Substrate screening approach for quasiparticle energies of two-dimensional interfaces with lattice mismatch

Two-dimensional (2D) materials are outstanding platforms for exotic physics and emerging applications by forming interfaces. In order to efficiently take into account the substrate screening in the quasiparticle energies of 2D materials, several theoretical methods have been proposed previously but are only applicable to interfaces of two systems' lattice constants with certain integer proportion, which often requires a few percent of strain. In this work, we analytically showed the equivalence and distinction among different approximate methods for substrate dielectric matrices. We evaluated the accuracy of these methods by applying them to calculate quasiparticle energies of hexagonal boron nitride interface systems (heterojunctions and bilayers) and compared the results with explicit interface calculations. Most importantly, we developed an efficient and accurate interpolation technique for dielectric matrices that made quasiparticle energy calculations possible for arbitrarily mismatched interfaces free of strain, which is extremely valuable for practical applications.

Figure 1

Atomic structures of 2D interfaces (a) $\mathrm{hBN}/\mathrm{Sn}{\mathrm{S}}_2$, (b) bilayer hBN with AB stacking, and (c) bilayer hBN with ${\mathrm{AA}}^{\prime }$ stacking. The green balls denote boron atoms; the white balls denote N atoms; the yellow balls denote sulfur atoms; and the silver balls denote Sn atoms.

Figure 2

Schematic diagram of (a) mapping between computed data points of substrate (orange dots) and computed data points of material (blue dots), where the reciprocal space $\mathbf{q}+\mathbf{G}$ grid from the substrate and $\tilde{\mathbf{q}}+\tilde{\mathbf{G}}$ grid from the material are overlapping; (b) bilinear interpolation of the black cross point at $P$ (from $\tilde{\mathbf{q}}+\tilde{\mathbf{G}}$ grid) from the four nearest data points $A,B,C,D$ (red cross) when $P$ is inside the boundary of the $\mathbf{q}+\mathbf{G}$ grid (orange dots); and (c) proximal interpolation with the only nearest one point $A$ when the interpolation point is $P$ at the boundary.

Figure 3

3D plot of in-plane diagonal elements of function ${\epsilon }^{-1}-\mathbb{1}$ of $\mathrm{Sn}{\mathrm{S}}_2$ substrate, with $\mathbf{G}$ vector subset $\left(G_z,G_z^{\prime }\right)=\left(0,0\right)$ in (a) full reciprocal space and (b) an enlarged portion of panel (a) that contains the irreducible Brillouin zone of the interpolated points (blue) in hBN $\tilde{\mathbf{q}}+\tilde{\mathbf{G}}$ subspace. The $q_i+G_i,i=x,y$ are in-plane reciprocal space Cartesian coordinate in atomic unit (${\mathrm{Bohr}}^{-1}$). The orange points are directly computed data points (“DIR CALC”) in $\mathrm{Sn}{\mathrm{S}}_2$ substrate $\mathbf{q}+\mathbf{G}$ subspace, while the blue points are interpolated points (“INTERP”) to (hBN) material $\tilde{\mathbf{q}}+\tilde{\mathbf{G}}$ subspace. Note that the orange points used for interpolating blue points in panel (b) are beyond the first Brillouin zone of the substrate $\mathbf{q}+\mathbf{G}$ subspace. The single point at zero is the head element of ${\epsilon }^{-1}-\mathbb{1}$, which is exactly zero for both material and substrate.

Figure 4

hBN direct band gap at $K$ from several interfaces with different approximations of substrate screening, compared with explicit interface calculations. The black dashed line is the direct band gap at $K$ of free-standing ML hBN. For each symbol in the figure, “DIR HS” with black cross denotes direct GW calculation of explicit heterostructure; “${\chi }_{\text{eff}}$-sum” with blue circle denotes sum of effective polarizability approach by Eq. (3) with ground-state inputs from free-standing ML hBN; “${\chi }_{\text{eff}}^{GSC}$-sum” method with red down triangle denotes “${\chi }_{\text{eff}}$-sum” method with additional eigenvalue corrections from ground state interface eigenvalues (“GSC”); “${\chi }_{\text{eff}}^{FWF}$-sum” with magenta up triangle denotes “${\chi }_{\text{eff}}$-sum” method with both ground-state eigenvalues and wave functions from interfaces; and “$\chi$-sum” denotes noninteracting “$\chi$-sum” method by Eq. (6).

Figure 5

GW results for hBN direct band gap at $K$ using ${\chi }_{\text{eff}}$-sum or equivalently ${\chi }_0$-sum method to examine the effect of diagonal approximations. “Full matrix” in black cross denotes full dielectric matrices without any diagonal approximation as reference; “in-plane ${\epsilon }^{-1}$-diag” in red downward-pointing triangle denotes diagonal approximation to in-plane elements of ${\epsilon }^{-1}$; “out-of-plane ${\epsilon }^{-1}$-diag” in dark blue right-pointing triangle denotes diagonal approximation to out-of-plane elements for ${\epsilon }^{-1}$; “in-plane $\epsilon$-diag” in magenta upward-pointing triangle denotes diagonal approximation to in-plane elements of $\epsilon$; and “out-of-plane $\epsilon$-diag” in light blue left-pointing triangle denotes diagonal approximation to out-of-plane elements of $\epsilon$.

Figure 6

GW band structure of hBN at hBN/stretched $\mathrm{Sn}{\mathrm{S}}_2$ interface, referenced to valence band maximum (VBM) by using “${\chi }_{\text{eff}}$-sum” method. The blue solid line is computed with commensurate $\mathbf{q}$-point sampling with the reciprocal space mapping approach, while the red and black dashed line results are computed with incommensurate $\mathbf{q}$-point sampling with the reciprocal-space linear-interpolation approach. Panel (a) shows both valence and conduction band edges; panel (b) shows only the conduction band edge close to $K$.

Figure 7

GW band structures of hBN with stretched $\mathrm{Sn}{\mathrm{S}}_2$ substrate (“hBN/stretched $\mathrm{Sn}{\mathrm{S}}_2$,” red dash-dotted line), hBN with $\mathrm{Sn}{\mathrm{S}}_2$ substrate with no strain (“$\mathrm{hBN}/\mathrm{Sn}{\mathrm{S}}_2$,” blue solid line), and compressed hBN with $\mathrm{Sn}{\mathrm{S}}_2$ substrate (“compressed $\mathrm{hBN}/\mathrm{Sn}{\mathrm{S}}_2$,” black dashed line), respectively.