Topological Dirac nodal loops in nonsymmorphic hydrogenated monolayer boron

The existence of a topological Dirac nodal loop is predicted for hydrogenated monolayer boron sheets with a nonsymmorphic symmetry. The three-center two-electron bonds in boron compounds restrict the electronic system to be insulating or semimetallic with a Dirac nodal loop. Two types of electronic structures are distinguished by the ${\mathbb{Z}}_2$ topological index and confirmed by first-principles total-energy calculations. The topological taxonomy, developed in this research, can be applied to other two-dimensional materials and to seek novel Dirac fermions.

Figure 1

(a) Atomic structure of $(5-7)-\alpha$-borophene monolayer, and two-dimensional Brillouin zone with symbols at the symmetry points. Top and side views of atomic structures of the HB layer (borophane) of (b) $(5-7)-{\alpha }_1-{\mathrm{B}}_{16}{\mathrm{H}}_{16}$ monolayer and (c) $(5-7)-{\alpha }_2-{\mathrm{B}}_{16}{\mathrm{H}}_{16}$ monolayer. Boron and hydrogen atoms are shown as blue and white balls, respectively, while the unit cell is shown by the solid line. Locations of the 5-membered rings are indicated with red pentagons. Bond drawings are defined by the interatomic distance.

Figure 2

(a) Atomic structure of $(5-6-7)-\gamma$-borophene, (b) the HB layer (borophane) of $\text{(5-6-7)-}{\gamma }_1-{\mathrm{B}}_{24}{\mathrm{H}}_{24}$, and (c) the HB layer (borophane) of $\text{(5-6-7)-}{\gamma }_2-{\mathrm{B}}_{24}{\mathrm{H}}_{24}$. Locations of the 5-membered and 6-membered rings are indicated with red pentagons and purple hexagons, respectively.

Figure 3

(a) $\alpha$-borophene as an example of the layer group No. 44. (b) $(5-7)-\alpha$-borophene before (translucent) and after (opaque) $C_{2x}$ operation. (c) $\alpha$-borophene translated by $\frac{1}{2}{\mathit{a}}_1+\frac{1}{2}{\mathit{a}}_2$ after (b). We can see that now the new configuration (opaque) is equivalent to the original one in (a) (shown as the translucent one).

Figure 4

Brillouin zone of the layer group No. 44.

Figure 5

Schematic picture of the band dispersion in the layer group No. 44. All bands on the zone boundary are doubly degenerate and the degenerate pairs have the same eigenvalues.

Figure 6

Electronic energy band dispersions [(a) and (c)] along the $k$ path in the Brillouin zone and spatial distributions of wave functions [(b) and (d)] of ${\psi }_1$ and ${\psi }_2$ bands at the $\mathrm{\Gamma }$ point for $(5-7)-{\alpha }_1-{\mathrm{B}}_{16}{\mathrm{H}}_{16}$ and $(5-7)-{\alpha }_2-{\mathrm{B}}_{16}{\mathrm{H}}_{16}$, respectively. $E_F$ is the Fermi energy. DP means Dirac point. The color of the wave functions corresponds to the sign of wave functions. The parity of mirror reflection symmetry with respect to the $x-y$ plane (layer plane) for ${\psi }_1$ and ${\psi }_2$ bands are labeled with $+$ and $-$ signs, respectively.

Figure 7

Energy band dispersion of (a) $\text{(5-6-7)-}{\gamma }_1-{\mathrm{B}}_{24}{\mathrm{H}}_{24}$ and (b) $\text{(5-6-7)-}{\gamma }_2-{\mathrm{B}}_{24}{\mathrm{H}}_{24}$. The parity of mirror reflection symmetry with respect to the $x-y$ plane (layer plane) for ${\psi }_1$ and ${\psi }_2$ bands are labeled with $+$ and $-$ signs, respectively. Two-dimensional Brillouin zone are shown with symbols at the symmetry points.

Figure 8

Phonon dispersion diagrams of layers of $(5-7)-{\alpha }_1-{\mathrm{B}}_{16}{\mathrm{H}}_{16}, (5-7)-{\alpha }_2-{\mathrm{B}}_{16}{\mathrm{H}}_{16}$, and $\text{(5-6-7)-}{\gamma }_1-{\mathrm{B}}_{24}{\mathrm{H}}_{24}$.

Figure 9

Simulation of atomic structure of the $(5-7)-{\alpha }_1-{\mathrm{B}}_{16}{\mathrm{H}}_{16}$ after 5.0 ps at 700 K.

Figure 10

Simulation of atomic structure of the $(5-7)-{\alpha }_2-{\mathrm{B}}_{16}{\mathrm{H}}_{16}$ after 5.0 ps at 700 K.

Figure 11

Structure of $\alpha$-borophene as an example of the layer group No. 44. The equivalent points atoms are indicated with same symbols (red point, blue circle, orange circle, red square and blue square). When a bond is placed at the corner of the unit cell (red point), another bond must be placed at the center of the unit cell (shown with another red point) to satisfy the symmetry of the layer group No. 44. When an atom is placed at the position indicated with blue circle, other three atoms must be placed at the position indicated with other three blue circles.

Figure 12

Schematic picture of the band dispersion in the layer group No. 44. Electronic bands of odd and even symmetries are labeled $+$ and $-$, respectively. A relation between a DNL and the symmetry points of the Brillouin zone for the cases (a) without and (b) with DNL. (c) A schematic drawing of DNL.

Figure 13

Schematic pictures of Brillouin zone, nodal lines and band dispersion for the system with [(a) and (b)] two nodal lines and ${\mathbb{Z}}_2=0$ and (c) no nodal line and ${\mathbb{Z}}_2=0$. (a) Two bands are inverted and as a result the ${\mathbb{Z}}_2$ is kept trivial. (b) The band inversion occurs only on some part of the line between the $\mathrm{\Gamma }$ point and the zone boundary. In this case, the symmetry of wave function on the $\mathrm{\Gamma }$ point is not changed by the band inversion. Thus it is essentially impossible to distinguished (b) and (c) by examining the $\mathrm{\Gamma }$ point.

Figure 14

(a) Atomic structure and (b) energy band dispersion of $\text{(5-6-7)-}{\gamma }_1-{\mathrm{B}}_{24}{\mathrm{H}}_{24}$. [(c) and (d)] Spatial distributions of wave functions of ${\psi }_1$ and ${\psi }_2$ bands at the $\mathrm{\Gamma }$ point. Locations of the 5-membered and 6-membered rings are indicated with red pentagons and purple hexagons, respectively. $E_F$ is the Fermi energy. The color of wave functions corresponds to the sign of wave functions.

Figure 15

(a) Atomic structure and (b) energy band dispersion of $\text{(5-6-7)-}{\gamma }_2-{\mathrm{B}}_{24}{\mathrm{H}}_{24}$. [(c) and (d)] Spatial distributions of wave functions of ${\psi }_1$ and ${\psi }_2$ bands at the $\mathrm{\Gamma }$ point. Locations of the 5-membered and 6-membered rings are indicated with red pentagons and purple hexagons, respectively. $E_F$ is the Fermi energy. The color of wave functions corresponds to the sign of wave functions.

Figure 16

(a) Atomic structure and (b) energy band dispersion of $\text{(5-6-7)-}{\gamma }_3-{\mathrm{B}}_{24}{\mathrm{H}}_{24}$. [(c) and (d)] Spatial distributions of wave functions of ${\psi }_1$ and ${\psi }_2$ bands at the $\mathrm{\Gamma }$ point. Locations of the 5-membered and 6-membered rings are indicated with red pentagons and purple hexagons, respectively. $E_F$ is the Fermi energy. The color of wave functions corresponds to the sign of wave functions.

Figure 17

(a) Atomic structure and (b) energy band dispersion of $\text{(5-6-7)-}{\gamma }_4-{\mathrm{B}}_{24}{\mathrm{H}}_{24}$. [(c) and (d)] Spatial distributions of wave functions of ${\psi }_1$ and ${\psi }_2$ bands at the $\mathrm{\Gamma }$ point. Locations of the 5-membered and 6-membered rings are indicated with red pentagons and purple hexagons, respectively. $E_F$ is the Fermi energy. The color of wave functions corresponds to the sign of wave functions.