ABOP_lammps

Collection of some analytical bond order potentials ( listed in the LAMMPS format) from literature.

Comparison between different formulas

Tersoff_1 in LAMMPS	Analytical bond-order potential (ABOP)
$$E=\frac{1}{2}\sum_{i}\sum_{j\neq i}V_{ij}$$	$$E=\sum_{i}\sum_{j>i}V_{ij}$$
$$V_{ij}=f_\mathrm{C}(r_{ij}+\delta)[f_\mathrm{R}(r_{ij}+\delta)+b_{ij}f_\mathrm{A}(r_{ij}+\delta)]$$	$$V_{ij}=f_\mathrm{C}(r_{ij})[V_\mathrm{R}(r_{ij})-\bar{b_{ij}}V_\mathrm{A}(r_{ij})]$$
$$f_\mathrm{C}(r)=\left{\begin{array}{ll}1, & r <R-D \\frac{1}{2}-\frac{1}{2}\sin\left(\frac{\pi}{2}\frac{r-R}{D}\right), & R-D<r<R+D \0,&r>R+D\end{array} \right.$$	$$f_\mathrm{C}(r)=\left{\begin{array}{ll}1, & r \leq R-D \\frac{1}{2}-\frac{1}{2}\sin\left[\frac{\pi}{2D}(r-R)\right], &
$$f_\mathrm{R}(r)=A\exp(-\lambda_{1}r)$$	$$V_\mathrm{R}(r_{ij})=\frac{D_0}{S-1}\exp\left[-\beta\sqrt{2S}(r_{ij}-r_{0})\right]$$
$$f_\mathrm{A}(r)=-B\exp(-\lambda_{2}r)$$	$$V_\mathrm{A}(r_{ij})=\frac{SD_0}{S-1}\exp\left[-\beta\sqrt{2/S}(r_{ij}-r_{0})\right]$$
$$b_{ij}=\left(1+\beta^{n}\zeta_{ij}^{n}\right)^{-\frac{1}{2n}}$$	$$\bar{b_{ij}}=\frac{b_{ij}+b_{ji}}{2}$$
	$$b_{ij}=(1+\chi_{ij})^{-1/2}$$
$$\zeta_{ij}=\sum_{k\neq i,j}f_\mathrm{C}(r_{ik}+\delta)g\left[\theta_{ijk}(r_{ij},r_{ik})\right]\exp\left[\lambda_{3}^{m}(r_{ij}-r_{ik})^{m}\right]$$	$$\chi_{ij}=\sum_{k(\neq i,j)}f_\mathrm{C}(r_{ik})g_{ik}(\theta_{ijk})\omega_{ijk}\exp[\alpha_{ijk}(r_{ij}-r_{ik})]$$
$$g(\theta)=\gamma_{ijk}\left[1+\frac{c^2}{d^2}-\frac{c^2}{d^2+(\cos\theta-\cos\theta_{0})^{2}}\right]$$	$$g_{ik}(\theta_{ijk})=\gamma_{ik}\left[1+\frac{c^2_{ik}}{d^2_{ik}}-\frac{c^2_{ik}}{d^2_{ik}+(h_{ik}+\cos\theta_{ijk})^{2}}\right]$$

$D_0$ and $r_0$: the bond energy and length of a dimer.

$\beta$ and $S$: fitting parameters that control the shape of the pair potential.

From the parameters $r_0$, $D_{0}$, $S$, $\beta$, $\alpha$, $\omega$, and $h$ given in the ABOP, we can obtain the parameters $A$, $B$, $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, $\gamma$, and $\cos\theta_{0}$ in the Tersoff_1 format of LAMMPS according to the followings: $\lambda_{1} = \beta\sqrt{2S}$, $\lambda_{2} = \beta\sqrt{2/S}$, $A=\frac{D_{0}}{S-1}\exp(\lambda_{1}r_{0})$, $B = \frac{SD_{0}}{S-1}\exp(\lambda_{2}r_{0})$, $\lambda_{3} = \alpha$ with $m=1$, $\cos\theta_{0} = -h $, $\gamma = \omega\gamma$, where $n$ = 1, $\beta$ = 1, $m$ = 1 in Tersoff_1 format of LAMMPS.

Data format in Tersoff file with a“.tersoff” suffix

Lines that are not blank or comments (starting with #) define parameters for a triplet of elements. The parameters in a single entry correspond to coefficients in the formula above:

Tersoff_1	element 1	element 2	element 3	$m$	$\gamma$	$\lambda_{3}$	$c$	$d$	$\cos\theta_{0}$	$n$	$\beta$	$\lambda_{2}$	$B$	$R$	$D$	$\lambda_{1}$	$A$
Note	the center of atom in a 3-body interaction	the atom bonded to the center atom	the atom influencing the 1-2 bond in a bond-order sense			1/distance units			can be a value < -1 or > 1			1/distance units	energy units	distance units	distance units	1/distance units	energy units
				1	$\omega\gamma$	$\alpha$			$-h$	1	1	$\beta\sqrt{2/S}$	$\frac{SD_{0}}{S-1}\exp(\lambda_{2}r_{0})$	$R$	$D$	$\beta\sqrt{2S}$	$\frac{D_{0}}{S-1}\exp(\lambda_{1}r_{0})$

• The $n$, $\beta$, $\lambda_2$, $B$, $\lambda_1$, and $A$ parameters are only used for two-body interactions.

• The $m$, $\gamma$, $\lambda_3$, $c$, $d$, and $\cos\theta_0$ parameters are only used for three-body interactions.

• The $R$ and $D$ parameters are used for both two-body and three-body interactions.

• The non-annotated parameters are unitless.

• The value of $m$ must be 3 or 1.

• The Tersoff potential file must contain entries for all the elements listed in the pair_coeff command. It can also contain entries for additional elements not being used in a particular simulation; LAMMPS ignores those entries.

• For a single-element simulation, only a single entry is required (e.g. SiSiSi). For a two-element simulation, the file must contain 8 entries (for SiSiSi, SiSiC, SiCSi, SiCC, CSiSi, CSiC, CCSi, CCC), that specify Tersoff parameters for all permutations of the two elements interacting in three-body configurations. Thus for 3 elements, 27 entries would be required, etc.

• The first element in the entry is the center atom in a three-body interaction and it is bonded to the second atom and the bond is influenced by the third atom.

  • Thus an entry for SiCC means Si bonded to a C with another C atom influencing the bond.
  • Thus three-body parameters for SiCSi and SiSiC entries will not, in general, be the same. The parameters used for the two-body interaction come from the entry where the second element is repeated.
  • Thus the two-body parameters for Si interacting with C, comes from the SiCC entry.

• The parameters used for a particular three-body interaction come from the entry with the corresponding three elements. The parameters used only for two-body interactions ( $n$, $\beta$, $\lambda_{2}$ , $B$, $\lambda_{1}$, and $A$) in entries whose second and third element are different (e.g. SiCSi) are not used for anything and can be set to 0.0 if desired.

W-C

The WC potential is taken from

Ref.1: N. Juslin, P. Erhart, P. Träskelin, J. Nord, K. O. E. Henriksson, K. Nordlund, E. Salonen, and K. Albe, Analytical interatomic potential for modeling nonequilibrium processes in the W-C-H system, J. Appl. Phys., 98, 123520(2005); DO: 10.1063/1.2149492.

Ref.2: M.V.G. Petisme, M.A. Gren, G. Wahnström, Molecular dynamics simulation of WC/WC grain boundary sliding resistance in WC-Co cemented carbides at high temperature, Int. J. Refract. Hard Met., 49, 75--80(2015); DOI: 10.1016/j.ijrmhm.2014.07.037.

Note: there is a typo for the "beta" value of C-C in Table 1 of Ref. 2, which is inconsistent with the one in Ref. 1.

Ga-N

The GaN potential is taken from

Ref.3: J. Nord, K. Albe, P. Erhart, and K. Nordlund, Modelling of compound semiconductors: analytical bond-order potential for gallium, nitrogen and gallium nitride, J. Phys. Condens Matter, 15, 5649 (2003); DOI: 10.1088/0953-8984/15/32/324.

Source 1: provided by openKIM. In this file, the n values of 'Ga Ga N', 'N Ga N', 'N N Ga', 'Ga N Ga' are set to 0.0.

Source 2: Provided by LAMMPS. In this file, the n values of 'Ga Ga N', 'N Ga N', 'N N Ga', 'Ga N Ga' are set to 1.0.

This difference in the setup of n doesn't affect the simulation results. It will give identical results.

Ga-As

The GaAs potential is taken from

Ref. 4: K. Albe, K. Nordlund, J. Nord, and A. Kuronen, Modeling of compound semiconductors: Analytical bond-order potential for Ga, As, and GaAs, Phys. Rev. B 66, 035205(2002); DOI: 10.1103/PhysRevB.66.035205.

Source 1: provided by openKIM. In this file, the n values of 'Ga Ga As', 'As Ga As', 'As As Ga', 'Ga As Ga' are set to 0.0.

Zn-O

The ZnO potential is taken from

Ref. 5: P. Erhart, N. Juslin, O. Goy, K. Nordlund, R. Müller, and K. Albe, Analytic bond-order potential for atomistic simulations of zinc oxide, J. Phys.: Condens. Matter 18 (2006)6585-6605; DOI: 10.1088/0953-8984/18/29/003.

Source 1: provided by openKIM. In this file, the n values of 'Zn Zn O', 'O Zn O', 'O O Zn', 'Zn O Zn' are set to 0.0.

Note: some discussions about the reliability of this Zn-O potential can be found on the forum of lammps.

Au

The Au potential is taken from

Ref. 6: M. Backman, N. Juslin, and K. Nordlund, Bond order potential for gold, Eur. Phys. J. B 85(2012)317; DOI: 10.1140/epjb/e2012-30429-y.

Pt-C

The PtC potential is taken from

Ref. 7: K. Albe, K. Nordlund, and R. S. Averback, Modeling the metal-semiconductor interaction: Analytical bond-order potential for platinum-carbon, Phys. Rev. B 65 (2002)195124; DOI: 10.1103/physrevb.65.195124.

Ti-Al-C

The Ti-Al-C potential for Ti₃AlC₂ MAX Phase and Ti-Si-C poential for Ti₃SiC₂ MAX Phase are taken from

Ref. 8: G. Plummer and G. J. Tucker, Bond-order potentials for the Ti₃AlC₂ and Ti₃SiC₂ MAX phases, Phys. Rev. B 100 (2019)214114; DOI: 10.1103/PhysRevB.100.214114.

Be-O

The Be-O potential is taken from

Ref. 9: J. Byggmästar, E. A. Hodille, Y. Ferro, and K. Nordlund, Analytical bond order potential for simulations of BeO 1D and 2D nanostructures and plasma-surface interactions, J. Phys.: Condens. Matter, 30, 135001(2018); DOI: 10.1088/1361-648X/aaafb3.

To describe repulsive short-range interactions more accurately, the universal repulsive Ziegler-Biersack-Littmark (ZBL) potential V_ZBL (r_ij) is used jointly with the original potential V_ij.

V'_ij = F(r_ij) V_ij + [1-F(r_ij)] V_ZBL

F(r_ij) is Fermi function, i.e., 1/[1+exp(-b_f (r_ij -r_f))]

Parameters	Be-Be	O-O	Be-O
bf	15.0	12.0	15.0
rf	0.8	0.5	0.8

Be-C

The Be-C potential is taken from

Ref. 10: C. Björkas, N. Juslin, H. Timko, K. Vörtler, K. Nordlund, K. Henriksson, and P. Erhart, Interatomic potentials for the Be-C-H system, J. Phys.: Condens. Matter, 21, 445002(2009); DOI: 10.1088/0953-8984/21/44/445002.

To describe repulsive short-range interactions more accurately, the universal repulsive Ziegler-Biersack-Littmark (ZBL) potential V_ZBL (r_ij) is used jointly with the original potential V_ij.

V'_ij = F(r_ij) V_ij + [1-F(r_ij)] V_ZBL

F(r_ij) is Fermi function, i.e., 1/[1+exp(-b_f (r_ij -r_f))]

Parameters	Be-Be	C-C	Be-C
bf	15.0	8.0	16.0
rf	0.8	0.6	0.7

W-N & W

The W-N potential is taken from

Ref. 11: J. Polvi, K. Heinola, and K. Nordlund, An interatomic potential for W–N interactions, Modelling Simul. Mater. Sci. Eng., 24, 065007(2016); DOI: 10.1088/0965-0393/24/6/065007.

To describe repulsive short-range interactions more accurately, the universal repulsive Ziegler-Biersack-Littmark (ZBL) potential V_ZBL (r_ij) is used jointly with the original potential V_ij.

V'_ij = F(r_ij) V_ij + [1-F(r_ij)] V_ZBL

F(r_ij) is Fermi function, i.e., 1/[1+exp(-b_f (r_ij -r_f))]

Parameters	W-W	N-N	W-N
bf	12.0	12.0	12.0
rf	1.3	0.5	0.4

The parameters for W-W are taken from

Ref. 12: T. Ahlgren, K. Heinola, N. Juslin, and A. Kuronen, Bond-order potential for point and extended defect simulations in tungsten, J. Appl. Phys. 107, 033516(2010); DOI:10.1063/1.3298466.

Be-W

The Be-W potential is taken from

Ref. 13: C. Björkas, K. O. E. Henriksson, M. Probst, and K. Nordlund, A Be–W interatomic potential, J. Phys.: Condens. Matter, 22, 352206(2010); DOI: 10.1088/0953-8984/22/35/352206.

To describe repulsive short-range interactions more accurately, the universal repulsive Ziegler-Biersack-Littmark (ZBL) potential V_ZBL (r_ij) is used jointly with the original potential V_ij.

V'_ij = F(r_ij) V_ij + [1-F(r_ij)] V_ZBL

F(r_ij) is Fermi function, i.e., 1/[1+exp(-b_f (r_ij -r_f))]

Parameters	Be-Be	W-W	Be-W
bf	15.0	12.0	13.0
rf	0.8	1.3	1.3

Fe

The Fe potential is taken from

Ref. 14: M. Müller, P. Erhart and K. Albe, Analytic bond-order potential for bcc and fcc iron-comparison with established embedded-atom method potentials, J. Phys.: Condens. Matter, 19, 326220(2007); DOI: 10.1088/0953-8984/19/32/326220.

Mo-Er

The Mo-Er potential is taken from

Ref. 15: Q. Q. Sun, T. Yang, L. Yang, S. M. Peng, X. G. Long, X. S. Zhou, X. T. Zu, and F. Gao, Analytical interactomic potential for a molybdenum–erbium system, Modelling Simul. Mater. Sci. Eng., 24, 045018(2016); DOI: 10.1088/0965-0393/24/4/045018.

Si & Ge

The Si and Ge potentials are taken from

Ref. 16: Brian Andrew Gillespie, Bond Order Potentials for Group IV Semiconductors, University of Virginia, 2009; URL: https://www2.virginia.edu/ms/research/wadley/Thesis/BGillespiePhD.pdf.

Fe-O

The Fe-O potential is taken from

Ref. 17: J. Byggmästar, M. Nagel, K. Albe, K. O. E. Henriksson and K. Nordlund, Analytical interatomic bond-order potential for simulations of oxygen defects in iron, J. Phys.: Condens. Matter, 31, 215401(2019); DOI: 10.1088/1361-648X/ab0931.

To describe repulsive short-range interactions more accurately, the universal repulsive Ziegler-Biersack-Littmark (ZBL) potential V_ZBL (r_ij) is used jointly with the original potential V_ij.

V'_ij = F(r_ij) V_ij + [1-F(r_ij)] V_ZBL

F(r_ij) is Fermi function, i.e., 1/[1+exp(-b_f (r_ij -r_f))]

Parameters	Fe-Fe	O-O	Fe-O
bf	2.9	12.0	10.0
rf	0.95	0.5	1.0