Migration Pathway Analysis

Steps for Migration Energy Calculation

  1. Define Initial and Final States:
  2. Perform geometry optimization on the initial and final configurations of the defect.

  3. Ensure both structures are fully relaxed.

  4. Generate Intermediate Images:

  5. Use tools like Quantum Espresso's neb.x or external tools (e.g., ASE or pymatgen) to interpolate between initial and final states.

  6. Create multiple intermediate images (e.g., 5-10 images) to accurately resolve the energy barrier.

  7. Perform NEB Calculation:

  8. Run NEB calculations to relax the images and calculate their total energies.

  9. Output the energy profile along the migration path.

  10. Extract Migration Energy:

  11. Plot the energy of each image along the migration path.
  12. The migration energy barrier is the difference between the maximum energy along the path and the energy of the initial state.

Key Equation

The migration energy barrier (( E_\text{mig} )) is calculated as:

[ E_\text{mig} = E_\text{max} - E_\text{initial} ]

Where: - ( E_\text{max} ): Energy of the highest-energy image along the NEB path. - ( E_\text{initial} ): Total energy of the initial state.

Python Script for Post-Processing NEB Data

You can post-process NEB output files to extract and visualize the energy profile:

import pandas as pd
import matplotlib.pyplot as plt

# Load NEB energy data
neb_file = "neb_results.csv"  # Replace with your NEB output file
df_neb = pd.read_csv(neb_file)

# Extract image number and energy
images = df_neb['Image']
energies = df_neb['Energy (eV)']

# Plot energy profile
plt.plot(images, energies, marker='o', linestyle='-', color='b')
plt.xlabel('NEB Image Number')
plt.ylabel('Total Energy (eV)')
plt.title('Migration Energy Pathway')
plt.grid()
plt.show()

# Calculate migration barrier
migration_barrier = max(energies) - energies.iloc[0]
print(f"Migration Energy Barrier: {migration_barrier:.3f} eV")

Output

The NEB analysis will produce:

  1. Energy Profile Plot:

  2. Shows the total energy of images along the migration pathway.

  3. Migration Energy Barrier:

  4. Quantifies the energy cost for defect migration.

Example System: Cr Migration in Sb₂Te₃

For Cr-doped Sb₂Te₃: - Initial state: Cr atom at an interstitial site. - Final state: Cr atom at a neighboring interstitial site. - The NEB calculation reveals the energy barrier for Cr migration, which is crucial for understanding its diffusivity and stability.

Tools Used

  • Quantum Espresso (neb.x): NEB calculation engine.
  • Matplotlib & Pandas: Post-processing and visualization of NEB data.

Conclusion

Migration energy analysis using the NEB method provides insights into defect mobility and energy barriers in materials. This analysis is particularly useful for understanding the thermoelectric performance and defect dynamics of systems like Cr-doped Sb₂Te₃.