Defect energies¶

vacancy_formation_energy, surface_energy and stacking_fault_energy relax the configurations of the structure builder with a calculator and give the energies of their defects.

import numpy as np
from mdapy.build import build_crystal
from mdapy.calculators import EAM
from mdapy.simulation import (
    stacking_fault_energy,
    surface_energy,
    vacancy_formation_energy,
)

copper = EAM.zhou(["Cu"])
a = 3.615
crystal = build_crystal("fcc", a, elements="Cu", repeat=(4, 4, 4))
vacancy_formation_energy(crystal, copper).energy  # eV, about 1.28

layers = build_crystal(
    "fcc",
    a,
    elements="Cu",
    orientation=[(1, 1, -2), (-1, 1, 0), (1, 1, 1)],
    repeat=(2, 3, 4),
)
surface_energy(layers, copper).energy  # J/m², the (111) surface

# the generalized stacking fault energy along [11-2] on (111)
curve = stacking_fault_energy(layers, copper, [a * np.sqrt(6) / 2, 0, 0], points=13)
curve.energy[4]  # J/m², the intrinsic stacking fault at 1/6 [11-2]

The curve, with plot.stacking_fault:

from mdapy.visualization import plot

with plot.style():
    fig, ax = plot.figure("single")
    plot.stacking_fault(curve, ax=ax)
    plot.save(fig, "images/stacking-fault-energy", formats=("png",))

Generalized stacking fault energy of copper

The energy of copper slipped along [11-2] on (111), over a full lattice vector: the unstable stacking fault at the first maximum, the intrinsic stacking fault at the local minimum a third of the way along, and at two thirds the highest energy, with the atoms of the two halves on top of each other.

Parameters¶

Function	Parameters
`vacancy_formation_energy(crystal, calculator, *, atom=0, cell=None, max_force=1e-4, max_steps=10000)`	the perfect `crystal`, the `atom` removed, and how the box relaxes (`cell` of minimize)
`surface_energy(crystal, calculator, *, axis=2, vacuum=15.0, max_force=1e-4, max_steps=10000)`	the periodic `crystal`, the box vector `axis` across the slab, and the `vacuum` in Å added along it
`stacking_fault_energy(crystal, calculator, direction, *, points=21, height=None, axis=2, relax=True, max_force=1e-4, max_steps=10000)`	the `crystal` with the plane of the fault normal to the Cartesian `axis`, the slip vector `direction` in Å, the number of `points` from 0 to the whole vector, the fractional `height` of the fault (default 0.5) and whether the atoms relax along the normal

max_force (eV/Å) and max_steps are those of each minimization, with FIRE; a minimization that does not converge raises an error. The crystal must be large enough that a defect does not meet its periodic images, and the slab thick enough that its surfaces do not feel each other. With a crystal periodic along the normal of the fault, the box moves with the upper half, so that the box holds a single fault (see slip); without it, the fault lies between two free surfaces, whose energy cancels in the difference.

Results¶

Result	Attributes
`VacancyResult`	`energy` (eV), `perfect` and `defect` energies (eV), the relaxed `system`
`SurfaceResult`	`energy` (J/m²), `bulk` and `slab` energies (eV), the `area` of one surface (Å²), the relaxed `system`
`StackingFaultResult`	`fraction` of the slip vector, `displacement` (P, 3) in Å, `energy` (P,) in J/m² relative to no slip, the `area` of the fault (Å²)

Errors¶

They raise InvalidParameter for a crystal of fewer than two atoms, a crystal that is not periodic for a surface, a vacuum that is not positive, fewer than two points, a plane of fault not normal to the axis, an unknown axis, and a minimization that does not converge.

Definition¶

With \(E_N\) the energy of the perfect crystal of \(N\) atoms and \(E_{N-1}\) that of the crystal with a vacancy, both relaxed,

\[ E_v = E_{N-1} - \frac{N-1}{N} E_N . \]

With \(E_\text{slab}\) the energy of the crystal opened by the vacuum, relaxed, and \(A\) the area of each of its two surfaces,

\[ \gamma_s = \frac{E_\text{slab} - E_\text{bulk}}{2A} . \]

The generalized stacking fault energy at the slip \(f\,\mathbf d\) is

\[ \gamma(f) = \frac{E(f\,\mathbf d) - E(0)}{A}, \]

each configuration relaxed along the normal of the fault only: a movable column keeps the atoms still along the plane, as LAMMPS fix setforce 0 0 NULL does. In fcc along \([11\bar2]\) the curve has the unstable stacking fault energy as its first maximum and the intrinsic stacking fault energy at \(\tfrac16[11\bar2]\).