MEAM Potential for Si-Ge

MEAM Potential for Au-Si

Xiaohan Zhang and Wei Cai

Created Mar, 2017, Last modified Mar, 2017

This tutorial explains how to specify the parameters for the Si-Ge MEAM potential in MD++. It starts with the parameters in pure Si and pure Ge potentials, then walks through SiGe cross potential, based on the reference: "A modified embedded atom method interatomic potential for alloy SiGe", Gregory Grochola, Salvy P.Russo, Ian K. Snook, Chemical Physics Letters 493 (2010) 57-60.

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Potential for Pure Elements

MEAM Potential for Si

We use the 'Siz' potential as those used in Kang, et al "Size and Temperature Effects on Brittle and Ductile Fracture of Silicon Nanowires", International Journal of Plasticity, 26, 1387 (2010" and "Brittle and Ductile Fracture of Semiconductor Nanowires – Molecular Dynamics Simulations", Philosophical Magazine, 87, 2169, (2007)." 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.

                                      Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_i}
      Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta_i^{(0)}}
    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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      1.60    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.

MEAM Potential for Ge

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 Ge 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

melting point

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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).

phase diagram

Use the following command to obtain the phase diagram of SiGe.

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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).