Charged Point Defect Calculations

Overview

This guide outlines the standard first-principles workflow for calculating point defect formation energies, charge transition levels, and equilibrium defect concentrations in periodic Density Functional Theory (DFT) calculations.

The Governing Thermodynamics

The formation energy of a defect $D$ in charge state $q$ is defined by the Zhang-Northrup formula:

$E^f[D^q]=E_{\text{tot}}[D^q]-E_{\text{tot}}[\text{bulk}]-{\sum }_i{n}_i{\mu }_i+q(E_{\text{VBM}}+E_F)+E_{\text{corr}}$

Symbol	Meaning
$E_{\text{tot}}[D^q]$	Total DFT energy of the defect supercell in charge state $q$
$E_{\text{tot}}[\text{bulk}]$	Total DFT energy of the pristine bulk supercell (same volume and atomic count)
$n_i$	Number of atoms of type $i$ added ($n_i>0$) or removed ($n_i<0$)
${\mu }_i$	Thermodynamic chemical potential of species $i$
$E_{\text{VBM}}$	Energy of the Valence Band Maximum of the bulk host crystal
$E_F$	Fermi energy referenced to $E_{\text{VBM}}$ ($0\le E_F\le E_{\text{bandgap}}$)
$E_{\text{corr}}$	Finite-size electrostatic corrections for periodic image charge interactions

Step-by-Step Computational Workflow

Step 1: Bulk Preparation & Band Structure

1. Lattice Optimization: Relax the pristine primitive unit cell to high precision.
2. Supercell Generation: Expand to a supercell (typically $\ge 64$–$128$ atoms) to minimize artificial defect–defect image interactions.
3. Electronic Structure: Compute the static bulk total energy ($E_{\text{tot}}[\text{bulk}]$) and locate the Valence Band Maximum ($E_{\text{VBM}}$).
4. Band Gap Check: Use hybrid functionals (e.g., HSE06) or DFT+$U$ if standard PBE underestimates the bandgap, as defect charge states depend on correct band edge alignment.

Step 2: Supercell Defect Generation & Relaxation

1. Construct Defect Geometry:
   • Vacancy: Remove host atom ($n_i=-1$).
   • Substitution: Remove host atom ($n_i=-1$), insert dopant ($n_j=+1$).
   • Interstitial: Insert atom ($n_k=+1$) at symmetric or Voronoi-tessellated sites.
2. Set Charge State ($q$): Adjust the total electron count (NELECT in VASP) relative to the default neutral cell.
3. Ionic Relaxation: Relax atomic positions in the defect cell for each charge state $q$, keeping the supercell volume fixed to the bulk host lattice constants.

Step 3: Chemical Potential Boundaries (${\mu }_i$)

1. Compute total energies for all elemental phases and competing secondary phases in the system.
2. Construct the phase stability diagram (convex hull) to determine the allowed chemical potential range (${\mu }_i={\mu }_i^{\circ }+\Delta {\mu }_i$).
3. Identify extreme growth limits (e.g., metal-rich vs. anion-rich conditions) at the vertices of the stability polytope.
4. For gas reservoirs (${\text{O}}_2$, ${\text{N}}_2$), map $\Delta {\mu }_{\text{gas}}(T,P)$ using experimental JANAF thermochemical tables.

Step 4: Electrostatic Finite-Size Corrections ($E_{\text{corr}}$)

Because periodic boundary conditions introduce unphysical image interactions between charged supercells:

1. Calculate Static Dielectric Tensor (${\mathbf{\epsilon }}_{\text{static}}$): Obtain electronic and ionic components using DFPT (${\mathbf{\epsilon }}_{\text{static}}={\mathbf{\epsilon }}_{\infty }+{\mathbf{\epsilon }}_{\text{ion}}$).
2. Apply Correction Scheme:
   • Freysoldt–Neugebauer–Van de Walle (FNV): For isotropic or mildly anisotropic systems, using planar-averaged electrostatic potentials.
   • Kumagai–Oba Scheme: For highly anisotropic systems, using atomic core-level potential alignment.
3. Potential Alignment ($\Delta V_{\text{PA}}$): Align the electrostatic potential of a reference atom far from the defect core with the corresponding atom in the bulk cell.

Post-Processing & Concentration Analysis

Thermodynamic Transition Levels $\epsilon (q/q^{\prime })$

The Fermi level position inside the bandgap where the ground state charge state switches from $q$ to $q^{\prime }$:

$\epsilon (q/q^{\prime })=\frac{E^f[D^q;E_F=0]-E^f[D^{q^{\prime }};E_F=0]}{q^{\prime }-q}$

Self-Consistent Equilibrium Fermi Level ($E_F^{\text{eq}}$)

Defect concentrations ($C_D$) follow Boltzmann statistics:

$C_D(q,E_F)=N_{\text{sites}}{g}_{D,q}\exp \left(-\frac{E^f[D^q;E_F]}{k_{B}T}\right)$

To find the true $E_F^{\text{eq}}$, solve the global charge neutrality equation numerically:

$p(E_F)-n(E_F)+{\sum }_D{\sum }_{q}q\cdot C_D(q,E_F)=0$

Where $n(E_F)$ and $p(E_F)$ are free electron and hole concentrations integrated over the bulk Density of States (DOS).

Recommended Automation Tools

Instead of parsing DFT output files manually, use standard open-source Python packages:

• doped: Complete workflow management built on pymatgen for defect structure generation, automated FNV/Kumagai corrections, and concentration calculations.
• pydefect: VASP-integrated tool for charge localization analysis, 3D potential alignment plateaus, and transition level plotting.
• pymatgen.analysis.defects: Core module for handling defect supercells and charge neutrality root-finding routines.

Key References

1. General Methodology & Review:

   • Freysoldt, C., Grabowski, B., Hickel, T., Neugebauer, J., Kresse, G., Janotti, A., & Van de Walle, C. G. (2014). First-principles calculations for point defects in solids. Reviews of Modern Physics, 86(1), 253.

2. Formulation of Charged Defects:

   • Zhang, S. B., & Northrup, J. E. (1991). Chemical potential dependence of defect formation energies in GaAs: Application to Ga self-diffusion. Physical Review Letters, 67(17), 2339.

3. Finite-Size Correction Schemes:

   • Freysoldt, C., Neugebauer, J., & Van de Walle, C. G. (2009). Fully ab initio finite-size corrections for charged-defect supercell calculations. Physical Review Letters, 102(1), 016402.
   • Kumagai, Y., & Oba, F. (2014). Electrostatics-based finite-size corrections for first-principles point defect calculations. Physical Review B, 89(19), 195205.

4. Band Filling & Potential Alignment:

   • Lany, S., & Zunger, A. (2008). Assessment of correction methods for finite-size effects in charged supercell calculations. Physical Review B, 78(23), 235104.

Related Pages

• VASP Overview
• Compiling VASP
• HPC Guide