A refined parameterization of the analytical Cd–Zn–Te bond-order potential

Abstract

This paper reports an updated parameterization for a CdTe bond order potential. The original potential is a rigorously parameterized analytical bond order potential for ternary the Cd–Zn–Te systems. This potential effectively captures property trends of multiple Cd, Zn, Te, CdZn, CdTe, ZnTe, and Cd_1-xZn_xTe phases including clusters, lattices, defects, and surfaces. It also enables crystalline growth simulations of stoichiometric compounds/alloys from non-stoichiometric vapors. However, the potential over predicts the zinc-blende CdTe lattice constant compared to experimental data. Here, we report a refined analytical Cd–Zn–Te bond order potential parameterization that predicts a better CdTe lattice constant. Characteristics of the second potential are given based on comparisons with both literature potentials and the quantum mechanical calculations.

Appendix

For the BOP formulation the total energy of a system is expressed as

$$ E=\frac{1}{2}{\displaystyle \sum_{i=1}^N{\displaystyle \sum_{j={i}_1}^{i_N}{\phi}_{ij}\left({r}_{ij}\right)}}-{\displaystyle \sum_{i=1}^N{\displaystyle \sum_{j={i}_1}^{i_N}{\beta}_{\sigma, ij}\left({r}_{ij}\right)}}\cdot {\varTheta}_{\sigma, ij}-{\displaystyle \sum_{i=1}^N{\displaystyle \sum_{j={i}_1}^{i_N}{\beta}_{\pi, ij}\left({r}_{ij}\right)}}\cdot {\varTheta}_{\pi, ij} $$

(4)

where ϕ _ij(r _ij) is a short-range two-body potential, β _σ,ij(r _ij) and β _π,ij(r _ij) are, respectively, σ and π bond integrals, Θ _σ,ij and Θ _π,ij are σ and π bond-orders. ϕ _ij(r _ij), β _σ,ij(r _ij), and β _π,ij(r _ij) are expressed in a general form as

$$ {\phi}_{ij}\left({r}_{ij}\right)={\phi}_{0, ij}\cdot {f}_{ij}{\left({r}_{ij}\right)}^{m_{ij}}\cdot {f}_{c, ij}\left({r}_{ij}\right) $$

(5)

$$ {\beta}_{\sigma, ij}\left({r}_{ij}\right)={\beta}_{\sigma, 0, ij}\cdot {f}_{ij}{\left({r}_{ij}\right)}^{n_{ij}}\cdot {f}_{c, ij}\left({r}_{ij}\right) $$

(6)

$$ {\beta}_{\pi, ij}\left({r}_{ij}\right)={\beta}_{\pi, 0, ij}\cdot {f}_{ij}{\left({r}_{ij}\right)}^{n_{ij}}\cdot {f}_{c, ij}\left({r}_{ij}\right) $$

(7)

where f _ij(r _ij) is a Goodwin-Skinner-Pettifor (GSP) radial function [23], and f _c,ij(r _ij) is a cutoff function (see [7] for formulation). Furthermore, Θ _σ,ij is given by:

$$ {\varTheta}_{\sigma, ij}={\varTheta}_{s, ij}\left({\varTheta}_{\sigma, ij}^{\left(1/2\right)},{f}_{\sigma, ij}\right)\cdot \left[1-\left({f}_{\sigma, ij}-\frac{1}{2}\right)\cdot {k}_{\sigma, ij}\cdot \frac{\beta_{\sigma, ij}^2\left({r}_{ij}\right)\cdot {R}_{3\sigma, ij}}{\beta_{\sigma, ij}^2\left({r}_{ij}\right)+\frac{\beta_{\sigma, ij}^2\left({r}_{ij}\right)\cdot {\varPhi}_{2\sigma}^i+{\beta}_{\sigma, ij}^2\left({r}_{ij}\right)\cdot {\varPhi}_{2\sigma}^j}{2}+{\zeta}_2}\right] $$

(8)

Where, Φ ⁱ_2σ and Φ ^j_2σ are local variables arising from electron hop paths. In addition, Φ ⁱ_2σ and Φ ^j_2σ have the same formulation but are merely evaluated for atoms i and j, respectively. Since only the product of β ²_σ,ij (r _ij) ⋅ Φ ⁱ_2σ is required for Eq. (8), the formulations are given as:

$$ {\beta}_{\sigma, ij}^2\left({r}_{ij}\right)\cdot {\varPhi}_{2\sigma}^i={\displaystyle \sum_{\begin{array}{l}k={i}_1\\ {}k\ne j\end{array}}^{i_N}{g}_{\sigma, jik}^2\left({\theta}_{jik}\right)}\cdot {\beta}_{\sigma, ik}^2\left({r}_{ik}\right) $$

(9)

where θ _jik is the bond angle at atom i spanning atoms j and k, and the function g _σ,jik(θ _jik) introduces angular-dependent contributions to the bonding resulting from the overlap of the hybridized atomic orbital. The three-body angular function is written as

$$ \begin{array}{ll}{g}_{\sigma, jik}\left({\theta}_{jik}\right)=\hfill & \frac{\left({b}_{\sigma, jik}-{g}_{0, jik}\right)\cdot {u}_{\sigma, jik}^2-\left({g}_{0, jik}+{b}_{\sigma, jik}\right)\cdot {u}_{\sigma, jik}}{2\cdot \left(1-{u}_{\sigma, jik}^2\right)}+\frac{g_{0, jik}+{b}_{\sigma, jik}}{2}\cdot \cos {\theta}_{jik}+\hfill \\ {}\hfill & \frac{g_{0, jik}-{b}_{\sigma, jik}+\left({g}_{0, jik}+{b}_{\sigma, jik}\right)\cdot {u}_{\sigma, jik}}{2\cdot \left(1-{u}_{\sigma, jik}^2\right)}\cdot { \cos}^2{\theta}_{jik}\hfill \end{array} $$

(10)

where g _σ,jik, b _σ,jik, and u _σ,jik are three-body-dependent parameters. The half full valance bond order is given by:

$$ {\varTheta}_{\sigma, ij}^{\left(1/2\right)}=\frac{\beta_{\sigma, ij}\left({r}_{ij}\right)}{\sqrt{\beta_{\sigma, ij}^2\left({r}_{ij}\right)+{c}_{\sigma, ij}\cdot \left[{\beta}_{\sigma, ij}^2\left({r}_{ij}\right)\cdot {\varPhi}_{2\sigma}^i+{\beta}_{\sigma, ij}^2\left({r}_{ij}\right)\cdot {\varPhi}_{2\sigma}^j\right]+{\zeta}_1}} $$

(11)

Equation (8) also requires knowing β ²_σ,ij (r _ij) ⋅ R _3σ,ij given by

$$ {\beta}_{\sigma, ij}^2\left({r}_{ij}\right)\cdot {R}_{3\sigma, ij}={\displaystyle \sum_{\begin{array}{l}k={i}_1\\ {}k,j=n\end{array}}^{i_N}{g}_{\sigma}\left({\theta}_{jik}\right)\cdot }{g}_{\sigma}\left({\theta}_{ij k}\right)\cdot {g}_{\sigma}\left({\theta}_{ik j}\right)\cdot {\beta}_{\sigma, ik}\left({r}_{ik}\right)\cdot {\beta}_{\sigma, jk}\left({r}_{jk}\right) $$

(12)

The symmetric band-filling function is expressed as the continuous function

$$ {\varTheta}_{s, ij}\left({\varTheta}_{\sigma, ij}^{\left(1/2\right)},{f}_{\sigma, ij}\right)=\frac{\varTheta_0+{\varTheta}_1+S\cdot {\varTheta}_{\sigma, ij}^{\left(1/2\right)}-\sqrt{{\left({\varTheta}_0+{\varTheta}_1+S\cdot {\varTheta}_{\sigma, ij}^{\left(1/2\right)}\right)}^2-4\left(-\varepsilon \sqrt{1+{S}^2}+{\varTheta}_0\cdot {\varTheta}_1+S\cdot {\varTheta}_1\cdot {\varTheta}_{\sigma, ij}^{\left(1/2\right)}\right)}}{2} $$

(13)

where

$$ \left\{\begin{array}{l}\varepsilon ={10}^{-10}\hfill \\ {}{\varTheta}_0=15.737980\cdot {\left(\frac{1}{2}-\left|{f}_{\sigma, ij}-\frac{1}{2}\right|\right)}^{1.137622}\cdot {\left|{f}_{\sigma, ij}-\frac{1}{2}\right|}^{2.087779}\hfill \\ {}S=1.033201\cdot \left\{1- \exp \left[-22.180680\cdot {\left(\frac{1}{2}-\left|{f}_{\sigma, ij}-\frac{1}{2}\right|\right)}^{2.689731}\right]\right\}\hfill \\ {}{\varTheta}_1=2\cdot \left(\frac{1}{2}-\left|{f}_{\sigma, ij}-\frac{1}{2}\right|\right)\hfill \end{array}\right. $$

(14)

The π bond-order Θ_π,ij used in Eq. (4) is expressed as

$$ \begin{array}{l}{\varTheta}_{\pi, ij}=\frac{a_{\pi, ij}\cdot {\beta}_{\pi, ij}\left({r}_{ij}\right)}{\sqrt{\beta_{\pi, ij}^2\left({r}_{ij}\right)+{c}_{\pi, ij}\cdot \left(\frac{\beta_{\pi, ij}^2\left({r}_{ij}\right)\cdot {\varPhi}_{2\pi}^i+{\beta}_{\pi, ij}^2\left({r}_{ij}\right)\cdot {\varPhi}_{2\pi}^j}{2}+\sqrt{\beta_{\pi, ij}^4\left({r}_{ij}\right)\cdot {\varPhi}_{4\pi }+{\zeta}_3}\right)+{\zeta}_4}}+\\ {}\frac{a_{\pi, ij}\cdot {\beta}_{\pi, ij}\left({r}_{ij}\right)}{\sqrt{\beta_{\pi, ij}^2\left({r}_{ij}\right)+{c}_{\pi, ij}\cdot \left(\frac{\beta_{\pi, ij}^2\left({r}_{ij}\right)\cdot {\varPhi}_{2\pi}^i+{\beta}_{\pi, ij}^2\left({r}_{ij}\right)\cdot {\varPhi}_{2\pi}^j}{2}-\sqrt{\beta_{\pi, ij}^4\left({r}_{ij}\right)\cdot {\varPhi}_{4\pi }+{\zeta}_3}+\sqrt{\zeta_3}\right)+{\zeta}_4}}\end{array} $$

(15)

where a _π,ij and c _π,ij are pair parameters, ζ₃ and ζ₄ are constants, and Φ ⁱ_2π , Φ ^j_2π , Φ _4π are local variables.

The β ²_π,ij (r _ij) ⋅ Φ ⁱ_2π,ij and β ⁴_π,ij (r _ij) ⋅ Φ _4π,ij terms used in Eq. (15) can be written as

$$ {\beta}_{\pi, ij}^2\left({r}_{ij}\right)\cdot {\varPhi}_{2\pi, ij}^i={\displaystyle \sum_{\begin{array}{l}k={i}_1\\ {}k\ne j\end{array}}^{i_N}\left[{p}_{\pi, i}\cdot {\beta}_{\sigma, ik}^2\left({r}_{ik}\right)\cdot { \sin}^2{\theta}_{jik}+\left(1+{ \cos}^2{\theta}_{jik}\right)\cdot {\beta}_{\pi, ik}^2\left({r}_{ik}\right)\right]} $$

(16)

$$ \begin{array}{l}{\beta}_{\pi, ij}^4\left({r}_{ij}\right)\cdot {\varPhi}_{4\pi, ij}=\frac{1}{4}{\displaystyle \sum_{\begin{array}{l}k={i}_1\\ {}k\ne j\end{array}}^{i_N}{ \sin}^4{\theta}_{jik}\cdot {\widehat{\beta}}_{ik}^4\left({r}_{ik}\right)}+\frac{1}{4}{\displaystyle \sum_{\begin{array}{l}k={j}_1\\ {}k\ne i\end{array}}^{j_N}{ \sin}^4{\theta}_{ij k}\cdot {\widehat{\beta}}_{jk}^4\left({r}_{jk}\right)}+\\ {}\frac{1}{2}{\displaystyle \sum_{\begin{array}{l}k={i}_1\\ {}k\ne j\end{array}}^{i_N}{\displaystyle \sum_{\begin{array}{l}k\prime =k+1\\ {}k\prime \ne j\end{array}}^{i_N}{ \sin}^2{\theta}_{jik}\cdot { \sin}^2{\theta}_{jik\hbox{'}}\cdot {\widehat{\beta}}_{ik}^2\left({r}_{ik}\right)\cdot {\widehat{\beta}}_{ik\hbox{'}}^2\left({r}_{ik\hbox{'}}\right)\cdot \cos \left(\varDelta {\psi}_{kk\prime}\right)}}+\\ {}\frac{1}{2}{\displaystyle \sum_{\begin{array}{l}k={j}_1\\ {}k\ne i\end{array}}^{j_N}{\displaystyle \sum_{\begin{array}{l}k\prime =k+1\\ {}k\prime \ne i\end{array}}^{j_N}{ \sin}^2{\theta}_{ij k}\cdot { \sin}^2{\theta}_{ij k\hbox{'}}\cdot {\widehat{\beta}}_{jk}^2\left({r}_{jk}\right)\cdot {\widehat{\beta}}_{jk\hbox{'}}^2\left({r}_{jk\hbox{'}}\right)\cdot \cos \left(\varDelta {\psi}_{kk\prime}\right)}}+\\ {}\frac{1}{2}{\displaystyle \sum_{\begin{array}{l}k\prime ={i}_1\\ {}k\prime \ne j\end{array}}^{i_N}{\displaystyle \sum_{\begin{array}{l}k={j}_1\\ {}k\ne i\end{array}}^{j_N}{ \sin}^2{\theta}_{jik\prime}\cdot { \sin}^2{\theta}_{ij k}\cdot {\widehat{\beta}}_{ik\hbox{'}}^2\left({r}_{ik\prime}\right)\cdot {\widehat{\beta}}_{jk}^2\left({r}_{jk}\right)\cdot \cos \left(\varDelta {\psi}_{kk\prime}\right)}}\end{array} $$

(17)

With

$$ {\widehat{\beta}}_{ik}^2\left({r}_{ik}\right)={p}_{\pi, i}\cdot {\beta}_{\sigma, ik}^2\left({r}_{ik}\right)-{\beta}_{\pi, ik}^2\left({r}_{ik}\right) $$

(18)

The β ⁴_π,ij (r _ij)  ⋅  Φ _4π,ij term contains four-body dihedral angles Δψ _kk ′ important in π bonding, and can be calculated as

$$ \cos \left(\varDelta {\psi}_{kk\prime}\right)=\left\{\begin{array}{c}\hfill \frac{2{\left( \cos {\theta}_{kik\prime }- \cos {\theta}_{jik\prime}\cdot \cos {\theta}_{jik}\right)}^2}{{ \sin}^2{\theta}_{jik}\cdot { \sin}^2{\theta}_{jik\prime }}-1\begin{array}{cc}\hfill \hfill & \hfill or\hfill \end{array}\hfill \\ {}\hfill \frac{2{\left(\frac{\overrightarrow{ ik\prime}\cdot \overrightarrow{ jk}}{\left|\overrightarrow{ ik\prime}\right|\cdot \left|\overrightarrow{ jk}\right|}+ \cos {\theta}_{ijk}\cdot \cos {\theta}_{jik\prime}\right)}^2}{{ \sin}^2{\theta}_{ijk}\cdot { \sin}^2{\theta}_{jik\prime }}-1\hfill \end{array}\right. $$

(19)

For more detailed discussion and descriptions of all equations, please see [7].

The BOP parameterization of CdTe can be done independently for elemental Cd, elemental Te, and finally for CdTe. As stated above, the ability to capture crystalline growth is a critical component of a high-fidelity interatomic potential. In general, a more transferrable (flexible for many phases) potential is more difficult to parameterize for capturing crystalline growth because the properties of various phases vary more dramatically with changes of the parameters.

Since the refined parameterization only updates the portions of the potential containing CdTe interactions many of the parameters remain consistent with the previous potentials [7, 8]. This particular fitting process includes a total of 40 parameters. However, many parameters can be fixed prior to the fitting process. ζ ₁-ζ ₄,r ₀, r _c, r ₁, r _cut, c _σ, a _π, f _σ, k _σ, g ₀ are all chosen before optimizing the remaining parameters (see [7] for details). This leaves 25 parameters to be determined.