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A Computational Framework for Automation of Point Defect Calculations
Anuj Goyal, Prashun Gorai, Haowei Peng, Stephan Lany, Vladan Stevanovic
TL;DR
Point-defect calculations require difficult interstitial-site generation and corrections for finite-size and electronic-structure errors. The paper develops an automated DFT framework that generates and relaxes defects and applies correction schemes, validating it on Si, ZnO, and In2O3. The framework recovers known defect structures and electronic properties while discovering interstitial configurations in complex crystals.
Problem
Interstitial sites and complex defect configurations are difficult to identify, while supercell finite-size effects and DFT band-gap errors complicate defect energetics.
Method
The framework automates defect generation, DFT relaxation, finite-size corrections, and GW-based band-edge corrections, using Voronoi tessellation to identify interstitial candidates.
Results
The framework recovers known intrinsic defect structures and electronic properties in Si, ZnO, and In2O3, while identifying interstitial configurations in complex structures.
Takeaways & Limitations
Automated point-defect analysis can support discovery of interstitials in complex crystal structures and prediction of defect formation energies using finite-size corrections.
Abstract
from arXiv · showhide
A complete and rigorously validated open-source Python framework to automate point defect calculations using density functional theory has been developed. The framework provides an effective and efficient method for defect structure generation, and creation of simple yet customizable workflows to analyze defect calculations. The package provides the capability to compute widely-accepted correction schemes to overcome finite-size effects, including (1) potential alignment, (2) image-charge correction, and (3) band filling correction to shallow defects. Using Si, ZnO and In$_2$O$_3$ as test examples, we demonstrate the package capabilities and validate the methodology.
I. INTRODUCTION
The paper presents an automated DFT framework for point-defect calculations, addressing difficult interstitial generation and finite-size and band-gap corrections. It validates the framework on Si, ZnO, and In2O3.
- Point-defect formation energies help determine semiconductor defect concentrations and properties relevant to microelectronics, optoelectronics, and thermoelectrics.
- Interstitial generation is challenging because complex multinary systems contain many possible sites and defects can adopt split or dumbbell configurations.
- The framework implements potential alignment, image-charge, and band filling corrections to address finite-size effects in supercell calculations.
- Its modular design supports additional correction schemes, supercell-size extrapolation, and fitted elemental-phase reference energies for chemical potentials.
- GW band-edge shifts correct defect formation energies across charge states, while DFT+GW provides a lower-overhead alternative to hybrid-functional defect calculations.
- The framework is illustrated and validated using Si, ZnO, and In2O3, covering 17 unique interstitial and vacancy structures in multiple charge states.
II. OVERVIEW OF THE AUTOMATED DEFECT FRAMEWORK
The automated framework combines defect-structure generation, DFT relaxation, and finite-size and band-gap corrections to compute defect formation energies.
- The workflow generates defect structures, relaxes defect supercells with DFT using PyLada, and determines finite-size and band-gap corrections.
A. Generate Defect Structures
Defect structures are generated from relaxed crystal inputs by creating vacancies or substitutions and sampling interstitial candidates with Voronoi tessellation.
- The workflow uses a fully relaxed primitive cell to create supercells containing vacancies, substitutions, or interstitial candidates.
- Vacancies and substitutions are generated by identifying occupied Wyckoff positions and removing or replacing the corresponding atom.
- Randomly displacing first-neighbor atoms by approximately 0.1 Å breaks site symmetry and helps capture non-symmetric defect configurations.
- Voronoi regions are computed for occupied Wyckoff positions, and symmetry-inequivalent vertices, face centers, and edge centers become interstitial candidates.
- Candidate-site counts increase as crystal symmetry decreases and structural complexity increases; In2O3 uses a 0.5 Å minimum tolerance to merge nearby sampled sites.
B. Perform Defect Calculations
Defect calculations use PAW-based DFT with material-specific exchange-correlation settings, followed by bulk characterization and charged-defect relaxations.
- Calculations use PAW-VASP with PBE, GGA for Si and In2O3, and spin-polarized GGA+U for ZnO with U(Znd) = 6 eV.
- The computational setup uses a 340 eV plane-wave cutoff, Monkhorst-Pack k-point sampling, and fully relaxed structures from the ICSD.
- The workflow relaxes bulk volume, cell shape, and ionic positions before calculating dielectric constants and GW properties on the relaxed primitive cell.
- Defects are created in bulk supercells and relaxed only in ionic positions across multiple charge states.
- Interstitial calculations first relax all candidates neutrally, then relax only unique interstitial structures in additional charge states.
C. Compute Defect Formation Energy
Defect formation energy is calculated from defect and host supercell energies, chemical potentials, Fermi energy, and finite-size corrections. The workflow distinguishes starting and relaxed interstitial structures across Si, ZnO, and In2O3.
- Defect formation energy is calculated from the total DFT energies of the defect and host supercells.
- The formation-energy expression includes a charge-state term qEF and a finite-size correction term Ecorr.
- Chemical potentials account for atoms added to or removed from the host supercell when forming the defect.
- Figure 3 compares the numbers of distinct starting and final DFT-relaxed interstitial structures in Si, ZnO, and In2O3.
1. Chemical Potential and Phase Stability
Chemical potentials represent atomic reservoir energies and are obtained from FERE reference energies. Formation enthalpies and thermodynamic stability conditions determine the allowed chemical-potential limits for the studied compounds.
- Chemical potentials µi reflect the energies of reservoirs for atoms involved in defect creation.
- The framework obtains reference chemical potentials from fitted elemental-phase reference energies, denoted µFERE_i.
- FERE energies provide formation enthalpies for competing phases needed to determine thermodynamic limits on ∆µi.
- The ZnO and In2O3 stability conditions are ∆µZn + ∆µO = ∆Hf(ZnO) and 2∆µIn + 3∆µO = ∆Hf(In2O3), respectively.
2. Electron Chemical Potential
The electron chemical potential is represented by the Fermi energy, referenced to the host VBM and bounded by the CBM. Defect formation energies incorporate potential alignment, image-charge, and band-filling corrections within the supercell framework.
- The Fermi energy measures the electron chemical potential, is referenced to the host VBM, and is bounded by the CBM.
- DFT band gaps are corrected using GW-derived shifts for the VBM and CBM, which adjust defect formation energies across charge states.
- Potential alignment restores the host VBM reference for charged-defect calculations and affects the Fermi-energy contribution to formation energies.Reference potentials are obtained from atomic-sphere-averaged electrostatic potentials at sites far from the defect.
- Image-charge correction removes spurious electrostatic interactions associated with charged defects and homogeneous compensating backgrounds.Its formulation depends on supercell size, dielectric constant, Madelung constant, and shape factor.
- Band-filling correction addresses Moss-Burstein-type effects caused by high defect concentrations in finite supercells.For shallow donors, it uses weighted k-point band occupations and host band-edge energies after potential alignment.
A. Silicon
The framework identifies multiple stable and metastable silicon self-interstitial and vacancy configurations, while calculated energetics and transition levels generally agree with prior studies.
- Self-interstitials: Three silicon self-interstitial structures—split, hexagonal, and tetrahedral—emerge from seven candidate sites, with neutral split lowest in formation energy at 3.25 eV.The hexagonal structure is 0.2 eV higher, while tetrahedral is about 0.33 eV higher than split.
- Self-interstitials: The hexagonal interstitial has a non-symmetric configuration 5 meV lower than the symmetric one but relaxes to tetrahedral geometry in charge states 1+ and 2+.The non-symmetric configuration is unstable despite its slightly lower energy.
- Vacancies: Silicon vacancies undergo charge-dependent structural reconstruction associated with reported Negative-U behavior.Negative-U behavior involves energy-lowering distortion upon addition of a second electron.
- Vacancies: Spin-polarized calculations yield higher-energy C2v and D2d vacancy configurations for neutral and 2- charge states, respectively.Using spin-polarized HSE calculations on DFT structures instead gives a direct (2+/0) charge transition.
- Energetics: Computed silicon defect formation energies and charge transition levels agree with reported GGA and HSE calculations, although GGA and GGA+GW transition levels differ noticeably.The difference is attributed mainly to band-edge positions predicted by self-consistent GW calculations.
B. Zinc Oxide
In ZnO, the framework recovers distinct interstitial and vacancy structures and characterizes their relative energetics, charge states, and transition levels consistently with earlier calculations.
- Interstitials: Three oxygen interstitial structures emerge from 17 starting possibilities, with split lowest, split* metastable, and octahedral substantially higher in energy.Split* is 0.21 eV above split and relaxes to split upon further relaxation; octahedral is about 1.8 eV higher than neutral split.
- Interstitials: Octahedral oxygen interstitials have symmetric and non-symmetric configurations, with symmetric 0.3 eV higher and stable only in the neutral charge state.Positive and negative charge states relax to the non-symmetric configuration, which is the lowest-energy octahedral structure.
- Electronic character: Octahedral oxygen interstitials act as deep acceptors, while zinc interstitials act as shallow donors with 2+ as the most stable charge state.Zinc interstitial formation energies reach 2.6 eV at the conduction-band minimum even under Zn-rich conditions.
- Vacancies: The oxygen vacancy undergoes a 2+ to 0 transition 0.45 eV below the conduction-band minimum, confirming reported Negative-U behavior.The zinc vacancy is a deep acceptor with (0/1-) and (1-/2-) levels at 1.16 and 1.58 eV above the valence-band maximum.
- Validation: The ZnO analysis confirms known interstitial and vacancy structures and produces defect energetics and electronic structures consistent with previous calculations.The framework is then applied to In2O3, whose primitive unit cell contains 40 atoms.
C. Indium Oxide
The framework identifies and evaluates diverse intrinsic defect structures in In2O3, including previously unreported oxygen configurations, while characterizing their electronic behavior and formation energies.
- In2O3 is a direct-band-gap semiconductor widely used as a transparent conducting oxide, with intrinsic defects receiving moderate experimental and theoretical attention.
- Among 22 initial oxygen-interstitial candidates, three relaxed structures occupy the general 48e Wyckoff position, with split oxygen the lowest-energy configuration.
- The split* oxygen interstitial is stable only in the neutral charge state and lies about 0.67 eV above the lowest-energy split interstitial, while the 4-fold coordinated structure is about 1.0 eV higher.
- Three indium-interstitial configurations were identified, with the lowest-energy structure placing indium at the 16c Wyckoff position in threefold coordination.
- Oxygen interstitials and indium vacancies produce deep gap states, whereas indium interstitials act as shallow donors with resonances above the CBM.
- The study reports consistency with prior In2O3 DFT calculations and demonstrates the framework’s potential to discover interstitial structures in complex crystals.