Layer dependence of defect charge transition levels in two-dimensional materials

Point defects in two-dimensional (2D) materials hold great promise for optoelectronic and quantum technologies. Their properties depend sensitively on the dielectric environment and number of 2D layers, but this has remained a challenge to include in first-principles calculations on account of the high computational cost. Recent first-principles techniques facilitate efficient prediction of substrate effects on defect charge transition levels in 2D materials, replacing the substrate by a continuum dielectric model. Here, we show that analogous continuum treatment of defect-free layers in multilayer 2D materials accurately predicts defect energies compared to explicit multilayer calculations, but at a small fraction of the computational cost. Applications of this technique to one- to five-layer and bulk hexagonal boron nitride reveal that defect ionization energies systematically decrease with an increasing number of layers and for defects in inner layers due to increased dielectric screening. Our results highlight the dominant role of electrostatic screening in the effect of the environment and the feasibility of tuning defect levels in 2D materials using material thickness and defect location within the material.

Figure 1

(a) Continuum model treatment of defects in an outer layer of five-layer hBN, specified by the dielectric function $\varepsilon \left(z\right)$. The layer containing the defect is placed at $z=0$, and the nearest peak of $\varepsilon \left(z\right)$ is $z_{dm}$ away. (b) Scheme for determining band edge positions of the five-layer system in the continuum approach, accounting for the rigid shift due to the electrostatic potential from the continuum model calculation and edge shifts ($\mathrm{\Delta }{E}_{\mathrm{CBM}}$, $\mathrm{\Delta }{E}_{\mathrm{VBM}}$) from (c) an explicit density-of-states calculation of the five-layer system. After aligning the core levels (gray dashed line) with that in 1L, the difference in the band edges (dashed red and blue lines) yields $\mathrm{\Delta }{E}_{\mathrm{CBM}}$ and $\mathrm{\Delta }{E}_{\mathrm{VBM}}$.

Figure 2

Convergence of GW calculations of the quasiparticle band gap of monolayer hBN with the number of bands included in the Coulomb hole summations $N_b$ and the energy cutoff for the dielectric matrix $S_{\mathrm{cut}}$ with an “infinite” number of empty bands for the dielectric matrix $N_c$ (approaching the number of plane waves) and a $12\times 12\times 1k$ mesh. Values at $S_{\mathrm{cut}}\to \infty$ are extrapolated using the 40–90 Ry results to infinite energy cutoff. The azure solid line represents the “final” extrapolated result at $S_{\mathrm{cut}}\to \infty$ and $N_b\to \infty$. The blue circle represents the band gap at the chosen parameters of $S_{\mathrm{cut}}$ = 40 Ry and $N_b=1700$.

Figure 3

Predicted formation energies of ${\mathrm{C}}_{\mathrm{B}}$, ${\mathrm{C}}_{\mathrm{N}}$, ${\mathrm{C}}_{\mathrm{B}}{\mathrm{V}}_{\mathrm{N}}$ and ${\mathrm{C}}_{\mathrm{N}}{\mathrm{V}}_{\mathrm{B}}$ in neutral and singly charged states ($+1$ for donors and $-1$ for acceptors) as a function of Fermi energy under (a) B-rich and (b) N-rich conditions. Colored lines and solid symbols denote formation energies and charge transition levels of these defects in monolayer (1L) hBN, while gray lines and open symbols denote them for the bulk hBN case. Black and gray vertical lines mark the corresponding band edge positions. Neutral defects shift by less than 0.1 eV, while charged defects are stabilized by $\approx 0.60$ eV from 1L to bulk, resulting in an $\approx 0.56$ eV reduction in acceptor ionization energies and $\approx 1.05$ eV reduction in donor ionization energies after accounting for small VBM and larger CBM shifts.

Figure 4

Charge transitions levels of ${\mathrm{C}}_{\mathrm{B}}$, ${\mathrm{C}}_{\mathrm{N}}$, ${\mathrm{C}}_{\mathrm{B}}{\mathrm{V}}_{\mathrm{N}}$, and ${\mathrm{C}}_{\mathrm{N}}{\mathrm{V}}_{\mathrm{B}}$ in 1L to 5L hBN for all possible nonequivalent placements (e.g., surface, inner, or central layer) of the defect shown in the lower panel. Ionization energies labeling the arrows are sorted from outermost to innermost layer placement of the defect. With respect to the vacuum level as the zero of the energy axis, the conduction band moves downwards with increasing levels much more substantially than the valence band, amplifying the ionization energy reduction of donors compared to acceptors.

Figure 5

(a) Band positions relative to the vacuum level from GW calculations. (b) Defect ionization energies relative to GW band edges as a function of the layer number. DFT ionization energies of ${\mathrm{C}}_{\mathrm{B}}$ and ${\mathrm{C}}_{\mathrm{N}}$ shown for comparison using dashed lines and half-filled triangles. GW increases all ionization energies, weakens the variation of the donor ionization energy with layers due to reduced variation of the CBM position, and strengthens the variation of acceptor ionization energy to greater variation of the VBM position.