Phonon-mediated superconductivity near the lattice instability in hole-doped hydrogenated monolayer hexagonal boron nitride

Employing the density functional theory with local density approximation, we show that the fully hydrogenated monolayer-hexagonal boron nitride (${\mathrm{H}}_2\mathrm{BN}$) has a direct band gap of 2.96 eV in the blue-light region, while the pristine $h$-BN has a wider indirect band gap of 4.78 eV. The hole-doped ${\mathrm{H}}_2\mathrm{BN}$ is stable at low carrier density ($n$) but becomes dynamically unstable at higher $n$. We predict that it is a phonon-mediated superconductor with a transition temperature ($T_c$) which can reach $\sim 31$ K at $n$ of $1.5\times {10}^{14}$ holes ${\mathrm{cm}}^{-2}$ near the lattice instability. The $T_c$ could be enhanced up to $\sim 82$ K by applying a biaxial tensile strain at 6% along with doping at $n$ of $3.4\times {10}^{14}$ holes ${\mathrm{cm}}^{-2}$ close to a new lattice instability.

Figure 1

DFT-optimized structure of an undoped ${\mathrm{H}}_2\mathrm{BN}$ monolayer: side (top panel) and top (bottom panel) views. The height (h) of the ${\mathrm{H}}_2\mathrm{BN}$ monolayer is 2.74 Å. Here, the height is defined as $\mathrm{h}=d_{\text{(B-H)}}$ $+$ $d_{\text{(N-H)}}$ $+$ $t_{\text{(BN)}}$, where $d_{\left(B\text{−}H\right)}$ is the bond length between B and H atoms, and $d_{\text{(N-H)}}$ is the bond length between N and H atoms, and $t_{\text{(BN)}}$ is the thickness of a reconstructed BN layer. Alternatively, the height of the monolayer can be defined as the difference in z-position of the two hydrogen atoms, $H_B$ and $H_{\text{N}}$, i.e., $Z\left(H_{\text{B}}\right)$ $Z\left(H_{\text{N}}\right)$, where $Z\left(H_{\text{B}}\right)$ and $Z\left(H_{\text{N}}\right)$ represent the z positions of hydrogen atoms bonded to boron and nitrogen atom, respectively. The ($1\times 1$) unit cell of the ${\mathrm{H}}_2\mathrm{BN}$ is shown by black lines (bottom).

Figure 2

Electronic band structures of monolayer (a) $h$-BN and (b) ${\mathrm{H}}_2\mathrm{BN}$ (undoped) along the high-symmetry directions in the first Brillouin zone. The chemical potential is set to zero. In inset, the $E_g$ represents the energy gap between the valence band maximum and the conduction band minimum.

Figure 3

(a) Electronic band structure along high-symmetry directions in the first Brillouin zone (left) and the electronic density of states (right) of the hole-doped ${\mathrm{H}}_2\mathrm{BN}$ with the carrier density of $1.0\times {10}^{14}$ holes ${\mathrm{cm}}^{-2}$. The Fermi level is set to zero. The color bar represents the projections of H s states to bands. For clarity of the effect of hole doping, the band structure for the small energy window around the Fermi level is also shown in inset. [(b) and (c)] Fermi surfaces in the first Brillouin zone of the doped ${\mathrm{H}}_2\mathrm{BN}$ with the carrier density ($n$) of (b) $1.0\times {10}^{14}$ and (c) $1.5\times {10}^{14}$ holes ${\mathrm{cm}}^{-2}$. In (b), the high-symmetry $k$-point directions ($K\text{−}\mathrm{\Gamma }\text{−}M\text{−}K$) are also depicted by arrow. The Fermi surfaces of strained ${\mathrm{H}}_2\mathrm{BN}$ with BTS of 6% are shown for $n$ of: (d) $2.0\times {10}^{14}$ and (e) $3.4\times {10}^{14}$ holes ${\mathrm{cm}}^{-2}$.

Figure 4

Comparison of phonon dispersion of the undoped ${\mathrm{H}}_2\mathrm{BN}$ (green trace) and hole-doped ${\mathrm{H}}_2\mathrm{BN}$ with carrier density of $1.0\times {10}^{14}$ holes ${\mathrm{cm}}^{-2}$ (red) for the selected frequency range up to 1280 ${\mathrm{cm}}^{-1}$, showing the Kohn anomaly [46].

Figure 5

Phonon frequency dispersion along high-symmetry directions of the first Brillouin zone (left) and phonon density of states (right) of the hole-doped ${\mathrm{H}}_2\mathrm{BN}$ with $n$ of $1.0\times {10}^{14}$ holes ${\mathrm{cm}}^{-2}$. The modes with $q<2k_F$ are shaded.

Figure 6

Phonon frequency dispersion of the strained and doped ${\mathrm{H}}_2\mathrm{BN}$ models under 6% BTS with the carrier densities of (a) $3.4\times {10}^{14}$ and (b) $3.5\times {10}^{14}$ holes ${\mathrm{cm}}^{-2}$. For the latter carrier density, the doped ${\mathrm{H}}_2\mathrm{BN}$ has negative frequencies around the $\mathrm{\Gamma }$ point, as shown in inset.

Figure 7

Eliashberg function of hole-doped ${\mathrm{H}}_2\mathrm{BN}$ with the carrier density of (a) $1.0\times {10}^{14}$, (b) $1.3\times {10}^{14}$ (green trace), and $1.5\times {10}^{14}$ holes ${\mathrm{cm}}^{-2}$ (magenta).

Figure 8

Projected density of states (PDOS) of (a) monolayer $h$-BN and (b) ${\mathrm{H}}_2\mathrm{BN}$ without doping. The chemical potential is set to zero. In (a), the B $2p_z$ state of monolayer $h$-BN is partly unfilled just above the chemical potential (pink trace) where the density of B $2p_z$ is relatively higher than that below the chemical potential. In (b), for the case of ${\mathrm{H}}_2\mathrm{BN}$, both B $2p_z$ and H $1s$ states dominantly contribute to the DOS just below the chemical potential, while the N and B $2p_{x,y}$ orbitals also contribute to the DOS. The contributions from $2p_x$ and $2p_y$ are added and shown as $2p_{x,y}$. Here, H1 and H2 are bonded to B and N atoms, respectively.

Figure 9

Projected density of states (PDOS) of hole-doped ${\mathrm{H}}_2\mathrm{BN}$ with a carrier density of $1.0\times {10}^{14}$ holes ${\mathrm{cm}}^{-2}$. Here, the PDOS is shown for the selected energy range from $-3.0$ to 5.6 eV. The Fermi level is set to zero. Within from $\sim -1.8$ to 0.2 eV, B $2p_z$ (magenta trace) and H1 s (green) states are strongly hybridized. The N $2p_{x,y}$ also have contributions to this hybridized state. The H1 atom which is bonded to B atom has similar PDOS near and at Fermi level to that coming from B $2p_z$ orbital.

Figure 10

Schematic representations of the selected optical phonon modes at the $\mathrm{\Gamma }$ point of Brillouin zone of the hole-doped ${\mathrm{H}}_2\mathrm{BN}$ with a hole carrier density of $1.0\times {10}^{14}$ holes ${\mathrm{cm}}^{-2}$. The green, pink, and blue arrows indicate the motions of hydrogen, boron, and nitrogen atom, respectively.

Figure 11

Phonon frequency dispersion of the doped ${\mathrm{H}}_2\mathrm{BN}$ with the carrier density of (a) $1.5\times {10}^{14}$ and (b) $1.6\times {10}^{14}$ holes ${\mathrm{cm}}^{-2}$. For the latter hole carrier density, the doped ${\mathrm{H}}_2\mathrm{BN}$ system has negative frequencies shown in inset.

Figure 12

Superconducting transition temperature for hole-doped ${\mathrm{H}}_2\mathrm{BN}$ with BTS=6% and carrier density of $3.4\times {10}^{14}$ holes ${\mathrm{cm}}^{-2}$ as a function of degauss for phonons.

Figure 13

Superconducting transition temperature for hole-doped ${\mathrm{H}}_2\mathrm{BN}$ with BTS=6% and carrier density of $3.4\times {10}^{14}$ holes ${\mathrm{cm}}^{-2}$ as a function of double delta smearing for e-ph interaction.

Figure 14

The variation of $T_c$ with ${\mu }^{*}$. In our work, we have chosen ${\mu }^{*}=0.13$, as indicated by the dashed line. The chosen ${\mu }^{*}$ can be regarded as Coulomb repulsion with the moderate strength.