Symmetry-guaranteed nodal-line semimetals in an fcc lattice

We demonstrate theoretically that nodal-line semimetals (NLSs) can be realized in an fcc lattice with orbitals belonging to the same irreducible representation, such as $\{p_x,p_y,p_z\}$ or $\{d_{xy},d_{yz},d_{zx}\}$ orbitals on every lattice site. The three orbitals are divided into two subgroups in terms of the parity with respect to the mirror reflections on high-symmetry planes of the fcc lattice, which, with rotation symmetry, endows symmetry-guaranteed NL passing through $W$ points in the Brillouin zone. Depending on the parameters, there also appears an accidental NL around the $\mathrm{\Gamma }$ point. We notice that the symmetry-guaranteed NL addressed in the present work can be found in band structures of elemental solids taking the fcc structure, such as Cu, Ag, Au, In, Ga, etc., as well as opal, which is an fcc photonic crystal of ${\mathrm{SiO}}_2$ spheres. Furthermore, we clarify that the fcc lattice of Si spheres exhibits a NL in a frequency band where no other photonic band exists, which provides a unique platform to realize topological NLSs under intensive search, and can be explored for achieving slow light.

Figure 1

(a) First Brillouin zone of the fcc lattice where one set of four $W$ points out of six sets is denoted by red dots which are connected by reciprocal lattice vectors. (b) $p_x-p_x$ and (c) $p_x-p_y/p_z$ hopping integrals between the nearest-neighbor sites of an fcc lattice. Wave functions with positive and negative signs are depicted in gray and green, respectively, whereas positive and negative hopping integrals are indicated in red and blue, respectively.

Figure 2

(a) and (b) symmetry-guaranteed NL (green curves) derived based on the TB model for the three $p$ orbitals with nearest-neighbor hopping and dispersions along the typical momentum directions. Hopping parameters are $t_1=0.8t_0$ and $t_2=1.2t_0$, satisfying $\tau <1$. (c) and (d) same as (a) and (b) except for $t_2=0.5t_0$, satisfying $\tau >1$, where accidental NLs (blue curves) appear additionally around the $\mathrm{\Gamma }$ point. (e) Dependence of the shape of NLs in the $k_z=0$ plane on hopping integrals in terms of $\tau =(t_0+t_1)/2t_2$ according to Eq. (8).

Figure 3

Band dispersions of an fcc lattice of dielectric spheres on the $k_z=0$ plane, where there are two sets of symmetry-guaranteed NLs (denoted by open circles) associated with the $p$ and $d$ orbitals and one set of NLs (denoted by dots) associated with their mixture. The blue and red curves are the bands with even and odd parity upon mirror reflection with respect to the $xy$ plane. Distributions of the electric field at the sphere center are displayed schematically for the $p$ and $d$ orbitals in the left and right insets, respectively (for details see the text). The radius of the spheres is $R=0.5a$, and the relative permittivity is ${\epsilon }_{\mathrm{fcc}}=11.8$, corresponding to Si [35]. (b) and (c) Schematic dispersions of $d$-like modes with and without the light cone, respectively. Purple (green) curves indicate the band belonging to the $E$ ($B_2$) representation of $\mathit{k}$-group $C_{4v}$ for the $\mathrm{\Gamma }X$ cut.

Figure 4

Band dispersion in the same cuts as Fig. 3 for a diamond lattice of dielectric spheres with the unit cell shown in the inset. The sphere radius is $r=\sqrt{3}/4a$, and the relative permittivity ${\epsilon }_{\mathrm{dia}}=6$, corresponding to lead glass [35].

Figure 5

(a) Heterostructure of the fcc and diamond lattices with the same lattice constant. (b) Band dispersions for $k_y=0$ of the structure shown in (a) with the projected BZ and NL in the $k_x-k_y$ plane displayed in the inset. (c) Intensity of the electric field in arbitrary units on the cross section of the heterostructure indicated by the dashed line in (a) for the almost-flat band at the projected $\mathrm{\Gamma }$ point. The thickness of the fcc (diamond) lattice is $34a$ ($16a$). All other parameters are the same as those in Fig. 3 and Fig. 4.

Figure 6

Same plot as Fig. 3 for the network of dielectric rods connecting all the nearest-neighbor sites of the fcc lattice. The dielectric constant and radius of rods are taken to be $\epsilon =11.56$ and $r=0.1a$ with lattice constant $a$.

Figure 7

(a) Symmetry-guaranteed NL in the $k_z=0$ plane and associated $\mathit{d}\left(\mathit{k}\right)$ vector. The black line indicates the BZ boundary. (b) Vortex formed by $d_1$ and $d_3$ components, with the green dots denoting two gap closing points and $k_{\parallel }$ denoting a momentum confined in the $k_z=0$ plane.

Figure 8

$\mathit{d}\left(\mathit{k}\right)$ vector on the $k_x=k_y$ plane for the system with SOC. $k_{\parallel }$ is the momentum along $\widehat{\mathit{x}}+\widehat{\mathit{y}}$.

Figure 9

Dispersion relations along several momentum directions (a) without and (b) with SOC calculated from Eq. (A6). In both cases hopping parameters ${\widehat{t}}_0$ are the same as those for Fig. 2, while SOCs are taken as $A=0.15t_0, B=0.1t_0$, and $C=0.2t_0$.

Figure 10

Schematics of energy dispersion along the path $k_y=0$ for a slab system with the surface in the $xy$ plane.