Three-body bond-order potential by Khor and Das Sarma (1988) v000

Anshul Chawla; Daniel S. Karls; Kok-Eam Khor; Sankar Das Sarma

doi:10.25950/b2df2fd7

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ThreeBodyBondOrder_KDS__MD_697985444380_000

Title A single sentence description.	Three-body bond-order potential by Khor and Das Sarma (1988) v000
Description	Based on the idea that bonding energies of many substances can be modeled by pairwise interactions moderated by the local environment, we propose a new universal interatomic potential for tetrahedrally bonded materials. This potential, which uses very few parameters, should be useful, particularly for surface studies.
Disclaimer A statement of applicability provided by the contributor, informing users of the intended use of this KIM Item.	In the source article, Khor-Das Sarma state that the summation of the three-body term is "to be carried out only over nearest-neighbor atoms whose bonds ik are nearest bonds of ij (i.e., when theta_jik-theta_i, is small)." Since no details are provided on how this was implemented in practice, this model driver does not include this constraint. This is likely to affect the predictions of the model away from the ground state.

Contributor	Anshul Chawla
Maintainer	Anshul Chawla
Implementer	Anshul Chawla Daniel S. Karls
Developer	Kok-Eam Khor Sankar Das Sarma
Published on KIM	2019
How to Cite	This Model Driver originally published in [1] is archived in OpenKIM [2-4]. [1] Khor KE, Das Sarma S. Proposed universal interatomic potential for elemental tetrahedrally bonded semiconductors. Phys Rev B. 1988;38(5):3318–22. doi:10.1103/PhysRevB.38.3318 [2] Chawla A, Karls DS, Khor K-E, Sarma SD. Three-body bond-order potential by Khor and Das Sarma (1988) v000. OpenKIM; 2019. doi:10.25950/b2df2fd7 [3] 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 [4] 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_697985444380_000
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.	ThreeBodyBondOrder_KDS__MD_697985444380_000
DOI	10.25950/b2df2fd7 https://doi.org/10.25950/b2df2fd7 https://commons.datacite.org/doi.org/10.25950/b2df2fd7
KIM Item Type	Model Driver
KIM API Version	2.0
Programming Language(s) The programming languages used in the code and the percentage of the code written in each one.	100.00% C

Models using this Model Driver

ThreeBodyBondOrder_KDS_KhorDasSarma_1988_C__MO_454320668790_000
ThreeBodyBondOrder_KDS_KhorDasSarma_1988_Ge__MO_216597146527_000
ThreeBodyBondOrder_KDS_KhorDasSarma_1988_Si__MO_722489435928_000

Files

CMakeLists.txt
LICENSE
README
ThreeBodyBondOrder.c
bondorder.inc
bondorder_aux.inc
kimprovenance.edn
kimspec.edn

Download

ThreeBodyBondOrder_KDS__MD_697985444380_000.txz	Tar+XZ	Linux and OS X archive
ThreeBodyBondOrder_KDS__MD_697985444380_000.zip	Zip	Windows archive

In 1984, G.C. Abell noted that the bond energies of a particular structure can be expressed using pairwise interactions which are modified to suitably incorporate the effect of the local environment. Tersoff introduced a Morse-like potential which utilized this concept. This potential was known to be quite successful in reproducing surface reconstruction energies but was inaccurate in some cases such as the Pandey defect reconstruction, i.e. cases wherein $$\pi$$ bonding was important. Furthermore, it wrongly predicted the bcc phase to be more stable than the diamond structure and was inapplicable to the graphitic structure of silicon. Dodsen, in 1987, tried to improve this potential by adding another parameter, but the results were insufficient. Also, the equilibrium atomic distance found from Dodsen's potential function was within 5 percent agreement with the density functional results given by Yin and Cohen.

Khor and Das Sarma based on the work of Abell and identified a linear relationship between $$\ln(D_e/Z)$$ and the equilibrium distance, $$R_e$$, and as well as between $$R_e$$ and $$\ln(Z)$$, where $$D_e$$ is the cohesive energy at equilibrium and $$Z$$ is the coordination number. Based on this concept they derived a bond-order potential function, consisting of ten parameters, that worked well in the case of C, Si, Ge and likewise also for tetrahedrally bonded semiconductors. The potential function has the following form:-

Energy of system = Cohesive Energy = 0.5 $$ (\sum_{ij} \phi^{\rm KDS}(r_{ij})) $$

$$
\begin{align*}
  \phi^{\rm KDS}(r_{ij}) &= A \left\{\exp[-\theta r_{ij}] - b^{\rm KDS}_{ij} \exp[-\lambda r_{ij}]\right\}
\end{align*}
$$

$$
\begin{align*}

b^{\rm KDS}_{ij} &= \frac{B_0}{(Z^{\rm KDS}_i)^\alpha}
[ 1+\sum_k g^{\rm KDS}(\theta_{jik}) ]

\end{align*}
$$

$$
\begin{align*}

g^{\rm KDS}(\theta_{jik})&= \cos\eta(\theta_{jik}-\theta_\circ)-1

\end{align*}
$$

$$
\begin{align*}

f_{\rm c}^{\rm KDS}(r_{ij}) &= \exp[-\beta(r_{ij}-R)^\gamma]

\end{align*}
$$

$$
\begin{align*}

Z^{\rm KDS}_i &= \sum_j \exp[-\beta(r_{ij}-R)^\gamma]

\end{align*}
$$

Here $$R$$ is the nearest neighbor distance in the lattice structure, for instance for the diamond lattice, $$ R=a\sqrt{3}/4 $$ where $$a$$ is the equilibrium lattice parameter. $$r_{ij}$$ is the distance between atoms $$i$$ and $$j$$, $$\theta_{jik}$$ is the angle between the bond between i, j and i, k atoms, $$\theta_{\circ}$$ is the equilibrium bond angle and $$Z_i$$ is the coordination number of the atom. All the summations in the above set of equations are carried over the nearest neighbors of atom i. In the above set of equations, only a simple angular factor has been introduced for simplicity.

$$\beta$$ and $$\gamma$$ were evaluated to give the correct coordination number of bcc and $$\beta$$-tin structures. $$\eta$$ was fitted to the bond bending force constant of the diamond structure. The rest of the parameters were evaluated by fitting to the slopes and intercepts of the graphs of $$D_e$$ vs $$R_e$$ and $$R_e$$ vs $$\ln(Z)$$ and the bulk modulus. The technique used to fit the parameters was not mentioned in the research paper.

The table below gives the values of the parameters for C, Si, and Ge.

![](/wimage/MD_697985444380_000/Anshul/FTABLE.png){:height="110px"}

Following are the equilibrium properties of the C, Si, and Ge lattices with diamond structure:

![](/wimage/MD_697985444380_000/Anshul/STABLE.png){:height="110px"}

Wiki Contributors

2019-05-09T11:09:13.168471	Anshul
2019-05-09T10:58:49.026513	tadmor
2019-05-08T22:38:40.917734	Anshul
2019-05-08T22:37:29.424426	Anshul
2019-05-08T22:19:13.144687	Anshul
2019-05-08T22:17:11.124360	Anshul
2019-05-08T22:16:11.133192	ramanish
2019-05-08T22:15:03.680988	ramanish
2019-05-08T21:55:25.770493	Anshul
2019-05-08T21:50:37.096147	Anshul
2019-05-08T21:44:51.223375	Anshul
2019-05-08T21:39:02.075094	Anshul
2019-05-08T21:30:37.047305	Anshul
2019-05-08T21:26:04.094030	Anshul
2019-05-08T21:25:44.082476	Anshul
2019-05-08T21:24:32.987513	Anshul
2019-05-08T21:23:14.650093	Anshul
2019-05-08T21:23:00.089223	Anshul