Method Overview

Disordered perovskites

We consider complex perovskites or perovskite solid solutions of the form ABO₃, where the A (or B) sublattice is occupied by a mixture of chemical species (e.g., Pb/Sr or Mg/Nb/Ti).

Monte Carlo sampling

At each MC step:

1. Select two A-sites (or B-sites) and propose a swap attempt of their species.

2. Evaluate the energy difference:

   \[\Delta E = E_{\mathrm{new}} - E_{\mathrm{old}}\]

3. Accept the move with Metropolis probability:

   \[\begin{split}P(\mathrm{accept}) = \begin{cases} 1, & \Delta E \le 0 \\ \exp(-\Delta E / k_B T), & \Delta E > 0 \end{cases}\end{split}\]

where \(T\) is the temperature and \(k_B\) is Boltzmann’s constant.

Practical notes

• The “energy” may come from an empirical potential, cluster expansion, machine-learning potential, or any callable energy model in principle.

• Swaps conserve overall composition.