Embedded atom method

GPUMD suppports two different analytical forms of embedded atom method (EAM) potentials. Using the form by Zhou et al. can simulate alloys with up to 10 atom types [Zhou2004], while using the form of Dai et al. the implementation only applies to systems with a single atom type [Dai2006].

Potential form

General form

The site potential energy is

\[U_i = \frac{1}{2} \sum_{j\neq i} \phi(r_{ij}) + F (\rho_i).\]

Here, the part with \(\phi(r_{ij})\) is a pairwise potential and \(F (\rho_i)\) is the embedding potential, which depends on the electron density \(\rho_i\) at site \(i\). The many-body part of the EAM potential comes from the embedding potential.

The density \(\rho_i\) is contributed by the neighbors of \(i\):

\[\rho_i = \sum_{j\neq i} f(r_{ij}).\]

Therefore, the form of an EAM potential is completely determined by the three functions: \(\phi\), \(f\), and \(F\).

Version from [Zhou2004]

The pair potential between two atoms of the same type \(a\) is

\[\phi^{aa}(r) = \frac{ A^a \exp[-\alpha(r/r_e^a-1)] } { 1+(r/r_e^a-\kappa^a)^{20} } - \frac{ B^a \exp[-\beta(r/r_e^a-1)] } { 1+(r/r_e^a-\lambda^a)^{20} }.\]

The contribution of the electron density from an atom of type \(a\) is

\[f^a(r) = \frac{ f_e^a \exp[-\beta(r/r_e^a-1)] } { 1+(r/r_e^a-\lambda^a)^{20} }.\]

The pair potential between two atoms of different types \(a\) and \(b\) is then constructed as

\[\phi^{ab}(r) = \frac{1}{2} \left[ \frac{ f^b(r) } { f^a(r) } \phi^{aa}(r) + \frac{ f^a(r) } { f^b(r) } \phi^{bb}(r) \right].\]

The embedding energy function is piecewise:

\[\begin{split}F(\rho) = \begin{cases} \sum_{i=0}^3 F_{ni} \left( \frac{\rho}{\rho_n}-1\right)^i & \rho < 0.85\rho_e \\ \sum_{i=0}^3 F_{i} \left( \frac{\rho}{\rho_e}-1\right)^i & 0.85\rho_e \leq \rho < 1.15\rho_e \\ F_{e} \left[ 1- \ln \left(\frac{\rho}{\rho_s}\right)^{\eta}\right] \left(\frac{\rho}{\rho_s}\right)^{\eta} & \rho \geq 1.15\rho_e \end{cases}\end{split}\]

Version from [Dai2006]

This is a very simple EAM-type potential, which is an extension of the Finnis-Sinclair potential. The function for the pair potential is

\[\begin{split}\phi(r) = \begin{cases} (r-c)^2 \sum_{n=0}^4 c_n r^n & r \leq c \\ 0 & r > c \end{cases}\end{split}\]

The function for the density is

\[\begin{split}f(r) = \begin{cases} (r-d)^2 + B^2 (r-d)^4 & r \leq d \\ 0 & r > d \end{cases}\end{split}\]

The function for the embedding energy is

\[F(\rho) = - A \rho^{1/2}.\]

File format

The potential file for the version from [Zhou2004] reads:

eam_zhou_2004 num_types <list of elements>
r_e f_e rho_e rho_s alpha beta A B kappa lambda F_n0 F_n1 F_n2 F_n3 F_0 F_1 F_2 F_3 eta F_e cutoff

There are num_types rows of parameters but here we have only written a single row above. The order of the rows should be consistent with the <list of elements> in the first line.

The last parameter cutoff is the cutoff distance, which is not intrinsic to the model. The order of the parameters is the same as in Table III of [Zhou2004]. For multi-component systems, GPUMD will use the largest cutoff for every atom type.

The potential file for the version from [Dai2006] reads:

eam_dai_2006 1 Element
A d c c_0 c_1 c_2 c_3 c_4 B

Here, Element is the chemical symbol of the element.