  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 ZrS$_2$ and HfS$_2$ monolayers and their van der Waals
heterostructure. Both materials have an indirect quasi-particle band gap, which
amounts to about 2.8 eV in ZrS$_2$ and to 2.6 eV in HfS$_2$. In both systems
the valence-band maximum is at $\Gamma$ and the conduction-band minimum at M.
Spin-orbit coupling induces a splitting of about 100 meV at the $\Gamma$ 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 eV and 0.8 eV. The ZrS$_2$/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 ZrS$_2$ or HfS$_2$ character, respectively. As a consequence, both the
electron and the hole components of the first exciton are localized in the
ZrS$_2$ monolayer with non-vanishing probability of finding the hole also in
the HfS$_2$ sheet.
