Understanding angle-resolved polarized Raman scattering from black phosphorus at normal and oblique laser incidences

The selection rule for angle-resolved polarized Raman(ARPR)intensity of phonons from standard grouptheoretical method in isotropic materials would break down in anisotropic layered materials(ALMs)due to birefringence and linear dichroism effects.The two effects result in depth-dependent polarization and intensity of incident laser and scattered signal inside ALMs and thus make a challenge to predict ARPR intensity at any laser incidence direction.Herein,taking in-plane anisotropic black phosphorus as a prototype,we developed a so-called birefringence-linear-dichroism(BLD)model to quantitatively understand its ARPR intensity at both normal and oblique laser incidences by the same set of real Raman tensors for certain laser excitation.No fitting parameter is needed,once the birefringence and linear dichroism effects are considered with the complex refractive indexes.An approach was proposed to experimentally determine real Raman tensor and complex refractive indexes,respectively,from the relative Raman intensity along its principle axes and incident-angle resolved reflectivity by Fresnel’s law.The results suggest that the previously reported ARPR intensity of ultrathin ALM flakes deposited on a multilayered substrate at normal laser incidence can be also understood based on the BLD model by considering the depth-dependent polarization and intensity of incident laser and scattered Raman signal induced by both birefringence and linear dichroism effects within ALM flakes and the interference effects in the multilayered structures,which are dependent on the excitation wavelength,thickness of ALM flakes and dielectric layers of the substrate.This work can be generally applicable to any opaque anisotropic crystals,offering a promising route to predict and manipulate the polarized behaviors of related phonons.

Numerical Solution of 3D Poisson-Nernst-Planck Equations Coupled with Classical Density Functional Theory for Modeling Ion and Electron Transport in a Confined Environment

We have developed efficient numerical algorithms for solving 3D steadystate Poisson-Nernst-Planck(PNP)equations with excess chemical potentials described by the classical density functional theory(cDFT).The coupled PNP equations are discretized by a finite difference scheme and solved iteratively using the Gummel method with relaxation.The Nernst-Planck equations are transformed into Laplace equations through the Slotboom transformation.Then,the algebraic multigrid method is applied to efficiently solve the Poisson equation and the transformed Nernst-Planck equations.A novel strategy for calculating excess chemical potentials through fast Fourier transforms is proposed,which reduces computational complexity from O(N2)to O(NlogN),where N is the number of grid points.Integrals involving the Dirac delta function are evaluated directly by coordinate transformation,which yields more accurate results compared to applying numerical quadrature to an approximated delta function.Numerical results for ion and electron transport in solid electrolyte for lithiumion(Li-ion)batteries are shown to be in good agreement with the experimental data and the results from previous studies.