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.
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)}}
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^{(2)}}
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^{(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
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 (R_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 E_i^0}
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 = alat for the diamond cubic structure.
= rozero will be important only for cross-potential. And note that this is the only different from Si4 line.
ibar is a setting used in the equation of state (EOS), and will be explained later.
MEAM Potential for Ge
We use the 'Ge' potential whose parameters are originally given in M. I. Baskes, 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 corresponding to 'Ge5' are given below. Most of these parameters correspond to Table III of Baskes PRB (1992), as shown 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 \beta_i^{(1)}} 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^{(2)}} 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^{(3)}} elt lat z ielement atwt alpha b0 b1 b2 b3 'Ge' 'dia' 4. 32 72.64 4.98 4.55 5.5 5.5 5.5
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 (R_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 E_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 A_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 t_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 t_i^{(1)}}
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 t_i^{(2)}}
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 t_i^{(3)}}
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 \rho_0^{\rm Si}}
alat esub asub t0 t1 t2 t3 rozero ibar
5.6575 3.85 1. 1.0 4.02 5.23 -1.6 1.35 0
Note that the nearest neighbor distance 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 R_i^0 } = alat 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 \times \sqrt{3}/4} for the diamond cubic structure.
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 \rho_0^{\rm Si}} = rozero will be important only for cross-potential.
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 (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 r_e}
)
delta(1,2) = 0.125 (related to 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_c}
, see below)
lattce(1,2) = b1
alpha(1,2) = 5.819 (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}
)
attrac(1,2) = 0.0
repuls(1,2) = 0.26 (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 \gamma}
)
Cmin(1,1,2) = 1.9 (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 C_{\min}(1,1,2)}
)
Cmin(1,2,1) = 0.95 (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 C_{\min}(1,2,1)}
)
Cmin(1,2,2) = 1.85 (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 C_{\min}(1,2,2)}
)
Cmin(2,2,1) = 1.0 (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 C_{\min}(2,2,1)}
)
Table 3 of Ryu and Cai (2010) gives 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_c ({\rm AuSi}) = 4.155} . This value is related to delta(1,2) through
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_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} .
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 \rho_0^{\rm Si} / \rho_0^{\rm Au}} = 1.48 because of the 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 \rho_0^{\rm Si}} and 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 \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).