Model driver for Tersoff-style potentials ported from LAMMPS v005

Tobias Brink; Aidan P. Thompson; David E. Farrell; Mingjian Wen; Jerry Tersoff; J. Nord; Karsten Albe; Paul Erhart; Kai Nordlund; James F. Ziegler; Jochen P. Biersack; U. Littmark

doi:10.25950/9a7dc96c

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Tersoff_LAMMPS__MD_077075034781_005

Title A single sentence description.	Model driver for Tersoff-style potentials ported from LAMMPS v005
Description	Model driver that supports parametrizations of the Tersoff-style three-body potential. It is ported from LAMMPS and supports the same format for parameter files. The commonly used variants by Tersoff and Nord/Albe/Erhart/Nordlund are supported, see source citations for details. Optionally, a Ziegler-Biersack-Littmark potential for short interatomic distances can be smoothly joined via a Fermi function. This functionality mirrors the tersoff/zbl variant in LAMMPS.
Disclaimer A statement of applicability provided by the contributor, informing users of the intended use of this KIM Item.	None
Content Origin	Ported from LAMMPS

Contributor	Tobias Brink
Maintainer	Tobias Brink
Implementer	Tobias Brink Aidan P. Thompson David E. Farrell Mingjian Wen
Developer	Jerry Tersoff J. Nord Karsten Albe Paul Erhart Kai Nordlund James F. Ziegler Jochen P. Biersack U. Littmark
Published on KIM	2021
How to Cite	This Model Driver originally published in [1-4] is archived in OpenKIM [5-7]. [1] Tersoff J. New empirical approach for the structure and energy of covalent systems. Phys Rev B. 1988;37(12):6991–7000. doi:10.1103/PhysRevB.37.6991 — (Primary Source) A primary source is a reference directly related to the item documenting its development, as opposed to other sources that are provided as background information. [2] Tersoff J. Modeling solid-state chemistry: Interatomic potentials for multicomponent systems. Phys Rev B. 1989;39:5566–8. doi:10.1103/PhysRevB.39.5566 — (Primary Source) A primary source is a reference directly related to the item documenting its development, as opposed to other sources that are provided as background information. [3] Nord J, Albe K, Erhart P, Nordlund K. Modelling of compound semiconductors: analytical bond-order potential for gallium, nitrogen and gallium nitride. J Phys: Condens Matter. 2003;15:5649. doi:10.1088/0953-8984/15/32/324 — (Primary Source) A primary source is a reference directly related to the item documenting its development, as opposed to other sources that are provided as background information. [4] Ziegler JF, Biersack JP, Littmark U. The Stopping and Range of Ions in Matter. New York: Pergamon; 1985. — (Primary Source) A primary source is a reference directly related to the item documenting its development, as opposed to other sources that are provided as background information. [5] Brink T, Thompson AP, Farrell DE, Wen M, Tersoff J, Nord J, et al. Model driver for Tersoff-style potentials ported from LAMMPS v005. OpenKIM; 2021. doi:10.25950/9a7dc96c [6] Tadmor EB, Elliott RS, Sethna JP, Miller RE, Becker CA. The potential of atomistic simulations and the Knowledgebase of Interatomic Models. JOM. 2011;63(7):17. doi:10.1007/s11837-011-0102-6 [7] Elliott RS, Tadmor EB. Knowledgebase of Interatomic Models (KIM) Application Programming Interface (API). OpenKIM; 2011. doi:10.25950/ff8f563a Click here to download the above citation in BibTeX format.
Funding	Not available

Short KIM ID The unique KIM identifier code.	MD_077075034781_005
Extended KIM ID The long form of the KIM ID including a human readable prefix (100 characters max), two underscores, and the Short KIM ID. Extended KIM IDs can only contain alpha-numeric characters (letters and digits) and underscores and must begin with a letter.	Tersoff_LAMMPS__MD_077075034781_005
DOI	10.25950/9a7dc96c https://doi.org/10.25950/9a7dc96c https://commons.datacite.org/doi.org/10.25950/9a7dc96c
KIM Item Type	Model Driver
KIM API Version	2.2
Simulator Potential Compatibility	LAMMPS : tersoff — full Additional settings file required that lists supported species. See README for details. LAMMPS : tersoff/zbl — full Additional settings file required that lists supported species and activates ZBL support. See README for details.

Previous Version	Tersoff_LAMMPS__MD_077075034781_004

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Tersoff_LAMMPS_AlbeNordlundAverback_2002_PtC__MO_500121566391_004
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Tersoff_LAMMPS_ByggmastarNagelAlbe_2019_FeO__MO_608695023236_000
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Tersoff_LAMMPS_PlummerTucker_2019_TiAlC__MO_736419017411_000
Tersoff_LAMMPS_PlummerTucker_2019_TiSiC__MO_751442731010_000
Tersoff_LAMMPS_Tersoff_1988T2_Si__MO_245095684871_004
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Tersoff_LAMMPS_Tersoff_1994_SiC__MO_794973922560_000
Tersoff_LAMMPS_ZhangNguyen_2021_MoSe__MO_152208847456_001

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## Description
This **Model Driver** utilizes the Tersoff potential, originally used for Silicon in its seminal publication by Tersoff in 1988. It can be used for both the T2 and T3 bond order models, which are simply different parameter sets for the form given below.  It also optionally supports a ZBL repulsion potential that can be superposed with the regular Tersoff potential forms.

## Parameters
Symbols used in the model (taking into account $$\alpha_{ij}$$ is set to zero for this implementation of the model, and the modifications given in the note at the bottom of this documentation):

$$ m, \gamma_{ijk}, \lambda_3, c, d, cos\theta_0, n, \beta, \lambda_2, B, R, D, \lambda_1, A $$

These correspond to the following variables in the code:

$$ \text{m, gamma, lambda3 (1/distance units), c, d, costheta0 (must be a valid value between -1 & 1), n,  beta, lambda2 (1/distance units), B (energy units), Rc (distance units), Dc (distance units), lambda1 (1/distance units), & A (energy units).}  $$

The parameters n, beta, lambda2, B, lambda1, & A are used for two-body interactions, while m, gamma, lambda3, c, d, costheta0 are only used for three-body interactions.

Additionally, the code requires three other variables to determine the elements being used:

$$ \text{Element 1}: \text{The center atom in a 3-body interaction} $$

$$ \text{Element 2}: \text{The atom bonded to the center atom} $$

$$ \text{Element 3}: \text{The atom influencing the 1-2 bond in a bond-order sense} $$

## Details
The interatomic potential is given as:

$$ E = \sum_i E_i =  {1 \over 2}{\sum_{i \neq j}V_{ij}}$$

Where $$E$$ is the total energy of the system, decomposed into site energy, $$E_i$$, and bond energy, $$V_{ij}$$. The indices $$i$$ and $$j$$ run through all atoms in the system, and $$r_{ij}$$ is the distance from atom $$i$$ to atom $$j$$.

$$V_{ij}$$ has the form:

$$ V_{ij} =  f_c(r_{ij})[a_{ij}f_R(r_{ij})+b_{ij}f_A(r_{ij})] $$

The functions $$f_A$$ and $$f_R$$ are the respective attractive and repulsive pair potentials:

$$ f_A = Ae^{-\lambda_1r} $$

$$ f_R = -Be^{-\lambda_2r} $$

Moreover, $$f_c$$ is the cutoff function with the form:

$$ f_c(r) = \begin{cases}
      1 &  \text{if  } r < R-D\\
      {1 \over 2}-{1 \over 2}sin[{\pi \over 2}{(r-R) \over D}]& \text{if  } R-D < r < R+D\\
      0 & \text{if  } r > R+D
    \end{cases} $$

The bond-order function, $$b_{ij}$$, is given as:

$$b_{ij} = (1+\beta^n\zeta^n_{ij})^{-1 / 2n} $$

Where $$\zeta_{ij}$$ is the coordination function, which gives the bond coordination of the environment:

$$ \zeta_{ij} = \sum_{k \neq i,j}f_c(r_{ik})g(\theta_{ijk})\exp[\lambda_3^3(r_{ij}-r_{ik})^3] $$

$$g(\theta_{ijk})$$ is as follows:

$$ g(\theta_{ijk}) =  1+ {c^2 \over d^2}- {c^2 \over [d^2+(h -cos\theta)^2]} $$

where $$\theta_{ijk}$$ is the bond angle between bonds $$i-j$$ and $$j-k$$.

Additionally, $$ a_{ij} $$ is given by the following expression:

$$ a_{ij} = (1+\alpha^n\eta^n_{ij})^{-1/2n} $$

where $$\eta_{ij}$$ is:

$$ \eta_{ij} = \sum_{k \neq i,j} f_c(r_{ik})\exp[\lambda_3^3(r_{ij}-r_{ik})^3] $$

$$\alpha$$ is generally small enough such that $$a_{ij} \simeq 1$$; however, for most of the work presented in the paper by Tersoff in 1988, as well as this model, $$\alpha=0$$ such that $$a_{ij}=1$$, and thus $$V_{ij}$$ reduces to the form:

$$ V_{ij} =  f_c(r_{ij})[f_R(r_{ij})+b_{ij}f_A(r_{ij})]. $$

### **Note:** 
This model driver is ported from LAMMPS, which uses a few function modifications to allow for commonly used variants of the Tersoff potential (as documented [here](http://lammps.sandia.gov/doc/pair_tersoff.html)). The differences are as follows:

$$\zeta_{ij}$$ is modified to use the factor m:

$$ \zeta_{ij} = \sum_{k \neq i,j}f_c(r_{ik})g(\theta_{ijk})\exp[\lambda_3^m(r_{ij}-r_{ik})^m] $$

Additionally, $$g(\theta)$$ used in this model driver is modified by the factor $$\gamma_{ijk}$$:

$$ g(\theta) = \gamma_{ijk}({1+ {c^2 \over d^2}- {c^2 \over [d^2+(cos\theta -cos\theta_0)^2]}}) $$

It can be noted that setting $$m = 3$$ and $$\gamma_{ijk} = 1$$ gives the original Tersoff form (from the 1988 papers) shown above.

Introduction

In an attempt to explore the ability of classical mechanics to replicate quantum-mechanical calculations, R. Biswas and D. R. Hamann developed a three body cluster potential for Si in 1987. In their article “New classical models for silicon structural energies” (Physical Review B, 36(12): 6434, 1987), Biswas and Hamann discuss two different potentials, referred to as ‘old’ and ‘new’ potentials in their original article. To address the shortcomings of their ‘old’ potential, namely its inability to capture properties of tetrahedral structure of Si, they introduce their ‘new’ potential, which is attractive for molecular-dynamics simulations given its short range. However, as a trade-off, the ‘new’ potential is not as efficient as the ‘old’ model in describing high-pressure transitions and bulk metallic Si structures. The authors discuss how their ‘new’ potential is able to model the formation energies of interstitials and vacancies, and can also give reasonable account of thermal properties and melting of Si. Our model driver implements this ‘new’ potential of Biswas and Hamann.

Functional form

The functional form of the two-body potential is

\[\phi^{\rm BH}_2(r_{ij}) = (A_1 \exp[-\lambda_1 r_{ij}^2] + A_2 \exp[-\lambda_2 r_{ij}^2]) f_{\rm 2c}^{\rm BH}(r_{ij})\] \[f_{\rm 2c}^{\rm BH} = \{1+\exp[(r-r_c)/\mu]\}^{-1}\]

The functional form of the three body potential is

\[\phi^{\rm BH}_3(r_{ij},r_{ik},\theta_{jik}) = [B_1 \exp[-\alpha_1 r_{ij}^2] \exp[-\alpha_1 r_{ik}^2] (\cos\theta_{jik}+\tfrac{1}{3})^2 + B_2 \exp[-\alpha_2 r_{ij}^2] \exp[-\alpha_2 r_{ik}^2] (\cos\theta_{jik}+\tfrac{1}{3})^3] f_{\rm 2c}^{\rm BH}(r_{ij}) f_{\rm 2c}^{\rm BH}(r_{ik})\]

Wiki Contributors

2021-10-01T18:51:25.454552	karls