The phonon confinement effect in two-dimensional nanocrystals of black phosphorus with anisotropic phonon dispersions

The RWL model for the phonon confinement effect in nanocrystals (NCs) had been found to result in deviations and limitations for crystals exhibiting obvious anisotropic phonon dispersions and modified models have been proposed to overcome these deficiencies. Here, we examine this issue in black phosphorus (BP), a typical anisotropic two-dimensional crystal exhibiting pronounced anisotropy in phonon dispersions. A detailed study is performed on the Raman spectra of BP NCs prepared by the ion implantation technique. With decreasing NC size, the peak positions of the three characteristic Raman modes, Ag1, B2g and Ag2 modes, remain almost unchanged, while the Ag1 and Ag2 modes show significant asymmetrical broadening tails towards higher- and lower-frequency sides, respectively. It is found that the RWL model based on one-dimensional phonon dispersion along Γ-Y and Γ-X axes in the Brillouin zone (BZ) cannot interpret the unusual frequency invariance and inhomogeneous line shape broadening of these three modes. However, after considering the contribution of two-dimensional anisotropic phonon dispersions from the whole BZ, the frequency and asymmetrical broadening of the Ag1 and Ag2 modes can be well reproduced. This study demonstrates that the RWL model can be applicable for crystals with anisotropic phonon dispersions once the phonons in the whole two-dimensional or three-dimensional BZ are properly taken into account, and provides a physically sound route into understanding the phonon confinement effect for anisotropic systems.