Three-center tight-binding potential model for C and Si

A tight-binding potential model which goes beyond the Slater-Koster two-center approximation and includes explicit three-center and crystal field expressions is presented. Using carbon and silicon as examples, we show that various bulk structures, surface reconstructions, and the structures of clusters and liquids of C and Si can be well described by the present three-center tight-binding model. These results demonstrate that three-center interaction and crystal field effect are very important for improving the transferability of tight-binding models in describing the structures and properties of materials over a broad range of bonding configurations.

Figure 1

Illustration of the 3c-TB Hamiltonian model. $H_{1s,2pz}^{2\mathrm{c}}=0,H_{1py,2pz}^{2\mathrm{c}}=0$ in the two-center approximation, but the three-center corrections are $H_{1s,2pz}^{3\mathrm{c}}\ne 0,H_{1py,2pz}^{3\mathrm{c}}\ne 0$; the crystal field effect $H^{cf}$ leads to lower $s$ and $p$ orbital energies and a splitting of $px,py$, and $pz$ energy levels.

Figure 2

3c-TB energy bands (dotted line) of C in the diamond, graphene, sc, and bcc structures compared with the DFT-PW91 (solid line) and DFTB (circle line) calculated results.

Figure 3

3c-TB energy bands (dotted line) of Si in the diamond, sc, bcc, and fcc structures compared with the DFT-PW91 (solid line) and DFTB (circle line) calculated results.

Figure 4

Binding energies per atom $\left(E_b\right)$ from the 3c-TB and DFTB calculations for C and Si compared with the DFT-PW91 calculated results. The DFT-PW91 energies were shifted in order to make the binding energies of the C and Si diamond structures consistent with the experimental data of $-7.37$ and $-4.67\mathrm{eV}$, respectively. For comparison, the DFTB energies were also shifted to make the binding energies of diamond structures at equilibrium bond lengths equal to the shifted DFT-PW91 values.

Figure 5

Binding energies per atom $\left(E_b\right)$ relative to free singlet C or Si atoms of the ${\mathrm{C}}_n$ and $\mathrm{S}{\mathrm{i}}_n$ clusters calculated at 3c-TB, compared with the calculated results at DFTB, and DFT-B3LYP/6-31G$\left(d\right)$ for C and DFT-PBEPBE/6-31G$\left(d\right)$ for Si. The absolute binding energies at DFT are much lower than the TB data, since in the data fitting of the 3c-TB parameters, the DFT energies were shifted upwards to make the binding energy of the C and Si diamond structures consistent with the experimental values $-7.37\mathrm{eV}$ (C) and $-4.67\mathrm{eV}$ (Si).

Figure 6

Scatter plot of the binding energies per atom from the TB and DFTB calculations versus those from the calculations at DFT-B3LYP for ${\mathrm{C}}_n$ and DFT-PBEPBE for $\mathrm{S}{\mathrm{i}}_n$ with $n$ denoted in Fig. 5.

Figure 7

3c-TB optimized structures of ${\mathrm{C}}_{8,14,20,32}$ and $\mathrm{S}{\mathrm{i}}_{7,10,20,32}$, compared with the DFT-B3LYP/6-31G$\left(d\right)$ optimized ${\mathrm{C}}_n$ and DFT-PBEPBE/6-31G$\left(d\right)$ optimized $\mathrm{S}{\mathrm{i}}_n$ cluster structures. The atoms in light color for $\mathrm{S}{\mathrm{i}}_{32}$ denote the embedded atoms.

Figure 8

Pair correlation function $g\left(r\right)$ for (a) liquid carbon and (b) liquid silicon obtained from the TBMD simulations using the 3c-TB potential model are compared with the first-principles simulated results [38, 39].