Layer response theory: Energetics of layered materials from semianalytic high-level theory

We present a readily computable semianalytic layer response theory (LRT) for analysis of cohesive energetics involving two-dimensional layers such as BN or graphene. The theory approximates the random phase approximation (RPA) correlation energy. Its RPA character ensures that the energy has the correct van der Waals asymptotics for well-separated layers, in contrast to simple pairwise atom-atom theories, which fail qualitatively for layers with zero electronic energy gap. At the same time, our theory is much less computationally intensive than the full RPA energy. It also gives accurate correlation energies near the binding minimum, in contrast to Lifshitz-type theory. We apply our LRT successfully to graphite and to BN, and to a graphene-BN heterostructure.

Figure 1

Scheme for division of the $z$ axis.

Figure 2

(a) Parallel-field layer polarizabilities $\text{Im}\left[{\alpha }_{xx}^{2\text{D,scr}}\left(\omega +i0\right)\right]$ deduced, via the correct equation [Eq. (30)], from vasp ${\varepsilon }_{xx}^{\text{macro}}$ data for solid BN with stretched lattice spacings $D=0.8\text{nm}$ (triangles) and $D=1.2\text{nm}$ (circles). The agreement for the two $D$ values is excellent, verifying that (30) correctly allows for the interaction between the layers. (b) As in (a), except that the wrong formula (31) (appropriate to ${\epsilon }_{zz}$ instead of ${\epsilon }_{xx}$, and reflecting a lack of interlayer interaction) was used. Now the cases $D=0.8$ and $1.2\text{nm}$ yield distinctly different results for $\text{Im}\left[{\left({\alpha }_{xx}^{2\text{D,scr}}\right)}^2\right]$, though one could still in principle obtain the correct ${\alpha }_{xx}^{2\text{D,scr}}\left(\omega +i0\right)$ by going to the $D\to \infty$ case, where the layers truly do not interact.

Figure 3

(a) Field-perpendicular layer polarizabilities $\text{Im}\left[{\alpha }_{zz}^{2\text{D,scr}}(\omega +i0)\right]$ deduced via Eq. (31), from vasp ${\varepsilon }_{zz}^{\text{macro}}$ data for solid BN with stretched lattice spacings $D=0.8\text{nm}$ (triangles) and $D=1.2\text{nm}$ (circles). The agreement for the two $D$ values is good. (b) As in (a) but with unjustified use of the field-parallel formula (30). Agreement between $D=0.8$ and $1.2\text{nm}$ is now not so good.

Figure 4

Interlayer correlation energy $U_c\equiv E_c\left(D\right)$ per atom (meV) of stretched bulk BN with layer spacing $D$. Short green dashes with pale blue crosses: RPA correlation energy from vasp. Red dots: asymptotic $D^{-4}$ result from Eq. (51). Dark green dash-dots: subasymptotic energy from Eqs. (64) and (50). Solid blue line: our semianalytic result $\left[O\right(Q)$ subasymptotics plus $O\left(Q^2\right)$ correction] from (39), (50), (69), and (70). Inset: $E_c$ vs $D^{-4} \left(\text{Å}{}^{-4}\right)$ for larger $D$ values, showing approximate proportionality $E_c\approx K{D}^{-4}$.

Figure 5

Interlayer correlation energy $U_c\equiv E_c\left(D\right)$ (meV/at) of stretched bulk BN using only the parallel response, showing the importance of including both perpendicular and parallel layer polarizability in the analytic work. Red dashed line (with light blue crosses at the numerical data points): full $E_c\left(D\right)$ from numerical RPA using vasp. Solid blue line: our best analytic fit, but with the perpendicular polarizability ${\alpha }_{zz}^{2\text{D,scr}}$ set to zero. Green dash-dots: our subasymptotic energy (64) with ${\alpha }_{zz}^{2\text{D,scr}}$ set to zero. Note the poor fit of our best theory to full RPA data here with ${\alpha }_{zz}^{2\text{D,scr}}$ set to zero, in contrast to the excellent fit in Fig. 4 where both ${\alpha }_{zz}^{2\text{D,scr}}$ and ${\alpha }_{xx}^{2\text{D,scr}}$ are included.

Figure 6

Total interlayer energy $U\equiv E_{\text{tot}}\left(D\right)$ of stretched bulk BN (meV/at), including exact exchange energy from vasp in all cases. Red dashes with blue-green crosses: with numerically exact RPA correlation energy from vasp. Solid blue line: with correlation energy from our best semianalytic theory (see also Fig. 4). Dark green dash-dots: with subasymptotic correlation energy from (64), (50), and (51).

Figure 7

Interlayer correlation energy $E_c\left(D\right)$ of stretched graphite (meV/at). Red dashes: RPA correlation energy from vasp, with individual data points shown by crosses. Blue solid line: our best semianalytic [subasymptotic + $O\left(Q^2\right)]$ theory from (73), (39), and (50). Green dash-dots: our subasymptotic theory. Inset: energy vs $D^{-3}$ for $D>1$ nm. See also Ref. [56].

Figure 8

Total interlayer energy $U\equiv E_{\text{tot}}\left(D\right)$ of stretched graphite (meV/at). Red dashes with crosses: numerically exact RPA energy including exact exchange energy, from vasp. Solid blue line: using correlation energy from our best semianalytic theory plus Hartree and exact exchange energy from vasp. Green dash-dots: our subasymptotic approximation.

Figure 9

Interlayer correlation energy $U_c\equiv E_c\left(D\right)$ from an infinite stretched stack $\cdots \text{BN-gr-BN-gr}\cdots$ of alternating graphene and BN layers. Red dashes with crosses: RPA correlation energy from vasp. Green dash-dots: our subasymptotic formula. Blue solid line: our best layer theory prediction through $O\left(Q^2\right)$ from (50), (39), (61), and (73).

Figure 10

Total interlayer energy $U\equiv E_{\text{tot}}\left(D\right)$ from an infinite stretched stack $\cdots \text{BN-gr-BN-gr}\cdots$ of alternating graphene and BN layers (meV/at), including exact exchange energy from vasp. Color scheme of graph as in Fig. 9.