Mapping of the energetically lowest exciton in bulk $1T$-HfS$_2$

By combining electron energy-loss spectroscopy and state-of-the-art computational methods, we were able to provide an extensive picture of the excitonic processes in $1T$-HfS$_2$. The results differ significantly from the properties of the more scrutinized group VI semiconducting transition metal dichalcogenides such as MoS$_2$ and WSe$_2$. The measurements revealed a parabolic exciton dispersion for finite momentum $\textbf{q}$ parallel to the $\Gamma$K direction which allowed the determination of the effective exciton mass. The dispersion decreases monotonically for momentum exchanges parallel to the $\Gamma$M high symmetry line. To gain further insight into the excitation mechanisms, we solved the ab-initio Bethe-Salpeter equation for the system. The results matched the experimental loss spectra closely, thereby confirming the excitonic nature of the observed transitions, and produced the momentumdependent binding energies. The simulations also demonstrated that the excitonic transitions for $\textbf{q}$ || $\Gamma$M occur exactly along that particular high symmetry line. For $\textbf{q}$ || $\Gamma$K on the other hand, the excitations traverse the Brillouin zone crossing various high symmetry lines. A particular interesting aspect of our findings was that the calculation of the electron probability density revealed that the exciton assumes a six-pointed star-like shape along the real space crystal planes indicating a mixed Frenkel-Wannier character.