Symmorphic Intersecting Nodal Rings in Semiconducting Layers

The unique properties of topological semimetals have strongly driven efforts to seek for new topological phases and related materials. Here, we identify a critical condition for the existence of intersecting nodal rings (INRs) in symmorphic crystals, and further classify all possible kinds of INRs which can be obtained in the layered semiconductors with Amm2 and Cmmm space group symmetries. Several honeycomb structures are suggested to be topological INR semimetals, including layered and “hidden” layered structures. Transitions between the three types of INRs, named as $\alpha$, $\beta$, and $\gamma$ type, can be driven by external strains in these structures. The resulting surface states and Landau-level structures, more complicated than those resulting from a simple nodal loop, are also discussed.

Figure 1

(a) An $\alpha$-type INR, and (b) a simple model band structure manifesting nodal links from the two mirror symmetries $A$ and $B$ in (a). The right and left halves of (b) correspond to the bands on the $k$ path $X\text{−}Y$ [straight arrow in (a)] and $X\text{−}Z$ (curved black arrows), respectively. Different colors (red and blue) represent different symmetry eigenvalues. (c) $\beta$-type and (d) $\gamma$-type INRs. Symmetry eigenvalues of the occupied band are shown in (a), (c), and (d).

Figure 2

$A{A}^{\prime }$ stacked honeycomb layered structure (a), its primitive cell (b) and top view (c). “Hidden” $A{A}^{\prime }$ stacked honeycomb layered network (d), its primitive cell (e) and top view (f). Both of the structures are made of two kinds of atoms $A$1 and $A$2. $t_1\sim t_7$ in (b) and (e) describe the hopping parameters of the structures.

Figure 3

Band structures based on Eq. (1) with different parameters: (a) $t_1=-1$, $t_2=-0.5$, $t_3=-1.2$, $t_4=t_5=0$, $t_6=0.55$, $t_7=0.25$; (b) $t_1=-2.0$, $t_2=-1.1$, $t_3=-1.2$, $t_4=t_5=0$, $t_6=0.25$, $t_7=0.15$; (c) $t_1=-0.1$, $t_2=-0.05$, $t_3=-1.2$, $t_4=t_5=0$, $t_6=1.0$, $t_7=0.55$. Other parameters are ${ϵ}_1=1.7$, ${ϵ}_2=-0.9$. All the values are in units of eV. Red dashed line in (a) depicts band structure for a single-layer semiconducting with the same parameters as (a) but $t_6=t_7=0$. (d)–(f) Arrangements of the topological INRs in reciprocal space corresponding to the band structures in (a)–(c), respectively. In (a)–(c), eigenvalues for the $A$- and $B$-mirror planes are shown, where the two mirror planes are illustrated in (d). Note that, in (f), the $A$ eigenvalues at $T$ and $Z$ are denoted in different symbols ($a{1}^{\pm }$ and $a{2}^{\pm }$, respectively), and the BZ is different from (d)–(e) because of changed unit cell parameters, which are used to mimic the DFT results discussed later.

Figure 4

Projected band structures of (a) single-layer and (b) stacked 3D layered BN [Fig. 2]. Projected band structures of “hidden” layered structure SiC with $\theta =89°$ (c), 80° (d), and 108° (e). Insets: (c) charge density of a state around the nodal point, indicating the bonds are similar to the $\pi$ bonds in graphene; (d) a primitive cell of SiC where He atoms are inserted into the holes.

Figure 5

(a)–(c) Topological surface states for $\alpha$-, $\beta$- and $\gamma$-type LNRs, respectively, on the [$010$] surface. Insets show the surface states regions (red shadows) in the BZ. (d)–(f) Same but for the [$\overline{1}10$] surface. Because the [010] slabs are terminated by two different surfaces, two different surface states appear in (a)–(c). However, the surfaces of the [$\overline{1}10$] slabs are the same, and thus the two surface states in (d)–(f) are degenerate.