First-principles insights into excitonic and electron-phonon effects in a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Zr</mml:mi> <mml:msub> <mml:mi mathvariant="normal">S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>/</mml:mo> <mml:mi>Hf</mml:mi> <mml:msub> <mml:mi mathvariant="normal">S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> van der Waals heterostructure

Motivated by the successful synthesis of isolated $\mathrm{Zr}{\mathrm{S}}_{2}$ and $\mathrm{Hf}{\mathrm{S}}_{2}$ transition metal dichalcogenide monolayers and inspired by their nearly identical lattice constants, we construct and investigate a vertical $\mathrm{Zr}{\mathrm{S}}_{2}/\mathrm{Hf}{\mathrm{S}}_{2}$ van der Waals (vdW) heterostructure. Using first-principles calculations based on density functional theory and many-body perturbation theory, we explore its electronic, optical, and excitonic properties, with particular emphasis on excitonic effects and temperature-dependent behavior. Based on the $\mathit{GW}$ method, the $\mathrm{Zr}{\mathrm{S}}_{2}/\mathrm{Hf}{\mathrm{S}}_{2}$ vdW heterostructure exhibits an indirect band gap of 2.60 eV with a type-I band alignment. The optical gap of the heterostructure is found to be 2.64 eV, with an exciton binding energy of 0.71 eV, both reduced compared to those in the isolated monolayers. We further investigate the temperature-dependent optoelectronic behavior of the heterostructure, considering electron-phonon coupling. A zero-point renormalization of 0.04 eV in the direct band gap is observed. While the direct band gap decreases monotonically with temperature from 0 to 400 K, the indirect band gap displays a nonmonotonic trend. As a result, the absorption spectrum undergoes a meaningful redshift with increasing temperature. At 300 K, the optical gap of the heterostructure is reduced to 2.51 eV and the exciton binding energy to 0.63 eV. Our findings highlight the important role of electron-phonon interaction in the optoelectronic response of $\mathrm{Zr}{\mathrm{S}}_{2}/\mathrm{Hf}{\mathrm{S}}_{2}$ vdW heterostructure, supporting its use in high-performance optoelectronic devices.