First-Principles-Based Method for Electron Localization: Application to Monolayer Hexagonal Boron Nitride

We present a first-principles-based many-body typical medium dynamical cluster approximation and density function theory method for characterizing electron localization in disordered structures. This method applied to monolayer hexagonal boron nitride shows that the presence of boron vacancies could turn this wide-gap insulator into a correlated metal. Depending on the strength of the electron interactions, these calculations suggest that conduction could be obtained at a boron vacancy concentration as low as 1.0%. We also explore the distribution of the local density of states, a fingerprint of spatial variations, which allows localized and delocalized states to be distinguished. The presented method enables the study of disorder-driven insulator-metal transitions not only in $h$-BN but also in other physical materials.

Figure 1

(a) Schematic of the TMDCA@DFT method. The DFT SCF solution is downfolded and used in the primary TMDCA SCF routine of a typical medium mapped by the disordered lattice. The cluster solver has also a secondary SCF routine to account for the response of electron interactions [see the Supplemental Material (SM) [22] for details]. (b) Illustration of a local density of states distribution $P[\rho ]$ that is approximately Gaussian (log-normal) for delocalized (localized) states. (c) The typical density of states, defined below, as a function of boron vacancy concentration $\delta$ and Hubbard-$U$ calculated at the Fermi level in monolayer hexagonal boron nitride. The typical density of states per unit cell [Fig. 2] ranging from 0 (blue) to $0.065{\mathrm{eV}}^{-1}$ (red) shows an insulator-metal transition at roughly $\delta \times U^4\approx 0.8{\mathrm{eV}}^4$. The dashed line is intended to give a rough estimate of the location of the transition in parameter space.

Figure 2

(a) The hexagonal structure of monolayer $h$-BN with a highlighted unit cell defined by the lattice vectors ${\overrightarrow{a}}_1$ and ${\overrightarrow{a}}_2$. (b) The DFT band structure (solid black bands) reproduced by the downfolded Hamiltonian $H_0$ (red dashed bands). The large band gap $E_g$ at the Brillouin zone corner points $K$ makes pristine $h$-BN an insulator.

Figure 3

TMDCA@DFT results of $h$-BN obtained for $N_c=8$ and $U=0$. (a) and (b) show the TDOS and the local density of states distribution, respectively, at three different $\delta$. For reference, (a) also includes the density of states of pristine $h$-BN obtained from DFT and the Fermi level (vertical dashed lines) from the TMDCA SCF calculations. (c) A comparison of the density of states obtained using the supercell approach with the TMDCA@DFT for $\delta =25\%$. (d) The formation energy as a function of boron and nitrogen vacancy concentrations.

Figure 4

TMDCA@DFT results of $h$-BN obtained for $N_c=8$ and $\delta =1.0\%$. (a) The TDoS for various $U$. At small $U$, impurity states develop near the valence band edge and as $U=2.5\mathrm{eV}$, these impurity states have hybridized with conduction band states, resulting in an insulator-metal transition. The vertical dashed line depicts the Fermi level. (b) The local density of states distribution at $U=2.5\mathrm{eV}$ and various energies. The distributions are characteristic of a disordered metal.