Notes¶

Several modifications of the original Tersoff potential have been proposed. One of them was introduced in [AG99] and is a combination of the Tersoff, Brenner and Tanaka potentials. For the sake of brevity we call it the Tersoff-Brenner potential.

The potential energy of the Tersoff-Brenner potential is given as the sum of attractive and repulsive potentials:

$\begin{split}U^{TB} =\sum_i \sum_{i < j} U_{ij}^{rep} - U_{ij}^{att}.\end{split}$

The repulsive interactions are the same as in the Tersoff potential

$U_{ij}^{rep} = f^{TB}_{ij}(r_{ij}) A_{ij} \exp(-\lambda_{ij} r_{ij})$

but a different tapering function $$f^{TB}_{ij}$$ is used:

(1)$\begin{split}f^{TB}_{ij} = \begin{cases} 1 & r_{ij}\leq R_{ij} \\ \frac{1}{2} - \frac{9}{16}\sin\left(\pi\frac{r_{ij} - 0.5(S_{ij} + R_{ij})}{S_{ij} - R_{ij}}\right) - \frac{1}{16}\sin\left(3\pi\frac{r_{ij} - 0.5(S_{ij} + R_{ij})}{S_{ij} - R_{ij}} \right) & R_{ij} < r_{ij} < S_{ij} \\ 0 & r_{ij}\geq S_{ij}. \end{cases}\end{split}$

The attractive potential $$U^{att}_{ij}$$ is given by the term

(2)\begin{split}\begin{align} U_{ij}^{att} &= \bar{b}_{ij} f^{TB}_{ij}(r_{ij}) B_{ij} \exp(-\mu_{ij} r_{ij}) \\ \bar{b}_{ij} &= \frac{1}{2} \left(b_{ij} + b_{ji} \right)\\ b_{ij} &= \left(1 + \left[\zeta_{ij}\right]^{\eta_{ij}} \right)^{-\delta_{ij}} \label{eq:TB:bij} \\ \zeta_{ij} &= \sum_{k\neq i,j} f_{ik}^{TB}(r_{ik}) g(\theta_{ijk}) \exp\left(\alpha_{ijk}\left[ (r_{ij} - R_{ij}^{(e)}) - (r_{ik} - R_{ik}^{(e)})\right]^{\beta_{ijk}} \right) \\ \end{align}\end{split}

The function $$g(\theta_{ijk})$$, where $$\theta_{ijk}$$ is the angle between the particles i-j and i-k, is either

(3)$g(\theta_{ijk}) = c_{ijk} + d_{ijk}(h_{ijk} - \cos(\theta_{ijk}))^2$

or

(4)$g(\theta_{ijk}) = a_{ijk} \left(1 + \frac{c_{ijk}^2}{d_{ijk}^2} - \frac{c_{ijk}^2}{d_{ijk}^2 + (h_{ijk} - \cos(\theta_{ijk}))^2}\right).$

The potential is activated in several steps.

To begin with, the pair-dependent parameters must be set by adding a TersoffBrennerPairPotential for each particle pair.

The parameters in the constructor of TersoffBrennerPairPotential map the the potential parameters as follows:

 a The parameter $$A_{ij}$$ b The parameter $$B_{ij}$$ lambda The parameter $$\lambda_{ij}$$ mu The parameter $$\mu_{ij}$$ re The parameter $$R^{(e)}_{ij}$$ that is only used if additional bond-order terms are activated r1 The parameter $$R_{ij}$$ in the taper function r2 The parameter $$S_{ij}$$ in the taper function

This sets all pairwise parameters except $$\eta_{ij}$$ and $$\delta_{ij}$$ which are set to 0, thereby deactivating all three-body interactions.

These parameters can be set by the TersoffBrennerBOPairPotential. Not that these parameters are used in a non-symmetric way, which means a TersoffBrennerBOPairPotential should be specified for each ij and ji.

The three-body parameters are set in a similar way. To use $$g(\theta)$$ from equation (3) you need to set up a TersoffBrennerTriplePotential. For the alternative form (4) the TersoffBrennerTriplePotential2 should be used. Note that particle_type1 denotes the type of the central particle during the angle calculation. The remaining parameters are the following:

 alpha The parameter $$\alpha_{ijk}$$ beta The parameter $$\beta_{ijk}$$ g_a The parameter $$a_{ijk}$$ (TersoffBrennerTriplePotential2 only) g_c The parameter $$c_{ijk}$$ g_d The parameter $$d_{ijk}$$ g_h The parameter $$h_{ij}$$

Another correction modifies the term $$b_{ij}$$ in equation (2) as follows:

$b_{ij} = \left(1 + [\zeta_{ij}]^{\eta_{ij}} + H_{ij}(N^{(1)}_{ij}, N^{(2)}_{ij}) \right)^{-\delta_{ij}}.$

$$H_{ij}$$ is an arbitrary 2D-function that is given as a bicubic spline. $$N^{(1)}_{ij}$$ and $$N^{(2)}_{ij}$$ are given by

\begin{split}\begin{align} N^{(1)}_{ij} &= \sum_{k\neq i,j\atop\textrm{type of }k\in P^{(1)}_{ij} } f_{ik}^{TB} \\ N^{(2)}_{ij} &= \sum_{k\neq i,j\atop\textrm{type of }k\in P^{(2)}_{ij} } f_{ik}^{TB}. \end{align}\end{split}

$$P^{(1)}_{ij}$$ and $$P^{(2)}_{ij}$$ are two lists of particle types that specify which interactions are taken into account for the calculation of $$N^{(1)}_{ij}$$ and $$N^{(2)}_{ij}$$. To enable this correction, a TersoffBrennerSplinePotential can be used where the parameters are defined as follows.

 particleType1 The particle type referred to as i particleType2 The particle type referred to as j activeTypes1 The type list $$P^{(1)}_{ij}$$ activeTypes2 The type list $$P^{(2)}_{ij}$$ x The x-coordinates of the grid of the spline function y The y-coordinates of the grid of the spline function f The spline values on the x-y-grid

Note that this potential acts in a non-symmetric way, meaning that it will only act on ij type pairs, but not on ji type pairs.

Finally, the term $$\bar{b }_{ij}$$ in equation (2) can be modified in the following way:

$\bar{b}_{ij} = \frac{1}{2} \left(b_{ij} + b_{ji}\right) + F_{corr}(N^{(t)}_{ij}, N^{(t)}_{ji}, N^{(conj)}_{ij}).$

The function $$F_{corr}$$ is an arbitrary 3D-function that is given as a tricubic spline. $$N^{(t)}_{ij}$$ is the coordination number of particle $$i$$, excluding $$j$$, i.e.

$\begin{split}N^{(t)}_{ij} = \sum_{k\neq i,j} f_{ik}^{TB}. \\\end{split}$

$$N^{(conj)}_{ij}$$ is defined as follows:

$N^{(conj)}_{ij} = 1 + \sum_{k\neq i,j\atop\textrm{type of }k\in P^{(conj)}_{ij} } f_{ik}^{TB} T_{ij}(N_{ki}^{(t)}) + \sum_{l\neq i,j\atop\textrm{type of }k\in P^{(conj)}_{ij} } f_{jl}^{TB} T_{ij}(N_{lj}^{(t)}).$

$$T_{ij}$$ is a new type of tapering function given by

$\begin{split}T_{ij}(x) = \begin{cases} 1, & x\leq L_{ij} \\ \frac{1}{2} + \frac{1}{2}\cos\left(\frac{\pi(x - L_{ij})}{U_{ij} - L_{ij}} \right), & L_{ij} < x < U_{ij} \\ 0, & x\geq U_{ij}. \end{cases}\end{split}$

$$P^{(conj)}_{ij}$$ is a list of particle types that specify which interactions are taken into account for the calculation of $$N^{(conj)}_{ij}$$. For this correction the TersoffBrennerCorrectionPotential can be used.

 particleType1 The particle type referred to as i particleType2 The particle type referred to as j activeTypes The type list among which neighbors of atoms i and j should be searched for. L The lower cutoff in $$T_{ij}$$ U The upper cutoff in $$T_{ij}$$ x The x-coordinates of the grid of the spline function. The same values are used for the y-coordinates z The z-coordinates of the grid of the spline function f The spline values on the x-y-z-grid

In contrast to the TersoffBrennerSplinePotential, this correction acts in a symmetric way between the types ij.

 [AG99] Cameron F Abrams and David B Graves. Molecular dynamics simulations of si etching by energetic cf 3+. Journal of applied physics, 86(11):5938–5948, 1999. URL: https://doi.org/10.1063/1.371637, doi:10.1063/1.371637.