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10.4.9. Two-band (and multi-band) EAM potentials with hybrid/overlay

Some embedded-atom-method (EAM) potentials add a second embedding term to the standard EAM energy expression in order to capture effects that a single density and embedding function cannot. The best known examples are the two-band (or double-band, “2BM”) models for Fe-Cr, which add an “s-band” embedding term on top of the usual “d-band” EAM so that the sign of the alloy mixing enthalpy changes with chromium concentration (Olsson), (Bonny).

This Howto shows that such a model can be run with the existing pair_style eam/fs and pair_style hybrid/overlay commands, without modifying or recompiling LAMMPS, and how to build the required potential files.

The two-band energy expression

In a two-band model the energy of atom i is

\[E_i = \frac{1}{2} \sum_{j \neq i} V(r_{ij}) + F^{d}\!\left(\rho^{d}_i\right) + F^{s}\!\left(\rho^{s}_i\right), \qquad \rho^{d}_i = \sum_{j \neq i} \phi^{d}(r_{ij}), \quad \rho^{s}_i = \sum_{j \neq i} \phi^{s}(r_{ij}) .\]

This is an ordinary EAM (pair term \(V\), d-band density \(\phi^{d}\), d-band embedding \(F^{d}\)) plus one extra embedding term \(F^{s}\) acting on a second, independent density \(\rho^{s}\). Grouping the terms,

\[E_i = \underbrace{\left[\,\frac{1}{2} \sum_{j} V + F^{d}(\rho^{d})\,\right]}_{\text{a standard eam/fs}} + \underbrace{\left[\,F^{s}(\rho^{s})\,\right]}_{\text{eam/fs with }V\equiv 0}\]

shows that the total energy is the sum of two EAM potentials. Adding energies, forces, and virials of two potentials acting on the same atoms is exactly what pair_style hybrid/overlay does.

Why the s-band file must be eam/fs

In the two-band models the second (s-band) density is non-zero only between unlike atoms; it vanishes between atoms of the same element (Bonny):

\[\phi^{s}_{AA} = \phi^{s}_{BB} = 0, \qquad \phi^{s}_{AB} = \phi^{s}_{BA} \neq 0 .\]

A density that depends on the element of both the source and the target atom can only be expressed in the Finnis-Sinclair form, i.e. with pair_style eam/fs, whose density functions \(\rho_{\alpha\beta}(r)\) are indexed per element pair. It cannot be written as an eam/alloy potential, which has a single density per element summed over all neighbors. Therefore the s-band file is always an eam/fs file in which the like-element density functions are set to zero and only the unlike-element density is non-zero. Its pair-potential (\(r\,\phi\)) section is set entirely to zero, because the pair term belongs to the d-band file.

The d-band file is an ordinary binary EAM (the pair term plus the d-band density and embedding). Because its density depends only on the neighbor’s element it could equally be written as eam/alloy; using eam/fs for both files keeps the recipe uniform.

The recipe

pair_style hybrid/overlay eam/fs eam/fs
pair_coeff * * eam/fs 1 FeCr.dband.eam.fs Fe Cr   # pair V + d-band F_d(rho_d)
pair_coeff * * eam/fs 2 FeCr.sband.eam.fs Fe Cr   # s-band F_s(rho_s), no pair

The numeric 1 and 2 after eam/fs are required because the same sub-style is used twice; see pair_hybrid. Each eam/fs instance keeps its own density and embedding-derivative arrays and performs its own communication, so the two densities \(\rho^{d}\) and \(\rho^{s}\) never mix. hybrid/overlay sums the two contributions, and because the s-band file has no pair term there is no double counting: the combined energy is exactly \(\tfrac{1}{2}\sum V + F^{d}(\rho^{d}) + F^{s}(\rho^{s})\).

The two files are independent eam/fs files and may use different radial grids and cutoffs; the neighbor list is built with the larger of the two cutoffs. Because both files store the same element masses, you may also set masses explicitly with the mass command to avoid any ambiguity.

This generalizes to more than two bands: overlay N eam/fs files to add N embedding terms. The same construction works for any model of the form “standard EAM plus one or more additive embedding terms.”

Building the s-band file

The helper script tools/eam_generate/two_band_fecr.py writes the s-band eam/fs file (cross-only density, \(F^{s}\) embedding, zero pair term) and can also emit an illustrative demo d-band file plus a ready-to-run input deck:

cd tools/eam_generate
# self-contained runnable demo (writes both files + a sample input):
./two_band_fecr.py --demo --prefix FeCr_demo

# real use: add an s-band to your own trusted Fe-Cr d-band eam/fs file:
./two_band_fecr.py --dband MyFeCr.eam.fs --preset olsson-vasp --prefix FeCr_2bm

The script implements the s-band density as the square of a 4s Slater function with a smooth cutoff and the s-band embedding as \(F^{s}(\rho)=c_1\sqrt{\rho}+c_2\rho^2+c_3\rho^4\), matching the forms used in the published models. It ships parameter presets taken from the literature, but note that the absolute normalization of \(\rho^{s}\) and the matching \(F^{s}\) coefficients are convention-sensitive: validate a preset against a known result (for instance the concentration dependence of the mixing enthalpy) before using it for production work. The default --demo parameters are self-consistent and illustrative only; they exercise the overlay mechanics but are not a published fit.

A worked check

Running the demo input produced by --demo and decomposing the energy with compute pair confirms the construction:

compute ed all pair eam/fs 1 epair   # d-band sub-style energy
compute es all pair eam/fs 2 epair   # s-band sub-style energy
thermo_style custom step pe c_ed c_es

The total potential energy is exactly c_ed + c_es. For a pure element the s-band energy c_es is identically zero, because the like-element s-density is zero, so \(\rho^{s}=0\) and \(F^{s}(0)=0\). Introducing unlike neighbors (an alloy) makes c_es non-zero: the s-band term samples the local concentration, which is the entire purpose of the model.

Restrictions and notes

The s-band embedding of these models typically has a \(\sqrt{\rho}\) term, whose slope diverges as \(\rho \to 0\). This is harmless here: LAMMPS tabulates \(F^{s}\) as a spline (finite at every grid point) and the force contribution is multiplied by the s-density derivative, which is identically zero for like pairs. A pure-element atom therefore contributes no spurious s-band force.
This Howto reproduces the two-band functional form exactly. Reproducing a specific published parameterization additionally requires the correct d-band/pair files (for Fe-Cr, the Mendelev iron potential plus the refit chromium and cross interactions) and validated s-band coefficients.

(Olsson) Olsson, Wallenius, Domain, Nordlund, Malerba, Physical Review B, 72, 214119 (2005).

(Bonny) Bonny, Pasianot, Terentyev, Malerba, Philosophical Magazine, 91, 1724 (2011).