MEAM Potential for Au-Si

MEAM Potential for Au-Si

Adriano Santana and Wei Cai

Created Aug, 2015, Last modified Sep, 2015

This tutorial explains how to specify the parameters for the Au-Si MEAM potential in MD++. It starts with the parameters in pure Au and pure Si potentials, then talks about the Au-Si cross potential.

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Potential for Pure Elements

Original MEAM Potential for Au

As an example, we first describe the original 'Au' potential whose parameters are given in M. I. Baskes, Phys. Rev. B 46, 2727 (1992).

The main parameters in the MEAM potential is specified in the meamf file. (In MD++, this file is in the potentials/MEAMDATA folder.) The lines correspond to 'Au' is given below. Most of these parameters correspond to Table III of Baskes PRB (1992), as shown below.

                                      $\alpha _{i}$      $\beta _{i}^{(0)}$    $\beta _{i}^{(1)}$  $\beta _{i}^{(2)}$  $\beta _{i}^{(3)}$
elt  lat   z    ielement   atwt      alpha    b0       b1     b2    b3   
'Au' 'fcc' 12.     79     196.967 6.34090112  5.449   2.20    6     2.20  

$(R_{i}^{0})$ $E_{i}^{0}$   $A_{i}$   $t_{i}^{(0)}$   $t_{i}^{(1)}$        $t_{i}^{(2)}$         $t_{i}^{(3)}$     $\rho _{0}^{\rm {Au}}$
 alat  esub  asub t0     t1          t2           t3     rozero  ibar
 4.07  3.93  1.04 1.0  1.58956328 1.50776392  2.60609758    1.      3

Note that the nearest neighbor distance $R_{i}^{0}$ = alat / ${\sqrt {2}}$.

$\rho _{0}^{\rm {Au}}$ = rozero will be important only for cross-potential.

ibar is a setting used in the equation of state (EOS). It selects the G(gamma) function in Eq (4) and (5) on the paper by BJ LEE: Phys. Rev. B 64, 184102 (2001)

While the functional form is quite different, the modulus is almost not affected by the choice of ibar.

New 2nn MEAM Potential for Au

We now explain the newer 2nn MEAM potential whose parameters are given by Lee, Shim and Baskes, Phys. Rev. B 68, 144112 (2003), and later modified by Ryu and Cai, J. Phys. Condens. Matter 22, 055401 (2010).

The main parameters in the MEAM potential are specified in the meamf file. (In MD++, this file is in the potentials/MEAMDATA folder.) The lines which correspond to 'AuBt' are given below.

                                      $\alpha _{i}$      $\beta _{i}^{(0)}$    $\beta _{i}^{(1)}$  $\beta _{i}^{(2)}$  $\beta _{i}^{(3)}$
elt  lat   z    ielement   atwt      alpha    b0       b1     b2    b3   
'AuBt' 'fcc' 12.     79      196.967 6.59815965 5.77   2.20   6.0   2.20  

$(R_{i}^{0})$ $E_{i}^{0}$   $A_{i}$   $t_{i}^{(0)}$   $t_{i}^{(1)}$        $t_{i}^{(2)}$         $t_{i}^{(3)}$     $\rho _{0}^{\rm {Au}}$
 alat  esub  asub t0     t1          t2           t3     rozero  ibar
 4.073 3.93 1.00 1.0    1.7        1.64         2.0       1.      3

Note that the nearest neighbor distance $R_{i}^{0}$ = alat / ${\sqrt {2}}$.

We can see that from 'Au' to 'AuBt', the following parameters are changed. The new parameters correspond to values given in Table I of Lee, Shim and Baskes, PRB (2003).

       $\alpha$          $\beta _{i}^{(0)}$  $A_{i}$     $t_{i}^{(1)}$        $t_{i}^{(2)}$         $t_{i}^{(3)}$
'Au'   6.34090112 5.449  1.04  1.58956328 1.50776392  2.60609758
'AuBt' 6.59815965 5.77   1.00    1.7         1.64       2.0

Note that in Table I of Lee et al. (2003), $t^{(1)}=2.90$, while in the meamf file, t1 = 1.7. This is because of the augt1 parameter. In meam_setup_done.F, there is a line

 t1_meam(:) = t1_meam(:) + augt1 * 3.d0/5.d0 * t3_meam(:)

This means that if augt1 = 1.0, then the true value of t1 is 1.7 + 0.6 * 2.0 = 2.9.

augt1 is specified in the AuSi2nn.meam file, as described below.

The AuSi2nn.meam file contains several lines that are relevant for the pure Au potential. The variables in Eq.(A.1) of Ryu and Cai JPCM (2010) are given in the parenthesis.

erose_form = 3
rc = 4.5
attrac(1,1) = -0.182  ($\gamma$)
repuls(1,1) = 4.0  ($\lambda$)
Cmin(1,1,1) = 0.8  ($C_{\rm {min}}$)
augt1 = 1

Note that we label the atomic species of Au as 1. The variable $d=0.05$ is hard coded in meam_setup_done.F (when repuls < 5.0).

MEAM Potential for Si

We use the 'Si4' potential whose parameters are originally given in M. I. Baskes, Phys. Rev. B 46, 2727 (1992), and later modified by Ryu and Cai, J. Phys. Condens. Matter 22, 055401 (2010).

The main parameters in the MEAM potential is specified in the meamf file. (In MD++, this file is in the potentials/MEAMDATA folder.) The lines correspond to 'Siz' is given below. Most of these parameters correspond to Table III of Baskes PRB (1992), as shown below.

                                      $\alpha _{i}$      $\beta _{i}^{(0)}$    $\beta _{i}^{(1)}$  $\beta _{i}^{(2)}$  $\beta _{i}^{(3)}$
elt  lat   z    ielement   atwt      alpha    b0       b1     b2    b3   
'Si4' 'dia' 4.     14     28.086     4.87     4.4     5.5    5.5   5.5   

$(R_{i}^{0})$ $E_{i}^{0}$   $A_{i}$   $t_{i}^{(0)}$   $t_{i}^{(1)}$        $t_{i}^{(2)}$         $t_{i}^{(3)}$     $\rho _{0}^{\rm {Si}}$
 alat  esub  asub t0     t1          t2           t3     rozero  ibar
 5.431 4.63  1.  1.0    3.13        4.47          -1.8      1.48    0

Note that the nearest neighbor distance $R_{i}^{0}$ = alat $\times {\sqrt {3}}/4$ for the diamond cubic structure.

$\rho _{0}^{\rm {Si}}$ = rozero will be important only for cross-potential.

ibar is a setting used in the equation of state (EOS), and will be explained later.

The modification made in Ryu and Cai JPCM (2010) is specified in the AuSi2nn.meam file. The variables in Eq.(A.1) of Ryu and Cai JPCM (2010) are given in the parenthesis.

erose_form = 3
rc = 4.5
attrac(2,2) = -0.36 ($\gamma$)
repuls(2,2) = 16.0  ($\lambda$)
Cmin(2,2,2) = 1.85  ($C_{\rm {min}}$)

Note that we label the atomic species of Si as 2.

Cross Potential between Au and Si

The parameters for the cross potential are specified in AuSi2nn.meam file. The lines relevant for the cross potential (i.e. between species 1 and 2) are shown below. The values correspond to Table 3 of Ryu and Cai, J. Phys. Condens. Matter, 22, 055401 (2010).

re(1,2) = 2.700    ($r_{e}$)
delta(1,2) = 0.125   (related to $E_{c}$, see below)
lattce(1,2) = b1
alpha(1,2) = 5.819     ($\alpha$)
attrac(1,2) = 0.0      
repuls(1,2) = 0.26     ($\gamma$)
Cmin(1,1,2) = 1.9     ($C_{\min }(1,1,2)$)
Cmin(1,2,1) = 0.95     ($C_{\min }(1,2,1)$)
Cmin(1,2,2) = 1.85     ($C_{\min }(1,2,2)$)
Cmin(2,2,1) = 1.0     ($C_{\min }(2,2,1)$)

Table 3 of Ryu and Cai (2010) gives $E_{c}({\rm {AuSi}})=4.155$. This value is related to delta(1,2) through

$E_{c}({\rm {AuSi}})=0.5*[E_{c}({\rm {Au}})+E_{c}({\rm {Si}})]-{\rm {delta}}(1,2)=0.5*(3.93+4.63)-0.125=4.155$.

$\rho _{0}^{\rm {Si}}/\rho _{0}^{\rm {Au}}$ = 1.48 because of the $\rho _{0}^{\rm {Si}}$ and $\rho _{0}^{\rm {Au}}$ values specified above.

Cmax = 2.8 is the default value.

Benchmark in MD++

Compile the code using the following command.

make meam-lammps build=R SYS=gpp

Use the following command to compute the equilibrium lattice constant and cohesive energy of pure Au (FCC). You can download the si-au.tcl from the link.

bin/meam-lammps_gpp scripts/work/si_au/si_au_benchmark.tcl 1

The results are

a0 = 4.07300759775 Angstrom
Ecoh = -3.92996804082 eV

Use the following command to compute the equilibrium lattice constant and cohesive energy of pure Si (DC).

bin/meam-lammps_gpp scripts/work/si_au/si_au_benchmark.tcl 0

The results are

 a0 = 5.43100051581 Angstrom 
 Ecoh = -4.63000000205 eV

Use the following command to compute the equilibrium lattice constant and cohesive energy of Au-Si (B1).

bin/meam-lammps_gpp scripts/work/si_au/si_au_benchmark.tcl 2

The results are

 a0 = 5.4 Angstrom 
 Ecoh = -4.155000000083061 eV

Impurity energy

Use the following command to compute the impurity of a Au atom in Si DC lattice.

bin/meam-lammps_gpp scripts/work/si_au/si_au_benchmark.tcl 4

The results depend slightly on the cell size

cell size, Eimp(eV)
3x3x3      3.914
4x4x4      3.968
5x5x5      3.987
10x10x10   4.005
20x20x20   4.008

The result in the paper (S. Ryu and W.Cai JPCM 22 055401 (2010), Table 2, is $E_{2}=3.968$ (eV) for a Au atom in Si DC crystal. So it seems that the result in JPCM (2010) corresponds to the 4x4x4 cell here.

Use the following command to compute the impurity of a Si atom in Au fcc lattice.

bin/meam-lammps_gpp scripts/work/si_au/si_au_benchmark.tcl 3

cell size, Eimp(eV)
2x2x2      0.639
3x3x3      0.660
4x4x4      0.665
5x5x5      0.667
10x10x10   0.669
20x20x20   0.669

The result in the paper (S. Ryu and W.Cai JPCM 22 055401 (2010), Table 2, is $E_{1}=0.636$ (eV) for a Si atom in Au FCC crystal. So it seems that for a Si in Au FCC crystal, the predicted results here using the 2x2x2 cell corresponds to the value in JPCM (2010).