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