PhD thesis, University of Cambridge, Christ’s College, 1998.

[88] W.C. Herring. Phys. Rev., 57:1169, 1940.

[89] L. Kleinman and D.M. Bylander. Phys. Rev. Lett, 48:1425, 1982.

[90] The ABINIT Group. www.abinit.org, 2012.

78 APPENDIX A - THE CONNECTION BETWEEN DFT AND ELECTRONEGATIVITY

The traditional electronegativity is a property of the state of a system, calculable in terms of the constants of physics from DFT[81]. More speciﬁcally, Parr et. al. assert that electronegativity is the chemical potential within DFT[82]. Beginning with the basic tenets of

DFT, the ﬁrst Hohenberg-Kohn theorem[46] allows the external potential Vext(~r) to be fully determined by the electron density, n(~r), whose normalization, in turn, delivers the total number of electrons, N, for the system: Z n(~r)d~r = N. (A.1)

These parameters set the hamiltonian for the system,

Hˆ Ψ = EΨ; (A.2)

Ψ represents the electronic wavefunction, Hˆ deﬁnes the energy of the system via the Schr¨odinger equation and E = E[n(~r)], a functional of n(~r), denotes this energy. Minimization of E with respect to n(~r), as stated in the second Hohenberg-Kohn theorem[46], provides the appropriate n(~r). Optimizing n(~r) gives the Euler equation:

δ(E − µn(~r)) = 0, (A.3)

where µ is the pertinent Lagrange multiplier and δ symbolizes the variation of the enclosed quantity. The Euler equation can be manipulated to state: δF V (~r) + HK = µ, (A.4) δn(~r)

with FHK [n] = T [n] + Vee[n]; the sum of the kinetic energy and the electron-electron inter-

δFHK action. (Here, δn(~r) denotes the functional derivative of FHK with respect to the density,

79 at ~r.) This, in fact, can be viewed as the DFT analog of the time-independent Schr¨odinger Equation. Rearranging the equation for µ gives δT µ = V (~r) + s . (A.5) eff δn

Parr et. al. [82] recognized that the Lagrange multiplier was, in fact, the partial derivative of the system’s energy with respect to N at ﬁxed external potential:

∂E µ = . (A.6) ∂N V (~r)

Furthermore, according to ﬁrst order wavefunction perturbation theory, the electron density can be expressed as a correction to the ground state energy due to changes in the external potential, with ﬁxed number of electrons:

δE n(~r) = . (A.7) δV (~r) N

It was then recognized that the energy of an atom can be approximated by a polynomial in the diﬀerence between the atom’s total number of electrons and its nuclear charge Z (m = N − Z), around m = 0. Assuming that this energy is continuous and diﬀerential (which can cause some problems!), the slope of the energy at m = 0 is perceived to be the electronegativity of the atom:

∂E χ = , (A.8) ∂N Z

and a connection is made between DFT and electronegativity[83]; namely

∂E µ = −χ = . (A.9) ∂N Veff (~r)

80 APPENDIX B - PSEUDOPOTENTIALS AND THE PAW METHOD

B.1 Pseudopotential and Linear Augmentation Methods

This appendix is present because my introduction to pseudopotentials and pseudopotential methods piqued my interest in the subject matter and led me to wish to apply this experi- ence in my current work. Previous research with which I was involved at various national laboratories emphasized the importance of basis sets and core, semicore and valence orbitals, and the exigency of their contributions to energies of the systems being studied. The (in many cases) overall improvement on calculational results found by using the PAW method to create pseudopotentials in place of the more mainstream norm-conserving pseudopotentials was also intriguing. I learned by trial and error how to determine which orbitals to include as semicore and which as valence, predominantly in an FP-LMTO code. Aware of controversy surrounding the 3d Ca orbitals regarding their inclusion at all in the Ca pseudopotential, I was motivated to develop, test and implement my own.

One of the chief hurdles to producing accurate electronic structure calculations is the ability to represent the wavefunctions’ rapid oscillations near the nucleus and their relative smoothness in the bonding regions in a computationally practical manner. Existing electronic structure codes address the problem in diﬀerent ways, with the two dominating calculational methods being linear methods and pseudopotential methods[67]. Following is a brief description of the evolution of the ways in which this challenge has been met.

Linear methods are based on the APW (Augmented Plane Wave)[61, 62] and KKR (Korringa, Kohn and Rostoker)[64, 84] approaches to solving the one-particle Schr¨odinger equation, a result of Density Functional Theory (DFT) being used to approximate a many-electron system; Andersen[85] was able to combine them to form the framework for the linear methods.

81 The pseudopotential method is based on norm-conserving ab-initio pseudopotentials as developed by Hamann, Schluter and Chiang[65], with subsequent alterations by Vanderbilt[86] to create ultrasoft pseudopotentials. The PAW (Projector Augmented Wave) method takes advantage of the most salient aspects of the linear and pseudopotential methods, merging the flexibility of the former with the formal ease of the latter. In fact, LAPW can be considered a particular instance of the more general PAW, and pseudopotentials can be viewed as ap- proximations to PAWs. P.E. Blöchl[67] pioneered the PAW scheme; Kresse and Joubert[68] clarified the connection to ultrasoft pseudopotentials, and refined the PAW method for implementation. In both methods, wavefunctions are radial solutions to the Schrödingerequation times the spherical harmonics inside a chosen sphere radius, represented by partial wave expansions, and an ‘envelope function’ expanded in Fourier series or plane waves or another appropriate basis outside the sphere radius.

The motivation behind the creation of pseudopotentials is that for a given atomic system, the highly localized core electrons and the strong Coulomb potential present conditions difficult to mimic computationally. Relying on the non-uniqueness of pseudopotentials, the premise behind their development is that the core electrons are essentially unaffected by the chemical environment of the atom, enabling replacement of the core electron and Coulomb potential with a much weaker effective potential, and substitution of the valence electron wavefunctions with pseudo wavefunctions that vary smoothly in the core region. Their usefulness is high accuracy, good transferability and low computational cost.

Validation for the above-mentioned substitutions follow from observing that the contribution to the binding energy of a given system from the core electrons is both large and relatively unchanging as isolated atoms are brought together to form, for example, a molecule or crystal. This implies that the core electron portion of the binding energy can be subtracted from the total binding energy, leaving the change in binding energy associated with the valence electrons (∆Eval), which is ultimately of interest, as a much greater percentage

82 of the total binding energy. This situation enhances the accuracy with which ∆Eval can be calculated. We also know that all atomic wavefunctions are mutually orthogonal, a consequence of being eigenstates of the atomic hamiltonian. The core electrons are localized near the nucleus, thus mandating that the valence electrons oscillate rapidly in the vicinity, to preserve orthogonality. This motion begets a large kinetic energy, which is approximately canceled by the large Coulomb potential, leaving the valence electrons more weakly bound than the core electrons. Hence the justiﬁcation of the pseudopotential approximation.

Two approaches, the operator approach and the scattering approach, lend additional rein- forcement to the pseudopotential approximation[52, 87]. The scattering theory description is perhaps most illustrative; its statement that for a given localized spherical potential, scat-

tering properties at any energy can be expressed via the phase shift ηl (), thus establishing properties of the wavefunction beyond the localized region, is a central concept in physics. It can be said that the purpose of a pseudopotential is to precisely replicate the scattering properties of a given system over a speciﬁed energy range in a useful manner[52].

Beginning with a plane-wave of wave vector ~k incident upon a spherical potential centered

at the origin and localized within a radius rc, we write the plane-wave in terms of spherical waves:

∞ ` ~ X X ` ∗ ˆ exp ık · ~r = 4π ı j` (kr) Ylm k Ylm (ˆr). (B.1) `=0 m=−`

The elastic scattering of these partial waves by the potential gives rise to a subsequent phase

shift δ`, signiﬁcant in its relation to the logarithmic derivative (Ll(E)) of the exact radial

1 2 solution for given ` and energy E = 2 k within the core, and evaluated on the surface of the core region:

d L (E) = log [R (r, E)] (B.2) ` dr ` r=rc R0 (r ,E) = ` c (B.3) R` (rc,E)

83 j0 (kr ) − tan (δ ) n0 (kr ) = k ` c ` ` c , (B.4) j` (krc) − tan (δ`) n` (krc)

where here, jl and nl symbolize the spherical Bessel and von Neumann functions and

ˆ ψlm (~r, E) = R` (r, E) Y`,m ~r . (B.5)

with ψ`,m as the core region solution to the Schr¨odinger Equation.

Recombining the phase-shifted spherical waves gives the total scattered wave, and a reduced phase shift η` can be deﬁned (here n` is an integer):

δ` = n`π + η`. (B.6)

The reduced phase shift has the same effect as δ`; factors of π have no effect since the scattering amplitude is proportional to exp (2ıδ`). η` can then be fixed between 0 and π.

The number of radial nodes in R` (r, E) is represented by the integer n`, which consequently denotes the number of core states with angular momentum `. The pseudopotential is then established as the potential whose complete phase shifts are the reduced phase shifts η` such that the radial pseudo wavefunction is nodeless and the pseudopotential has no core states. The scattering eﬀect of the potential is equivalent to that of the original. In order that the pseudopotential be accurate over a range of energies, we must enforce matching of the energy-dependent phase shifts to ﬁrst order in energy. The `-dependence of the pseudopotential demonstrates its nonlocal character.

The operator path mirrors the OPW method[88]. Representing the atomic hamiltonian as ˆ H, corresponding core states and core energies as {|χni} and {n} respectively, and working with one speciﬁc valence state |ψi with energy E, a pseudostate can be created:

core X |ψi = |φi + an|χni, (B.7) n

Applying the orthogonality requirement, the expansion coeﬃcients can be extracted:

84 h|χm|ψi = 0 = hχm|φi + am (B.8) and

core X |ψi = |φi − |χnihχn|φi. (B.9) n

Using this form in the Schr¨odinger Equation Hˆ |ψi = E|ψi gives the Schr¨odinger Equation ˆ for the smooth wavefunction |φi, which includes an energy dependent, nonlocal potential Vnl in the hamiltonian:

core ˆ X Vnl = (E − En) |χnihχn|φi. (B.13) n

We see that the energies of the pseudo state and the valence state are equivalent. The range ˆ of VNL is confined to the core region and is repulsive, mitigating a portion of the Coulomb potential and leading to a weaker pseudopotential. When part of a solid or molecule, the interaction of the atom will cause eigenstate energies to vary; as long as this fluctuation is negligible compared to the difference in energy between the core and valence states, the ˆ approximation fixing E in VNL as the atomic valence eigenvalue is suitable. The aim is to ensure that the pseudopotential imitates the true potential to first order in E. The norm- conserving constraint is key to accomplishing this intention.

Prior to treating norm-conserving pseudopotentials, a closer look at the pseudopotential itself is in order. Its `-dependence attests to its nonlocality (diﬀerent angular momentum

85 ˆ states, or partial waves are scattered diﬀerently by the potential). In general, Vnl can be written in semilocal form, in which it is nonlocal in angular variables but local in the radial variable:

` ˆ ˆ X X ˆ VSL = Vloc + |`miδV`h`m| (B.14) ` m=−`

ˆ with |φ`mi an eigenstate of the atomic pseudo hamiltonian. VKS performs the same way on ˆ the reference state as does the original VSL, yet instead of the projections (matrix elements) ˆ scaling as the square of the number of basis states, they scale linearly. The hδV`φ`mk are the projectors that act on the wavefunction. A potential hazard of which to be aware is the ˆ ˆ appearance of ghost states, which occurs when Vloc is attractive and the δV repulsive; they manifest as discontinuous logarithmic derivatives of the PS wavefunctions when there is none present in the AE logarithmic derivative. Adjustments in reference energies and generation schemes can help to avoid these.

Norm-conserving pseudopotentials substitute a pseudo wavefunction (PS) for the all-electron valence wavefunction (AE) inside some core radius. (The treatment of core wavefunctions is similar, should a relaxed-core implementation be desired. The PAW method is exact within the frozen-core approximation, but upon implementation, is subject to two further approx- imations; that of a plane-wave energy cutoff, and a finite number of AE, PS wavefunctions and projectors. The latter approximation affects the completeness of the set of partial waves, and therefore the orthogonality of the valence wavefunctions to the core states. A modification of the definition of the transformation from PS to AE states addresses this concern.) The AE wavefunction is a full one-electron wavefunction, obtained by solving the Kohn-

86 Sham equations. The PS is soft and nodeless, and within the core radius, its norm must be equivalent to that of the AE. Outside the core radius, the wavefunctions are the same. For the norm-conserving pseudopotential scenario to have transferability, the core radius of a system must be extended beyond the maximum of the outermost AE wavefunction in order for the charge distribution and moments of the AE wavefunctions to be replicated by the PS wavefunctions. This results in an undesirably large amount of plane waves in the basis set for systems with highly localized orbitals. Vanderbilt’s modification to create ultrasoft pseudopotentials was to eliminate the norm-conserving constraint, incorporating the resulting charge density difference between the AE and PS constructions as an augmentation in the core region. These augmentations in turn need to be pseudized for the sake of computational efficiency. Core radii were then required only to be about half the distance between ions; this improvement, however, came at a cost to the construction of the pseudopotentials themselves. The charge augmentations added another cutoff radius; in effect, the number of parameters needed to specify the pseudopotential increased, along with the complexity of building the pseudopotentials[67].

An advantage of the PAW method is that it avoids the charge augmentation pseudization, by involving only the full AE wave functions and potentials[68]. Blöchl[67] introduces a linear transformation from the PS wavefunctions to the AE wavefunctions, thus allowing physical quantities expressed as expectation values of some operator on PS wavefunctions to be interpreted as physical AE quantities, including the total energy functional. The transformation maps the physically relevant wavefunctions in the Hilbert space of all wavefunctions orthogonal to the core states to pseudo wavefunctions in a new pseudo Hilbert space. The transformation is chosen such that it reflects specific atomic traits, and differs from the identity by the sum of local, atom-centered contributions.

Let τ represent the linear transformation, |Ψi = τ|Ψ˜ i, where |Ψ˜ i denotes the PS wavefunc- P tion, |Ψi the AE wavefunction. τ is speciﬁed by τ = 1 + R τˆR, whereτ ˆR are the local,

87 atom-centered contributions.τ ˆR’s domain is the augmentation region surrounding the atom

ΩR; this augmentation region is analogous to the core region for pseudopotentials and the muffin-tin for the linear methods. Using the knowledge that every PS and AE wavefunction ˜ can be expanded in partial waves (represented by |φii and |φii respectively), and invoking the linearity of τ, the expansion coefficients of the PS and AE wavefunctions must be equivalent. The linearity of τ also requires that the coefficients be linear functionals of the PS wavefunctions, and thus can be written as a scalar product of the PS wavefunction and some ˜ fixed function, the latter of which is designated the projector function |pii [67]. The electron density is then derived from the projector form of the AE wavefunction, and the total energy as a functional of the electron density. Following Blöchl[67], we have:

˜ X ˜ |Ψi = ci|φi in ΩR, (B.16) i ˜ ˜ |φii = τ|φii and |Ψi = τ|Ψi. (B.17)

We then have, in ΩR,

A requirement for the pseudo wavefunction is that the pseudo wavefunction be identical to its one-center expansion, placing the restriction on the projector functions that

˜ |φiihp˜i| = 1, (B.21) which leads to

˜ hp˜i|φii = δij. (B.22)

88 This gives an explicit form for τ and the recipe for the extraction of the AE wavefunctions from the PS wavefunctions:

X ˜ τ = 1 + |φii − |φii hp˜i| (B.23) i

and

˜ X ˜ ˜ |Ψni = |Ψni + |φii − |φii hp˜i|Ψni. (B.24) i

In contrast, the APW methods, upon which the linear methods are based, match value and energy derivative of the envelope functions at the sphere boundary to those of the partial waves inside the sphere. Because of the energy dependence, the matching results in nonlinear equations, which are computationally demanding since each eigenstate must be solved for individually. For this reason, the APW methods (here the MTO is exempliﬁed) were linearized, as summarized below[52]:

˙ Augmentation functions are defined to be linear combinations of Ψ(Eν, r) and Ψ(Eν, r), the radial function and its energy derivative, evaluated at a specific energy Eν. These functions essentially define a basis that allows calculations of all states in an interval around this chosen energy. An LMTO basis function is then defined in terms of κ, and functions that act

2 ~ 2 similarly to spherical Bessel and Neumann functions, with ( 2m )κ = − V0. From these an energy-independent LMTO orbital is obtained, that connects smoothly with the interstitial region and incorporates tail cancellation to lowest order. Using this basis, eigenvalues are determined via a variational expression with the full Hamiltonian. One assumption upon which the LMTO method is based is that of a ﬂat potential in the interstitial region. The ‘full potential’ is accommodated by generalizing the envelope function using Hankel functions or power laws. The hallmarks of the LMTO method are that the muﬃn-tin orbitals are generated from the Kohn-Sham hamiltonian and that calculations can satisfactorily be performed with a minimal basis set[52].

89 In order to successfully search for the mechanism behind the buckling of the CaAl2Si2 class of ternary semiconductors, it is crucial to choose an electronic structure code capable of providing an integrated theoretical and computational framework. We have selected ABINIT [43– 45], an open-source electronic structure package, for its ﬂexibility and proﬁciency. ABINIT’s main program uses pseudopotential methods and a planewave basis within the framework of Density Functional Theory (DFT) to calculate total energy, electron density and electronic structure of a given system of electrons and nuclei. Since the PAW method is considered state of the art for pseudopotentials, in that its convergence is fast (comparable to that of USPPs) and at any given point, the AE wavefunction can be recovered[90], we have also chosen to use PAWs within ABINIT. One attractive feature of ABINIT is the ability to generate one’s own PAWs[47] for use in one’s calculations. This is a fairly intricate process, and it is expected that a portion of this work will be spent on creating a PAW potential for Ca. Standard PAWs will be used for Al and Si as they are simple materials. For Ca, although it lacks 3d orbitals in its ground state, the inclusion of 3d orbitals in the Ca PAW can have a

significant effect on the structural properties[39], and transport properties[35] of CaAl2Si2. As well, the treatment of the 3s and 3p semicore states as core or valence will affect the performance of the pseudopotential; according to Huang, the density of states of CaAl2Si2 reveals a hybridization of Ca 3d and (Al, Si) 2p states in the valence band area, as well as a 3d character at the bottom of the conduction band[39], underscoring the importance of 3d orbital inclusion in the Ca PAW. All of these considerations have been taken into account in the construction of the Ca PAW, as detailed below.

B.2 Initial PAW Dataset

There are several ingredients needed and a set procedure to follow to create a PAW dataset. In this discussion, the ABINIT[43–45, 90] help ﬁles and user guide provide the basis, with support from N. Holzwarth[70, 71] and G. Kresse[68].

90 Beginning with the designation of the chemical species and its atomic number Z, the all- electron (AE) atomic problem is solved self-consistently with a certain exchange-correlation function, for a chosen electron conﬁguration. The latter can be tailored to its expected form in the pertinent solid; for example, in this work, the ground state of the Ca atom is adopted: 1s22s22p63s23p64s23d0 for its stability. The core electrons and valence electrons are selected; here 3s2, 3p6, 4s2 and 3d0 are identiﬁed as valence electrons, and the core density computed.

The core density is such that within a radius rc it is smooth, and matches the core density outside that radius. The valence wavefunctions are used in the PAW basis.

Next, the number of partial waves and projectors is determined, and thus the size of the basis.

Partial waves (|φii), either bound (valence) or unbound (solutions to the wave equation for a given ` at arbitrary reference energies) are chosen for inclusion in the basis. Pseudo partial ˜ waves kφii and their corresponding projectors (|p˜ii) are generated; the pseudo partial waves

PS are solutions to the PAW hamiltonian, constructed by using a local pseudopotential Vloc created from the pseudization of the eﬀective potential of the AE hamiltonian, constrained to equal Veff outside a matching radius rvloc. It should be mentioned the at this juncture,

PS Vloc is screened; to obtain the unscreened bare pseudopotential, the exchange-correlation and Hartree potentials must be subtracted. The PS partial waves must match the partial waves outside of a radius rc. The constraints on the projector functions have previously been described. Both PS partial waves and projectors are then orthogonalized, using a method ˜ such as Vanderbilt’s procedure[66]. The |φii, |φii and |p˜ii all comprise the augmentation region, whose radius is deﬁned as rP AW . Then a compensation charge density is built, needed to regain the total charge of the atom, which was impacted when the norm-conserving condition was dropped. This is described by an analytic shape function inside a radius rshape, itself inside the PAW sphere. And ﬁnally the PAW dataset is tested.

91 B.3 Ca PAW Dataset

As explained in the main body of work, the norm-conserving FHI GGA pseudopotentials were predominately the constructions used to approximate the exchange-correlation energies for the calculations performed. When PAW potentials were used, in instances for which we felt the slight improvement in accuracy that might be gained from them outweighed computational expense, the Ca PAW created by Torey Semi was put together with Si and Al PAWs from the ABINIT pseudopotential database. Below, Figure B.1 and Figure B.2 are some test output run for Semi’s Ca PAW potential; Figure B.3 and Figure B.4 are tests run on some PAW potentials supplied by the ABINIT database. Their presence is to oﬀer an idea of the kind of examination used to gauge the viability of a PAW potential.

92 Ca 1s Partial Waves, Projector

Projector: Bessel Partial Wave Valence: 3s 3p 4s 3d PS Partial Wave 1.5 Vloc: Bessel Projector rPAW: 1.9 1.5 1.4 1.9 rc: 1.5s,p 1.9d

0.5 amplitude (arbitrary units)

-0.5 0 0.5 1 1.5 2 radius (au)

Figure B.1: AE and PS wavefunctions for Ca 1s state are equivalent after ﬁrst maximum. Projector function exists only for r < rc.

93 Ca Logarithmic Derivative \ell = 1 GGA

50 AE PAW

-50 Projector: Vanderbilt Valence: 3s 3p 4s 3d Only one d orbital -100 rPAW, rc: 1.9 au Logarithmic Derivative Abinit Database (NH)

-150

-200 -4 -2 0 2 4 E (Ry)

Figure B.2: Scattering properties.

94 Logaritmic Derivative \ell = 2 GGA

1000

AE Natalie PAW 800 valence: 3s 3p 4s 3d proj: vanderbilt rPAW, rc: 1.9 au 600

400

logarithmic derivative 200

-200 -4 -2 0 2 4 E(Ry)

Figure B.3: Non-alignment of AE and PAW logarithmic derivatives indicate that Ca scattering properties are not reproduced exactly by this PAW dataset. Eﬀorts to add an additional d state were met with resistance.

95 Ca \ell = 2 Logarithmic Derivative Additional d State (Eref = 3 Ry) NH 200

Projector: Vanderbilt AE PAW 150 Valence: 3s 3p 4s 3d Two d orbitals rPAW, rc: 1.9 au Abinit Database (NH+ d) 100

50 Logarithmic Derivative 0

-50

-4 -2 0 2 4 E (Ry)

Figure B.4: Additional d state in Ca PAW with Eref = 3 Ry reproduces scattering properties more accurately for ` = 2.