Energetics of interlayer binding in graphite: The semiempirical approach revisited

We have developed a semiempirical method to obtain interlayer binding energy of graphite in the previous work [M. Hasegawa and K. Nishidate, Phys. Rev. B 70, 205431 (2004)]. In the present paper, we revisit this approach and develop an improved method, in which ab initio calculations based on the density functional theory (DFT) are also corrected through an empirical atom-atom van der Waals (vdW) interaction. The local density approximation (LDA) and generalized gradient approximation (GGA) are used in the DFT calculations. The parametrized damping function introduced to modify the asymptotic atom-atom vdW interaction is more flexible than the previous ones and covers a wider range of possibility in correcting for the approximate DFT calculations. The damping function is determined empirically by imposing the condition that the experimental interlayer spacing, in-plane lattice constant, and $c$-axis elastic constant are reproduced. We also require consistency between the LDA- and GGA-based methods ($\mathrm{LDA}+\mathrm{vdW}$, $\mathrm{GGA}+\mathrm{vdW}$) as the theoretically motivated necessary condition. The interlayer binding energy obtained by this method is $60.4\mathrm{meV}$/atom at $T=0\mathrm{K}$. The result of $\sim 54\mathrm{meV}$/atom at room temperature corrected by the thermal effect is consistent with the most recent experiment, $52\pm 5\mathrm{eV}$/atom [R. Zacharia et al., Phys. Rev. B 69, 155406 (2004)]. The atom-atom vdW interaction obtained by the present semiempirical method favorably corrects for the overbinding and underbinding nature of the LDA and GGA, respectively, in the in-plane energetics of graphite. That interaction also provides a useful starting point for the studies of energetics of other graphitic systems such as fullerenes and carbon nanotubes.

Figure 1

(Color online) Interlayer interaction energies $U\left(d\right)$ at the experimental interlayer spacing, $d=d_{\mathrm{expt}}$, as functions of ${\lambda }_0$ [the parameter of $f_{32}\left(r\right)$]. For each ${\lambda }_0$, $U\left(d\right)$ was determined by the physical conditions, (i)–(iii), described in the text. Both results for $U\left(d\right)$ ($\mathrm{LDA}+\mathrm{vdW}$ and $\mathrm{GGA}+\mathrm{vdW}$) coincide with each other at ${\lambda }_0=5.467$, thereby satisfying the theoretically motivated requirement (iv), and is equal to $-60.4\mathrm{meV}$/atom.

Figure 2

(Color online) Interlayer interaction energies as functions of the interlayer spacing obtained by the LDA (closed circles) and GGA (open circles) calculations and by the semiempirical methods, $\mathrm{LDA}+\mathrm{vdW}$ (solid line) and $\mathrm{GGA}+\mathrm{vdW}$ (dashed line), both being almost indistinguishable from each other.

Figure 3

(Color online) Damping functions $f_{\text{damp}}\left(r\right)$ given by Eq. (9) (with the parameters in Table ) in the $\mathrm{LDA}+\mathrm{vdW}$ (solid line) and $\mathrm{GGA}+\mathrm{vdW}$ (dashed line) methods. Also shown is the intrinsic part, $f_{32}\left(r\right)$ (dotted line).

Figure 4

(Color online) C-C vdW interaction potentials ${\varphi }_{\mathrm{vdW}}\left(r\right)$ given by Eq. (7) in the $\mathrm{LDA}+\mathrm{vdW}$ (solid line) and $\mathrm{GGA}+\mathrm{vdW}$ (dashed line) methods. Also shown are $-(C_6∕r^6)f_{32}\left(r\right)$ (dotted line) and the asymptotic form $-C_6∕r^6$ (dash-dot line). In (b), the vertical scale is expanded 50 times.

Figure 5

(Color online) Interlayer C-C interaction potentials ${\varphi }_{\mathrm{DFT}}\left(r\right)$ fitted to the calculated $U_{\mathrm{DFT}}\left(d\right)$ in the LDA (closed circles) and GGA (open circles) and the total potential, $\varphi \left(r\right)={\varphi }_{\mathrm{DFT}}\left(r\right)+{\varphi }_{\mathrm{vdW}}\left(r\right)$, obtained by the $\mathrm{LDA}+\mathrm{vdW}$ (solid line) and $\mathrm{GGA}+\mathrm{vdW}$ (dashed line) methods, both being almost indistinguishable from each other. Also shown is the asymptotic form $-C_6∕r^6$ (dotted line).