  Molecular hydrogen (H$_2$) production by electrochemical hydrogen evolution
reaction (HER) is being actively explored for non-precious-metal based
electrocatalysts that are earth-abundant and low cost like MoS$_2$. Although it
is acid-stable, its applicability is limited by catalytically inactive basal
plane, poor electrical transport and inefficient charge transfer at the
interface. Therefore, the present work examines its bilayer van der Waals
heterostructure (vdW HTS). The second constituent monolayer Boron Phosphide
(BP) is advantageous as an electrode material owing to its chemical stability
in both oxygen and water environments. Here, we have performed first-principles
based calculations under the framework of density functional theory (DFT) for
HER in an electrochemical double layer model with the BP monolayer, MoS$_2$/BP
and MoSSe/BP vdW HTSs. The climbing image nudged elastic band method (CI-NEB)
has been employed to determine the minimum energy pathways for Tafel and
Heyrovsky reactions. The calculations yield that Tafel reaction shows no
reaction barrier. Thereafter, for Heyrovsky reaction, we have obtained low
reaction barrier in the vdW HTSs as compared to that in the BP monolayer.
Subsequently, we have observed no significant difference in the reaction
profile of MoS$_2$/BP and MoSSe/BP vdW HTSs in case of high coverage (25 %) and
1/3 H$^+$ concentration (conc.). However, in the case of small coverage (11 %)
and 1/3 H$^+$ conc., MoSSe/BP shows feasible Heyrovsky reaction with no
reaction barrier. Finally, on comparing the coverages with 1/4 H$^+$ conc., we
deduce high coverage with low conc. and low coverage with high conc. to be apt
for HER via Heyrovsky reaction path.
