ANNNI Stacking Faults¶

Estimates intrinsic and extrinsic stacking fault energies of an FCC-forming composition from the relaxed energies of three bulk polytypes, using the second-order axial next-nearest-neighbor Ising (ANNNI) model.

Overview¶

Unlike USFE, which shears a supercell rigidly across a slip plane and evaluates the resulting energy landscape directly, the ANNNI approach never builds a faulted supercell at all. A stacking fault is a local change in the sequence of close-packed planes, and the ANNNI model treats each plane as an Ising-like spin whose interactions with nearby planes are inferred from the energies of small-period stacking sequences: FCC (ABCABC...), HCP (ABAB...), and DHCP (ABACABAC..., a doubled HCP-like period). To second order in the axial interactions, the intrinsic and extrinsic stacking fault energies can be written purely in terms of the (bulk) energies of these three polytypes. No explicit fault geometry or supercell is needed. ANNNIStackingFaultAnalyzer builds all three structures for a given composition via SqsGenerator (through ANNNIStackingFaultTransformation), relaxes the FCC cell, and evaluates HCP/DHCP at the same volume per atom as the relaxed FCC cell so all three energies are compared at a common density.

Theory¶

Let \(E_{\text{fcc}}\), \(E_{\text{hcp}}\), and \(E_{\text{dhcp}}\) be the relaxed energies per atom of the FCC, HCP, and DHCP polytypes, all evaluated at the FCC equilibrium volume per atom. Let \(a\) be the conventional FCC lattice parameter recovered from that volume (\(V_{\text{atom}} = a^3/4\), i.e. 4 atoms per conventional cubic cell), and

\[ A = \frac{\sqrt{3}}{4}\,a^2 \]

the area per atom of an FCC \(\{111\}\) plane. The second-order ANNNI formulae for the intrinsic stacking fault energy (ISFE) and extrinsic stacking fault energy (ESFE) are:

\[ \gamma_{\text{ISFE}} = \frac{E_{\text{hcp}} + 2 E_{\text{dhcp}} - 3 E_{\text{fcc}}}{A} \]

\[ \gamma_{\text{ESFE}} = \frac{4 \left( E_{\text{dhcp}} - E_{\text{fcc}} \right)}{A} \]

Both are reported in eV/Å². An intrinsic fault corresponds to a single missing plane in the FCC stacking sequence (locally HCP-like across one plane); an extrinsic fault corresponds to a single inserted plane (locally HCP-like across two planes, i.e. a thin DHCP-like slab). This is why ISFE mixes all three polytypes while ESFE only contrasts DHCP against FCC.

References¶

Denteneer, P. J. H., & van Haeringen, W. (1987). Stacking-fault energies in semiconductors from first-principles calculations. Journal of Physics C: Solid State Physics, 20(32), L883-L887. https://doi.org/10.1088/0022-3719/20/32/001