Effect of flexural phonons on the hole states in single-layer black phosphorus

Flexural thermal fluctuations in crystalline membranes affect the band structure of the carriers, which leads to an exponential density-of-states (DOS) tail beyond the unperturbed band edge. We present a theoretical description of this tail for a particular case of holes in single-layer black phosphorus, a material which exhibits an extremely anisotropic quasi-one-dimensional dispersion ($m_y\mathrm{/}{m}_x\gg 1$) and, as a result, an enhanced Van Hove singularity at the valence band top. The material parameters are determined by {\it ab initio} calculations and then are used for quantitative estimation of the effect of two-phonon (flexural) processes have on the charge carrier DOS. It is shown that unlike the isotropic case, the physics is determined by the phonons with wavevectors of the order of $q^{\ast }$, where $q^{\ast }$ determines the crossover between harmonic and anharmonic behavior of the flexural phonons. The spectral density of the holes in single-layer black phosphorus at finite temperatures is calculated.