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We investigate the effect of charge carrier interaction with surface opticalphonons on the band properties of monolayer black phosphorus induced by polarsubstrates. We develop an analytical method based on the Lee-Low-Pines theoryto calculate the spectrum of Fr\"ohlich type continuum Hamiltonian in thelong-wavelength limit. We examine the modification of a band gap andrenormalization of effective masses due to the substrate-related polaroniceffect. Our results show that an energy gap in supported monolayer blackphosphorus is enlarged depending on a particular substrate and the interlayerdistance, $z$. Among the substrate considered, the largest gap broadening at $z=2.5$ \AA{} is observed for the Al$_{2} $O$_{3} $ substrate, which is found tobe $ \sim 50$ meV. Carrier-phonon coupling also renormalizes the effectivemasses which is more pronounced along the zigzag direction. Anisotropy of theeffective masses becomes stronger by the influence of the polaronic effectcorresponding to direction-dependent carrier-phonon coupling. We conclude thatsubstrate phonons have a non-negligible effect on the static band properties ofmonolayer black phosphorus, which may be further exploited in its experimentaland theoretical studies.

Polaronic effects in monolayer black phosphorus on polar substrates