Similarly Dr. Stephanov's lessons in E&M brought my ...... [59] C. J. Johnson, E. E. Eissler, S. E. Cameron, Y. Kong, S. Fan, S. Jovanovic, and K. G.. Lynn ...
Application of ab-initio Methods to Grain Boundaries and Point Defects for Poly-CdTe Solar Cells By Christopher Buurma B.S., Ashland University, 2007
THESIS Submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Chicago, 2015. Chicago, Illinois
Defense Committee: Sivalingam Sivananthan, Chair and Advisor Maria K Y Chan, Argonne National Laboratory Christoph Grein, Physics Robert Klie, Physics Wyatt Metzger, National Renewable Energy Laboratory
ii |Acknowledgement
Acknowledgement I find this part of my thesis to be the most difficult to write, as there have been so many wonderful people involved in my life who have blessed me and brought me to this point I would feel remiss to leave someone out. First I would like to thank the various funding sources for my dissertation work who support fundamental sciences and the public interest including primarily the DoE Bridge Project DOE-EE00005956, the national renewable energy laboratory (NREL), and the use of the Center for Nanoscale Materials (CNM) at Argonne National Laboratory (ANL) under contract DE-AC02-06CH11357. Certainly without the support of my advisor, Dr. Siva, my dream of completing a PhD would not have been possible. His faith in me as a capable student and scientist from the very beginning pushed me to live up to his expectation. Without the rare and generous opportunities he has provided me, I could not have climbed this mountain. Thank you Siva. From my initial interest in physics in high school to the completion of my final class curriculum at the University of Illinois, I have had an array of excellent teachers who truly cared about the students they educated. Dr. Rodney Michael during my undergraduate studies at Ashland University taught me with a passion for his field, endless patience, kindness and would accept nothing less than 100% during my studies and would support me through my challenging class load. Dr. Kerkez and Dr. Ajwa taught me with equal vigor the principles of computing so that I would never again call a variable 𝑥, and certainly without Dr. Dense and Dr. Swain I would not have developed my love for the beauty of mathematics and the abstract. Previous to my graduate studies I had the pleasure of working alongside wonderful scientists, engineers and technicians at First Solar and EPIR. My early career in research could have only happened with the support of Dr. Scotty Gilmore who was both a wonderful manager but also both him and his wife have been great friends for many years. Our early research team with Roger Green, Brady Johnson, John Christiansen, Dmitriy Marinskiy, and Igor Sankin was a great first research family to have. Especially when combined with all of the other great people we worked with in adjoining groups. During my graduate studies, I shall certainly never forget Dr. Ogut whose clear mastery of quantum mechanics and solid state physics left no question without a deep answer and was always available to help me with my education at any level. Similarly Dr. Stephanov’s lessons in E&M brought my technical skill in this arena beyond where I thought possible. Joining the microphysics lab and meeting such great students there proved to have indelible impact: Brian Stafford, Jin Park, Brian Kaster, Eric Colegrove, and Xin Zheng in particular helped me a great deal with their friendship and in
Christopher Buurma | iii providing excellent discussions to challenge me. My study group for the qualifier exam helped prepare me for the hardest written test I have ever taken; thank you Brian, Dustin, Eric, Natalie, and Jayashri. Previous to, and concurrently with my graduate studies I enjoyed a stimulating work environment at EPIR Technologies whose staff scientists and engineers challenged me to not only bring my best technical expertise but to apply them in a real-world setting to make a lasting impact with our research efforts. Dr. Jim Garland who helped prepare this document and Dr. Wei Gao with his exemplary patience and understanding were wonderful mentors when I started there and continue to be great friends. Dr. Silviu Velicu has always been a great person to sound ideas off of and get good guidance. My dissertation work itself has been marked with so many wonderful people as well. Dr. Maria Chan, my unofficial advisor, has from the beginning been a wonderful mentor and pushed me to new heights of understanding. Without her consistent support and guidance this work certainly would not have been possible. Dr. Robert Klie’s leadership proved invaluable to keep us moving forward and not lose sight of the big picture. I’d also like to thank Tadas Paulauskas, Dr. Alper Kinaci and Dr. Fatih Sen for their insightful discussions and assistance into my work as well as their friendship. I of course cannot neglect my friends and family who have always been supportive of my passion for not only science and knowledge but also for stewardship to the earth and the people in it. I have found great comfort, especially in dark times, by looking heavenward and asking myself what Jesus would suggest, and how I can truly love my neighbor and myself. Both my mother and father supported my scientific interest from a young age, even if it meant leaving the family farm and moving away to a big city to follow my dreams. My sister and brother have also been great friends since childhood, and I only hope I can make them proud. Most of all I would like to thank my wife of now 11 years, Nina Buurma. Throughout my life she has been with me, in the good times and the bad, and brought me such joy I cannot express it in so many words. She saw the man I could become and with her patience, love, care, understanding, kindness, and support I have come so far. For my best friend for so long, I wish that the kindness she pours out so readily be returned to her ten-fold more. We did it my love. We did it.
iv |Table of Contents
Table of Contents Acknowledgement ........................................................................................... ii 1)
Introduction ............................................................................................ 1
1.1) CdTe Solar Cells ..................................................................................... 1 2)
First-Principles or Ab-initio Formalism............................................ 4
2.1) Theoretical Treatment of Bulk Solids .................................................... 4 2.2) Born-Oppenheimer approximation ......................................................... 6 2.3) Density Functional Theory ..................................................................... 7 2.4) Exchange, Correlation and Basis functions ......................................... 12 3)
Role of Defects in CdTe Semiconductors ........................................ 13
3.1) First-principles DFT Calculation of Defect Energy Levels .................. 13 3.2) Generation-recombination Current ...................................................... 16 4)
Defect classification in CdTe ............................................................. 20
4.1) Zero-dimensional Point Defects ............................................................ 20 4.1a) Point-defect Diffusion ..................................................................... 20 4.1b) Semiconductor Impurity States ..................................................... 21 4.2) 1D Dislocation Line Defects .................................................................. 22 4.2a) Edge and Screw Dislocations.......................................................... 23 4.2b) Elastic Properties ............................................................................ 24 4.3) 2D Grain Boundaries and Interfaces ................................................... 25 4.3a) Grain-boundary Definition - Macroscopic ...................................... 26 4.3b) Crystal Symmetry in Grain Boundaries ........................................ 28 4.3c) Grain-boundary Definition – Microscopic ...................................... 31 4.3d) Grain-boundary Models .................................................................. 32 4.3e) Low-angle Grain Boundaries.......................................................... 34 4.3f) High-angle Grain Boundaries ........................................................ 34 4.4) 3D Precipitates and Voids .................................................................... 36 5)
Results of This Work ............................................................................ 38
5.1) Bulk CdTe results.................................................................................. 38 5.1a) Bulk CdTe ....................................................................................... 38 5.1b) Native Point Defects in bulk CdTe ................................................. 41 5.1c) Defect pairs in bulk CdTe ............................................................... 44 5.1d) Bulk recombination lifetimes calculated using a tight-binding Hamiltonian ................................................................................................ 48
Christopher Buurma | v 5.2) (60°)(111)(111) 𝛴3 coherent twins ......................................................... 51 5.2a) Computational methods .................................................................. 53 5.2b) Pure Twins ...................................................................................... 54 5.2c) Twins with point defects ................................................................. 57 5.2c,i) Cd vacancies (VCd) ..................................................................... 58 5.2c,ii) Cd interstitials (CdI) ................................................................. 59 5.2c,iii) Te-on-Cd antisites (TeCd) .......................................................... 62 5.2c,iv) Other intrinsic point defects ..................................................... 64 5.2d) Twins with defect pairs ................................................................... 69 5.3) Creating GB interfaces with the Grain-boundary Genie...................... 72 5.3a) Grain boundary generation algorithm ........................................... 73 5.3b) From vicinal-CSL to GB interface .................................................. 74 5.3c) Further reduction perpendicular to the GB plane ......................... 78 5.3d) Grain-boundary Genie Interface .................................................... 80 5.4) Bicrystal Example Structures ............................................................... 81 5.4a) 𝚺𝟑 twin (60°)(111)(111) ................................................................... 82 5.4b) Mirror GB Plane (0)(100)(0,0,-1) ......................................... 82 5.4c) Half-rotated Twin (30°)(111)(111) ................................................. 84 5.4d) (90°)(110)(110) ................................................................................. 85 5.4e) Dual Interface (35.2)(110)(10-1) ......................................... 87 5.4f) 5° Mutual Tilt of (0º)(1.05,0.95,0.95)(1.05,1.05,0.95) .......... 88 5.4g) 4° mutual tilt of (0º)(11,0.05)(1,1,-0.05)............................... 90 5.5) Early Sampling of the GB Parameter Space Using pseudo-potentials 91 5.5a) Low-energy structures from metals ............................................... 93 5.5b) Near the Mackenzie Peak ............................................................... 94 5.5c) Asymmetric- tilt GBs ...................................................................... 98 5.6) Supporting software and database ..................................................... 100 5.6a) Structure Handler ......................................................................... 100 5.6b) VASP Helper ................................................................................. 101 5.6c) Defect Buddy ................................................................................. 102 5.6d) VASP Janitor and SQL Database................................................. 103 6)
Conclusions .......................................................................................... 105
7)
Cited Literature .................................................................................. 107
8)
Vita ......................................................................................................... 115
vi |Table of Contents LIST OF TABLES TABLE I.
II. III. IV. V. VI.
PAGE
Final energies per formula unit for the stable CdTe compound, metallic Cd and semimetallic Te. These values correspond well to those found by the Materials Project........................................................................ 40 Formation energies of native point defects at three different chemical potentials. .............................................................................. 43 Defect energy levels for native point defects in CdTe ......................... 44 Summary of lifetime calculation results .............................................. 50 Formation energies of all native point defects investigated along twins ...................................................................................................... 69 Formation energies of notably low-formation-energy point-defect pairs. ...................................................................................................... 72 LIST OF FIGURES
FIGURE 1. The
2. 3.
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Shockley-Queisser detailed balance limit, with CdTe highlighted in red. Loss mechanisms for the reduction of the theoretical efficiency down from 100% are listed. ...................................2 Breakdown of departures from the theoretical maximum for the 18.7% champion solar cell.[5] ...................................................................2 The conventional unit cell of CdTe. This binary species possesses a tetrahedral face-centered cubic symmetry and belongs to the space group 216 denoted as 𝐹43𝑚. .....................................................................5 A conceptual overview of Density Functional Theory. The real world problem is mapped to a non-interacting system, solved, and then remapped to the real system to find the total energy while bypassing the need to solve for the electron wavefunctions in the real system. .............................................................................................10 Screw and edge dislocation in a cubic simple lattice. [3].......................23 Schematic representation of shear strain. The deformation of points B, C and D can be seen as a change in angle from that of a square to that of a parallelogram. ......................................................................25 Interface-plane scheme for a (30°)(110)(111) GB. First, identify the two grain facing directions 𝑛1 and 𝑛2, then rotate the two faces along the same axis, lastly provide mutual rotation about that axis by 𝜃. .........................................................................................................27 Misorientation Scheme. Grains begin facing 𝑏 and then the second is rotated away from the first along axis 𝑜 by angle 𝜙 ..........................27
Christopher Buurma | vii 9. Projection of crystallographic directional unit sphere onto a plane, 10.
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providing a pole diagram. ...................................................................... 29 Pole figures with the center at (111) showing symmetrically equivalent points (left) and the 24 symmetrically equivalent regions (right) for CdTe and other fcc structures. ............................................. 30 Snapshot of the 3D grain boundary region wedge in the fundamental zone. Any line segment in this space can correspond to a single grain boundary. The first point is constrained to the surface, while the second point may dip into the structure. The length of the line in the XY projection is the tilt angle, while in the YZ projection this is the twist angle. The overall length is the misorientation angle. ............................................................................. 31 Depiction of commonly applied grain boundary models. Low-angle GBs are treated with dislocation theory first, and then after the decomposition of the Burger’s vectors, continuum theory is used. For GBs with high symmetry, either a coincident site lattice (CSL) model is used, or is slightly modified as a Vicinal CSL. Alternatively, one can look at the coincidence of lattice points and not atoms, and modify for continuous deformations using the displacement-shift complete lattice (DSC). ........................................... 33 Grain boundary classification schemes commonly found in the literature. Low-angle grain boundaries are described in terms of their Burger’s vectors pointing along, or perpendicular to the interfacial plane. High-angle tilts without a twist component comprise a series of asymmetric-tilt GBs (ATGB) while those with a twist component are either pure twists, fully symmetric twists resulting in a CSL-like structure, or comprise ‘the rest’ being mixed or random-angle grain boundaries. ....................................................... 33 The interfacial energy of tilt GBs in Ni as a function of the tilt angle 𝜉 and the degree of symmetry 𝜂 between the grain directions and high symmetry lines.[45] ....................................................................... 36 Final energy/atom of the ionic relaxations using different k-point grid sizes and centering. An initial relaxation was performed, and then a second relaxation was performed using the final state of the first relaxation as the new initial state was done to ensure there was no local minimum............................................................................ 39 Band structure of CdTe as calculated from Kohn-Sham DFT. The energy is expressed in eV. The solid lines are for the occupied valence-band states, while the dashed lines are for the unoccupied conduction-band states. The Kohn-Sham gap is 0.61 eV. Grey lines are drawn to indicate (𝛤, 0) .................................................................... 40 Defect formation energies for native point defects in bulk CdTe (color online): VTe, CdTe, TeCd, VCd, CdI, TeI. The thermodynamic Fermi level is indicated in grey. ............................................................ 42
viii |Table of Contents 18. Defect charge transition levels for native point defects in bulk 19.
20. 21.
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CdTe.........................................................................................................42 Low precision screening calculations for defect pair binding energies in bulk CdTe. Formation energy is plotted against the 1 st and 2nd defect in the pairing and the two chemical potential exteremes are listed along the top within each grouping. Each defect pair is also listed with its multiple stable separation distances. These are for charge neutral structures and the expected charge state will reduce or increase the formation energy appropriately. ..........................................................................................45 Defect formation energy diagram for the 6 lowest energy defect pairs. Binding energies of these pairs are also quite high. ...................46 Defect charge states for point defect pairs. The two stoichiometric defects possess doping states near the conduction band while the other four have many gap states throughout the band. ........................46 Binding energy and formation energy of Frenkel and Schottky pairs. Solid lines are the bound pairs, while dashed lines are the sum of the two defects independently. The dotted line gives the total binding energy, corresponding to the right y-axis. .......................47 Binding energy diagram for VCd + TeI. The pair slightly separated shows a lower overall formation energy than if both defects occupied the same lattice site as a TeCd. The relative rarity of an isolated TeI suggests that TeI may be more readily found in proximity to a VCd....................................................................................48 A series of twins in poly-CdTe. Both extrinsic and intrinsic stacking faults are present, as well as grain terminating twins, and lamellar twins. .......................................................................................................52 A lamellar twin with EDS overlayed in the box revealing the atomic species. .....................................................................................................52 Examples of model twin structures. (Left) An A4B1 isolated stacking fault and (right) an A3B3 lamellar twin. Atoms located at the twin boundary plane region are highlighted. .................................................53 Examples of model twin structures. Top left: A7B1 stacking fault. Top right: A6B6 lamellar (repeating) twin. Bottom: A19B1 isolated stacking fault...........................................................................................55 The formation energy per atom as a function of strain for a variety of different twin n and m values as well as for bulk CdTe. ...................55 Interfacial energy as a function of average distance between twin boundaries in the computational cell. A3B3 is indicated for reference. .................................................................................................56 Minimum interfacial energy found over all strain as a function of bulk columns (n) and twinned columns (m) for an AnBm structure. Changes in energy are very small, as seen from the distribution inset. ........................................................................................................56
Christopher Buurma | ix 31. Kohn-Sham density of states for all twin structures calculated
32. 33. 34.
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above. No gap states are revealed indicating twin boundaries free of point defects should be electrically benign. ........................................... 57 Inequivalent defect positions along an A4B1 isloated stacking fault ... 58 Inequivalent defect positions along an A3B3 lamallar twin.................. 58 Formation energy diagram for VCd. Only a very minor energy drop is observed between those found in the bulk and those along either A3B3 or A4B1 twins.................................................................................. 59 Defect energy transition level diagram for VCd. Doping states are revealed as expected at 15 meV and 270 meV with no change between bulk and twinned material.. .................................................... 59 CdI location along an A3B3 interface. The interstitial is stable exactly on the interfacial plane and equidistant from the nearest 3 Cd and Te atoms. ................................................................................................. 60 Defect formation energy diagram for CdI. Interstitials on the GB plane show reduced formation energy of its neutral state, and increased formation energy of its charged states. Bulk calculations and the deeper positions along twins are indistinguishable. ............... 61 Defect energy transition levels for CdI. Bulk calculations and those of the defect deeper in the quasi-bulk region are nearly the same. As the defect moves closer to the GB plane, transition states move more towards the valence band while the charge neutral state is lowered in formation energy. ................................................................. 61 Atomic relaxation of a TeCd defect in the bulk (left) which is isotropic and when on a A3B3 twin (right) which cannot relax isotropically. ........................................................................................... 62 Defect formation plot for TeCd. Bulk calculations show much higher formation energies than when the defect is along isolated twins, and when along periodic twins the energy is even smaller due to the anisotropic relaxation of the TeCd defect interacting with the GB plane........................................................................................................ 63 Defect energy transition levels for TeCd both in the bulk, along A3B3 and along A4B1 twins. Gap states tend to shift downward towards the valence band, reducing the performance even further than when found in p-type bulk material. ..................................................... 63 Three interstitial positions for TeI in the bulk. Tetrahedral (left), Octahedral (center), and Trigonal Planar (right). ................................ 64 Preferred location of the TeI along an A3B3 twin interface from two perspectives. The Te atom is more strongly attracted to the Cdheavier side, displacing some nearby Te atoms. This places it in a tetrahedral configuration when surrounded by Cd atoms, but on the trigonal planar site when surrounded by Te atoms. ............................ 64 TeI formation energy diagram indicating both the tetrahedral and trigonal planar sites found in the bulk. Along twins the trigonal
x |Table of Contents
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51. 52. 53.
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planar site is always preferred. The neutral charge state for Te I near twin structures is greatly reduced for both types of twins and at all positions. Immediately adjacent to the isolated stacking fault is the most preferable TeI location. ........................................................65 TeI defect energy levels. The bulk trigonal site possesses fewer gap states, however when along twins many gap states are removed entirely, leaving two amphoteric compensating doping states. ............65 VTe formation energy diagram. Formation energies are always mildly lower when along twins than in the bulk for this native point defect. Additionally, when the material is more p-type, VTe exhibits a preferential positive charge state rather than the neutral state found in the bulk, as well as much lower formation energies. ..............66 Defect charge states for VTe. The newly preferred +2 charge state creates new gap states, and further from the interface a +3 state appears even closer to the valence band. ...............................................67 CdTe formation energy diagram. At the Fermi levels of interest, there is no change compared to the bulk. For a more n-type material, CdTe will form more readily upon twins who possess a more favorable -1 and -2 charge state. ...................................................67 Defect energy levels for CdTe. Inthe more favorable -1 and -2 charge states along twins, those transitions move downward into a double transition at +1/-1. Position 2 along A3B3 is somewhat less favorable than immediately on the GB plane, or deeper within the bulk. .........................................................................................................68 Formation energy diagram for Schottky defect pairs along twins. Energies always decrease as the spacing decreases with the A 4B1 isolated stacking fault having a near bulk-like configuration, while on A3B3 the ionic positions are anisotropic. ...........................................70 Formation energy diagram for Frenkel pairs along twins. ...................71 Formation energy diagram of the remaining defect pairs: (VCd + CdTe), (CdI + VTe), (VCd + TeCd) and (VCd+TeI). .......................................72 Coincident points (black) and the bounding points for an asymmetric tilt GB from a square lattice. In this example, the interfacial plane corresponds with grain 1 exactly. The minimal parallelepiped is in green while the forced orthorhombic cell is highlighted in red and blue. ...................................................................76 An asymmetric tilt GB of a square lattice, now with the interfacial plane not along the axis of rotation. Coincident points along the interfacial plane are chosen first, and then those nearest to this plane are selected to form the repeating parallelepiped. ......................77 Bounding cuboids for two rotated regular parallelipipeds which respect the interfacial plane. Bounding with unperterbed regular parallelepipeds produces a yet smaller structure but is harder to visualize. ..................................................................................................79
Christopher Buurma | xi 56. Side and top views of Figure 55. Edge components which are used 57. 58. 59.
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to construct the interface repeating vector 𝑟........................................ 80 Screenshot of the grain-boundary genie interface. ............................... 81 Bi-crystal of a twin. ..................................................................... 82 XRD and STEM images layered with EDS of this interface. As the mirror rotation leaves it as a Te-Te interface, an additional layer of Te is discovered creating a unique orientation at the interfacial plane........................................................................................................ 83 Model atomic structure for this interface after DFT relaxation which is stoichiometric. The observed bi-crystal requires elemental replacements to form the Te-Te bond. ................................................... 83 Normalized density of Kohn-Sham states for the mirror plane GB. A near-continuum of states was observed from the midgap up, extending into the conduction band. .................................. 84 At left, XRD images of the bi-crystal for this interface. At right, an early atomic model. ................................................................................ 85 Early atomic relaxation of this interface. .............................................. 85 STEM and XRD images of this interface. An amorphous region is observed at the interfacial plane, which may be due to experimental challenges, or may naturally occur due to the gross mismatch of the two lattices. ............................................................................................. 86 Early atomic model of the (90º)(110)(110). The spacing between periodic interfaces was insufficient and atomic rearrangement can be observed throughout the second grain. ............................................. 87 XRD, STEM on the left, with FFT noise filtered on the right showing this interface. ........................................................................... 88 Atomic model before ionic relaxation. ................................................... 88 STEM and XRD images for the mutual tilt of the interface. .... 89 Model atomic structure after relaxation. A Cd dislocation core is observed with nearby regions otherwise bulk-like. For this structure, exploitation of the interfacial plane’s periodic image was made easier as the mutual rotation caused the displacement vector 𝑟 to be very small. ................................................................................... 89 Site-projected DOS for the atoms near the Cd-core (red) and the Tecore (blue). Bulk DOS is provided for reference in black. Te-core has a notable peak at 0.29eV. ....................................................................... 90 STEM image, XRD and a strain map of a 4° mutual tilt bi-crystal of . The dislocation core is very localized as the continuous crystalline behavior is quickly recovered. ............................................. 91 Model atomic structure before (left) and after (right) using pseudopotentials[79] to find a closer atomic structure before DFT relaxation. ............................................................................................... 91 Grain boundary parameter space. All possible GBs which are symmetrically inequivalent are shown in black. Those which are
xii |Table of Contents
74. 75.
76. 77. 78.
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viewable under STEM are in red, while those with low tilt angles suitable for bi-crystals are in blue. Green represents those both viewable and able to be fabricated. ........................................................92 Top left: (0)(311)(-311) a CSL 𝛴11 grain boundary structure for CdTe. ..................................................................................................93 𝛴11 GB after ionic relaxation. Each quasi-bulk region is pinched together nearer the interfacial plane straining the local bonds. This strain is likely the cause of the additional gap states observed in the K-S DOS. .......................................................................94 Kohn-Sham normalized density of states for the above structure........94 The Mackenzie distribution with the peak highlighted. .......................95 Example structures with misorientations near the Mackenzie peak. Top left: (41.6º)(111)(432) with a misorientation of 𝜙 = 4223º. Top right: (41.9º)(111)(17.16.12) with 𝜙 = 4223º. Bottom: (43.6º)(810)(37,7,6) with 𝜙 = 44º ............................................................96 Atomic structure for this high tilt/twist GB. STEM channeling conditions are fulfilled for this interface, allowing it to be observed if it was to be fabricated..........................................................................97 Normalized Kohn-Sham DOS for the above structure. .........................97 Atomic model for the pure twist interface ..................................98 Kohn-Sham Density of states for the above interface. No defect states are revealed. .................................................................................98 Left: (0º)(110)(881) interface with a 5 degree tilt. Right: (0º)(111)(876) with a 6.6 degree tilt. .......................................................99 Model atomic structure for the (0º)(110)(100) asymmetric tilt GB. ......99 Normalized Kohn-Sham DOS for the above structure. .......................100 Structure handler interface. .................................................................101 Variability diagram generated for a series of calculations. ................102 Screenshot of Defect Buddy showing a series of defect energy calculations for bulk CdTe. ...................................................................103 Database topology for the VASP SQL database. Primary and foreign keys are linked and highlighted ..............................................104
Christopher Buurma | xiii LIST OF ABBREVIATIONS CSL
Co-incident site lattice
DFT
Density functional theory
DOS
Density of states
EDS
Energy dispersive spectroscopy
GB
Grain boundary
GR
Generation-recombination
K-S
Kohn-Sham
SRH
Shockley-Read-Hall
STEM
Scanning tunneling electron microscope
TBH
Tight-binding Hamiltonian
VASP
Vienna ab-initio simulation package
1) Introduction Throughout human history, the capture and use of energy sources has defined and empowered times of progress both technologically and socially. From the early exploitation of rivers and waterways to transport people and goods to the development of fossil fuels and nuclear power for electric grids which began a new era in human society, energy technology heralds the way. The next age in our history will also be defined by our ability to invent and implement new energy technologies from fusion power to wind and solar, all of which have a place in the modern landscape. Solar power in particular promises to be infinitely renewable with the rising sun each day, removes any dependence on input fuel costs, provides peak power during the periods of peak demand, and does not produce contaminants or byproducts during operation. The sun itself produces such vast quantities of power for the earth that if only a miniscule fraction is captured each day it could power the entire modern world. Any imagined future for energy sources must include solar power in its tapestry. From this promising technology some of the most encouraging advancements are in thin-film CdTe-based solar cells. These devices consume very little material and energy to fabricate, can be produced rapidly and cheaply, and have excellent solar energy conversion efficiencies. Present-day lab-scale efficiencies are greater than 20%, and module efficiencies are above 17% [1] all while maintaining a very low and still dropping cost of $0.45/watt installed and financed [2], [3].
1.1) CdTe Solar Cells These CdTe thin-film photovoltaic cells now are second in market share only to Si cells, due to CdTe’s higher efficiencies and lower production costs. Much of these latest improvements in conversion efficiency have been attributed to improvements in the short-circuit current density and fill-factors which now approach their theoretical limit, suggesting a near-optimal photon collection. The remaining obstacles to further improving the efficiencies of these devices towards their theoretical limits concern improvements in the fill factor (FF) and, more importantly, open-circuit voltage (Voc) which is still far from its theoretical limit [4]. (See Figure 1 and Figure 2.)
2 |Introduction
Figure 1. The Shockley-Queisser detailed balance limit, with CdTe highlighted in red. Loss mechanisms for the reduction of the theoretical efficiency down from 100% are listed.1
18.7% Efficiency Champion Cell Percentage of Theoretical Maxima 96.81%
90.22%
68.19% 59.56%
Eff
Voc
Jsc
FF
Figure 2. Breakdown of departures from the theoretical maximum for the 18.7% champion solar cell.[5] 1
http://sjbyrnes.com/sq.html
Christopher Buurma | 3 Minority carrier recombination outside of the quasi-neutral region can severely limit the open circuit voltage, and play a powerful role in determining the fill factor. The carrier lifetime is primarily limited by defect mediated, non-radiative, Shockley-Reed-Hall processes[6], [7]. Thus improvements in the FF and Voc, and ultimately efficiency, can be accomplished through defect mitigation to prevent recombination of carriers as they transit the device. Typical CdTe solar cells are formed from rapid-deposition techniques such as closed-space sublimation (CSS), or high-rate vapor transport deposition (HRVDT). Both processes produce a poly-crystalline CdTe photo-absorbing film only a few microns in thickness. This absorber is then matched into a hetrojunction, often with CdS, and then is contacted through a transparent conductive oxide and a metallic back contact[8]. The rapid and low-cost deposition method produces a CdTe absorber riddled with defects of all types, including point defects, structural defects, dislocations, grain boundaries, precipitates and voids. Many techniques have been invented over the years to mitigate these defects, most notably a CdCl2 treatment, which has numerous beneficial effects [9]. Each component of the final CdTe device has been studied and optimized by academic and industrial scientists, resulting in very high optical light collection within the device. Two final effects hold back device performance: non-radiative recombination and a work-function contact mismatch. Non-radiative recombination within the CdTe grain boundary network limits the open-circuit voltage (Voc), and to a lesser extent the fill-factor (FF). The work-function mismatch between available materials and the CdTe absorber causes significant forward resistance and a reduction in the conversion efficiency by further reducing the FF. While both problems must be addressed, the scope of this work is to address the CdTe grain boundary network in an attempt to increase the minority carrier lifetimes within the device toward their theoretical limits. Many grain boundaries exist which can harm carrier transport by either directly providing defect states in the bandgap that provide nonrdiative recombination pathways either directly through their atomic misalignment, or indirectly by attracting and forming point defects. Bringing this technology forward to widespread use as a powerful energy supply will require a better and more fundamental understanding of these defects and of how to mediate and exploit them. Providing theoretical guidance on defect formation, passivation, and electrical transport through modern theoretical and computational techniques can help to bring CdTe solar-cell efficiencies closer to their limit and allow such cells to demonstrate their true potential.
4 |First-Principles or Ab-initio Formalism
2) First-Principles or Ab-initio Formalism The tool of choice for the theoretical study of CdTe in this work is that of first-principles or ab-initio calculations. In this formalism, no appeal to experimental measurements is required beyond model validation, no measured inputs are needed, and no fitting is performed. It is indeed from the first-principles of quantum mechanics that all of this work is derived. Beginning along this path, we start with the primary theory of crystalline solids.
2.1) Theoretical Treatment of Bulk Solids There is an amazing degree of regularity in the structure of solids. Crystalline solids are ubiquitous in nature, especially in minerals, metals, rocks, and other geological formations. Atoms in a crystalline solid tend to arrange in a regular 3D pattern which is simply repeated in all dimensions. There is a rich array of the various finite possible arrangements, all of which can involve different atomic nuclei interacting with one another and with their electrons in unique ways, giving a nearly limitless array of combinations of lattices and atomic species, notwithstanding departures from their pure states. The reason for the crystallization of most solids has not been formally proven, however the commonly accepted hypothesis is that the crystalline state represents the lowest possible energy structure for a solid. Given that nature tends to bring objects to their lowest-energy state, it is logical to infer that given enough time and sufficiently low temperature, nearly all nuclei would eventually form into crystalline solids. Essentially, each atom is locally optimized for the lowest energy which gives rise to long-range order over systems whose nuclei do not interact strongly. Due to this periodic arrangement of atoms the study of the properties of solids, and thus most objects, can be done through a careful understanding of single small repeating unit cells. This single cell represents the bulk of a solid, repeating nearly infinitely in all directions. The structure of an example unit cell for CdTe is shown in Figure 3.
Christopher Buurma | 5
Figure 3. The conventional unit cell of CdTe. This binary species possesses a tetrahedral face-centered cubic symmetry and belongs to the space group 216 denoted as 𝐹4̅3𝑚. While it is true that near a surface or interface this ideal arrangement is broken, these regions contain only a very small minority of the total number of atoms in a solid. For example: if a solid box of 1 cm3 consists of 1023 atoms, then the perimeter atoms number only approximately 1016, (because there are (1023 ⁄𝑎3 )2/3 atoms/cm2, and only 6a2 cm2 total surface area), less than 0.000013% of the total number of atoms in the box. Given the immense importance of the bulk atoms on the properties of both ideal unit cells and defectuous unit cells, one can reveal a vast majority of material properties simply from first-principles considerations. It is remarkable that such a wide number of observed properties for many different materials such as conductivity, magnetism, even color and mechanical strength can all be derived from the same physical equation. This equation governs the interaction of the nuclei with themselves and with their electrons. This fundamental equation for solids can be readily derived from the Schrodinger equation.
̂ Ψ = 𝐸Ψ 𝐻 The Hamiltonian contains 5 contributions:
(1)
6 |First-Principles or Ab-initio Formalism
The kinetic energy of the electrons The kinetic energy of the nuclear ions The ion-ion interactions The electron-ion interactions The electron-electron interactions
Thus the Hamiltonian is prepared as follows: Assume that there are 𝐼 ions and 𝑖 electrons in a system, each located at positions 𝑹𝑰 and 𝒓𝒊 , each with mass 𝑀𝐼 and 𝑚𝑖 , with an effective ionic charge of 𝑍. The Hamiltonian for both ions and electrons can be constructed similarly; thus we construct all-electron Hamiltonian (see 2.3 for other methods). This yields a Hamiltonian of the form
̂=∑ 𝐻 𝐼,𝑖
−ℏ2 𝜵2𝑹𝑰 𝜵2𝒓𝒊 + ( ) + 𝑉𝑖𝑜𝑛↔𝑖𝑜𝑛 + 𝑉𝑖𝑜𝑛↔𝑒 + 𝑉𝑒↔𝑒 2 𝑀𝐼 𝑚𝑖
(2)
with
𝑍𝐼 𝑍𝐽 1 𝑉𝑖𝑜𝑛↔𝑖𝑜𝑛 = ∑ 2 |𝑹𝐼 − 𝑹𝐽 |
(3)
𝐼≠𝐽
as well as the ‘external’ ionic potential felt by the electrons given by
𝑉𝑖𝑜𝑛↔𝑒 = − ∑ 𝑖,𝐼
𝑍𝐼 |𝒓𝑖 − 𝑹𝐼 |
(4)
and finally, the most complex component being
𝑉𝑒↔𝑒
1 𝑒2 = ∑ 2 |𝒓𝑖 − 𝒓𝑗 |
(5)
𝑖≠𝑗
It is at this stage, in principle, that all behaviors can be calculated. It is also at this stage that the monumental nature of the task to be performed in attempting to do so is clearly seen. One must solve an eigenvalue problem, whose size is on the order of not only the number of atoms (e.g. 1023), but also the number of electrons (e.g. 20*1023). The solution of such a large set of coupled equations is far from practical. Indeed, it would be foolish to begin there, as there are many unexploited symmetries that can lead to a variety of simplifications.
2.2) Born-Oppenheimer approximation The first and most common approximation is known as the BornOppenheimer approximation. First, note that the mass of the atomic nucleus is far greater than the mass of an electron. Indeed, even in the most extreme case of hydrogen, the mass of the single proton that forms the nucleus
Christopher Buurma | 7 exceeds the electron mass by 3 orders of magnitude! Thus, the nuclei in a solid can only respond slowly to changes in the electron configuration, while the electrons can respond rapidly to changes in the ionic positions. This means that the relatively slow motion of nuclei leads to an adiabatic change in the electron wavefunctions relative to the nuclear positions. As a result, only the instantaneous configurations of the nuclei matter for the quantum mechanics of the electrons. The nuclear positions essentially become fixed parameters. Under this approximation, the nuclear components and the electronic components are separable. In the nuclear portion, because the nuclear are quite localized and heavy in comparison with the electrons, they may be treated classically. The nuclear locations can be treated as fixed parameters in solving for the electron wave functions at each step, and will change adiabatically during any nuclear motion. This greatly simplifies two of the three potentials in our current Hamiltonian: 𝑁 𝑁𝑖𝑜𝑛
𝑁
𝑍𝐼 𝑉𝑖𝑜𝑛↔𝑒 = − ∑ ∑ = ∑ 𝑉𝑖𝑜𝑛 (𝒓𝑖 ) |𝒓𝑖 − 𝑹𝐼 | 𝑖 𝐼 𝑖 𝑍𝐼 𝑍𝐽 1 𝑉𝑖𝑜𝑛↔𝑖𝑜𝑛 = ∑ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 2 |𝑹𝐼 − 𝑹𝐽 |
(6)
(7)
𝐼≠𝐽
It seems as though the reduction of the problem to a solvable form is almost complete, however, the electron-electron effect must still be accounted for. This is the most complex and beautiful part of the problem containing the richest physics and encompasses a number of observed effects: the many-body correlation effects of electrons dragging other electrons around during their motion, non-locality effects considering localized or de-localized wave functions, the exchange potential regarding the anti-symmetric nature of electrons, and other phenomena. Indeed it is this interaction that constitutes much of the ‘glue’ in solids. Without electrons and their intricate dance, many familiar materials would not enjoy stable bonds to form.
2.3) Density Functional Theory There are many ways to approach the problem of the electron-electron interaction. The simplest is to treat it with the mean-field Hartree approximation [10] in which each electron sees only the self-consistent average Coulomb potential from all of the other electrons, called the Hartree potential VH(ri), thus accounting only for average effects. A somewhat more sophisticated approach is arrived at by noting that electrons are fermions, and thus have anti-symmetric wavefunctions. This brings us to the HartreeFock[10] approximation by including electronic exchange, but still reduces the dynamic many-body quantum mechanical electron-electron Hamiltonian
8 |First-Principles or Ab-initio Formalism to a single-electron Hamiltonian, the Hartree-Fock Hamiltonian HHF, in which each electron sees only the sum of a self-consistent average Coulomb potential from all of the other electrons and a self-consistent average exchange potential from all of the other electrons. Thus, this approximation also treats only average effects. While this level of approximation is quite powerful at predicting many behaviors, it is insufficient for this work as it is often inapplicable to insulators and semiconductors. In addition, Cd and Te ions have strongly interacting and delocalized d-electrons, making careful treatment of these electron interactions important for CdTe. In principle, taking the electronelectron interaction into account directly in an all-electron model could be accomplished[11]. However, due to the computational complexity of the solution this would restrict such a study to smaller unit cells, whereas this effort will focus on interfaces and point defects, which will require computations on large supercells. The computational complexity required for large systems was dealt with in the work presented here by using density functional theory (DFT). DFT is well suited to this task as it maintains a strong computational efficiency even for the study of large supercells, while still capturing many features of the full electron-electron interaction. Popular functionals used in this formalism often capture the bulk state accurately, as well as painting a good description of the valence band and the occupied states. The strength of this technique is to provide very accurate total energy results rapidly, allowing many related phenomena to be extracted. The primary limitations of this technique include a poorer description of elastic constants and notably a failure to generate accurate electron bandenergy diagrams. The pillar of this method maps the true electron system into a fictitious system with the same total-energy functional of the electron density, and thus allows the computation of very accurate total-system energy levels, at the expense of the accurate determination of single particle excitations. In this formalism, one begins with a simple observation. If the total ‘external’ potential is known, the many-body wave functions Ψ can be derived. With Ψ in hand, this allows the easy determination of the charge density, 𝑛(𝒓). Perhaps with a shift in paradigm, the exact wave functions could be discarded and more directly arrive at the final energy. Is this charge density is unique? The answer given by the Hohenberg-Kohn theorem[12] is yes, 𝑛(𝒓) is unique. Proof: Let us begin with the total external potential, called 𝑉. Suppose that there are two potentials 𝑉 and 𝑉 ′ , both of which lead to the same 𝑛(𝑟). First require that 𝑉 and 𝑉′ are not trivially different, so that
Christopher Buurma | 9
𝑉 − 𝑉 ′ ≠ 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
(8)
̂ and 𝐻 ̂ ′ be the two corresponding Hamiltonians, with ground state Now, let 𝐻 solutions Ψ and Ψ′ giving energies 𝐸 and 𝐸′. That is
̂ Ψ = 𝐸Ψ and 𝑉′ → 𝐻 ̂ ′Ψ′ = 𝐸′Ψ′ 𝑉→𝐻
(9)
By the variational principle, one may use any other wavefunction which yields an upper bound on the energy instead. Thus
̂ |Ψ′⟩ 𝐸 ≤ ⟨Ψ′|𝐻
(10)
with the equality only holding if Ψ = Ψ′. Now express this as
̂ + 𝑉 ′ − 𝑉′|Ψ′⟩ ⟨Ψ′|𝐻
(11)
̂ + 𝑉′ = 𝐻 ̂ ′ + 𝑉, 𝐻
(12)
yet
Since the kinetic energy portions are the same. Now the upper bound on the energy is given by
̂ ′|Ψ′⟩ + ⟨Ψ′|𝑉 − 𝑉′|Ψ′⟩, 𝐸 ≤ ⟨Ψ′|𝐻
(13)
𝐸 ≤ 𝐸′ + ∫ 𝑛(𝒓)[𝑉(𝒓) − 𝑉′(𝒓)] 𝑑 3 𝑟
(14)
Now, begin at the opposite application of the variational principle. That is,
̂ |Ψ⟩ 𝐸′ ≤ ⟨Ψ|𝐻′
(15)
which ultimately leads to the similar expression,
𝐸′ ≤ 𝐸 + ∫ 𝑛(𝒓)[𝑉 ′ (𝒓) − 𝑉(𝒓)] 𝑑 3 𝑟
(16)
𝐸 ≤ 𝐸 ′ ≤ 𝐸,
(17)
This implies that
and the equality can hold only if our wave functions and potentials are the same. Thus, any given potential can lead only to one unique charge density for that potential. This is the first part of the Hohenberg-Kohn theorem. This theorem now piques our interest. If the charge density and the external potential are oneto-one unique, then can the energy be found using only the charge density and not the wave functions? If so, this would provide a formally equivalent approach to solving the many-body Hamiltonian encountered previously. Again, the answer is yes, such a mapping exists [13].
10 |First-Principles or Ab-initio Formalism This gives us the two primary tenets of the theory. (I) (II)
There cannot be two external potentials different by more than a constant that give rise to the same charge density ⃗⃗⃗⃗ 𝑛0 (𝑟) 𝑛0 (𝑟) → 𝑉𝑒𝑥𝑡 (𝑟) uniquely ⃗⃗⃗⃗ A universal functional for the total energy, 𝐸[𝑛0 (𝑟)] can be defined valid for any external potential. For any particular external potential, the exact ground-state energy for the system’s global minimum of this functional and the density which minimizes the functional is the exact ground state density.
So, what is the form of this functional? It must have the form
̂ |Ψ⟩ = 𝑇[𝑛] + 𝐸𝑥𝑐 [𝑛] + ∫ 𝑑3 𝑟𝑉𝑒𝑥𝑡 (𝑟)𝑛(𝑟) (18) 𝐸[𝑛0 (𝑟)] = ⟨Ψ|𝐻 This leads us closer to being able to solve the electron-electron Hamiltonian. However this formulation has now only recast the problem into that of finding the kinetic energy functional 𝑇[𝑛] and the exchange-correlation functional 𝐸𝑥𝑐 [𝑛] which can describe the electron-electron interaction accurately. For a fully interacting system, this is again an intractable problem. The Kohn-Sham ansatz[14] provides a brilliant solution to this problem. Let’s now move from our realistic system of interacting electrons to that of a noninteracting system with the same charge density 𝑛(𝑟). This new auxiliary imaginary system can now be readily solved using treatment of noninteracting electrons. However, by the Hohenberg-Kohn theorem, the total energy functional 𝐸[𝑛] must be the same for both systems. Thus, our auxiliary system will have the same ground state energy as the real system. A schematic conceptual example of this procedure is given in Figure 4.
Figure 4. A conceptual overview of Density Functional Theory. The real world problem is mapped to a non-interacting system, solved, and then remapped to the real system to find the total energy while bypassing the need to solve for the electron wavefunctions in the real system. What has been lost in this construction is the electron wavefunctions. Our auxiliary system now contains fictitious fermions whose only requirement is
Christopher Buurma | 11 that they have the same charge density as the interacting electrons in the real system. Working in the auxiliary system, now write the following: 𝐸[𝑛(𝑟)] = 𝑇𝑛𝑜𝑛 [𝑛(𝑟)] + 𝐸𝐻𝑎𝑟𝑡𝑟𝑒𝑒 [𝑛(𝑟)] + ∫ 𝑑 3 𝑟 𝑛(𝑟) 𝑉𝑒𝑥𝑡 (𝑟) + 𝐸𝑥𝑐 [𝑛(𝑟)],
𝑇𝑛𝑜𝑛 [𝑛(𝑟)] = ∑𝑁 𝑖=1 ⟨ϕi | 𝐸𝐻𝑎𝑟𝑡𝑟𝑒𝑒 [𝑛(𝑟)] =
−ℏ2 2𝑚𝑒
𝜵2𝑖 |ϕi ⟩
and
(19) (20)
𝑞2 𝑛(𝑟)𝑛(𝑟′) ∫ ∫ 𝑑3𝑟 𝑑3𝑟 ′ 2 |𝑟 − 𝑟′|
(21)
where 𝑇𝑛𝑜𝑛 is now the kinetic energy of the non-interacting system, the second term is simply the non-interacting Hartree energy, the third term is the electron’s response to the stationary ions and its ‘external’ potential, and the last term is the sum of all remaining unknowns: the exchange and correlation energy. As such, all many-body effects are now contained within this last term and the rest is formally exact and solvable. Next, one must minimize this functional using Lagrange multipliers, 𝜖𝑖 . Consider a variation of the Kohn-Sham wavefunctions 𝜙 → 𝜙 + 𝛿𝜙 in which 𝛿𝑛(𝑟) = 𝛿𝜙 ∗ 𝜙. By the conservation of particle number,
∫ 𝑑 3 𝑟𝛿𝑛(𝑟) = ∫ 𝛿𝜙 ∗ 𝜙 = 0.
(22)
Thus by the variation of the functional through a functional derivative, 𝛿𝐸[𝑛] 𝛿𝜙∗ (𝑟 )
=
𝛿𝑇𝑛𝑜𝑛 𝛿𝜙∗ (𝑟 )
+[
𝛿𝐸𝑒𝑥𝑡 [𝑛] 𝛿𝑛(𝑟 )
+
𝛿𝐸𝐻𝑎𝑟𝑡𝑟𝑒𝑒 [𝑛] 𝛿𝑛(𝑟 )
+
𝛿𝐸[𝑛]
𝛿𝑛(𝑟 )
𝛿𝑛(𝑟
𝛿𝜙 ∗ (𝑟 )
] )
.
(23)
Giving the coupled Kohn-Sham equations, first a Schrödinger-like equation:
[−
ℏ2 2𝑚𝑒
∇2 + 𝑉𝑒𝑓𝑓 ] 𝜙𝑖 (𝑟) = 𝜖𝑖 𝜙𝑖 (𝑟),
(24)
then, a new definition for the effective potential energy, 𝑛(𝑟 )
𝑉𝑒𝑓𝑓 = 𝑉𝑒𝑥𝑡 + 𝑞 2 ∫ 𝑑 3 𝑟 |𝑟
−𝑟 ′ |
+
𝛿𝐸𝑥𝑐 [𝑛(𝑟 )] 𝛿𝑛(𝑟 )
,
(25)
and finally with the definition that the Kohn-Sham wavefunctions 𝜙𝑖 must be complete.
𝑛(𝑟) =
∑
|𝜙𝑖 (𝑟)|2
(26)
𝑖,𝑜𝑐𝑐𝑢𝑝𝑖𝑒𝑑
Equations 24, 25 and 26 are known as the Kohn-Sham equations. All that is left now is to solve our coupled equations in a given basis, and to find an approximate exchange-correlation functional which embodies the behaviors we intend to model.
12 |First-Principles or Ab-initio Formalism
2.4) Exchange, Correlation and Basis functions The problem has now been recast, and in its formally exact derivation all quantities for our system are known analytically with one exception: the 𝐸𝑥𝑐 [𝑛(𝑟)]. There are many ways to complete this remaining functional including the original Thomas-Fermi approximation, or a local density approximation. In this work a powerful and proven functional developed by Perdew, Burke, and Ernzerhof [15] will be used. This functional is a member of the class of functionals known as the generalized gradient approximations (GGA). These are an extension of the local density approximation in which it is assumed that the local charge density (and the nearby gradients of that density) at each point in space are largely responsible for the behavior in that region. This functional is computationally very efficient, but treats correlation poorly. Since CdTe is not dominated by correlated electrons however, GGA functionals describe their behavior admirably and has been used by other authors previously with great success[16], [17]. The chief success of this functional is its ability to fulfill many exact criterion of an exchangecorrelation functional while being fully expressible in terms of fundamental constants. With a powerful exchange-correlation functional now ready for use, one is left to find a basis for our Kohn-Sham equations. Plane waves provide a complete basis which is impartial to the atomic system studied while capturing the desired behavior quickly. Since in practice one cannot use an infinite set of functions, the series must at one point be truncated and the solution computed with a partial basis. While using a large enough set of plane waves will eventually yield the correct solution, a more clever construction of our basis functions can be used to greatly improve computational efficiency. In this work, we will be using the projector-augmented wave (PAW) method[18]. The principal challenge in creating an efficient basis is dealing with the orthogonality of ionic core electron states to the states of nearby valence electrons. This orthogonality requirement can often cause the wavefunctions to oscillate violently near an ionic core, which can cause many unneeded computations. The PAW method is based on other augmented plane wave methods which rather than terminating the core, intermediate and valence zones abruptly at a given radius, instead allow them to overlap, using a localized projector function.
Christopher Buurma | 13
3) Role of Defects in CdTe Semiconductors Defects play a crucial role in the operation of CdTe solar cells. Indeed, for all semiconductors, crystallographic defects give rise to states which are normally absent between the occupied valence-band states and energetically separated conduction-band states (the band gap). These states result in two major effects regarding cell operation. First, the presence of these defectinduced states near the band edges give rise to mobile carriers that are easily excited and manipulated, allowing for the creation of a p-n junction and a built-in potential which can be used to collect photogenerated carriers, leading to power generation. Second, defect states away from the band edges and nearer to the local Fermi level provide recombination sites into which an otherwise mobile and excited carrier in the band may drop and then experience Shockley-Read-Hall (SRH) [6], [7] recombination. SRH effects will reduce the total density of electrons transiting the device, reducing the quasiFermi levels in the device and ultimately leading to a reduction in the generated voltage and power. Also, the defect site itself may provide a scattering center for mobile carriers which will reduce the mobility of the carriers in the region, which in turn will reduce the total generated power.[10]
3.1) First-principles DFT Calculation of Defect Energy Levels In DFT formalism, these states can be predicted by discovering the chargestate energy transition level. This is the electron energy level in which two differently charged structures have the same total energy. In this work, defect formation energy and charge state calculations follow the formalism of Wang and Zunger [19], Kohan, et al [20] and Schultz[21]. This method will be briefly summarized here. The formation energy of a defect 𝐷 in a charge state 𝑞 is given by [20] Δ𝐻𝐷,𝑞 (𝐸𝑓 , 𝛼, 𝑞) = [𝐸𝐷𝑒𝑓𝑒𝑐𝑡 (𝛼, 𝑞) − 𝐸𝑃𝑢𝑟𝑒 ] + [∑ Δ𝑁𝛼 𝜇𝛼 + 𝑞𝐸𝐹 ]
(27)
𝛼
Here, 𝐸𝐷𝑒𝑓𝑒𝑐𝑡 is the total energy of the defectuous structure, 𝐸𝑃𝑢𝑟𝑒 is the energy of the defect-free host structure, 𝛼 represents the atomic species of the defect(s), 𝑁𝛼 is the number of atoms removed from the perfect structure, 𝑞 is the charge state of the defect and 𝐸𝐹 is the Fermi level. For intrinsic CdTe defects this can be rewritten in the simple form Δ𝐻(𝛼, 𝑞) = 𝐸𝐷𝑒𝑓𝑒𝑐𝑡 (𝛼, 𝑞) − 𝐸𝑃𝑢𝑟𝑒 + Δ𝑁𝐶𝑑 𝜇𝐶𝑑 + Δ𝑁𝑇𝑒 𝜇 𝑇𝑒 + 𝑞𝐸𝑓
(28)
The total free energy (𝐹 = 𝑈 − 𝑇𝑆) of both defectuous and defect-free structures is readily available using the aforementioned ab-initio methods, and the number of missing or removed atoms also is readily available. The
14 |Role of Defects in CdTe Semiconductors remaining complexity arises in computing the chemical potentials of the Cd and Te atoms. We consider two rather extreme cases for native point defects, the Te-rich limit in which semi-metallic tellurium is the Te reservoir and the Cd-rich limit in which metallic Cd is the Cd reservoir. In these two limits total energies can be obtained for the metallic Cd and semi-metallic Te pure chemical potentials, 𝜇𝛼0 . In the Cd-rich limit, the most likely chemical destination for a Cd atom is Cd metal, while a recently liberated Te atom would pair with the excess Cd and form a new lattice site. In this limit: 0 𝜇𝐶𝑑 = 𝜇𝐶𝑑 and
0 0 𝜇 𝑇𝑒 = 𝜇𝐶𝑑𝑇𝑒 − 𝜇𝐶𝑑
(29)
However, in the Te-rich limit the reverse would be encountered: 0 𝜇𝐶𝑑 = 𝜇𝐶𝑑𝑇𝑒 − 𝜇0𝑇𝑒
and
𝜇 𝑇𝑒 = 𝜇0𝑇𝑒
(30)
Calculations of chemical potentials in the range between these two limits can be interpolated linearly for each species. With the results so far, formation energies can be calculated for a range of possible defects, however the above has not addressed the case in which they possess a net charge 𝑞. Many defects can and often will have a net charge associated with them, requiring that charged-defect calculations be performed. Simply adding a net charge to the computational supercells would result in a formally divergent solution. Thus, a background jellium charge is added to prevent divergence. After relaxation, the total energy for the charged supercell is obtained, however now one must correct for the interaction between the jellium and the net charge. Spurious contributions of the screened image charge to the defect supercell are removed using the firstand third-order corrections of Makov and Payne [22] and the adjustment for semiconductors proposed by Lany and Zunger [23]. A potential energy realignment can also be performed to ensure that these charged supercells are representative of an isolated net charge. In this approach, the energy of the electron added to or taken from the structure must be taken into account. This electron energy, 𝐸𝑓 , known as the electron chemical potential or Fermi energy, is simply an energy variable bounded by the conduction- and valance-band edges. Assuming the structure to be extrinsically doped, this variable can be taken as any value from approximately the valence-band maximum to approximately the conductionband minimum representing the propensity to find the extra electrons or holes needed to form such a defect. This seemingly free parameter however can be determined in the absence of extrinsic defects. We invoke a charge neutrality condition over any macroscopic region [24], which ensures that as negatively charged defects begin forming they are automatically compensated by positively charged
Christopher Buurma | 15 defects which can now form more easily. These two effects compete until they equilibrate which ultimately leads to the as-grown or ‘nominally undoped’ Fermi level in CdTe. To find this thermodynamic Fermi level, first compute the concentration of each defect in each charge state. That is,
𝐶𝐷,𝑞 (𝐸𝑓 , 𝜇𝛼 , 𝑇) = 𝑁𝑠𝑖𝑡𝑒 exp (−
Δ𝐻𝐷,𝑞 (𝐸𝑓 , 𝜇𝑎 ) + Δ𝑆 ) 𝑘𝑇
(31)
where 𝑁𝑠𝑖𝑡𝑒 is the concentration of possible defect sites, given by the multiplicity of its Wychoff position, and where entropic contributions to the defect density, Δ𝑆 are neglected and set to 0. The total charge of the system, 𝑄, is the sum of the defect charge concentrations and the free carrier concentrations. That is
𝑄(𝐸𝑓 , 𝜇𝑎 , 𝑇) = ∑𝐷 ∑𝑞 𝑞𝐶𝐷,𝑞 − 𝑛 + 𝑝, where ∞
𝑛 = ∫𝐸𝑐 𝑓(𝐸, 𝐸𝑓 , 𝑇)𝑔(𝐸)𝑑𝐸 and 𝐸
𝑣 𝑝 = ∫−∞ (1 − 𝑓(𝐸, 𝐸𝑓 , 𝑇)) 𝑔(𝐸)𝑑𝐸 ,
(32) (33) (34)
where 𝑛 is the electron concentration in the conduction band, and 𝑝 is the hole concentration in the valence band. Then during CdTe growth at its synthesis temperature, 𝑇𝑆𝑦𝑛 , demand that the crystal be charge neutral:
𝑄(𝐸𝑓 , 𝜇𝑎 , 𝑇𝑆𝑦𝑛 ) = 0
(35)
Since in the previous equation 𝐶𝐷,𝑞 has only one ‘free’ parameter, 𝐸𝑓 , this allows this value to be fixed to the one observed during growth. After growth, it is assumed that these physical defect concentrations do not change (easily); however, their charge states may change. This gives the concentration of any defect 𝐷 by summing over all of its charge states.
𝐶𝐷 = ∑ 𝐶𝐷,𝑞
(36)
𝑞
Now, consider the population of defect states at equilibrium, or cell operation temperature, 𝑇𝑒𝑞 . Then, ‘freeze-in’ the defect concentrations, 𝐶𝐷 , yet allow the charge states within each defect 𝐷 change in relative concentration until macroscopic charge neutrality is again achieved. This provides the final Fermi energy, 𝐸𝑓 , that is most probable after synthesis. With this in hand, defect formation energies are computed. This is a substantial accomplishment, but is not sufficient to characterize the defect for solar-cell applications. Of primary interest is the energy level of the defect which can cause generation/recombination to occur outside the
16 |Role of Defects in CdTe Semiconductors depletion region. Such events can greatly reduce the carrier lifetime during operation and thus reduce cell conversion efficiency. A charge transition, or ionization, occurs between two charge states 𝑞 and 𝑞’ when their formation energies are the same at a given 𝐸𝑓 . This provides the defect energy level at which an electron or hole can be captured. This is computed by solving for 𝜇𝑒 in the equations:
Δ𝐻𝐷,𝑞 (𝐸𝑓 ) − Δ𝐻𝐷𝑞′ (𝐸𝑓 ) = 0 𝐸𝑓 = 𝜖𝐷𝑞∗ =
𝐸𝐷,𝑞 − 𝐸𝐷,𝑞′ − 𝐸𝑉𝐵𝑀 𝑞′ − 𝑞
(37) (38)
Equations 37 and 38 can be solved iteratively using a midpoint algorithm. Then, all defect formation energies can be computed, and the relative concentration and energy levels of each defect becomes available.
3.2) Generation-recombination Current With all of the defect energy levels in hand, one may compute their effect on solar-cell performance. The primary effects are that of a reduction in V oc and FF. Voc is reduced through a reduction in electron and hole charge densities, which leads to a reduction in the quasi Fermi level and hence the voltage; FF is reduced by the added local impedance near the maximum power point. In general, we would begin by solving classical electron transport through Poisson’s equation computationally [25], [26]. This amounts to a driftdiffusion model in which the diffusion of carriers away from areas of high density is compensated by the drift current caused by the built-in potential of two oppositely doped regions. That is expressed by coupling Poisson’s equation to the continuity equations for electrons and holes separately:
∇2 𝑉 = −
𝜌(𝑉) 𝜖𝑠
(39)
and
⃗ 𝜙𝑛 ⃗⃗⃗𝑗𝑒 = 𝜇𝑒 𝑛∇
(40)
⃗ 𝜙𝑝 𝑗ℎ = 𝜇ℎ 𝑝∇ ⃗⃗⃗
(41)
for electrons and
for holes, where V is the electrostatic potential, 𝜙 is the quasi Fermi potential, 𝜌 is the charge density, 𝜖𝑠 is the dielectric constant, 𝐽 is the current density, 𝜇 is the mobility, 𝑛 is the number density of electrons and 𝑝 is the number density of holes. Such methods are readily implemented in a number of codes [27]–[29].
Christopher Buurma | 17 However, analytic solutions to Poission’s equation with these effects in mind can provide good insight. An early and common derivation will lead to the model of a parallel Generation-Recombination (GR) current alongside the normal diffusion current. While the diffusion current is the natural result of these equations, giving the Shockley model of a diode, corrections must be made to account for recombination in the depletion region. The derivation of GR current from the equation by Sah [30] is performed in many places using different methods. Begin with the assumption that there is one trapping energy level, 𝐸𝑡 , caused by defects found evenly distributed spatially throughout the depletion width. The generation-recombination rate associated with this level is given by
𝑈=
𝑞(𝜙𝑝 − 𝜙𝑛 ) 𝑛𝑖 sinh 2𝑘𝑇 √𝜏𝑝 𝜏𝑛 −(𝐸𝑖 + 𝑞(𝜙𝑝 − 𝜙𝑛 )⁄2 𝜏𝑝 𝑞(𝜙𝑝 − 𝜙𝑛 ) 𝜏𝑝 𝐸 − 𝐸𝑖 cosh ( ) + ln √𝜏 + exp cosh ( 𝑡 + ln √𝜏 ) 𝑘𝑇 2𝑘𝑇 𝑘𝑇 𝑛 𝑛
(42)
where, 𝑛𝑖 is the intrinsic carrier concentration, 𝜏𝑛,𝑝 is the lifetime in the n or p region, 𝐸𝑖 is the intrinsic or Fermi energy, 𝐸𝑡 is the localized energy of the trap state (for a single energy state), 𝜙𝑛,𝑝 is the quasi Fermi level for electrons or holes, 𝑞 is the elementary charge, 𝑘 is the Boltzmann constant, and 𝑇 is the absolute temperature in K. Now the current contribution of these GR centers is given by the spatial integral of this rate over the depletion width, where the traps are located. Notice that for a short-base diode in which the device becomes fully depleted at an earlier voltage since the physical device is too small, one instead must integrate over the device length. If the depletion width and physical width are appreciable in size, one can use a similar technique invoking the coth(1⁄𝑊 ) term used in diffusion lifetime derivations[31]. 𝐽𝐺𝑅 = 𝑞 ∫
𝑊𝑑
𝑈𝑑𝑥
(43)
0
Begin by observing that the difference in quasi Fermi levels, 𝜙𝑝 − 𝜙𝑛 , is approximately the voltage dropped across the depletion region, 𝑉. Assume this approximation holds throughout the region and thus that this term can be pulled out of the integral. The only remaining spatially varying term is the cosh term in the denominator of U. We further assume that the recombination lifetimes are not spatially dependent and that there are no thermal gradients or variations in trap energy across the depletion region. Then we arrive at the result 𝐽𝐺𝑅 =
𝑊𝑑 𝜏𝑝 𝜏𝑝 𝑞𝑉 −(𝐸𝑖 + 𝑞𝑉 ⁄2 𝑞𝑉 𝐸𝑡 − 𝐸𝑖 sinh ( ) ∫ 𝑑𝑥/ {cosh ( ) + ln √ + exp cosh ( + ln √ )} 2𝑘𝑇 𝑘𝑇 𝜏 2𝑘𝑇 𝑘𝑇 𝜏 𝜏 𝜏 𝑛 𝑛 √ 𝑝 𝑛 0
𝑞𝑛𝑖
(44)
18 |Role of Defects in CdTe Semiconductors It is at this stage that most derivations diverge. Sah proceeds to show limiting cases of this equation by taking the quasi Fermi levels in the cosh term to either be trivial, dominant or competing. Sze[31] and many others discard this cosh term and look only at limiting cases of high forward or reverse bias (and indeed replace sinh(…) with exp(…) - 1)). The remaining integral is of the form 𝑥=𝑊𝑑
∫
𝑥=0
𝑑𝑥 cosh 𝐴(𝑥) + 𝐶
(45)
with
𝐴(𝑥) = −
𝐸𝑖 (𝑥) + 𝑞 (𝜙𝑝 (𝑥) + 𝜙𝑛 (𝑥))⁄2 𝑘𝑇
𝐶 = exp
𝜏𝑝 + ln √ 𝜏𝑛
𝜏𝑝 𝑞𝑉 𝐸𝑡 − 𝐸𝑖 cosh ( + ln √ ) 2𝑘𝑇 𝑘𝑇 𝜏𝑛
(46)
(47)
The value of C (called 'b' by Sah) can be found from environmental parameters and phenomenological parameters alone and is presumed to be spatially invariant over the depletion region. The value of 𝐸𝑡 can be derived from first-principles calculations, as in this work, or from experimental observations and literature values. At this stage, derivations again can diverge from using a Taylor series expansion of cosh, using semi-infinite integration limits, or by re-parameterizing the integral. The approximation of 𝜙𝑛,𝑝 (𝑥) is the remaining challenge. Taking the intrinsic Fermi level to be varying linearly across the depletion region, make the substitution:
𝐸𝑖 (𝑥) − 𝑞
𝜙𝑝 (𝑥) + 𝜙𝑛 (𝑥) 𝑉𝑏𝑖 − 𝑉 𝑊𝑑 =𝑞 (𝑥 − ) 2 𝑊 2
(48)
Now,
𝐴(𝑥) =
𝜏𝑝 𝑞 𝑉𝑏𝑖 − 𝑉 𝑥 + ln √ = 𝐴𝑥 + 𝐵 𝑘𝑇 𝑊 𝜏𝑛
(49)
gives an integral of the form 𝑊𝑑
∫
0
whose analytic solution is
𝑑𝑥
1 cosh(𝐴𝑥 + 𝐵) + 𝐶
(50)
Christopher Buurma | 19
2
𝐶−1 𝐴𝑥 + 𝐵 tanh−1 (√ tanh ( )) 𝐶+1 2 𝐴√𝐶 2 − 1
𝑥=𝑊𝑑
(51)
𝑥=0
Combining all of the above expressions leads to the familiar form
𝐽𝐺𝑅 =
𝑞𝑛𝑖 √𝜏 𝑝 𝜏 𝑛
sinh (
𝑞𝑉 2𝑘𝑇 𝑊𝑑 𝑓(𝐶, 𝑉) ) 2𝑘𝑇 𝑞 𝑉𝑏𝑖 − 𝑉
(52)
but where the 𝑓(𝑏) in the familiar expression has been replaced by 𝑓(𝐶, 𝑉) and where fewer assumptions have been made over different bias regions. Here, 𝑥=𝑊𝑑
𝑓(𝐶, 𝑉) =
1 √1 − 𝐶 2
tanh−1
𝜏𝑝 1 𝑞 𝑉𝑏𝑖 − 𝑉 √1 − 𝐶 tanh ( ( 𝑥 + ln √ )) 1+𝐶 2 𝑘𝑇 𝑊𝑑 𝜏𝑛
(53) 𝑥=0
The effect of defect energy levels can be readily quantified in this analytic form, or to be even more precise, a direct solution of Poisson’s equation can be performed with fewer assumptions. As expected, as the trap energy becomes closer to the Fermi level, recombination rates increase greatly. This increase in recombination then causes GR current to overtake diffusion current causing a reduction in FF. As the density of trap states increases, the overall Voc is reduced, because of the reduced quasi Fermi levels. Another way to use defect energy levels to predict the impact of defects on solar-cell performance is to compute the recombination lifetime quantummechanically using the electron wavefunctions. An early set of these calculations was accomplished in this work in collaboration with the Stanford Research Institute[32].
20 |Defect classification in CdTe
4) Defect classification in CdTe The defects within any crystal fall into a variety of different classifications, each of which can be treated differently. Given the profound importance of defects in the operation and design of CdTe solar cells, special attention must be paid to the effects of each class of defects within a cell in order for the occurrence and effects of the defects to be controlled effectively. A detailed knowledge of the environmental conditions that lead to defect formation, activation, passivation, and migration will provide powerful weapons to control and tailor any solar cell or semiconductor system. Defects are primarily classified by their spatial extent. The simplest defect, a zero-dimensional or point defect, represents a single atomic error within the lattice. Dislocations follow, with a 1D spatial extent throughout the crystal lattice (or grain). Grain boundaries, surfaces, and interfaces comprise the 2D defect space and are the focus of the study presented here. Lastly, 3D voids and precipitates are included for completeness.
4.1) Zero-dimensional Point Defects Zero-dimensional, atomic-scale or point defects (PD) are defects localized on a single atomic site on the lattice or an adjoining interstitial location. PDs are classified as: vacancies (atoms removed from the lattice), substitutionals (impurity atoms or antisite atoms, such as a Te atom on a Cd atomic site), and interstitials (extra atoms placed within the lattice but not on lattice sites). The cause of these defects may come from the crystal itself in the form of native point defects and may occur even during ideal crystal growth. Extrinsic defects are those caused by non-native atoms such as dopants, passivants or contaminants. PDs are arguably the most important defects for semiconductors in that they tend to occur in larger quantities, are easier to control, and can readily modify the band structure through the addition of defect states within the band gap. Many extended defects, such as 1D or 2D defects, come with high formation energy barriers or possess too many detrimental states within the band gap to be effective at doping safely. This makes extended defects more important for defect migration and the local attraction/repulsion of nearby PDs and for controlling the mechanical properties of the solid, while PDs primarily control its electronic properties.
4.1a)
Point-defect Diffusion
Point defects represent a deviation from the ideal lattice, and thus bring a change in total energy by their very presence. This perturbation on the lattice also effects the local diffusion of atoms in this region, including the defect itself, as new bonds can be broken and formed different than in an ideal
Christopher Buurma | 21 lattice. The diffusion of point defects is quite important in semiconductor development, Of special importance is the diffusion together of pairs of PDs that may then annihilate, such as in the case of a native vacancy and an interstitial or of a pair of antisite atoms, such as a Cd antisite and a Te antisite. Defects may also migrate from a safe position to a dangerous one, giving rise to a defect energy level, or vice versa returning to a safe one and essentially removing that defect’s electrical influence; e.g. a Te interstitial (Tei) replacing a Cd vacancy (VCd) on the lattice to generate an antisite. Other defects such as dopants may be introduced intentionally; their diffusion instead results in a change in the doping density profile., It may even lead to the collection of dopant atoms in other defectuous regions, which in turn may give rise to more defect states within the band gap or to compensating doping states that undermine doping efforts. The changes in partial concentration at a time 𝑡 are governed by a continuity equation describing the conservation of particles:
𝐽 = −∇r 𝑐(𝑟, 𝑡) .
(54)
Mass conservation then leads to the result, 𝑦𝑖𝑒𝑙𝑑𝑠 𝑑 𝑑 (55) 𝑐(𝑟, 𝑡) + 𝐷(𝑇)𝐽(𝑟, 𝑡) = 0 → 𝑐(𝑟, 𝑡) = 𝐷(𝑇)∇2 𝑐(𝑟, 𝑡) 𝑑𝑡 𝑑𝑡 This equation is known as Fick’s law, which is in general rather difficult to solve. The term D(T) is the diffusion constant or diffusivity. Diffusing processes are often taken as a random walk as material diffuses from regions of high concentration to those of low concentration. Such solutions can be found computationally.
4.1b)
Semiconductor Impurity States
The effects of these defect sites on the operation of CdTe devices enter in two major ways: doping and carrier-lifetime reduction which has been previously explained in detail. Doping is accomplished by successfully introducing extrinsic atoms as substitutional impurities that either donate or accept electrons. If these donors or acceptors have energy levels near the valance band minimum or conduction band maximum they can contribute to conduction without introducing midgap energy transitions within the material. The the donors or acceptors are then thermally activated at higher temperatures. The resultant conduction band electrons or valence band holes then shift the band edges relative to the Fermi level and contribute to conduction in the device. These ionized carriers under operation will drift through the semiconductor and maintain charge neutrality. Electrons excited to the conduction band will follow the electric field while holes will travel in an opposite direction along the valence band edge. By placing oppositely doped semiconductors into
22 |Defect classification in CdTe contact these mobile carriers will readily diffuse across the junction leaving behind the charged ions that contributed them to the lattice. These charged ions in turn generate a built-in potential which eventually competes for and cancels out the diffusion of carriers leading to a built-in electric field near the interface. This built-in potential is the basic building block of transistors, optical sensors and switches, as well as an acceleration mechanism for photogenerated carriers in solar cells. However if a defect introduces states near the center of the band gap, these states allow for optical transitions and provide scattering centers for carriers during their drift or diffusion during operation. By allowing an electron in the conduction band to transition easily to a lower energy level through emission of a photon this provides a path towards radiative recombination. The electron drops to the valence band and can easily find a similarly ionized hole traveling in the opposite direction and annihilate. Similarly, an electron or hole may be captured in an attractive scattering center and held in this state until the opposite carrier is captured or is within a small energy difference to the captured carrier and the two again annihilate. These generation-recombination effects are an important source of deviations from ideal device performance; they directly contribute to a loss in carrier lifetime in the device which directly reduces its operational efficiency.
4.2) 1D Dislocation Line Defects The first class of spatially extended defects is that of dislocations. As these are not the focus of this work, they will be treated here only briefly; more detailed treatments are available elsewhere [33] [34]. These extended defects appear in both single-crystal CdTe and within the grains of polycrystalline CdTe and make major contributions to their electronic and mechanical properties. As with PDs, dislocations often introduce a series of localized states within the bandgap, and they provide a ready avenue for the diffusion of all types of PDs. The dislocations contain 1D core regions that consist of a series of misbonded Cd and Te atoms that generate or terminate an extra atomic plane. Again similar to PDs, dislocations also can migrate through a solid, and two dislocations can combine to annihilate one another, just as interstitials and vacancies can annihilate one another. Even when crystal growth is rigorously controlled, dislocations often still result and must be dealt with using post-growth treatments. In polycrystalline materials dislocations are even more likely to be formed, and due to the presence of grain boundaries fewer methods are available to manage and mitigate their effects (See Section 5.4).
Christopher Buurma | 23
4.2a)
Edge and Screw Dislocations
Dislocations are characterized by two quantities, the unit vector in the direction of the dislocation and the Burger’s vector, which is constructed geometrically by comparing the dislocated crystal lattice to that of an ideal crystal with no dislocations. To find this vector, begin by tracing a closed loop (a Burger’s circuit) around a dislocation on both sides of the structure. Taking the same path in an ideal crystal will reveal an additional unit-cell vector, which is the Burger’s vector, as is illustrated in Figure 5 below. One-dimensional dislocations are classified as belonging to one of two different types: ‘edge’ and ‘screw’ dislocations or any linear combination of the two. Figure 5 below gives a schematic diagram of both types and shows their corresponding Burger’s vectors. Edge dislocations are easy to visualize as two lattices having an interface that terminates between two adjacent lines of lattice points on a third lattice, not at those lattice points, with the stress being alleviated through a small dislocation at a plane in between them. Screw dislocations instead consist of a displacement of the lattice in which the Burger’s vector lies parallel to the dislocation rather than perpendicular to it, as is the case for an edge dislocation.
Figure 5. Screw and edge dislocation in a cubic simple lattice. [3] Dislocations in principle have no endpoints and either extend the entire length of a crystal, or form a closed loop or circuit with other dislocations. However, in real systems these dislocations eventually do end abruptly, normally with another defect, e.g. meeting another dislocation in a perpendicular direction or a point defect. From this one can deduce that
̂ 𝑒𝑑𝑔𝑒 ⋅ 𝒃𝑒𝑑𝑔𝑒 = 0 𝑎𝑛𝑑 𝒏 ̂ 𝑠𝑐𝑟𝑒𝑤 × 𝒃𝑠𝑐𝑟𝑒𝑤 = 0 𝒏
(56)
24 |Defect classification in CdTe Additionally, compound dislocations having Burger’s vectors equal to the vector sum of the Burger’s vectors of the composite pieces can be created easily. In this way, if the vector sum of all Burger’s vectors of a set of dislocations vanishes then such an extended defect can be bounded within a finite region of the crystal. Similarly, dislocations may annihilate one another if their Burger’s vectors sum to zero and they are coincident at the same location. Lastly, dislocations may disassociate into combinations of dislocations whose total has an overall lower interfacial energy. This can often be seen as terracing along a grain boundary, or with two common dislocation lines in proximity to cancel out portions of their Burger’s vector, as in a Lomer-Cottrell or Hirth-Lock dislocation [35]. The ability of dislocations to migrate through a crystal is thus of special interest, because two or more dislocations may be able to migrate to the same location and annihilate one another, making their their electronic contribution vanish. Unlike non-isovalent PDs which require the breaking of at their ideal lattice sites for their generation, dislocations involve breaking and reforming nonideal bonds that are already weakened by being deformed and strained. Since this energy is relatively smaller, dislocations tend to migrate easily. This migration of dislocations is the main mechanism governing the plastic deformation of solids. In CdTe dislocations often lead to dangerous midgap states that can cripple electronic performance. One method of dealing with such dislocations is to encourage their migration and annihilation through cyclical annealing processes in which the sample is repeatedly brought to a high temperature for a short time. Dislocations (and other defects) will migrate rapidly in an attempt to annihilate and find a lower energy configuration, until the sample is cooled to room temperature and the process is repeated. Other techniques involve ‘gettering’ of these defects away from important regions of a device by making intentional geometric changes in the crystal away from the regions of high current density in the device. Still other methods rely on passivation of dislocations by saturating dangling bonds and relieving stress in the crystal by introducing extrinsic interstitials such as hydrogen that readily bond to the dislocations.
4.2b)
Elastic Properties
Many elements of elasticity theory can be used to describe concisely the energies, forces, stresses and strains associated with dislocations. Details of many such behaviors can be found in ref [34] but some features will be summarized here. For example, an edge dislocation can be viewed as a compression in which tension is built up along the Burger’s vector. Such a distortion results in a displacement vector field within the solid leading to internal stress and strain.
Christopher Buurma | 25 A dislocation then can be thought of as the source of internal stress in a solid, and in most cases linear elasticity theory captures the physics quite well. In this spiri,t define a displacement field u and more importantly the derivative of this field in various directions, which generates the symmetric stress tensor 𝜎 ⃡ via the outer product ⨂.
2𝜎 ⃡ = ∇ ⨂ 𝒖 + [∇ ⨂ 𝒖]𝑻
(57)
Similarly, according to Hooke’s law the strain tensor ε is assumed to be linearly related to the stress tensor by a matrix a.
𝜎 ⃡=𝑎 ⃡ 𝜖⃡
(58)
The normal compressive or expansive strain elements lie along the diagonal, corresponding to translations along each direction, while shear strain is described by the off-diagonal elements corresponding to a deformation in which the area is kept constant As is shown schematically in Figure 6.
Figure 6. Schematic representation of shear strain. The deformation of points B, C and D can be seen as a change in angle from that of a square to that of a parallelogram. A result from this treatment is that the elastic energy of the dislocation is the product of stress and strain tensors summed over all components and then integrated over the volume of interest. The displacement field is often treated by a widely used model developed by Peierls and Nabarro involving a continuum approach. In the PN model, the elastic energy of a dislocation core must be balanced with the energy cost of introducing the displacement.
4.3) 2D Grain Boundaries and Interfaces The next defect type, and the primary focus of this work, is that of 2D extended defects. These occur either as free surfaces, interfaces between unlike materials, or grain boundaries between two grains of the same material. In single-crystal CdTe and its related alloy compounds, surface troleffects and interfaces comprise an important focus for device engineering. It is in these areas that can often limit overall performance, once point
26 |Defect classification in CdTe defects are under control. Despite this disadvantage, polycrystalline solar cells can be fabricated much more quickly and cheaply than their single crystal counterparts. Furthermore much more engineering development has been made on poly-CdTe cells, providing a series of passivation and currentcollection enhancement processes that exploit the existence of grain boundaries, making a potential source of weakness benign or even beneficial. It is in the spirit of improved grain boundary engineering that future significant improvements in poly-CdTe solar cells can be realized. While photon absorption methods are mature and provide photocurrents near their theoretical limits, cell photovoltages and resistances remain primary barriers to achieving conversion efficiencies close to their theoretical limits. Any grain boundary can be defined with five macroscopic and three microscopic degrees of freedom. There are many varied definitions and types for these, however in this work the interface-plane scheme will be adopted with a brief description of the other methods and of the conversion between the two terminologies.
4.3a)
Grain-boundary Definition - Macroscopic
In the interface-plane scheme, a grain boundary between two grains is defined by unit vectors describing the interfacial plane of each grain, and a twist angle that represents a rotation in the interface plane. For convenience, and especially in face-centered cubic (fcc) materials, Miller indices are used to represent the two grain directions, giving rise to the following notation:
(𝜃)(ℎ𝑘𝑙)1 (ℎ𝑘𝑙)2 = (𝜃, 𝑛⃗1 , 𝑛⃗2 )
(59)
where 𝜃 is the twist angle and (ℎ𝑘𝑙)𝑛 is the interfacial plane of grain 𝑛. This gives easily the tilt angle between grain 1 and 2, 𝜑, by
cos 𝜑 =
(ℎ𝑘𝑙)1 ⋅ (ℎ𝑘𝑙)2 |(ℎ𝑘𝑙)1 ||(ℎ𝑘𝑙)2 |
(60)
Another common grain-boundary representation that is used is the so-called misorientation representation. In this formalism the misorientation axis 𝑜 and angle 𝜙 represent the rotation required to bring grain 1 and 2 into alignment. Lastly, a boundary plane is defined by (ℎ𝑘𝑙)𝑏 and is easiest to picture as being the direction of only grain 1 towards the boundary plane. Sometimes thus it is labeled as (ℎ𝑘𝑙)𝐴 and one may also include for convienience the resulting direction of the second grain (ℎ𝑘𝑙)𝐵 which is then more similiar to the interface-plane scheme, which has this convenience without over-parameterization, but not without its own flaws [36]. Thus, in the misorientation scheme, one arrives at an equivalent definition
(𝜙)[ℎ𝑘𝑙]𝑜 (ℎ𝑘𝑙)𝑏 = (𝜙, 𝑜, 𝑏⃗)
(61)
Christopher Buurma | 27 These two methods are illustrated in Figure 7 and Figure 8 below:
Figure 7. Interface-plane scheme for a (30°)(110)(111) GB. First, identify the two grain facing directions 𝑛⃗1 and 𝑛⃗2 , then rotate the two faces along the same axis, lastly provide mutual rotation about that axis by 𝜃.
Figure 8. Misorientation Scheme. Grains begin facing 𝑏⃗ and then the second is rotated away from the first along axis 𝑜 by angle 𝜙 One may easily convert between the two schemes[37] by using the misorientation matrix [𝓜]:
[𝓜] = cos 𝜙 [𝑰] + sin 𝜙 [𝑜]× + (1 − cos 𝜙) [𝑜⨂𝑜]
(62)
where [𝑜]× is the cross-product matrix of 𝑜, ⨂ is the tensor product, and [𝑰] is the identity matrix, . Then, using the notation above, one can go from the misorientation representation to the interface-plane representation,
𝑏⃗ = (ℎ𝑘𝑙)𝑏 = 𝑛⃗1 = (ℎ𝑘𝑙)1
(63)
[𝓜]𝑏⃗ = 𝑛⃗2
(64)
28 |Defect classification in CdTe
cos 𝜃 =
2(cos 𝜙 + 1) −1 cos 𝜑 + 1
(65)
and similarly can find the misorientation matrix from the tilt and twist matrices,
[𝓜] = [𝚯][𝝋]
(66)
where
[𝚯] = cos 𝜃 [𝑰] + sin 𝜃 [𝑛⃗1 ]× + (1 − cos 𝜃) [𝑛⃗1 ⨂𝑛⃗1 ],
(67)
[𝝋] = cos 𝜃 [𝑰] + sin 𝜃 [𝑛⃗1 × 𝑛⃗2 ]× + (1 − cos 𝜃) [(𝑛⃗1 × 𝑛⃗2 )⨂(𝑛⃗1 × 𝑛⃗2 )],
(68)
and 𝑜 and 𝜙 can be extracted from [𝓜] by
𝑜 = (ℳ32 − ℳ23 ), (ℳ13 − ℳ31 ), (ℳ21 − ℳ12 ),
(69)
2 cos 𝜙 + 1 = Tr[𝓜].
(70)
One last piece of information needed to fully describe grain boundaries in a binary polar system such as CdTe is the identification of the atomic species, Cd or Te, which dominates the interfacial plane. Some interfacial plane directions are non-polar, such as , however many others such as and have a terminating species. This is normally indicated by a letter A or B for the 1st or 2nd atom, e.g. (111A) is the (111) surface which is terminated by Cd. When it comes to grain boundary interfaces, it is extremely unfavorable energetically for both grain–boundary faces to be dominated by the same species. Thus, this letter is often neglected as a (60)(111𝐴)(111𝐴) grain boundary is very unfavorable due to the Cd-Cd bonding, while (60)(111𝐴)(111𝐵) is extremely favorable. Unless otherwise specified, here the GB interfacial planes will be assumed to be composed of different species.
4.3b)
Crystal Symmetry in Grain Boundaries
There is one additional complexity, or rather a simplification, in these definitions. Working with a material system with various degrees of symmetry such as CdTe each vector direction can be expressed as any symmetrically equivalent direction. In this way, the directions can be folded back upon one another, misorientation angles and grain directions can be reduced to the smallest symmetrically equivalent rotation. Such a classification in which the symmetry has been invoked to reduce the problem size is called the disorientation of the grain boundary. CdTe possess a face-centered cubic or zincblende structure noted as 𝐹4̅3𝑚 with space group 216. This space group has a tetrahedral 24-fold symmetry with the identity, eight 𝐶3 operations (rotations of 120° about [111] equivalent
Christopher Buurma | 29 axes), three 𝐶2 operations (rotations of 180° about [100] equivalent axes), six 𝑆4 operations (rotations of 90 about [100] axes) and six 𝜎 operations (reflections about [110] equivalent axes). Due to symmetry, this produces an equivalent grain-boundary after exchanging any of the Miller indices, placing them in any order, and/or reflecting the grain boundary. Thus, by convention any [ℎ𝑘𝑙] grain direction for CdTe is listed with ℎ ≥ 𝑘 ≥ 𝑙 ≥ 0. Exploitation of this symmetry will fold these five macroscopic degrees of freedom back into what is called the fundamental zone in which all values within are symmetrically inequivalent, and those on the border represent high-symmetry boundaries. A nice way of visualizing this reduction in the parameter space is through pole figures and their reduction. Begin by taking the unit vector 𝒑 which describes the direction of a plane in 3D. These directions trace out a unit sphere. Now, project this unit sphere onto a plane with the center chosen as some preferred direction (usually [100] or [111]) as shown in Figure 9.
Figure 9. Projection of crystallographic directional unit sphere onto a plane, providing a pole diagram. Now, the stereographic projection of this circle represents all possible directions in 3D. Invoking the crystal symmetry upon this surface creates a series of regions which are symmetrically equivalent. Each region can thus be folded onto another by using one of the 24 symmetry operations. This leaves one region, the standard notation or fundamental zone, as all that is needed to fully describe any direction in CdTe. See Figure 10.
30 |Defect classification in CdTe
Figure 10. Pole figures with the center at (111) showing symmetrically equivalent points (left) and the 24 symmetrically equivalent regions (right) for CdTe and other fcc structures.2 Thus, at long last we have arrived at the fundamental zone for CdTe directions. The misorientation rotation can be then thought of as a traversal on a pole figure in which the distance on the figure represents the misorientation required to bring both grains back into alignment with the two endpoints of the line segment being the two crystal face directions. Thus a grain boundary may be visually represented in the interface-plane scheme by the two points at the ends of a line segment within the fundamental zone, accompanied by a twist angle. This twist angle can also be viewed as new dimension in the reduced zone, which now reveals a small wedge shape from a cylinder in 3D. Since this line segment is undirected, this represents the last symmetry to be invoked to reduce the parameter space as a grain boundary (𝜃)[ℎ𝑘𝑙]1 [ℎ𝑘𝑙]2 is the same interface as (−𝜃)[ℎ𝑘𝑙]2 [ℎ𝑘𝑙]1 . This twist angle is also limited by symmetry as the perpendicular axis to the GB plane only has meaningful rotation within the fundamental zone of that axis, as is shown in Figure 11.
2
http://solidmechanics.org/Text/Chapter3_12/Chapter3_12.php
Christopher Buurma | 31
Figure 11. Snapshot of the 3D grain boundary region wedge in the fundamental zone. Any line segment in this space can correspond to a single grain boundary. The first point is constrained to the surface, while the second point may dip into the structure. The length of the line in the XY projection is the tilt angle, while in the YZ projection this is the twist angle. The overall length is the misorientation angle. In this way, many high-symmetry GBs can be identified. Pure tilt GBs are those line segments on the surface of the wedge in which the grain is tilted away from the GB plane. As a special case, a symmetric tilt is one in which the line segment lies on the border. Pure twist is similarly any line segment which only drops down the wedge with no change in the XY projection. Lastly, a mixed or random angle GB is any other line segment within this region. Misorientation angles are simply the length of such line segments. There is one last effect of symmetry to consider. The final wedge introduced above suggests that any mutual twist over any two boundary planes will always be unique. Yet, by applying the symmetry operators of CdTe, one finds that the maximum disorientation of any two grains is found to be at most 64°. Similarly one must apply symmetry operators over the tilt/twist combinations to find the minimal misorientation at each line segment. This has the effect of reducing the wedge shape containing the points representing the grain boundaries to be studied into a cropped irregular tetrahedron having an even smaller volume.
4.3c)
Grain-boundary Definition – Microscopic
Lastly, and oft-neglected, are the microscopic degrees of freedom of a grain boundary. These are given by the three components of the minor translation
32 |Defect classification in CdTe ⃗ . This between the lattices of the two grains, and are represented by a vector 𝑇 vector is bounded by the lattice vectors, thus each component 𝑇1 must lie within the range from 0 inclusive to 1 exclusive. Thus,
⃗ = 𝑇1 𝑎 + 𝑇2 𝑏⃗ + 𝑇3 𝑐 𝑇
(71)
with 𝑎,𝑏⃗ and 𝑐 as the three primitive lattice vectors for CdTe with 𝑇𝑖 ∈ [0,1). Notice that this microscopic rearrangement has two major effects. First, any translation perpendicular to the GB plane represents a volume expansion or reduction caused by the interface. As the two grains move into one another, atomic overlap becomes a crucial factor as open areas of the interface may be filled in. Similarly, moving away from one another can make high atomic density interfaces more relaxed. Next, translations parallel to the GB plane can arrive at lower energy interfaces than those without in a similar way by adjusting the local atomic density more towards that of a stoichiometric primitive CdTe region. While the macroscopic degrees of freedom describe geometrically the arrangement of each quasi-bulk region on either side, the microscopic degrees have a much larger impact on the total energy and electrical activity of the region. If two grains are brought into contact during grain growth, the microscopic degrees of freedom will be adjusted to lower the overall energy of the interface. It is this strong energy dependence that is perhaps why this is often neglected during experimental measurements of microtexture [38]. It is simply presumed that the microscopic translation will find the minimal ⃗ which provides the energy structure and that each GB observed has that 𝑇 lowest energy.
4.3d)
Grain-boundary Models
With GBs suitably defined and with their parameter space reduced by symmetry, we are ready to apply physical models to help elucidate their behavior. The usefulness of any structural GB model should be assessed in terms of its ability to predict and analyze experimental results. In this vein, there are two primary model types: low-angle grain boundaries, and highangle grain boundaries. This is largely due to the success and simplicity of the former. Low angle GBs are readily treated with dislocations and continuum theory, [39] while high-angle GBs are often studied in the context of highly symmetric interfaces or through a broken-bond model [40]. Breakdowns of the various models and limits are pictured in Figure 12 and Figure 13.
Christopher Buurma | 33
Misorientation Angle?
Dislocation Theory
High Symmetry?
No
Yes Continuum Theory
Coincident site lattice
O-Lattice
Vicinal CSL
DSC Lattice
Irregular Tetrahedron
Broken-Bond
Space-filling polyhedral
Imperfect co-incidence
Figure 12 Depiction of commonly applied grain boundary models. Low-angle GBs are treated with dislocation theory first, and then after the decomposition of the Burger’s vectors, continuum theory is used. For GBs with high symmetry, either a coincident site lattice (CSL) model is used, or is slightly modified as a Vicinal CSL. Alternatively, one can look at the coincidence of lattice points and not atoms, and modify for continuous deformations using the displacement-shift complete lattice (DSC).
Misorientation Angle?
Tilt and Twist?
Edge Dislocations
Screw Dislocations
Twist Angle?
Asymmetric Tilt
GB Plane?
Both Dislocations
Symmetric Twist
‘The Rest’
Symmetric Tilt
Figure 13. Grain boundary classification schemes commonly found in the literature. Low-angle grain boundaries are described in terms of their Burger’s vectors pointing along, or perpendicular to the interfacial plane. High-angle tilts without a twist component comprise a series of asymmetric-tilt GBs
34 |Defect classification in CdTe (ATGB) while those with a twist component are either pure twists, fully symmetric twists resulting in a CSL-like structure, or comprise ‘the rest’ being mixed or random-angle grain boundaries.
4.3e)
Low-angle Grain Boundaries
Among the most powerful GB models is that of the low-angle GB model first put forth by Read and Shockley [39]. If the angle is low enough, the misorientation can be accommodated by spatially separated dislocation-core arrays. In this way, the GB plane is composed of each dislocation line arranged along another perpendicular line. Each dislocation core in the array has the same Burger’s vectors, which are an edge dislocation for a pure tilt GB and a screw dislocation for a pure twist GB, and a mixed dislocation for a mixed GB. This allows the exploitation of dislocation theory and of the previous section involving 1D defects. The misorientation angle 𝜙 is related to the size of the Burger’s vector and to the spacing between them, 𝐷, by
sin
𝜙 |𝒃| = 2 2𝐷
(72)
It is in this expression that the ‘low-angle’ requirement enters. If the spacing between dislocations is too small, then the two dislocation core radii begin to interact with one another and ultimately the spatial separation of the cores is lost. Experimentally, the isolated nature of each dislocation begins to disappear around 𝜙~14°. That is, when the dislocation core radius and the distance between dislocations are on the order of 4|𝒃|. Remaining thus safely within the low-angle limit, the energy can be readily calculated from the previous dislocation section. Electrical behavior, doping, point defect formation, and all related behaviors maintain their dislocationcore nature and simply occur at regular intervals along the GB plane. Thus, studies of low-angle GBs are essentially studies of arrays of similar dislocation cores organized in a line. These dislocations are thus restricted over a finite subset of possible arrangements depending on the orientation of the vectors with respect to the lattice [41].
4.3f)
High-angle Grain Boundaries
As the dislocation cores overlap, new atomic arrangements begin to emerge. These more complex structures extend fully over the GB plane. Historically, high angle GBs are studied through a variety of models which attempt to capture the macroscopic and microscopic nature of the interface. Most of these models are characterized by exploiting a degree of symmetry along the interface.
Christopher Buurma | 35 The first and most pervasive models are based on the coincident-site lattice (CSL), O lattice (OL), and displacement-shift-complete (DSC) lattice models [42], [43]. These provide crystallographic extensions of the dislocation model which can be used to describe and compare both sides of the GB and largely ignore the location of the GB plane and any microscopic reorientation. These models were originally not intended for predicting physical properties but as a geometric tool for classification[44]. Regardless, there has been a mild correlation between the interfacial energy, and the density of coincident lattice sites. In this model, an atom is coincident if its displacement vector from the misorientation matrix has all integer multiples of the lattice vectors. Thus, one can trace its atomic position from the reference of either grain, meaning that this atom persists in either region regardless of the interface. If the density of such points is high, it suggests that the region near the interface is only mildly perturbed from its preferred orientation. If the density is rather low, then the CSL model fails to predict behaviors as other effects now dominate. These are characterized by Σ where Σ is the reciprocal of the density of coincident points per unit volume. CSL models have been widely used for many years as experimental evidence for a large array of metals (Cu, Ni, Ti, Al) shows many low Σ boundaries are very prevalent. The next series of models are based upon bond formation and breaking along the interface[40]. These models are powerful in their analytic capabilities to correlate a given atomic arrangement with its energetic and electronic properties. However, these models lack predictive capability as the topology of the interfacial energy and band structure as a function of the GB degrees of freedom is not smooth, but is sharply peaked in many regions, as is shown in Figure 14.
36 |Defect classification in CdTe
Figure 14. The interfacial energy of tilt GBs in Ni as a function of the tilt angle 𝜉 and the degree of symmetry 𝜂 between the grain directions and high symmetry lines.[45] Intriguingly, the topology of these GBs is material-independent for a variety of monatomic fcc metals, depending only on a scaling factor between the various systems. This suggests a deeper relationship between the geometric properties of grains, their atomic misorientation, and their physical properties. Lastly, high-angle grain boundaries can also be characterized by the volume and shape of their interfacial region. An irregular tetrahedron possesses 5 degrees of freedom and likewise, a given GB orientation gives rise to a volume between the two grains resembling an irregular tetrahedron along the interface. The last 3 degrees of freedom involving translation can be added, taking us away from tetrahedral and into more general polyhedra.
4.4) 3D Precipitates and Voids The last type of defect which is encountered is that of precipitates and voids. These fully-3D defects exist within the lattice and provide the most detrimental behavior. Precipitates tend to form if the local chemical environment forces new compounds to begin forming, taking these atoms from lattice sites and migrating towards the newly formed precipitate. Over time, or if given a heat treatment, these precipitates themselves will begin to diffuse through the lattice causing mechanical deformation, local straining, misbonding at interfaces, and other damaging effects. Due to the detrimental nature of such defects, many semiconductor processing regimens
Christopher Buurma | 37 use strict control over the atoms they are exposed to during device fabrication. Failure to purify material sources and growth chambers or eliminate airborne contaminants will result in many foreign atoms arriving on the surface. This will at best provide point defects and at worst collect within the material and create a chemical disassociation from the lattice. Some precipitates diffuse out entirely and are removed during processing. While this can remove the offending foreign atoms, it leaves behind voids and vacancies. The lattice vacancies will similarly diffuse and collect at void sites, in essence creating a concave free surface within the material. This new surface brings with it surface states, broken and dangling bonds, and an area where freely diffusing atoms can be chemisorbed and physisorbed.
38 |Results of This Work
5) Results of This Work The major research goal of this work is to provide theoretical guidance for experiments in grain-boundary engineering for poly-CdTe solar cells. First, an analysis of bulk CdTe was performed to set a baseline for comparison with future grain boundary (GB) structures. This includes bulk point-defect and defect-pair analysis as well as fully ab-initio recombination lifetime calculations. This was followed by a detailed analysis of the most prevalent GB structure, the Σ3 coherent twin. Next, a new algorithm was developed and implemented to generate fully periodic yet minimal GB structures suitable for DFT calculations in a fully automated way. Then, atomistic structures were created for a suite of bi-crystal interfaces fabricated by the University of Texas at Dallas. Finally, an early sampling of the GB space was performed with pseudo-potential calculations to find candidates for a more detailed DFT analysis. In parallel with these efforts a database and a suite of software was developed to assist in data analysis, data processing, visualization and algorithm implementation.
5.1) Bulk CdTe results Before any grain boundary calculations can be performed, a proper evaluation of the methods and techniques to be used must be complete to ensure that the ab-initio results closely resemble experimental observations. In addition, GB results are far less informative without comparative calculations to the bulk state. In light of this, a series of validation calculations were performed on pure bulk CdTe, metallic Cd, and semimetallic Te. After bulk calculations had converged properly, native point defects alone and in pairs were explored. Additionally, recombination lifetimes were calculated by blending both DFT-based defect energy levels and electronic wavefunctions calculated using a tight binding Hamiltonian (TBH).
5.1a)
Bulk CdTe
As previously mentioned in Section 3, PBE [15]-GGA exchange-correlation functionals were used in conjunction with the PAW [18] basis to perform periodic DFT calculations. These were performed using the Vienna ab-initio simulation package (VASP) [46], [47]. The filled d shells of the Cd ion were included as valence electrons. The initial structure is that of the conventional CdTe unit cell containing 4 atoms of Cd and 4 of Te. In order to ensure the Brillouin zone is properly sampled, a series of calculations was performed with different k-point grid spacings and offsets. The meshes were generated using the Monkhorst-Pack [48] method and included both Γ centered, and off-centered grids. The results of these initial bulk calculations are shown in Figure 15. They are labeled by “Gamma 𝑛” for
Christopher Buurma | 39 an 𝑛 × 𝑛 × 𝑛 k-point grid centered at Γ = (0,0,0); the “Monk 𝑛” are similarly spaced but not centered at zero. The ionic relaxations converged to a difference of less than 1 meV/atom. The plane-wave energy cutoff was set to 343 eV, and increases in this cutoff yielded no appreciable change.
Figure 15. Final energy/atom of the ionic relaxations using different k-point grid sizes and centering. An initial relaxation was performed, and then a second relaxation was performed using the final state of the first relaxation as the new initial state was done to ensure there was no local minimum. The comparative bulk case was then selected to be the Γ-centered 8 × 8 × 8 kpoint grid result with a 343 eV cutoff energy. This yielded a bond length of 6.609 Å which is less than 2% away from the experimental value of 6.486 Å. As expected, the Kohn-Sham bandgap of 0.61 eV is significantly smaller than the experimental bandgap of 1.5 eV. To generate a Kohn-Sham band diagram, a non-self-consistent calculation was performed to iterate over the k-points along the high-symmetry lines of the irreducible Brillouin zone. While the Kohn-Sham states in the conduction band are dubious at best, the occupied valence-band states tend to resemble those of the real electron band dispersion well, as is shown in Figure 16.
40 |Results of This Work
Figure 16. Band structure of CdTe as calculated from Kohn-Sham DFT. The energy is expressed in eV. The solid lines are for the occupied valence-band states, while the dashed lines are for the unoccupied conduction-band states. The Kohn-Sham gap is 0.61 eV. Grey lines are drawn to indicate (𝛤 , 0) With bulk CdTe results completed and within acceptable ranges, bulk metallic Cd and semi-metallic Te calculations were needed. These were used to determine their chemical potentials with respect to the stable binary CdTe. Metallic Cd was converged to within 5 meV/atom while Te was within 1 meV/atom. These were obtained with k-point grids of 11×11×6 and 7×7×6 for the primitive Cd and Te cells respectively. The final energies obtained are listed in Table 1. Table 1. Final energies per formula unit for the stable CdTe compound, metallic Cd and semimetallic Te. These values correspond well to those found by the Materials Project3.
Element(s) CdTe Cd Te
3
www.materialsproject.org
eV/ Formula unit This Work Materials Project -4.979 -4.982 -0.904 -0.914 -3.140 -3.142
Christopher Buurma | 41
5.1b)
Native Point Defects in bulk CdTe
With suitable bulk calculations completed and chemical potentials readily available, point-defect calculations were now performed. These were completed using the methodology previously described in Section 4. Point defects were placed in bulk supercells of CdTe and their formation energies and charge transition levels were computed. Native point defects were placed on their respective lattice sites, with interstitial defects placed in the tetrahedral, octahedral, and trigonal-planar [49], [50] sites surrounded by the opposite atomic species. The CdTe bulk supercells contained 64 and 216 atoms and were (13.2 Å)3 and (19.8 Å)3 in size, respectively Figure 17 shows the point defect formation energies (Δ𝐻) in bulk CdTe as a function of 𝐸𝑓 with respect to the valence band maximum (VBM) at the Cdrich and Te-rich limits. The corresponding charge transition levels are shown in Figure 18. Point defect formation energies for neutral charge states were found to agree with previous authors using similar methods [16]. The formation energy of the Cd vacancy (Vcd-2) was found to be very low in both chemical potential extremes, with a doping state at 0.09 and 0.28 eV above the VBM. The Cd interstitial (CdI+2) naturally compensates Vcd-2 in the Cd rich condition, while in the Te rich regime both CdI+2 and TeCd+2 appear in similar concentrations. Of these three most prevalent defects, the VCd-2 and TeCd+2 defects both show defect energy levels deeper in the band gap that can be electrically active for SRH recombination, making them detrimental to solar cell performance. The other three native point defects, the tellurium vacancy (VTe) and interstitial (TeI) and the cadmium-on-tellurium anti-site (CdTe) have formation energies too high to be found in appreciable quantities.
42 |Results of This Work
Figure 17. Defect formation energies for native point defects in bulk CdTe (color online): VTe, CdTe, TeCd, VCd, CdI, TeI. The thermodynamic Fermi level is indicated in grey.
Figure 18. Defect charge transition levels for native point defects in bulk CdTe. From the aforementioned methods, the thermodynamic Fermi levels of the bulk system were found to be 0.79 eV for the Cd rich limit, and 0.35 eV for
Christopher Buurma | 43 the Te rich limit, both relative to the VBM. This is consistent with previous results obtained using PBE [16], but differs substantially from the result obtained from HSE [51] . While Ref [51] predicts that Cd-rich CdTe is n-type and Te-rich CdTe would be neutral, experimentally as-grown CdTe grown by Closed Space Sublimation or by more controlled methods such as Molecular Beam Epitaxy reveal a p-type Fermi level, which is consistent with the results obtained from PBE. The implications of these results match previous predictions [52]. The density of TeCd defects should be minimized; thus, avoiding growth in a Te rich environment is preferred. Also, no intrinsic doping can achieve high enough doping levels suitable for solar cells, requiring the use of extrinsic dopants. However, to encourage proper extrinsic p-doping and take advantage of the low formation energy of VCd one also must not approach the full Cd-rich limit, requiring a delicate balance to be struck. Table 2. Formation energies of native point defects at three different chemical potentials.
Formation Energy (eV) Cd Rich Intermediate Te Rich Defect
Ef = 0.79 eV Ef = 0.56 eV Charge Bulk Bulk
Ef = 0.35 eV Bulk
VCd
-2
1.37
1.37
1.31
CdI
+2
1.37
1.37
1.42
TeCd
+1,+2
4.02
2.83
1.49
VTe
0
1.83
2.30
2.77
TeI (Tetrahedral)
-1,+1
4.22
3.76
3.11
TeI (Trigonal Planar)
+1,+2
3.89
3.05
2.17
+2
1.89
2.36
2.88
CdTe
44 |Results of This Work Table 3. Defect energy levels for native point defects in CdTe
Fermi Energy of Transition (eV) Defect
+2/+1
+1/0
VCd
0/-1
-1/-2
0.09
0.28
CdI
0.86
1.11
1.43
TeCd
0.58
0.97
1.37
VTe
1.08
1.42
TeI (Tetrahedral)
0.28
0.53
0.78
1.06
TeI (Trigonal Planar)
0.69
0.87
1.03
1.25
CdTe
0.85
1.04
1.20
1.42
5.1c)
Defect pairs in bulk CdTe
As there are 15 possible native point defect pairs, a set of rapid calculations was performed to screen all possible native point defect pairs. This is done for all symmetrically inequivalent positions within smaller 64-atom supercells. These screening calculations were performed with only 10 ionic relaxation steps ending with a difference in final energy on the order of 20meV. The result of this screening was used to select the defect pairs that exhibit strong binding energies and low relative formation energies for further investigations. Defect formation energies follow the formalism already established. Defect pair binding energy is defined as the difference between the sum of the formation energy of the two defects separately and the formation energy of the two defects in proximity at some separation distance 𝑥,. That is,
𝐸𝑏𝑖𝑛𝑑𝑖𝑛𝑔 = (𝐸𝐷𝑒𝑓𝑒𝑐𝑡 1 + 𝐸𝐷𝑒𝑓𝑒𝑐𝑡 2 ) − 𝐸𝐷𝑒𝑓𝑒𝑐𝑡 1+𝐷𝑒𝑓𝑒𝑐𝑡 2 (𝑥).
(73)
Thus, a positive binding energy implies that the two defects would prefer to be within proximity to one another at some stable distance, 𝑥, while a negative binding energy implies the independent defects are more stable and preferred. Some defect pairs which were too close and unstable would often recombine into single point defects (e.g. VTe and TeCd into a simple VCd) or annihilate
Christopher Buurma | 45 (e.g. CdI and VCd). The six pairs with the lowest stable formation energies and expected charge states were then further analyzed with accompanying structural optimization and all charge state calculations within 64 atom supercells. The screening calculation results are pictured in Figure 19.
Figure 19. Low precision screening calculations for defect pair binding energies in bulk CdTe. Formation energy is plotted against the 1st and 2nd defect in the pairing and the two chemical potential extremes are listed along the top within each grouping. Each defect pair is also listed with its multiple stable separation distances. These are for charge neutral structures and the expected charge state will reduce or increase the formation energy appropriately. This screening revealed that 6 defect pairs are likely to have strong binding energies and low formation energies and be present in appreciable quantities in CdTe: (VCd + VTe)0 Schottky pairs, (VCd + CdTe)0 i.e. a separated VTe, (VCd + TeCd)-2, (VCd + CdI)0 Frenkel pairs, (VCd + TeI)0 i.e. a separated TeCd, and (CdI + VTe)+2 i.e. a separated CdTe. These 6 structures were then analyzed with higher precision atomic relaxations and their various charge states calculated as a function of the Fermi level. The Frenkel and Schottky pairs were analyzed using larger 216 atom supercells, while the remaining 4 were kept in 64 atom supercells. The formation energy diagram for these 6 pairs is shown in Figure 20 and the charge state transitions in Figure 21. As the distance between these defects decreases, the binding energy increases until the two naturally move to the same lattice site (if possible) returning to a single point defect. As such, only calculations and stable distances for the defect pair configurations are pictured and analyzed.
46 |Results of This Work
Figure 20. Defect formation energy diagram for the 6 lowest energy defect pairs. Binding energies of these pairs are also quite high.
Figure 21. Defect charge states for point defect pairs. The two stoichiometric defects possess doping states near the conduction band while the other four have many gap states throughout the band. Consider first the formation energies alone and not the relative binding energy for each pair. In the Cd rich regime, the formation energies of both the
Christopher Buurma | 47 (VCd + VTe)0 Schottky pairs and the (VCd + CdI)0 Frenkel pairs are only 0.28 eV higher than the formation energy of the more common separate VCd and CdI, allowing them to form readily from the available VCd defects. (VCd + CdTe)0 and (CdI + VTe)0 are not far off, being only 0.44 eV greater than the Schottky pairs. Alternatively, in the Te-rich regime, the (VCd + TeI)+2 pair will actually form just as often as the CdI defect alone to be a charge compensating defect which will lead to many such pairings if there is available Te. This pairing however subsequently prefers a charge neutral state if the material becomes more ntype, allowing CdI to bring the Fermi level back down. In this regime, the (VCd + VTe)0 again arises as being 0.36 eV from the more common defects. The binding energies of the Schottky and Frenkel pairs are shown in Figure 22. These defects are stoichiometric, charge neutral, and consist of point defects with very low formation energies. As expected, these pairs bind very strongly over all Fermi levels and bind the strongest in p-type material. For Schottky pairs, in the Cd-rich limit the binding energy is 1.5 eV, and in the Te-rich limit (which is naturally more p-type) the binding is even stronger, at 2.5 eV. Frenkel pairs have the somewhat smaller binding energy 0.9 eV in both regimes.
Figure 22. Binding energy and formation energy of Frenkel and Schottky pairs. Solid lines are the bound pairs, while dashed lines are the sum of the two defects independently. The dotted line gives the total binding energy, corresponding to the right y-axis.
48 |Results of This Work The VCd + TeI pair also bonds very strongly with a neutral preferred charge state (Figure 23). As the two constituents of this pair would normally each prefer a net negative charge, these two would normally repel one another until stabilized by equidistant surrounding charges on the lattice such as a TeCd. However, the bound pair at a variety of stable distances is actually more preferred than if both occupied the same lattice site (TeCd) despite their mutual repulsion when isolated outside of the lattice. This surprising effect is likely due to the local lattice strain on either side of the interstitial and vacancy which distort lattice bonds nearby. The two defects in proximity cause local compressive and tensile strain to restore balance.
Figure 23. Binding energy diagram for VCd + TeI. The pair slightly separated shows a lower overall formation energy than if both defects occupied the same lattice site as a TeCd. The relative rarity of an isolated TeI suggests that TeI may be more readily found in proximity to a VCd. Thus when the prevalent VCd binds with either a VTe or a TeI, a favorable pair is formed. These pairs also show states moved away from the midgap and towards the band edges when compared to their constituents.
5.1d) Bulk recombination lifetimes calculated using a tightbinding Hamiltonian The defect energy levels of these defects can provide qualitative guidance towards experimental design and interpretation, however alone they lack predictive capability for recombination lifetimes. As recombination lifetime is the primary loss mechanism for the reduction of Voc in poly-CdTe solar cells,
Christopher Buurma | 49 it would be very desirable to build upon these defect energy levels by providing scattering cross-sections and recombination rates. Yet, to our knowledge, the capture cross sections for electrons and holes in CdTe have never been calculated from the underlying band structure for a given defect level and defect density. Because of the great importance of lifetimes, many direct and indirect measurements of recombination lifetimes have been performed using either single or two-photon time-resolved photoluminescence[53], [54], and photoconductive decay[55]. While the measured lifetimes in semi-insulating bulk CdTe grown from the melt range from 100 ns[54], [56]–[58], to 1 second[59], the lifetimes are considerably lower (only 1- 10 ns) in doped CdTe [60], [61]. Similarly, the lifetimes in low p-doped poly-CdTe grown by closed space sublimation (CSS) is typically 1-2 ns[54], which is far shorter than the 1 𝜇s observed in other II-VI materials such as HgCdTe[62]. Recent measurements of ~20 ns[52] lifetimes in a stoichiometric poly-CdTe sample grown under Cd-rich conditions suggests that lifetimes are limited by point defects. The measured carrier concentrations are not high and hence the Auger and radiative recombination mechanisms will be less effective than the defect-mediated Shockley-Read-Hall (SRH) mechanism. The SRH recombination lifetime is the primary metric for comparing the bulk absorber regions of different CdTe solar cell device architectures. However, nearly all models to analyze the solar cell performance treat the SRH capture cross section as an adjustable parameter.[28], [63]. The computation of the capture cross section has proven historically to be very difficult. The proper treatment of local phonon modes and defect energy level broadening alone could consume significant effort. Moreover, all methods require the computation of the overlap matrix element of the electron wavefunction with the states newly created by the defect to determine the transition rate from Fermi’s golden rule. A powerful method used previously for III-V semiconductors is to use a tightbinding Hamiltonian (TBH)[64]. In this formalism the electronic states are assumed to be determined by only nearest neighbor interactions up to a certain range. While this technique can be effective in determining electron wavefunctions, it is traditionally very poor at determining defect energy levels. However, under a Green’s function formalism, the perturbing potential can be set based upon the defect energy levels obtained from DFT. This then is used to compute a more representative band structure and electronic wavefunctions for the system at hand, allowing for the computation of recombination lifetimes. By extending the nearest-neighbor interaction out to the 10 nearest neighbors, even somewhat longer-range behaviors can be properly accounted for.
50 |Results of This Work As the focus of this work is on DFT, the TBH and Green’s function calculations were performed by our collaborating partner at SRI, as it is beyond the scope of the effort performed for this thesis. However, the joint results will be summarized here with permission; for the formalism and computational methodology please see Ref. [32] . The full band structures were obtained and used in the calculations of the Fermi levels of doped CdTe, which were required for the calculations of rates and lifetimes. For a given defect, defect energy level, and defect density, the recombination lifetimes 𝜏𝑝0 for n-CdTe and 𝜏𝑛0 for p-doped CdTe were calculated. For these calculations two defects were considered; TeCd and VCd,, with the energy levels at 0.9 eV and 0.8 eV, respectively (above the valence band edge). While the above DFT calculations do not reveal a 0.8 eV defect energy level, this was included as a comparative study, as it was found by previous authors [65]. The defect density was varied from 1012 cm-3 to 1015 cm-3, and it was found that 𝜏𝑛0 and 𝜏𝑝0 are perfectly linear in 1/𝑛𝑡 , justifying the assignment of capture cross sections as defined from previous authors [6], [7] and reproduced below. ∗ 1/𝜏𝑛(𝑝)0 = 𝜎𝑛(𝑝) 𝑛𝑡 √3𝑘𝐵 𝑇/𝑚𝑒(ℎ)
(74)
Furthermore, 𝜏𝑛0 (and 𝜏𝑝0 ) for various p (and n) doping concentrations was calculated from 1010 cm-3 to 1018 cm-3 and found to be fairly constant, given that the intrinsic carrier concentration at 300 K is only 1.25 x 106 cm-3. With the known experimental electron (0.096 m0) and hole (0.566 m0) effective masses, the capture cross sections were obtained from Eq. 1. The calculated values of minority carrier lifetimes and the capture cross sections at room temperature are given in Table 4 for the defects considered. Table 4. Summary of lifetime calculation results
In p-doped CdTe (commonly used in solar cells), the lifetimes are seen to be comparable irrespective of whether VCd or TeCd is dominant. If VCd is dominant, our calculations predict a lifetime of 228 ns in p-CdTe at 1012 cm-3 defect density. The thermodynamic calculations at 300oC by Berding [65] predict a Vcd density in the mid 1014 cm-3 at the typical Te-rich growth conditions (of about 10-8 Torr or 1.3x10-11 atm of equivalent Cd overpressure). For a VCd density in the mid 1014 cm-3, our predicted lifetime was ~1 ns, in
Christopher Buurma | 51 reasonable agreement with those measured in doped single crystal CdTe [53], [54]. Although the growth conditions are slightly different in closed-space sublimination (CSS), this also is in reasonable agreement with the short lifetimes (1 to 2 ns) observed within the grains of CSS-grown poly-CdTe. As seen in Table 4 the capture cross sections for the mid-gap states are ~10-13 cm-2 which is much different from the values of 10-12 cm-2 (for electrons) and 10-15 cm-2 (for holes), most commonly used [63] in AMPS software. Furthermore, there is an interesting observation from Table 4. Contrary to intuition, the capture cross section for electrons is seen to be smaller for the defect level (0.9 eV) which is closer to the conduction band than for the defect level (0.8 eV) which is farther from the conduction band. But the capture cross section for holes is larger for the level at 0.9 eV than for the level at 0.8 eV. The CdTe conduction band is found to be mostly s-like and the valence band is almost entirely p-like. The dipole matrix element is largest between s- and p-like states. At mid-gap, the defect state has nearly equal s- and pcharacter. As it moves closer to the conduction band, it gets more s-content and less p-content. Since the conduction band is also s-like, the dipole matrix element, and thus the rate of electron capture, becomes smaller. The electron lifetime increases from 228 ns to 442 ns for p-doped CdTe, as shown in Table 4. For the same reason, the hole absorption rate and cross section increase, resulting in decreased hole lifetimes. For p-doped (n-doped) materials, the defect states in the lower (upper) half of the band gap affect the lifetime more than the states in the mid or upper (lower) half of the gap. Stated differently, in doped materials, the defect levels which are closer to the Fermi level affect the lifetimes more dramatically than those from the mid-gap states.
5.2) (60°)(111)(111) 𝚺𝟑 coherent twins The first structure analyzed in our grain-boundary analysis was that of the most common interface found in both single-crystal and polycrystalline CdTe: the Σ3 twin. In poly-CdTe, many different planar defects abound both at terminating grains and within grains. While random-angle grain boundaries can be a major source of recombination in poly-CdTe solar cells [66], coherent Σ3 twins are far more prevalent owing to their very low interfacial energy [67]–[70]. They appear commonly within grains [71], [72] with a density near 30/µm [73], and typically extend throughout the grain. As shown in Figure 24, these twins can appear either as a single stacking fault or as multiple binary columns in a mirror image. It is also common to observe a single column stacking fault near a random-angle grain boundary before it terminates. These twins have traditionally been seen as benign; however, they can play an important role in the formation of point defects near such an
52 |Results of This Work interface, causing carrier recombination when they are at or near such a twin.
Figure 24. A series of twins in poly-CdTe. Both extrinsic and intrinsic stacking faults are present, as well as grain terminating twins, and lamellar twins.
Figure 25. A lamellar twin with EDS overlayed in the box revealing the atomic species. There have been other DFT studies on point defects in bulk CdTe, carried out using PBE [16] and hybrid HSE [51] functionals. Some twin boundary structures have also been investigated using DFT [74], confirming the expected low interfacial energy of these twins, and formation energies of
Christopher Buurma | 53 neutral native point defects similar to those found in the bulk. Since twin interfaces represent regions where bulk-like isotropic relaxation of atomic positions often is not possible, it remains possible that when decorated with point defects they will exhibit different electronic properties than their bulk equivalents. The effects of point-defect pairs, whether near twin boundaries or in the bulk, however, have not previously been comprehensively surveyed. In this study, a systematic examination and analysis is presented of all native point defects and point-defect pairs, of all possible charge states, both in the bulk and at various locations near twin boundaries. Our results suggest that native point defects with high formation energies in the bulk have reduced formation energies along twins in their respective charge states, even those with detrimental defect energy levels. Additionally, defect pairs are shown to bind less strongly when a twin plane lies between them.
5.2a)
Computational methods
In CdTe, coherent Σ3 twins are formed by having a series of rotated binary columns in an otherwise bulk continuum. These defect structures can be labeled using two indices, n and m, where n is the number of bulk CdTe binary columns and m is the number of twinned CdTe columns. The structural models are labeled as AnBm, and two examples are shown in Figure 26. Three primary limits are identified. When 𝑛 = 𝑚, this is known as a lamellar or repeating twin which simply oscillates between two bulk-like regions. As a special case, if 𝑛 = 𝑚 = 1 this is a hexagonal closed-packed or sub-Wurtzite phase. If 𝑛 ≫ 𝑚, this represents a stacking fault with 𝑚 = 1 being intrinsic and 𝑚 ≥ 2 for extrinsic. Lastly, in the limit of 𝑛, 𝑚 ≫ 1 this describes a perfect twin interface where on either side it is an otherwise perfect bulk. In this way, a wide variety of twin structures were created and analyzed with varying 𝑛 and 𝑚. For studies of point defects on twins, two primary representative structures were analyzed in this study: an A4B1 intrinsic stacking fault, and an A3B3 lamellar twin.
A
A
A
B
A
A
A
A
B
B
B
Figure 26. Examples of model twin structures. (Left) An A4B1 isolated stacking fault and (right) an A3B3 lamellar twin. Atoms located at the twin boundary plane region are highlighted.
54 |Results of This Work These twin calculations followed a similar methodology to that described in section 6.1a. Periodic DFT calculations were performed using the Perdew, Burke and Ernzerhof (PBE) [15] generalized gradient approximation (GGA) exchange correlation functional with the projector augmented wave (PAW) [46] method as implemented in the Vienna ab-inito simulation package (VASP) [47]. The filled d shells of the Cd ions were again considered as valence electrons in these calculations. The A3B3 lamellar-twin supercell contained 144 atoms in a 14 Å x 16.2 Å x 23 Å volume, while the A4B1 isolated stacking fault twin contained 120 atoms in a 14 Å x 16.2 Å x 19.2 Å volume. Electronic self-consistency and ionic relaxations were converged to 1eV and 0.1 meV in total energy per cell, respectively. GB structures were expanded perpendicular to the GB plane until a minimal energy was found. Again, Monkhorst-Pack [48] -centered kpoint grids of 3×3×3, 2×2×2, and 3×2×2 were used for these 64, 216, and twin supercells respectively. A k-point convergence was completed for this set similar to that in section 5.1a. The kinetic-energy cutoff for the plane wave basis set was 343 eV for consistency. The total energy was converged with respect to all calculation parameters to 0.1 meV/atom. Chemical potentials were evaluated using the bulk metallic Cd, and bulk semi-metallic Te cells found previously representing the two extremes for atomic reservoirs.
5.2b)
Pure Twins
The total energies were calculated for twin boundary models AnBm with all combinations of n, m ∈ {1-12, 15, 18, 21, 30}. An example of some of these interfaces is pictured in Figure 27. The interfacial energies of the twin boundaries were computed from the energy per unit area introduced by the twin boundaries. Because the energy differences between the supercells with and without twin boundaries are small (e.g. 1.8 meV/atom for A3B3), it is necessary to compare the total energies from bulk supercells with the same size and shape and the same k-point grids with and without a twin boundary, rather than use the total energy per formula unit from the calculation of a CdTe primitive cell.
Christopher Buurma | 55
Figure 27. Examples of model twin structures. Top left: A7B1 stacking fault. Top right: A6B6 lamellar (repeating) twin. Bottom: A19B1 isolated stacking fault. The total energies were calculated as a function of supercell size perpendicular to the twin boundaries, and the results suggest that semiinfinite twins neither accommodate nor create local strain. However, it has been seen previously that dislocation terminated twins do create local strain fields [75]. See Figure 28.
Figure 28. The formation energy per atom as a function of strain for a variety of different twin n and m values as well as for bulk CdTe. Twin interfacial energies were found to be quite small, on the order of 4-6 mJ/m2 (± 1mJ/m2). These values are in agreement with other publications on twins in CdTe and similar material systems [68]. As 𝑛 and 𝑚 increase, the interfacial energy drops linearly as a function of inter-twin separation (Figure 29 and Figure 30) until at 𝑛, 𝑚 ≥ 21 the values approach zero within
56 |Results of This Work the uncertainty of the calculations. The decrease in interfacial energies with larger twin separation points to a long-range electrostatic or elastic repulsion between twin boundaries. The repulsion is extremely mild, equivalent to 7.6 atm, which does not prevent the clustering of twins as is often observed in STEM [73]. Inspection of the Kohn-Sham density of states (DOS) reveals that these twins should be electrically benign and will not inhibit performance if they remain defect free, as is shown in Figure 31.
Figure 29. Interfacial energy as a function of average distance between twin boundaries in the computational cell. A3B3 is indicated for reference.
4mJ/m²
7.5
Figure 30. Minimum interfacial energy found over all strain as a function of bulk columns (n) and twinned columns (m) for an AnBm structure. Changes in energy are very small, as seen from the distribution inset.
Christopher Buurma | 57
Figure 31. Kohn-Sham density of states for all twin structures calculated above. No gap states are revealed indicating twin boundaries free of point defects should be electrically benign.
5.2c)
Twins with point defects
Once the baseline was established for a series of different AnBm twin structures, two representatives were selected for point-defect calculations. Namely, the A4B1 isolated stacking fault, and the A3B3 lamallar twin. Native point defects were placed on their respective lattice sites, with interstitial defects placed in the tetrahedral, octahedral, and trigonal-planar [49], [50] sites surrounded by unlike atomic species. Defects may be placed along these structures in a number of symmetrically inequivalent positions as shown in Figure 32 and Figure 33. For the A4B1 stacking fault, lattice site defects may be placed along the twin column itself (Site 1 in Figure 32), at the closest site on the nearby bulk (2-3), or deeper within the quasi-bulk region (4-5). Along the A3B3 lamellar twin, lattice defects may be placed on the boundary plane (Site 1 in Figure 33), or deeper within the quasi-bulk of each grain (2 and 3). Symmetry of the structure and the finite size of the supercell limits the number of possible defect sites to be explored.
58 |Results of This Work
4 5
2
4
1
3
3
1 2
5 Figure 32. Inequivalent defect positions along an A4B1 isloated stacking fault
3
3 2
1
1
2 3 3
Figure 33. Inequivalent defect positions along an A3B3 lamallar twin 5.2c,i) Cd vacancies (VCd) As one of the most dominant point defects in CdTe, VCd was first explored. The formation energy diagram of VCd is shown in Figure 34. This defect when found along or near an A4B1 or A3B3 twin shows nearly identical behavior to that in the bulk. No major change in either defect formation energies or defect energy levels (Figure 35) is revealed, regardless of proximity to the GB plane or of the type of twin.
Christopher Buurma | 59
Figure 34. Formation energy diagram for VCd. Only a very minor energy drop is observed between those found in the bulk and those along either A3B3 or A4B1 twins.
Figure 35. Defect energy transition level diagram for VCd. Doping states are revealed as expected at 15 meV and 270 meV with no change between bulk and twinned material. 5.2c,ii) Cd interstitials (CdI) Another dominant point defect in CdTe which naturally compensates VCd-2 is CdI+2. This defect’s formation energy diagram is shown in Figure 37. In the
60 |Results of This Work bulk, interstitials may be stable either at an octahedral or tetragonal site and may be surrounded by like (cation for CdI) or unlike (anion) atoms. Of these possibilities, the unlike-tetragonal configuration shows the lowest formation energy in the bulk and was first explored along twins. Placing the interstitial directly on the grain boundary plane between both position 1 sites in both twin types shows a reduction in formation energy at the neutral charge state, and a slight increase for the +2 and +1 states. As the defect moves deeper into the quasi-bulk region its formation energy approaches that of the pure bulk calculation. For the A4B1 this transition is immediate, whereas for A3B3 it is more gradual as the defect moves from site 1, to 2 and then 3. The Δ𝐻 for the neutral state is reduced by 0.47 eV compared to the bulk. At the Te-rich Fermi level, which is slightly p-type, Δ𝐻 is instead increased by 0.28 eV as CdI now prefers the new higher-energy charge states. In contrast, the Cd-rich Fermi level is immediately at a series of charge state transitions on the twin structures, keeping their formation energies near each other, with a slight preference to A3B3 over bulk, and bulk over A4B1. In solar cells, the absorber will likely be p-doped, and at p-doped Fermi levels, CdI+2 is shown to be slightly more favorable in bulk than when placed along a twin. The non-isotropic relaxation of CdI along a twin causes more gap states to emerge showing a mild performance impact when they are close to the doping level [32].
Figure 36. CdI location along an A3B3 interface. The interstitial is stable exactly on the interfacial plane and equidistant from the nearest 3 Cd and Te atoms.
Christopher Buurma | 61
Figure 37. Defect formation energy diagram for CdI. Interstitials on the GB plane show reduced formation energy of its neutral state, and increased formation energy of its charged states. Bulk calculations and the deeper positions along twins are indistinguishable.
Figure 38. Defect energy transition levels for CdI. Bulk calculations and those of the defect deeper in the quasi-bulk region are nearly the same. As the defect moves closer to the GB plane, transition states move more towards the valence band while the charge neutral state is lowered in formation energy.
62 |Results of This Work 5.2c,iii) Te-on-Cd antisites (TeCd) During bulk growth in the Te rich limit, TeCd+2 can also be a dominant defect, occurring as often, or even more often, than CdI+2. The behavior of TeCd along twins is markedly different than that in the bulk owing to the lack of isotropic relaxation at the interface (Figure 39 and Figure 40). When along twins, and at the thermodynamic Fermi level, the formation energy drops by 0.8 eV and 0.5 eV for the A3B3 and A4B1 twins respectively, with no dependence on the proximity to the GB plane. Also, the preferred charge state of this defect is TeCd0 and not the expected TeCd-2, which is preferred in the bulk at the same Fermi level. This preferred charge state occurs over a broader range of Fermi levels on the A3B3 than on the A4B1 twin. On the A4B1 twin, defect states are moved closer to midgap and away from deep-donor levels with a notably more detrimental effect (Figure 41). A defect being located near A3B3, however, moves the defect energy levels even further downward and ultimately reveals both a deep donor and a deep acceptor state, which allows the defect to remain charge neutral. The combined effect of lowering the energy of formation and the pushing of defect energy levels downward towards the Fermi level would have a mild detrimental effect on solar cell performance in a p-type absorber [32].
Figure 39. Atomic relaxation of a TeCd defect in the bulk (left) which is isotropic and when on a A3B3 twin (right) which cannot relax isotropically.
Christopher Buurma | 63
Figure 40. Defect formation plot for TeCd. Bulk calculations show much higher formation energies than when the defect is along isolated twins, and when along periodic twins the energy is even smaller due to the anisotropic relaxation of the TeCd defect interacting with the GB plane.
Figure 41. Defect energy transition levels for TeCd both in the bulk, along A3B3 and along A4B1 twins. Gap states tend to shift downward towards the valence band, reducing the performance even further than when found in p-type bulk material.
64 |Results of This Work 5.2c,iv) Other intrinsic point defects CdTe, TeI, and VTe tend to occur in far lower concentrations than their counterparts, but they will be discussed here for completeness. TeI shows a remarkable preference to form along twins, with H 1.5 eV less for A4B1 and 1.3 eV less for A3B3 twins, compared to that in bulk (Figure 44). This preference follows from the atomic relaxation of the TeI, which shows a strong preference to instead settle at the trigonal planar site, equidistant from 3 Cd atoms on the face of the tetrahedron (See Figure 42). Bulk calculations of the trigonal planar site for TeI indicate it is stable at both neutral and p-doped Fermi levels. Additionally, it is in fact lower in energy than the tetrahedral site for those Fermi levels. This behavior was not observed for CdI and warrants further investigation. In the Te rich limit for the equilibrium 𝐸𝑓 , this significant drop in formation energy along twins places TeI within 300 meV of the lowest formation energy defects: CdI and VCd. The lower formation energy would allow TeI to form in appreciable quantities and supports the formation of lower energy TeCd defects as well.
Figure 42. Three interstitial positions for TeI in the bulk. Tetrahedral (left), Octahedral (center), and Trigonal Planar (right).
Figure 43. Preferred location of the TeI along an A3B3 twin interface from two perspectives. The Te atom is more strongly attracted to the Cd-heavier side, displacing some nearby Te atoms. This places it in a tetrahedral configuration when surrounded by Cd atoms, but on the trigonal planar site when surrounded by Te atoms.
Christopher Buurma | 65
Figure 44. TeI formation energy diagram indicating both the tetrahedral and trigonal planar sites found in the bulk. Along twins the trigonal planar site is always preferred. The neutral charge state for TeI near twin structures is greatly reduced for both types of twins and at all positions. Immediately adjacent to the isolated stacking fault is the most preferable TeI location.
Figure 45. TeI defect energy levels. The bulk trigonal site possesses fewer gap states, however when along twins many gap states are removed entirely, leaving two amphoteric compensating doping states.
66 |Results of This Work The VTe defect shows very different behavior along twins than in the bulk. Along both twin types, the overall formation energy is lower for all charge states by 0.2 to 0.3 eV, as shown in Figure 46. Both structures can also more readily accommodate the +2 charge state, giving rise to lower formation energy and a preference for this state for p-doped Fermi levels. On A3B3 this defect also shows a positional dependence, where when near the GB plane it shows a strong preference for the +2 charge state, and as the defect is further away, only the +3 charge state is preferred, and then eventually the original neutral state found in the pure bulk. When along A4B1 a new lower neutral state is observed for the two locations equidistant from the intrinsic stacking faults at positions 4 and 5. This effect however would only come into play for Fermi levels of either undoped Cd-rich material, or when n-doped. These new charge states provide new gap states either near the midgap for +2 or closer to the valence band for +3 (Figure 47).
Figure 46. VTe formation energy diagram. Formation energies are always mildly lower when along twins than in the bulk for this native point defect. Additionally, when the material is more p-type, VTe exhibits a preferential positive charge state rather than the neutral state found in the bulk, as well as much lower formation energies.
Christopher Buurma | 67
Figure 47. Defect charge states for VTe. The newly preferred +2 charge state creates new gap states, and further from the interface a +3 state appears even closer to the valence band. Lastly, the CdTe defect located at twin boundaries is very bulk-like, particularly at the Fermi levels of interest (Figure 50). The -1 and -2 charge states become more favorable when placed near either twin as compared to the bulk, with the A3B3 position 2 slightly less favorable than others. Trap states also show a minor shift towards the valence band edge (Figure 49).
Figure 48. CdTe formation energy diagram. At the Fermi levels of interest, there is no change compared to the bulk. For a more n-type material, CdTe will form more readily upon twins who possess a more favorable -1 and -2 charge state.
68 |Results of This Work
Figure 49. Defect energy levels for CdTe. Inthe more favorable -1 and -2 charge states along twins, those transitions move downward into a double transition at +1/-1. Position 2 along A3B3 is somewhat less favorable than immediately on the GB plane, or deeper within the bulk. In summary, single point defects found along twins will have a mildly detrimental effect to solar cell performance. Of the three most prevalent native point defects: VCd, TeCd and CdI; two show slightly lower formation energies over the Fermi levels of interest (TeCd and CdI) while moving gap states closer to the valence band and midgap. Of the less prevalent, two others (CdTe and TeI) are more favorable along twins, and in fact have fewer gap states than their bulk counterparts; however, only the TeI falls low enough to form in appreciable quantities. A summary table of these results is available below in Table 5.
Christopher Buurma | 69 Table 5 Formation energies of all native point defects investigated along twins Formation Energy (eV) Defect
Charge
Location 1 2 -2 3 VCd 4 5 +2, +1, 0 Near 1 CdI +2, +1 Near 2 & 3 Near 3 & 4 +1, 0 1 2 3 TeCd 0 4 5 1 +2, 0 2 +3, 0 3 VTe +2 4 +2 5 -1 Tetrahedral +1, 0 Near 1 TeI Near 2 & 3 0 Near 3 & 1 +2, +1 1 2 CdTe +1 3 4
5.2d)
Cd Rich, Ef = 0.79 eV 3 3 4 1 Bulk AB AB 1.28 1.28 1.29 1.26 1.37 1.27 1.27 1.25 1.26 1.29 1.29 1.37 1.49 1.40 3.06 3.47 3.18 3.48 4.02 3.13 3.52 3.52 1.64 1.65 1.66 1.50 1.51
1.83
1.65 1.66 1.67
4.22
unstable
unstable
2.65 2.57 2.57 1.91 1.91 1.91
2.45
3.89
1.89
Intermediate, Ef = 0.56 eV 3 3 4 1 Bulk AB AB 1.28 1.28 1.29 1.26 1.37 1.27 1.27 1.25 1.26 1.55 1.54 1.37 1.73 1.40 2.13 2.54 2.25 2.55 2.83 2.19 2.58
2.30
3.76
unstable
unstable
2.18 2.10 2.10 2.44 2.45 2.45
1.98
3.05 1.90 1.90 1.92 1.91
2.52 2.11 1.78 2.13 1.78 1.80
1.76 2.12 2.14
2.36
Te Rich, Ef = 0.35 eV 3 3 4 1 Bulk AB AB 1.22 1.22 1.23 1.20 1.31 1.22 1.22 1.20 1.20 1.72 1.70 1.42 1.84 1.45 1.19 1.45 1.31 1.47 1.49 1.26 1.47
2.77
3.11
unstable
unstable
1.71 1.63 1.63 2.97 2.97 2.97
1.51
2.17 2.46 2.46 2.47 2.46
1.38 2.57 1.83 2.59 1.84 1.85
1.82 2.59 2.58
2.88
2.98 2.98 2.99 2.99
Twins with defect pairs
In order to determine if the presence of twin boundary planes in these structures affects the binding energy, prevalence, and properties of common defect pairs, combinations of intrinsic point defects were also investigated. All defect pair combinations at all possible symmetrically inequivalent distances along were first explored using a rapid screening calculation in 64-atom bulk supercells. These screening calculations were performed with only 10 ionic relaxation steps ending with a difference in final energy on the order of 20 meV. The result of this screening was used to select the defect pairs that exhibit strong binding energies and low relative formation energies for further investigations. Some defect pairs which were too close and unstable would often recombine into single point defects (e.g. VTe and TeCd into a simple VCd) or annihilate (e.g. CdI and VCd). The six pairs with the lowest stable formation energies and expected charge states were then further analyzed with accompanying structural optimization and all chargestate calculations within 64 atom supercells.
70 |Results of This Work The same 6 defect pairs analyzed above were also studied when present along representative twin structures. The A3B3 lamallar twin structure was studied with all 6, however only the two stoichiometric pairs, Frenkel and Schottky, were placed along the isolated stacking fault A4B1. The pairs were each placed on either side of the interfacial plane and relaxations where the two defects occupied the same lattice site were discarded. The Schottky pair (VCd+VTe)0 shows a slightly higher formation energy when along A3B3, and nearly the same for A4B1 (Figure 50). As expected, the lowest energy configuration is obtained when both defects are placed on the twin plane, however the energy was lower and closer to that of the bulk when they are present on the isolated twin A4B1. As the pair is moved deeper into the quasi-bulk regions both away from each other, and also from the GB plane, the energies increase. The A3B3 structure also exhibits a higher overall energy than would be found in the bulk, even when placed next to one another on the twin plane. The anisotropic relaxation of nearby atoms prevents them from achieving a nearer bulk-like configuration.
Figure 50. Formation energy diagram for Schottky defect pairs along twins. Energies always decrease as the spacing decreases with the A4B1 isolated stacking fault having a near bulk-like configuration, while on A3B3 the ionic positions are anisotropic. The other stoichiometric and charge neutral pair, the (VCd + CdI)0 Frenkel pair, also shows a somewhat higher formation energy when along twins than in the bulk (Figure 51); the previous trend however is reversed. Now, the lamellar twin A3B3 shows the most bulk-like behavior when the pair is placed closest to the GB plane, with A4B1 showing more anti-binding behavior even
Christopher Buurma | 71 when situated nearest. Placing the pairs deeper within the bulk regions on either side of the GB plane has the effect of further raising the formation energy both due to separation distance and to the interaction with the GB plane.
Figure 51. Formation energy diagram for Frenkel pairs along twins. The VCd + CdTe pair also shows a slight anti-binding behavior, raising the formation energy when present along twins by 0.23 eV. The last three pairs (CdI + VTe), (VCd + TeCd) and (VCd+TeI); however show bulk-like behavior, with only mild variations. See Figure 52.
72 |Results of This Work
Figure 52. Formation energy diagram of the remaining defect pairs: (VCd + CdTe), (CdI + VTe), (VCd + TeCd) and (VCd+TeI). In summary the interfacial plane, while very similar to that of the bulk, does still inhibit normal pair binding and can force different atomic arrangements which can be less favorable. This anti-binding behavior persists through a variety of different pair spacing and arrangements and will approach bulklike when placed deeper within the quasi-bulk region on each side of the twin. A summary table of defect pair formation energies is given in Table 6. Table 6. Formation energies of notably low-formation-energy point-defect pairs. Formation Energy (eV) Cd Rich, Ef = 0.79 eV Defect Pair
Alternate Name
Charge
VCd+CdI
Frenkel
0
VCd+VTe
Schottky
0
VCd+CdTe
Near VTe
0,+2
VCd+TeI
Near TeCd
0
VCd+TeCd TeI + 2 (VCd)
-2, 0
CdI+VTe
0,+2
Near CdTe
Stable Distance (Å) 5.48 10.12 2.86 4.81 2.86 2.86 4.67 (1-1) 4.67 (1-2) 3.3
Bulk
3 3
AB
1.793 2.307 1.706
1.829 2.356 2.087 2.410
1.809 3.161 4.394 1.884
4 1
AB
2.227 2.483 1.645 1.676
Intermediate, Ef = 0.56 eV Bulk
3 3
AB
1.793 2.307 1.706
1.829 2.356 2.087 2.410
1.588 3.064 4.461 4.278
1.884 2.226 3.235
1.885
2.723
4 1
AB
2.227 2.512 1.645 1.676
Te Rich, Ef = 0.35 eV Bulk
3 3
AB
1.793 2.307 1.706
1.829 2.356 2.087 2.410
2.056 2.129 3.292 3.315
1.942 1.291 1.833
2.523 1.195 2.096 1.913
2.446
2.997
2.969
4 1
AB
2.227 2.512 1.645 1.676
5.3) Creating GB interfaces with the Grain-boundary Genie With the most obvious GB structure comprehensively studied, more general grain boundaries now will be examined. In order to properly sample the GB
Christopher Buurma | 73 space and to ensure that representative model structures can be created and visualized quickly the Grain Boundary Genie code was developed. This code serves two primary purposes. The first is an interactive visualization of an interface, allowing for manual control and fine-tuning of a wide variety of options. This allows the user to ensure that the structure is representative of a related STEM image and that STEM channeling conditions are obeyed, that the structure supports periodic boundary conditions or is a properly terminated slab, that the spacing between the periodic interfaces is sufficient, and that overlap regions and translation vectors are tuned properly. Secondly, it provides an automated scriptable method for generating GBs from only their degrees of freedom. Other optional features exist to facilitate generation including preferred grain width, structure geometry, maximum allowable size, atom merger and removal based on proximity, required atomic density, stoichiometry and so forth. This provides a powerful tool to the research community to either manually tune an atomic structure for detailed analysis, or to automate a large batch of calculations to generate a broader spectrum analysis. This code, and others developed for this project are openly available on the UIC microphysics website: http://www.uic.edu/depts/mplab/mplhome.html
5.3a)
Grain boundary generation algorithm
Creating a grain boundary interface directly is rather straightforward. Begin with the two primitive (or conventional) unit cells. Rotate each grain to the interfacial plane desired and then apply the mutual twist (e.g., use the interface-plane scheme). Traditionally this structure is then simply extended in each other dimension periodically and then terminated on each side by pseudo-hydrogen creating a slab calculation. However, this method becomes cumbersome to use because slab calculations have many spurious errors associated with the new vacuum region which must be introduced. In addition, calculations involving charged defects become exceedingly challenging, making point-defect analysis far more difficult. The principal challenge then is to create the interface, and then demand that periodic boundary conditions take effect. The new lattice which then subsumes the two smaller primitive lattices on each side of the interface is called the overlattice. This imposes a number of constraints on the atomic structure. First, the periodic boundary between grain 2 and grain 1 must be a mirror image of the original boundary between grains 1 and 2. This has the effect of canceling out any Burger’s vectors from low-angle GBs and restoring each periodic grain in the next replication. Next, demand that the other two periodic boundaries parallel to the GB plane be of sufficient size so as to allow them to each repeat within the sub-lattice on either side of the
74 |Results of This Work interface. Lastly, the total volume and number of atoms of this structure must be minimized so as to require the least computational resources to perform the required computations. Note that one need not demand that this grain boundary structure be a member of the same crystallographic group, and it may indeed by triclinic. With a high-symmetry structure, these constrains can be easily filled, and are underdetermined. With many symmetric tilts and with those of the coincident site lattice with low Σ the directions parallel to the GB plane are the same in either grain, making the only challenge that of finding the minimal width to accommodate both sides. Similarly, by demanding that the displaced lattice have integer coefficients of lattice vectors (a CSL boundary or O-lattice) the size can be readily determined from the magnitude of the displacement in each direction. These methods have been applied previously in different codes [76].The remaining and more difficult problem is that of treating low-symmetry, more mixed or ‘random’ GBs. This remaining difficulty will be treated in two steps. First, ensure that the two periodic directions parallel to the GB plane obey periodic boundary conditions by invoking a near-CSL structure. Then, the direction perpendicular to the GB plane may be further reduced by recognizing that the GB plane need not replicate exactly along the overlattice vectors in that direction, but may be offset by any vector which is periodic within the minimally spanned volume of the rotation and still correspond to the same interface. These two steps will be explained in detail below.
5.3b)
From vicinal-CSL to GB interface
First, model the general interface as a ‘vicinal’ or ‘near–CSL’ (or near Olattice) grain boundary (see Section 5). Begin with the primitive unit cell of CdTe, grain 1, and perform a misorientation rotation of these lattice points into grain 2. Then iterate over grain 1 radially to a sufficient distance and compute the displacement vector 𝑑 between grain 1 and grain 2 at each point. Now check to see if 𝑑 is expressible as (near) integer multiples of the primitive lattice vectors. If it is, then it is coincident or nearly coincident with grain 2. By providing a tolerance of co-incidence, in effect a slight local strain is introduced mutually on each grain to conform within the periodic boundary. This tolerance can be set quite low so as to not affect the results (700 mJ/m²) as well as some questionable atomic rearrangements. Sadly, experimental difficulties also occurred in studying this bicrystal, resulting in
86 |Results of This Work an amorphous region at the interface and making guidance from experiment problematic. Given the gross mismatch between the two interface planes it is perhaps reasonable for the amorphous region to have formed to lower the overall energy, rather than having the crystal attempt to conform to a crystal lattice in near proximity to the GB on either side. See Figure 64.
Figure 64. STEM and XRD images of this interface. An amorphous region is observed at the interfacial plane, which may be due to experimental challenges, or may naturally occur due to the gross mismatch of the two lattices.
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Figure 65. Early atomic model of the (90º)(110)(110). The spacing between periodic interfaces was insufficient and atomic rearrangement can be observed throughout the second grain.
5.4e)
Dual Interface (35.2)(110)(10-1)
A combination of a 35.2° tilt and a 60° twist, this interface represents a misorientation of 85 degrees around the [321] direction. This yields an unchanged interfacial plane crystallographically speaking ( the plane). Since no pure twist can arrive at this orientation, it produces a unique atomic arrangement. The original atomic model for this interface proved too large, and some additional strain will need to be added to allow for a more rapid computation of its properties.
88 |Results of This Work
Figure 66. XRD, STEM on the left, with FFT noise filtered on the right showing this interface.
Figure 67. Atomic model before ionic relaxation.
5.4f)
5° Mutual Tilt of (0º)(1.05,0.95,0.95)(1.05,1.05,0.95)
A series of mutual tilt grain boundaries were also fabricated. The first one was a plane with an off-cut on either side of the interface such that the reformed bi-crystal had a total tilt angle of 5°. This is a low-angle GB, and after relaxation should produce a series of dislocation cores along the interface at regular intervals.
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Figure 68. STEM and XRD images for the mutual tilt of the interface.
Figure 69. Model atomic structure after relaxation. A Cd dislocation core is observed with nearby regions otherwise bulk-like. For this structure, exploitation of the interfacial plane’s periodic image was made easier as the mutual rotation caused the displacement vector 𝑟 to be very small.
90 |Results of This Work The formation of the dislocation cores in this structure yield many midgap states. Since this structure is periodic, both the Cd-core (pictured above) and the Te-core on the periodic border were simulated in tandem. The dangling and wrong-bonds at the interface provide a series of states in the midgap. However, in such a calculation for a periodic structure it is difficult to determine which core is responsible for which states. Site-projected DOS for the regions near each core suggest that the Cd-cores contain many states in the gap, and that the Te-cores possess one notable state at 0.29 eV. Further analysis of this structure is needed to ensure that the computations contain no spurious contributions. See Figure 70.
Figure 70. Site-projected DOS for the atoms near the Cd-core (red) and the Tecore (blue). Bulk DOS is provided for reference in black. Te-core has a notable peak at 0.29eV.
5.4g)
4° mutual tilt of (0º)(11,0.05)(1,1,-0.05)
This is another low-angle mutual tilt GB with a non-polar interface plane. STEM images of this bi-crystal reveal the regular Burger’s vectors at periodic intervals, which is expected if the interface is well formed. There appears to be substantial local strain near the GB, and the crystals on each side of the GB are realigned by the GB. Early pseudopotential calculations have been performed on a model structure, however DFT results are still pending. See Figure 71 and Figure 72.
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Figure 71. STEM image, XRD and a strain map of a 4° mutual tilt bi-crystal of . The dislocation core is very localized as the continuous crystalline behavior is quickly recovered.
Figure 72. Model atomic structure before (left) and after (right) using pseudopotentials[79] to find a closer atomic structure before DFT relaxation.
5.5) Early Sampling of the GB Parameter Space Using pseudopotentials As was seen in Section 4, in a random polycrystal, grain orientations should follow a Mackenzie distribution[80]. In CdTe this is confirmed to be the case before any post-growth treatment. After growth a variety of methods can be
92 |Results of This Work applied to change this distribution to one more favorable, most notably the CdCl2 treatment[67], [81], [82]. Since the GB parameter space is vast, any computational search for grain boundaries must be constrained. As optional additional constrains, if atomic structures can be found which also allow for STEM challenging conditions along a major viewing axis it would be preferable to fabricate those structures for visual verification. Additionally, if the two interface planes can be near to an offcut angle of available substrates, it would accelerate the process of acquiring and bonding a bi-crystal. See Figure 73.
Figure 73. Grain boundary parameter space. All possible GBs which are symmetrically inequivalent are shown in black. Those which are viewable under STEM are in red, while those with low tilt angles suitable for bicrystals are in blue. Green represents those both viewable and able to be fabricated. There are three primary categories of interest which contain good candidates: Low-energy interfaces from metals (CSL), those near the Mackenzie peak,
Christopher Buurma | 93 and asymmetric-tilt grain boundaries. Representative structures from each of these categories were created, as well as a random sampling of a wider parameter space. However, efforts are ongoing to analyze the vast data from these structures. Example atomic structures are listed below.
5.5a)
Low-energy structures from metals
There is a wealth of knowledge on the interfacial energy of metals [33], [45], [83]–[88]. Many authors have now found a series of experimentally verified low-energy interfaces for fcc metals. As described in Section 4, these trends occur between different metal species as well, with only a scaling factor between them. Early atomic structures were then created to resemble these well-known low energy interfaces. All of these happen to also be those of the CSL lattice model. Each of these are listed below.
Figure 74. Top left: (0)(311)(-311) a CSL 𝛴11 grain boundary structure for CdTe. Top right: (0)(331)(-311) a CSL 𝛴19𝐴 GB. Bottom left: (0)(211)(-211) a CSL 𝛴3 GB.
94 |Results of This Work Bottom Right: (0)(221)(22-1) a CSL 𝛴9 GB. In particular, the Σ11 interface reveals a number of strained bonds after relaxation. As a result, the Kohn-Sham DOS reveals a series of defect states in the band gap which can hamper performance. This structure has the high interfacial energy of 784 mJ/m². See Figure 75.
Figure 75. 𝛴11 GB after ionic relaxation. Each quasi-bulk region is pinched together nearer the interfacial plane straining the local bonds. This strain is likely the cause of the additional gap states observed in the K-S DOS.
Figure 76. Kohn-Sham normalized density of states for the above structure.
5.5b)
Near the Mackenzie Peak
Since our Poly-CdTe represents nearly an ideal polycrystal, it is likely that most random-angle GBs would have a total misorientation near the Mackenzie peak[80]. This corresponds to a misorientation of ~44.5 degrees. See Figure 77.
Christopher Buurma | 95
Figure 77. The Mackenzie distribution with the peak highlighted. Since these structures are difficult to create while being relatively small, a large series of candidates was generated automatically. This space was then searched for those with a small number of atoms (>500) suitable for a DFT calculation. As an additional constraint, it is required that the two grain sides could be resolved with STEM channeling conditions with only a small angle of difference on either side. This search revealed 3 strong candidates with others still under consideration as new methods to search for these representatives are under development. See Figure 78.
96 |Results of This Work
Figure 78. Example structures with misorientations near the Mackenzie peak. 2 Top left: (41.6º)(111)(432) with a misorientation of 𝜙 = 42 3 º. Top right: 2
(41.9º)(111)(17.16.12) with 𝜙 = 42 3 º. Bottom: (43.6º)(810)(37,7,6) with 𝜙 = 44º Two example structures from near the Mackenzie peak are summarized below. First the (32.8º)(110)(1,0.53,-0.42) interface reveals a combination tilt/twist behavior with no higher symmetry (Figure 79) and a high interfacial energy of 849 mJ/m². The interfacial area shows many missing and wrong bonds in the structure on either side. Despite this microstructural change, the K-S DOS reveals no defect states in the band gap (Figure 80).
Christopher Buurma | 97
Figure 79. Atomic structure for this high tilt/twist GB. STEM channeling conditions are fulfilled for this interface, allowing it to be observed if it was to be fabricated.
Figure 80. Normalized Kohn-Sham DOS for the above structure. Next is the pure twist (48º)(210)(210) interface. This structure shows only minor atomic rearrangements from ionic relaxation, and maintains a clean KS DOS in the bandgap. Despite this, it has a very high interfacial energy of 802mJ/m².
98 |Results of This Work
Figure 81. Atomic model for the pure twist interface
Figure 82. Kohn-Sham Density of states for the above interface. No defect states are revealed.
5.5c)
Asymmetric- tilt GBs
This class of GBs has received special attention in the past when studying metals as they mediate many mechanical behaviors[87]. From a simulation perspective, this class is also easy to generate and easy to analyze, however depending on the total misorientation angle the structures can be quite large. Again, the parameter space search was reduced to search for those with STEM channeling conditions and those near enough to a bi-crystal plane
Christopher Buurma | 99 available for fabrication. Two example structures are pictured below in Figure 83.
Figure 83. Left: (0º)(110)(881) interface with a 5 degree tilt. Right: (0º)(111)(876) with a 6.6 degree tilt. In particular, the tilt boundary from 110 to 100 was created and analyzed using DFT. Its atomic structure shows many broken bonds and misalignments which can explain the numerous states in the band gap revealed by DOS calculations. This structure had an interfacial energy of 659mJ/m². See Figure 84 and Figure 85.
Figure 84. Model atomic structure for the (0º)(110)(100) asymmetric tilt GB.
100 |Results of This Work
Figure 85. Normalized Kohn-Sham DOS for the above structure.
5.6) Supporting software and database As part of this effort custom software routines were written to quickly create atomistic structures, prepare VASP input files, analyze VASP output files, extract defect formation energies, solve routine analytic equations, visualize outputs, and organize related data into a relational database structure. These tools greatly reduced the amount of time needed to manage data input and output, as well as to look for physical trends in different behaviors more easily and intuitively. These routines possess a user-friendly interface to allow those unfamiliar with command-line interfaces and programming methodologies to analyze results more collaboratively. These codes are openly available on the UIC microphysics lab website. http://www.uic.edu/depts/mplab/mplhome.html
5.6a)
Structure Handler
The structure handler code was designed to allow easy generation and manipulation of many atomistic structures as well as related VASP input files. This code was designed to interface with a relational SQL database to facilitate quick retrieval of past structures and past input files. Structure manipulation features include bulk padding of any structure, building of supercells, cropping, resizing, rotating, translating, removal of atoms based on proximity, point defect addition, and point defect pair addition. This code includes an interactive visual user-interface to visualize the atoms or to output them to other visualization software. See Figure 86.
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Figure 86. Structure handler interface. Structure Handler allows for the direct output of VASP input files. INCAR files used for configuring calculation setting are managed with quick look-up and cross references for different calculation flags. Commonly-used parameters can be set as defaults and new batches of related calculations can be prepared in a fully automated way. For example: after a structural relaxation of a point-defect in a bulk structure, all related files needed for computing and relaxing the relevant charge states for this defect can be automatically output for ready submission to the computing cluster Carbon. This code also handles generation of k-point grids and writing KPOINTS files either by invoking VASP’s automated routine, or through manual k-point selection. This allows for easy submission of band-structure calculations that trace along high-symmetry points in CdTe. POTCAR files are similarly read in from a directory and defaults are pre-loaded based on the atomic species found in the structure. README files are also created with comments on the calculation intent, and any extra information not contained in the VASP submission files.
5.6b)
VASP Helper
Once calculations are completed, a vast amount of data is ready for analysis. VASP Helper parses, plots, organizes the output data and optionally stores it in a relational database with the VASP Janitor (See Section 5.6d). Parsed calculation data can easily be plotted from one of the various tabs along the top, giving a quick visual representation of the electronic and ionic relaxations, the atomic structure and animations of its relaxation, the electronic density of states (DOS) as well as projected DOS, parameter variability, bivariate plots, defect formation energies, and interfacial energies.
102 |Results of This Work Electronic and ionic relaxations can be displayed directly as energy vs iteration numbers, or the atomistic structure can be animated showing the ionic displacements resulting from the relaxation. Final energy values are obtained and can be plotted alongside other calculations for quick comparison. The total DOS can be plotted directly, or as an orbital-projected DOS along each atom. The atomic structure and these atomically localized DOS are integrated together, allowing for selection of specific atoms and viewing the contribution of these atoms to the DOS (see Figure 87). Band diagrams can be output as well, providing non-self-consistent calculations that evaluate eigenvalues at high-symmetry points are performed. Parameter variability can be quickly displayed and is grouped by any value, revealing trends in group behavior. Additionally, bivariate plots are possible by plotting any numeric parameter against any other, and then fitting the plots with a trendline, quadratic, or any other analytic function.
Figure 87. Variability diagram generated for a series of calculations.
5.6c)
Defect Buddy
One powerful feature of VASP helper is to populate a point-defect database from output files. This database is then queried by Defect Buddy which can provide a wide variety of features to assist in analysis. A set of defect-free structures can be selected, populating a listing of related defects found in the database. For each selected defect, the minimal energy over the set of charge states for a given chemical potential and Fermi level is then plotted. Charge transition levels are similarly extracted and are indicated on the plot. With the defect energy levels extracted, they are also easily plotted with respect to each defect type and reorganized. Makov-Payne
Christopher Buurma | 103 [22], potential realignment, and Lany-Zunger [23], [89] corrections can also be added to any set of defect calculations. As described in Section 5, self-consistent Fermi levels are also calculated by enforcing the macroscopic charge neutrality condition. Chemical potentials can be readily changed, plotting a smooth transition between the two extremes. The atomic structure of any defect in the database can be readily queried and visualized for comparison as well as animating the relaxation. The Defect Buddy module also readily plots binding energies for pairs of defects, and can output data directly in a tabular text format, such as that shown in Figure 88.
Figure 88. Screenshot of Defect Buddy showing a series of defect energy calculations for bulk CdTe.
5.6d)
VASP Janitor and SQL Database
On the backend of the various tools previously mentioned is a relational SQL database. Such a database gives powerful control to manage the large quantity of data from each VASP calculation. The database is organized by a set of tables for each output file as well as a parent table storing details of the calculation itself. A unique ID is generated from each file from its MD5 hash (or SHA-256 hash). The unique ID of the calculation itself is the hash of the OUTCAR file. Structure IDs however are generated differently. Since any translation of the structure leaves it essentially the same, the hash of a POSCAR file would not correspond properly to two identical structures. Instead the following value is computed. The ionic position vectors both as provided, and re-centered at the geometric center is computed and summed.
104 |Results of This Work Then these two summands are reduced to the nearest integer in the base of a 61 character alphabet (numerals, upper and lowercase) and then truncated to a total length of 6 characters for each summand. This gives over 100 billion combinations for an ID, while still being generated rapidly, with a low probability of collision and is invariant to vector translation. This has the added benefit of discarding extremely high precision values in favor of their approximate value, removing any dependence on the machine epsilon or rounding errors based on the data types.
Figure 89. Database topology for the VASP SQL database. Primary and foreign keys are linked and highlighted The VASP Janitor was written to passively and recursively parse through any directory to find and import VASP files into this database. This has an added benefit of deleting incomplete files, compressing those which are beyond a certain size, and reducing and indexing the drive space needed to store files.
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6) Conclusions Results of bulk CdTe calculations provide two major benefits. First is to build confidence in subsequent efforts as bulk results match that of previous authors, ensuring that computational artifacts and errors are minimized with suitable precision. Next, an exhaustive study of binding energies of native point defects reveals that the stoichiometric Schottky pairs, and to a lesser extent the Frenkel pair, should be abundant in CdTe. The VCd+TeI pair surprisingly is more preferred than if they occupied the same lattice site as a TeCd which is also more preferred than the two defects spatially separated. The bound pair shows no midgap states and only a deep acceptor state, as opposed to the performance impacting TeCd and TeI. An exhaustive study of the most prevalent extended defect in CdTe, the Σ3 twin, has revealed a number of interesting trends. Pure twins are electrically inactive, show no residual strain or change in the bulk modulus, and the very low interfacial energies decay from 5mJ/m² down to 0 over 30 atomic columns of separation. Most native point defects along these twins exhibit either unchanged bulklike or a reduced formation energy. Most notably is the TeCd antisite which is significantly more preferred yet has its states pushed closer towards the band edges. Also, the TeI defect which is far more preferred along twins than in the bulk similarly has its midgap states pushed toward the band edges. When considering defect pairs however, the interfacial plane of a twin boundary greatly reduces the binding energy of commonly bound pairs such as Schottky and Frenkel pairs, making such a binding over a twin interface unlikely. A novel algorithm and software implementation was developed to generate grain boundary structures rapidly and efficiently. This algorithm generates structures from their degrees of freedom directly, satisfies periodic boundary conditions, has a minimal number of atoms, is computationally efficient, and requires a suitable separation distance between periodic boundary planes. This implementation has a wide array of future uses in both CdTe and other material systems. Using the GB Genie code representative atomic structures were generated to correspond to bi-crystals fabricated at the University of Texas at Dallas. These structures resembled that of the STEM images obtained for these interfaces, and some highly symmetric interfaces reveal electrically active behavior and higher interfacial energies than anticipated. Other interfacial energies and atomic relaxation studies are still ongoing. The GB Genie code has also enabled a study random-angle GBs which exhibit a low degree of symmetry. Studies of these early structures reveal that relative atomic coordination at the interface plays a major role in electrical activity for solar cell performance and interfacial energy. Highly disordered
106 |Conclusions interfaces actually have been revealed to be electrically benign to solar cell performance. Significant progress has been made to enhance grain boundary engineering for CdTe solar cells. A complete bulk CdTe analysis has been completed with the notable addition of defect pairs and first principles recombination lifetime computations. These results both support previous work by other authors, and measurements of >10ns lifetimes in CdTe devices. A comprehensive study of the most common interface, Σ3 twins was completed suggesting that they are mildly detrimental overall. A powerful new tool has been developed to generate grain boundary structures by exploiting all symmetries and creating a minimal size. Sample structures have been created to correspond to bi-crystals and a larger sampling of the GB parameter space, of which the analysis is ongoing. Lastly in parallel with this effort a series of interactive software tools was developed for use by the community. While the ultimate goal of improving CdTe grain boundary networks remains unsolved, many steps have been taken to improve the fundamental knowledge of CdTe interfaces and their effect on solar cell performance. What remains now is to analyze more thoroughly other atomic structures of such interfaces and investigate their related effects and impact on solar cell performance as well as future passivation methods.
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8) Vita Christopher Buurma received his B.S in physics and mathematics with a minor in computer science from Ashland University in 2007 and he is currently the Director of Computation and Modeling at EPIR Technologies. As a computational physicist by training he has considerable knowledge of many programming languages and computational concepts along with diverse experience in various aspects of photovoltaics including atomistic, abinitio, device, system modeling and simulation, as well as computer algorithm design, image/data processing, regression, statistical analysis, equation solution, and other aspects of computational physics. At First Solar in 2008, and then at EPIR since 2009 Buurma has developed sophisticated modeling and fitting routines for both IR and solar applications to extract current contributing mechanisms and parameters to predict limitations and optimize PV performance from a variety of test data. A complete diode device, optical and systems modeling package was developed (Infrared Measurement Analyzer) to directly apply the developed models to test data in a robust and intuitive format. Simulation and analysis tools for hyperspectral imaging and atmospheric modeling were also incorporated into IRMA. He is the lead developer of both SoMA and IRMA analysis codes which blend characterization data and physical models with a reverse-modeling algorithm to automatically analyze measurement data. Other custom analysis routines and GUIs were also developed by Mr. Buurma to quickly utilize such fits and compare parameters across large datasets and various measurements to determine statistical significance in the presence of practical significance. Buurma has written, lead, directed and succeeded on a variety of DoD contracts while at EPIR including research on hyperspectral image capture and analysis, long-range detection of chemicals and biological aerosols, MEMS Fabry-Perot imagers, and more. Buurma’s experience in photovoltaic systems, together with his mathematical and computational capabilities have given him a unique perspective in understanding photovoltaic device physics and experimental data analysis. Graduate studies have focused on fundamental materials physics and simulations. The goal is to enhance and improve knowledge gained from available experiments by pairing them directly with accurate material and device simulations. Buurma has given a number of talks and published papers relating to both first-principles studies and PV device physics. Chronological Publication List: 1. R. Banai, C. Blissett, C. Buurma, E. Colegrove, P. Bechmann, J. Ellsworth, M. Morley, S. Barnes, C. Lennon, C. Gilmore, R. Dhere, J. Bergeson, M. Scott, and T. Gessert, “Polycrystalline CdTe solar cells on
116 |Vita buffered commercial TCO-coated glass with efficiencies above 15%,” 2011 37th IEEE Photovolt. Spec. Conf., pp. 003410–003414, Jun. 2011. 2. E. Colegrove, R. Banai, C. Blissett, C. Buurma, J. Ellsworth, M. Morley, S. Barnes, C. Gilmore, J. D. Bergeson, R. Dhere, M. Scott, and T. Gessert, “High-Efficiency Polycrystalline CdS/CdTe Solar Cells on Buffered Commercial TCO-Coated Glass,” J. Electron. Mater., vol. 41, no. 10, pp. 2833–2837, May 2012. 3. P. Boieriu, S. Velicu, R. Bommena, C. Buurma, C. Blisset, C. Grein, S. Sivananthan, and P. Hagler, “High operation temperature of HgCdTe photodiodes by bulk defect passivation,” vol. 8631, p. 86311J–86311J– 20, Feb. 2013. 4. F. Troni, R. Menozzi, E. Colegrove, and C. Buurma, “Simulation of Current Transport in Polycrystalline CdTe Solar Cells,” J. Electron. Mater., Aug. 2013. 5. C. Buurma, P. Boieriu, R. Bommena, and S. Sivananthan, “Applications of the Infrared Measurement Analyzer: Hydrogenated LWIR HgCdTe Detectors,” J. Electron. Mater., Sep. 2013. 6. P. Boieriu, C. Buurma, R. Bommena, C. Blissett, C. Grein, and S. Sivananthan, “Effects of Inductively Coupled Plasma Hydrogen on Long-Wavelength Infrared HgCdTe Photodiodes,” J. Electron. Mater., vol. 42, no. 12, pp. 3379–3384, Sep. 2013. 7. C. Buurma, S. Krishnamurthy, and S. Sivananthan, “Shockley-ReadHall lifetimes in CdTe,” J. Appl. Phys., vol. 116, no. 1, p. 013102, Jul. 2014. 8. S. Fahey, P. Boieriu, C. Morath, D. Guidry, L. Treider, R. Bommena, J. Zhao, C. Buurma, C. Grein, and S. Sivananthan, “Influence of Hydrogenation on Electrical Conduction in HgCdTe Thin Films on Silicon,” J. Electron. Mater., vol. 43, no. 8, pp. 2831–2840, Apr. 2014. 9. J. H. Park, C. Buurma, S. Sivananthan, R. Kodama, W. Gao, and T. a. Gessert, “The effect of post-annealing on Indium Tin Oxide thin films by magnetron sputtering method,” Appl. Surf. Sci., vol. 307, pp. 388– 392, 2014. 10. T. Paulauskas, C. Buurma, E. Colegrove, Z. Guo, S. Sivananthan, M. K. Y. Chan, and R. F. Klie, “Atomic-resolution characterization of the effects of CdCl2 treatment on poly-crystalline CdTe thin films,” Appl. Phys. Lett., vol. 105, no. 7, p. 071910, Aug. 2014. 11. T. Paulauskas, C. Buurma, E. Colegrove, B. Stafford, Z. Guo, M. K. Y. Chan, C. Sun, M. J. Kim, S. Sivananthan, and R. F. Klie, “Atomic scale study of polar Lomer–Cottrell and Hirth lock dislocation cores in CdTe,” Acta Crystallogr. Sect. A Found. Adv., vol. 70, no. 6, pp. 524– 531, Sep. 2014. 12. S. Velicu, C. Buurma, J. D. Bergeson, T. S. Kim, J. Kubby, and N. Gupta, “Miniaturized imaging spectrometer based on Fabry-Perot
Christopher Buurma | 117 MOEMS filters and HgCdTe infrared focal plane arrays,” in SPIE Proceedings, 2014, vol. 9100, p. 91000F. 13. C. Buurma, T. Paulauskas, C. Sun, M. Kim, R. Klie, M. K. Y. Chan, and S. Sivananthan, “An ab-initio study of native point defects and defect pairs along twin boundaries in CdTe,” 2014. Presentations and Talks 1. C. Buurma, M. K. Y. Chan, Z. Guo, R. Klie, and S. Sivananthan, “First principles calculations for the formation energy and density of states of twin and microtwin defects in poly-CdTe,” in II-VI Conference, 2013, pp. 4–7. 2. C. Buurma and J Garland, “Enhancing the efficiency of Si solar cells with CdZnTe,” in UIC Research Forum, 2013. 3. C. Buurma, M. K. Y. Chan, T. Pauluaskas, R. Klie, and S. Sivananthan, “Stacking faults and lamellar twins with intrinsic point defects in poly-crystalline CdTe analyzed by Density Functional Theory,” in American Physical Society, 2014. 4. C. Buurma, M. K. Y. Chan, T. Pauluaskas, R. Klie, and S. Sivananthan, “An ab-initio study of micro and lamellar twins with local intrinsic point defects in poly-CdTe,” in Materials Research Society, 2014. 5. R. F. Klie, S. Sivananthan, M. Kim, and M. K. . Chan, “How grain boundaries affect the efficiency of poly-CdTe solar-cells A fundamental atomic-scale study of grain boundary dislocation cores using CdTe bicrystal thin films,” in DOE Sunshot Grand Challenge Summit, 2014. 6. C. Buurma, T. Paulauskas, Z. Guo, E. Colegrove, W. Gao, M. Kim, S. Sivananthan, M. K. Y. Chan, and R. F. Klie, “An ab-initio study of twins within grains of poly-CdTe,” in NREL PV Workshop, 2014. 7. C. Buurma and M K Y Chan, “DFT calculations of pure and defectdecorated coherent twins in CdTe,” in Hands-on Summer School: Electronic Structure Theory for Materials and (Bio)molecules, 2014. 8. C. Buurma, F. Sen, M. K. Y. Chan, R. Klie, and S. Sivananthan, “A Computational Investigation of Random Angle Grain Boundaries for CdTe Solar Cells,” in American Physical Society, 2015. 9. C. Buurma, F. G. Sen, T. Paulauskas, C. Sun, M. Kim, S. Sivananthan, R. F. Klie, and M. K. Y. Chan, “Creation and Analysis of Atomic Structures for CdTe Bi - crystal Interfaces by the Grain Boundary Genie,” in 42nd IEEE Photovoltaic Specialists Conference, 2015, pp. 3– 5. 10. C. Buurma, “From first principles to device modeling,” in 4th workshop on thin film solar cells and their science and technology, 2015.
