Universal polarization energies for defects in monolayer, surface and bulk hexagonal boron nitride : A finite-size fragments GW approach

We study defect energy levels in hexagonal boron-nitride with varying number of layers using a fragment many-body $GW$ formalism, taking as examples the paradigmatic carbon-dimer and $C_BV_N$ defects. We show that a single layer can be fragmented in polarizable finite-size areas reproducing faithfully the effect of the dielectric environment, dramatically facilitating the study at the many-body level of point defects in the dilute limit. The evolution of defect energy levels from the monolayer to a $n$-layer system due to increased screening, labeled polarization energies, follow a simple $({\Delta P}/n + P_{\infty})$ behavior. The coefficients $\Delta P$ and $P_{\infty}$ are found to be close-to-universal, with opposite signs for holes and electrons, characterizing mainly the host and the position of the defect (surface or bulk), but hardly the defect type. Our results rationalize the evolution of defect energy levels with layers number, allowing to safely extrapolate results obtained for the monolayer to few-layers, surface or bulk \textit{h}-BN. The present many-body fragment approach further opens the door to studying disordered 2D layers.