Electronic and hyperbolic dielectric properties of $\mathrm{Zr}{\mathrm{S}}_2/\mathrm{Hf}{\mathrm{S}}_2$ heterostructures

In this paper we investigate the electronic and optical dielectric properties of lateral and vertical heterostructures composed of two-dimensional (2D) $\mathrm{Zr}{\mathrm{S}}_2$ and $\mathrm{Hf}{\mathrm{S}}_2$ monolayers based on density-functional theory. First, we show that the bulk and monolayer $\mathrm{Zr}{\mathrm{S}}_2$ and $\mathrm{Hf}{\mathrm{S}}_2$ as well as the vertical ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_m/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_n$ heterostructures are indirect band-gap semiconductors, while the lateral heterostructures exhibit an indirect to direct band-gap transition. Then we demonstrate that the optical properties of the bulk and monolayer $\mathrm{Hf}{\mathrm{S}}_2$ and $\mathrm{Zr}{\mathrm{S}}_2$ are strongly anisotropic; for the bulk $\mathrm{Hf}{\mathrm{S}}_2$ and $\mathrm{Zr}{\mathrm{S}}_2$, the in-plane components of the dielectric function are negative in a certain frequency band, where they can work as naturally hyperbolic metamaterials. Interestingly, the vertical heterostructures also possess a hyperbolic region, whose position and width can be tunable with the thickness ratio of constituents. It is also found that the $\left(\mathrm{Zr}{\mathrm{S}}_2\right)/\left(\mathrm{Hf}{\mathrm{S}}_2\right)$ vertical heterostructures can enhance spontaneous emission and about 100-fold improvement of the Purcell factor is obtained. These results prove the feasibility of 2D material heterostructures to realize tunable hyperbolic metamaterials; the heterostructures present a promising opportunity for the practical applications in light-generation technologies.

Figure 1

Brillouin zone with high-symmetry $k$ points used in band-structure calculations, where $1T-M{X}_2$ ($M=\mathrm{Zr}$, Hf; $X=\mathrm{S}$) unit-cell structure is also shown; the green and yellow spheres denote $M$ and S atoms, respectively.

Figure 2

Band structures of bulk $\mathrm{Zr}{\mathrm{S}}_2$, bulk $\mathrm{Hf}{\mathrm{S}}_2, \mathrm{Zr}{\mathrm{S}}_2$ monolayer, and $\mathrm{Hf}{\mathrm{S}}_2$ monolayer. The blue arrows indicate the band gaps and the horizontal dashed lines represent the Fermi level.

Figure 3

Top and side views of atomic configurations of ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_m-{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_n$ lateral heterostructure monolayers with zigzag interlines, where $m=6$ and $n=6$. Green, sand, and yellow balls indicate Zr, Hf, and S atoms, respectively. (b) 2D Brillouin zone of the monolayer LH with high-symmetry points.

Figure 4

Calculated band structures of the periodic lateral heterostructures ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_m-{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_n$ at PBE level. (a) ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_1-{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_1$, (b) ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_2-{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_2$, (c) ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_3-{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_3$, (d) ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_6-{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_6$, and (e) ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_9-{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_9$, where (c) is the calculated projected band structure; the blue and red lines represent the $\mathrm{Zr}{\mathrm{S}}_2$ and $\mathrm{Hf}{\mathrm{S}}_2$ components of the bands, respectively. The blue arrows indicate the band gaps and the horizontal dashed lines represent the Fermi level.

Figure 5

Calculated projected band structure of the $V$-infinite heterostructures ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_1/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_1$; the top and side views of the $\mathit{AA}$ stacking heterostructure are shown in the inset. (b) The band gap of different $V$-infinite heterostructures ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_m/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_n$, where $m=n$ is an integer.

Figure 6

Calculated band structures of the $V$-infinite heterostructures ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_1/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_3$ (a), ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_1/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_5$ (b), and ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_1/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_7$ (c), respectively.

Figure 7

Calculated projected band structure of the $V$-finite heterostructures ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_1/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_1$ constructed by $\mathrm{Zr}{\mathrm{S}}_2$ and $\mathrm{Hf}{\mathrm{S}}_2$ monolayer; the inset shows the side views of the $\mathit{AA}$ stacking heterostructures. (b) Band gap of different $V$-finite heterostructures ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_m/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_n$ formed by thin stacks of $\mathrm{Zr}{\mathrm{S}}_2$ and $\mathrm{Hf}{\mathrm{S}}_2$.

Figure 8

Real parts of the two principal components of permittivity of bulk $\mathrm{Zr}{\mathrm{S}}_2$, bulk $\mathrm{Hf}{\mathrm{S}}_2$, monolayer $\mathrm{Zr}{\mathrm{S}}_2$, and monolayer $\mathrm{Hf}{\mathrm{S}}_2$, respectively, where ${\varepsilon }_{\parallel }$ represents in-plane component and ${\varepsilon }_{\perp }$ represents out-of-plane component. The shaded region shows the hyperbolic regions.

Figure 9

Real parts of ${\varepsilon }_{\parallel }$ (${\varepsilon }_{xx}$,${\varepsilon }_{yy}$) and ${\varepsilon }_{\perp }$(${\varepsilon }_{zz}$) of the periodic lateral heterostructures ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_1-{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_1$ and ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_4-{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_4$, respectively.

Figure 10

${\varepsilon }_{\parallel }$ of the $V$-infinite heterostructures ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_1/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_1, {\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_3/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_3, {\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_6/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_6$, and ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_9/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_9$ (a), where the shaded region shows the hyperbolic regions. (b) Hyperbolic regions of different $V$-infinite heterostructures ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_m/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_n$. (c) ${\varepsilon }_{\parallel }$ of the $V$-infinite heterostructures ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_1/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_3$ and ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_1/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_5$; for comparison, ${\varepsilon }_{\parallel }$ of ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_1/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_1$ is also plotted. (d) Effective ${\varepsilon }_{\parallel }$ of the composited $\mathrm{Zr}{\mathrm{S}}_2/\mathrm{Hf}{\mathrm{S}}_2$ materials with different thickness ratios based on EMT, where the olive dots represent the calculated zeros of ${\varepsilon }_{\parallel }$ for ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_1/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_1, {\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_1/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_3$, and ${\left(\mathrm{Zr}{\mathrm{S}}_2\right)}_1/{\left(\mathrm{Hf}{\mathrm{S}}_2\right)}_5$, respectively.

Figure 11

(a) Purcell factor for the dipoles perpendicular ($F_{p\perp }$, short dotted line) and parallel ($F_{p//}$, short dashed line) to the surface as depicted in the insets (the blue dot represents a dipole); the solid line represents the Purcell factor $F_{\mathrm{iso}}$ for an isotropic dipole which is averaged from $F_{p\perp }$ and $F_{p//}$. The dipole is in air 5 nm above the $\left(\mathrm{Zr}{\mathrm{S}}_2\right)/\left(\mathrm{Hf}{\mathrm{S}}_2\right)$ multilayer HMM with$d_{\mathrm{HMM}}=20\mathrm{nm}$. (b) Purcell factor $F_{p\perp }$ as a function of wavelength with different $d$ and ${\varepsilon }_1$.