Content:
EAM Potentials
Basic Theory
Modified EAM
Angular-dependent Potentials
The energy in potentials of the Embedded Atom type consists of two parts, a pair potential term specified by the function Φ(r) representing the electrostatic core-core repulsion, and a cohesive term specified by the function F(n) representing the energy the ion core gets when it is "embedded" in the "Electron Sea". This Embedding Energy is a function of the local electron density, which in turn is constructed as a superposition of contributions from neighboring atoms. This electron transfer is specified by the function ρ(r).
The embedding function Fi(n) depends on the type of the embedded atom, the transfer function ρj(r) depends on the type j of the donating atom, whereas the pair potential Φij(r) depends on the types i and j of both atoms involved.
The Embedded Atom Method was suggested by Daw and Baskes (M. S. Daw and M. I. Baskes, Phys. Rev. B 29, 6443 (1984); S. M. Foiles, M. I. Baskes, and M. S. Daw, Phys. Rev. B 33, 7983 (1986) ) as a way to overcome the main problem with two-body potentials: the coordination independence of the bond strength, while still being acceptable fast (about 2 times slower than pair potentials).
Ideas from the Density Functional Theory or the Tight Binding formalism may lead to the following form for the total energy:
Etot = ∑i,jN Φij(rij) + ∑iN F(nhi)
nhi = ∑j ρatij(rij)
While an identification of the pair potential term Φij(rij) with the electrostatic core-core repulsion, and of the cohesive term F(nhi) with the energy the ion core gets when it is "embedded" in the local electron density nhi may be tempting, it is nevertheless without physical justification. Due to invariance properties of the EAM potential, a embedding energy term linear in the "electron" density can be described by pair interactions, thus shifting contributions between embedding and pair energy. So an isolated consideration of either part is not possible - physical relevance only lies in the combination of both.
The local electron density is constructed as a superposition of contributions ρatij(rij) from neighboring atoms.
Also belonging to this analytical form are models like the glue model, the Gupta model and the Finnis-Sinclair potentials.
The Modified Embedded Atom Method was formulated by Baskes (M.I.Baskes, Phys.Rev.B 46 (1992) 2727, M.I.Baskes, Mater.Chem.Phys. 50 (1997) 152). It consists of a generalization of the EAM potentials by including angular terms.
The total energy for the MEAM potentials have a form similar to the EAM potentials:
Here Fi is the embedding energy function, rhoi is the electron density at the position of atom i, phiij is a pair potential, and Sij is the screening function defined below.
For MEAM potentials, the electron density is given by
where the parameters t(s)i are weight factors and the quantities rho(s)i, s=0,...,3, are partial electron densities defined by
The functions L(s)(z) are Legendre polynomials, L(0)(z) = 1, L(1)(z) = z, L(2)(z) = z2-1/3, L(3)(z) = z3-3/5z. The atomic electron density functions from atom j at a distance rij from atom i are
Here, f0i, r0i, and beta(s)i are parameters, fc is a cutoff function, and Sij is the screening function.
The many-body screening function Sij between atoms i and j is defined as the product of the screening factors Sikj due to all other neighbor atoms k of ij, and
Skij is given by
where C1kij and C0kij are parameters and the function gc is defined as
The cutoff function fc used in the atomic electron density functions is given by
where rcij is a cutoff length and Delta_rij is the width of the cutoff region.
The implementation of MEAM is tedious, but can be found in simulation packages such as LAMMPS, IMD etc.
ADP Potentials
ADP stands for "angular dependent potentials". The formalism of ADP was proposed by Yuri Mishin et al..(Y.Mishin, M.J. Mehl, D.A. Papaconstantopulous, Acta Mater. 53, 4029, 2005), and was developed based on the idea that the angular components of atomic interactions must be taken into account to calculate the total energy and force. Two penalty functions are added to the conventional EAM formula to account for angular contributions, one for the dipole contribution and another for the quadrupole contribution.
EMT Potentials