Electronic states and modulation doping of hexagonal boron nitride trilayers

We study the stabilities and geometric and electronic properties of hexagonal boron nitride $\left(h\text{−}\mathrm{BN}\right)$ trilayers by using first-principles electronic-structure calculations within the framework of the density functional theory. From the results of total-energy calculations, we reveal the relative stabilities for various stacking sequences of $h$-BN trilayers. We also show that energy-band structures as well as spatial distributions of wave functions at the valence-band maximum (VBM) and the conduction-band minimum (CBM) strongly depend on the stacking sequences of the $h$-BN trilayers. We further investigate the effects of substitutional doping of a carbon atom on the electronic properties of the $h$-BN trilayers. In several stacking sequences of the C-doped $h$-BN trilayers, we find that the C-atom dopant can be spatially separated from the carrier transport layers associated with the VBM or the CBM, suggesting the possibility of realizing conduction channels only weakly disturbed by the C-atom impurity in $h$-BN trilayers. Interestingly, these donor states spatially separated from the CBM state are found to become rather shallow. This theoretical finding of “atomically thin modulation doping” using the $h$-BN layers may open an important way to design future layered electronic device materials.

Figure 1

Nomenclature of stacking sequence for $h$-BN layers: (a) $a$, $b$, and $c$ layers and (b) $a^{\prime }$, $b^{\prime }$, and $c^{\prime }$ layers. The dashed line denotes the unit cell of the monolayer sheet. The green and gray balls represent B and N atoms, respectively.

Figure 2

Schematic view of three different stacking sequences of $h$-BN trilayers: (a) ${\mathrm{aa}}^{\prime }a$, (b) abc, and (c) ${\mathrm{acb}}^{\prime }$. Dashed lines are guides to the eye to indicate relative positions of atoms within the layer and between the adjacent layers.

Figure 3

Energy-band structures of the pristine $h$-BN trilayers. The Fermi level is set to zero. In each case, blue and red arrows indicate the conduction-band minimum and valence-band maximum, respectively. Six sequences [(a) to (f)] possess indirect gaps, while three sequences [(g), (h), and (j)] possess direct gaps. The ${\mathrm{aa}}^{\prime }b$ trilayer in (i) possesses nearly direct gap.

Figure 4

Isosurfaces of squared wave functions (a) at the CBM and (b) at the VBM for the 10 stable stacking sequences. Isosurface values of the electron density is set to 0.01 electron/${\mathrm{bohr}}^3$.

Figure 5

Energy-band structures of the $h$-BN trilayer with (a) ${\mathrm{C}}_{\mathrm{B}}$ and (b) ${\mathrm{C}}_{\mathrm{N}}$. The Fermi level is set to zero.

Figure 6

Isosurfaces of squared wave functions of the second-lowest unoccupied state at five points along the ${\mathrm{T}}^{\prime }$ line (from the K point to the M point) in the $ac{b}^{\prime }$ trilayer. Isosurface values of the electron density is set to 0.01 electron/${\mathrm{bohr}}^3$.

Figure 7

Energy-band structures of $\overline{a}c{b}^{\prime }$, $ac{b}^{\prime }$ (undoped), and $\underline{a}bc$ obtained by using the 8 $\times$ 8 supercell. The Fermi level is set to zero.

Figure 8

Isosurfaces of the squared wave functions at (a) the CBM and donor states of three stacking sequences and (b) the acceptor and VBM states of seven stacking sequences. In these trilayers, the conducting layer(s) and the doped layer are spatially separated. The isosurface value of the electron density is set to 0.00016 electron/${\mathrm{bohr}}^3$. In each panel, the side view of isosurfaces on one supercell (8 $\times$ 8) is shown with the projected position of ${\mathrm{C}}_{\mathrm{B}}$ (${\mathrm{C}}_{\mathrm{N}}$), indicated by the arrow.