Raman Tensor of Layered Black Phosphorous

： Black phosphorous is an orthorhombic with a strong Raman anisotropy in the basal and cross plane. However, up to now, almost all the studies on anisotropy of black phosphorous have focused on basal plane but neglected cross plane. Here, we performed a systematic angle-resolved polarized Raman scattering on the basal and cross plane of black phosphorous and obtained its integral Raman tensors. In addition, we discovered that Raman intensity ratio ( 𝐼 !! : 𝐼 "" : 𝐼 ## ) of 𝐴 $% mode is 256:1:5 when the polarization direction of incident light is along different crystal axes. According to first-principle calculated results, we confirmed that the strong Raman anisotropy is due to larger differential polarizability of 𝐴 $% mode along a -axis. This phenomenon is also observed in the 𝐴 $& mode. Monolayer: Vibrational


Introduction
Recently, much attention has been focused on directional selective optoelectronic devices [1][2][3][4][5], of which the high directional charge and energy transfer characteristic are important indictors to evaluate performance of devices. The directional selective characteristic of optoelectronic device originates from anisotropy of atomic arrangement. Based on group theory, black phosphorous (BP) belongs to D2h space group, which suggests that BP has different atomic arrangements along zigzag and armchair directions [6][7][8][9]. Therefore, BP is one of the excellent candidates to fabricate directional selective optoelectronic devices.
Raman polarization characteristic, as an important branch of anisotropy, is determined by Raman tensor, which is a key index to evaluate Raman scattering intensity [10][11][12]. In order to obtain integral Raman tensor, Angle-resolved polarized Raman (APR) spectroscopy as a powerful and undamaged tool was performed, which has been used in our previous work where we studied the Raman polarization characteristic of traditional anisotropic crystal, such as AlN, GaN and ZnO [13][14][15][16]. BP, as an anisotropic layered material, have attracted much attention in the aspects of optical and electrical polarization characteristics. However, almost all the studies on the anisotropy of BP put focus on basal plane rather than cross plane so far [17][18][19][20][21][22][23], which causes a lack of integral Raman tensor. This blank can be filled by studying the anisotropy of cross plane.
In this work, we systematically analyzed the APR spectra of cross and basal planes of BP and discovered that $ % and $ & modes have strong Raman anisotropies along different crystal axes. Raman scattering intensity ratio of $ % mode along different where a，b，c，d and f represent amplitudes of Raman tensor element, ( 0 " , 0 & , is phase angle of Raman tensor element.
For // , the polarization direction vectors of incident and scattering light can be written as where denotes the angle between the a-axis of BP and the polarization direction of incident (scattering) light. Based on Eq. (1), the Raman scattering intensity expressions of $ % ， $ & and &$ modes are given by where differences between Raman tensor elements a and c.
For // , the polarization direction vectors of incident and scattering light can be written as = : where represents the angle between the b axis of BP and the polarization direction of incident light. Based on Eq. (1), the Raman scattering intensity expressions of $ % ， $ & and ($ % modes can be written as By the definition of Raman tensor element, Raman tensor element -4 5 is given by the derivative of susceptibility -4 with regard to atom position [10], thus -4 where can also be written as can be defined as the differential polarizability with Raman scattering intensity directly reflected. For $ % mode, when incident light propagates along b-axis, the differential polarizability along a-axis is larger than that along c-axis, determining a larger Raman scattering intensity when the polarization direction of incident light is parallel to a-axis. In addition, the susceptibility -4 is also related to relative permittivity -4 ( -4 = -4 − 1). Thus, Raman tensor element can be transformed into More importantly, the relative permittivity -4 can be calculated by first-principle. The Raman tensors of $ % and $ & modes can be obtained by calculating the change of relative permittivity before and after corresponding vibration. Here, we calculated the

Raman tensors of various vibration modes via Vienna Ab-initio Simulation Package
(VASP) [24][25][26][27][28][29][30][31][32][33][34][35] and open source package Phonopy. During the density functional perturbation theory (DFPT) calculation, a 3 × 3 × 3 supercell was adopted to obtain the force constants. A projector augmented wave (PAW) basis set with 500 cutoff was used to expand the electronic wave functions, and Perdew-Burke-Ernzerhof functional was used as pseudopotentials. The relaxation of electron will be stopped until the free energy change between two steps is smaller than 10 GH , and a high-density k-mesh of 50 × 1 × 50 was performed for self-consistent calculation to obtain inductance coefficient tensors, which were shown in Table I Besides, the phase angle of Raman tensor element can also be calculated via firstprinciple. As we elaborated above, Raman tensor element is proportional to the derivative of the relative permittivity -4 with respect to spatial position of atoms.
However, relative permittivity -4 is usually composed of a real part -4 I and an imaginary part -4 II , making Raman tensor element be divided into real and imaginary parts. Thus, the expression of Raman tensor element can be written as (10) Raman tensor, test environment and the property of materials will also affect the Raman scattering intensity. It seems difficult to find an acceptable reason to explain the large difference between experimental and theoretical values of Recently, a birefringence theory [36,37] has been proposed to elaborate the Raman phase difference in transparent crystal, which was confirmed in the materials such as AlN, GaN [14,38,39]. It is generally believed that due to a nearly negligible penetration depth, the theory is invalid in narrow bandgap materials. However, this understanding may be incomplete. For opaque optical crystals, the birefringence effect where !7 and #7 denote refractive index along x and z direction respectively, and !-and #-are extinction coefficient along x and z direction respectively.
Considering the birefringence effect, the theoretical value of Raman phase difference Although BP is an opaque material relative to 488 nm and its penetration depth is very small, the effect of birefringence on the modulation of Raman phase difference cannot be ignored.

Conclusions
In this work, we analyzed APR spectra of basal and cross plane of BP systematically,