Electronic and optical excitations of two-dimensional ${\mathrm{ZrS}}_2$ and ${\mathrm{HfS}}_2$ and their heterostructure

In a first-principles study based on density functional theory and many-body perturbation theory, we investigate the electronic properties and the optical excitations of ${\mathrm{ZrS}}_2$ and ${\mathrm{HfS}}_2$ monolayers and their van der Waals heterostructure. Both materials have an indirect quasiparticle band gap, which amounts to about 2.8 eV in ${\mathrm{ZrS}}_2$ and to 2.6 eV in ${\mathrm{HfS}}_2$. In both systems the valence-band maximum is at $\mathrm{\Gamma }$ and the conduction-band minimum at M. Spin-orbit coupling induces a splitting of about 100 meV at the $\mathrm{\Gamma }$ point in the valence band, while it does not affect the conduction band. The optical absorption spectra are dominated by excitonic peaks with binding energies between 0.6 and 0.8 eV. The ${\mathrm{ZrS}}_2/{\mathrm{HfS}}_2$ heterobilayer exhibits a peculiar type-I level alignment with a large degree of hybridization between the two monolayers in the valence band, while the conduction bands retain either ${\mathrm{ZrS}}_2$ or ${\mathrm{HfS}}_2$ character, respectively. As a consequence, both the electron and the hole components of the first exciton are localized in the ${\mathrm{ZrS}}_2$ monolayer with nonvanishing probability of finding the hole also in the ${\mathrm{HfS}}_2$ sheet.

Figure 1

Top and side view of the 1T structure of monolayer ${\mathrm{ZrS}}_2$ and ${\mathrm{HfS}}_2$. The unit cell, indicated by the red line, has lattice parameter $a$; $d$ is the distance between the upper and the lower S atoms in the monolayer. Metal atoms are marked in blue and sulfur atoms in orange.

Figure 2

Band structures of monolayer ${\mathrm{ZrS}}_2$ (top) and ${\mathrm{HfS}}_2$ (bottom) without SOC (left panels) and with SOC (right panels). Dashed red (solid blue) lines indicate $G_0{W}_0$ (DFT) results. The Fermi energy ($E_F$) is set to zero in the midgap. Inset of panels (b) and (d): Valence-band maximum around $\mathrm{\Gamma }$ showing the spin-orbit splitting.

Figure 3

Total QP density of states (TDOS, gray area) of monolayer ${\mathrm{ZrS}}_2$ (top panel) and ${\mathrm{HfS}}_2$ (bottom panel). Atom-projected contributions from the dominant states are shown by colored lines according to the legend.

Figure 4

BSE spectra of monolayer (a) ${\mathrm{ZrS}}_2$ and (e) ${\mathrm{HfS}}_2$, given by the in-plane component of the imaginary part of the dielectric function [Eq. (6)]. Vertical bars denote the oscillator strengths of the main excitons. (b)–(d) k-space analysis of the excitons forming the three main peaks in (a). The same is shown for ${\mathrm{HfS}}_2$ in panels (f)–(h). The radius of the red circles is indicative of the relative weight of the electronic states contributing to the respective excitation.

Figure 5

Real-space distributions of the degenerate pair of excitons at the $\mathrm{\Gamma }$ point in monolayer ${\mathrm{HfS}}_2$ (top and side views shown). Hole (electron) distributions are shown in pink (gray). The position of the fixed coordinate is marked by a circle of the corresponding color. The unit cell is marked in green in the top views.

Figure 6

(a) Ball-and-stick representation of the ${\mathrm{ZrS}}_2/{\mathrm{HfS}}_2$ heterostructure in the AA stacking configuration, with blue, green, and orange spheres denoting Zr, Hf, and S atoms, respectively. (b) Total and atom-projected QP density of states of the ${\mathrm{ZrS}}_2/{\mathrm{HfS}}_2$ heterostructure (only the most relevant contributions are shown).

Figure 7

Character-resolved band structures of the ${\mathrm{ZrS}}_2/{\mathrm{HfS}}_2$ heterostructure computed (a) without SOC and (b) with SOC. Degenerate (a) and split-off bands (b) at $\mathrm{\Gamma }$ are highlighted in the insets. The QP correction is included through a scissors shift of 0.91 eV.

Figure 8

(a) BSE spectrum of the ${\mathrm{ZrS}}_2/{\mathrm{HfS}}_2$ heterostructure represented by the in-plane component of the imaginary part of the dielectric function with the most relevant excitonic peaks labeled according to their character. (b, c) Real-space distribution of the two degenerate excitons at the $\mathrm{\Gamma }$ point: upper (lower) panels depict the hole (electron) distribution with the fixed electron (hole) position indicated by the circles. The color code for atoms and isosurfaces is the same as in Figs. 6 and 5, respectively.