Raman tensor of layered black phosphorus

Black phosphorus has a strong Raman anisotropy on the basal and cross planes due to its orthorhombic crystal structure. However, almost all the studies on black phosphorus’ anisotropy focus on basal plane with the cross plane neglected. Here, we performed a systematic angle-resolved polarized Raman scattering on both the basal and cross planes of black phosphorus and obtained its integral Raman tensors. It is discovered that when the polarization direction of excitation light is along different crystal axes, the Raman intensity ratio (Ixx : Iyy: Izz) of Ag1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {A}_g^1 $$\end{document} mode is 256:1:5. Besides, via calculation, it is confirmed that the strong Raman anisotropy mainly comes from different differential polarizability alone different directions. This phenomenon is also observed when it comes to the Ag2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {A}_g^2 $$\end{document} mode.

In this work, we systematically analyzed APR spectra of cross and basal planes of BP and discovered that A 1 g and A 2 g modes have strong Raman anisotropies. Raman intensity ratio of A 1 g mode along different crystal axes (I xx : I yy : I zz ) is up to 256:1:5, and that of A 2 g mode is 10:1:5. Via ab-initio calculation, we confirmed that it is the different differential polarizability along different crystal axes that determines anisotropic Raman intensity of A g mode.

Experimental details
Here, a backscattering geometry Renishaw spectrometer (inVia Reflx) was adopted to measure APR spectra of BP, where 488 nm laser was equipped as excitation light and focused on the basal or cross plane through 50× long focus quartz lens. Measured BP sample is a commercial single crystal with the size of 2 mm, and the orientations of crystal axes are shown in Fig. 1a and b, where the basal and cross planes are corresponding to (010) and (100) planes respectively. During the measurement, the polarization directions of excitation light and scattering light were fixed and remained the same orientation, and layered BP sample was rotated in the step of 10°. Besides, to prevent local overheating from decomposition of sample, laser power density and acquisition time were set as 5 mW and 1 s respectively. In addition, integration number was set as 10 times to smooth experimental curves.

Results and discussion
Two representative Raman spectra corresponding to basal and cross planes are plotted in Fig. 1c and d. Collected APR spectra of the three modes (A 1 g (363 cm − 1 ), A 2 g (470 Based on the definition of Raman intensity, B 2g mode will disappear in the Raman scattering process of cross plane, and B 1 3g mode cannot be observed in the Raman scattering process of basal plane. Therefore, B 2g and B 1 3g modes can be used as a criteria to distinguish crystal plane. Based on the APR spectra of basal and cross planes, the Raman intensities of each mode at different rotation angles can be obtained. As shown in Fig. 2b, d, f and h, when the polarization orientation of excitation light locates on the basal plane, the Raman intensity ratio (I xx : I zz ) of A 1 g mode is 11.1, and that of A 2 g mode is 1.94. When the polarization orientation of excitation light locates on the cross plane, the Raman intensity ratio (I zz : I yy ) of A 1 g mode is 23.1 and that of A 2 g mode is 4.9. Combining the APR spectra of basal plane with those of cross plane, the Raman intensity ratios of A 1 g and A 2 g modes along different crystal axes can be derived, which are 256:1:23 and 9.6:1:4.9. It is obvious that there are strong Raman anisotropies of A 1 g and A 2 g modes, which can be elaborated with Raman selection rule and the definition of Raman intensity [22,23].
Specifically, according to classical Raman scattering theory, Raman intensity I for Raman active mode is given by [12].
where e i and e s denote polarization direction vectors of excitation and scattering light respectively. To explore the polarization characteristic of BP, e i is parallel to e s throughout the whole measurement. R denotes Raman tensor, represented by a 3 × 3 matrix. According to group theory, the Raman tensors format of A 1 g , A 2 g , B 2g and B 1 3g modes can be written as [12,23].  4) and (6), and the phase differences between Raman tensor elements a and c, that is the ϕ ac , of A 1 g and A 2 g modes are 42°and 79°for basal plane. For cross plane, the phase differences between Raman tensor elements b and c of A 1 g and A 2 g modes are 0°and 50°respectively where a, b, c, d and f represent the amplitudes of Raman tensor element, ϕ (ϕ a 1 , ϕ a 2 , ϕ b 1 , ϕ b 2 , ϕ c 1 and ϕ c 2 ) is the phase angle of Raman tensor element.
When the polarization direction of excitation and scattering light locates on the basal plane, e i and e s can be written as.
where θ denotes the rotation angle between the a-axis of BP and the polarization orientation of excitation (scattering) light. Based on Eq. (1), the function relations of Raman intensity of A 1 g , A 2 g and B 2g modes with respect to rotation angle θ are given by where ϕ a 1 c 1 ð¼ ϕ a 1 − ϕ c 1 Þ and ϕ a 2 c 2 ð¼ ϕ a 2 − ϕ c 2 Þ represent Raman phase differences between Raman tensor elements a and c.
When the polarization direction vectors of excitation and scattering light locate on the cross plane, e i and e s can be written as.
where φ denotes the rotation angle between the b axis of BP and the polarization orientation of excitation light. Based on Eq. (1), the function relations of Raman intensity of A 1 g , A 2 g and B 2g modes with respect to rotation angle φ are given by where ϕ b 1 c 1 ð¼ ϕ b 1 − ϕ c 1 Þ and ϕ b 2 c 2 ð¼ ϕ b 2 − ϕ c 2 Þ represent Raman phase differences between Raman tensor elements b and c.
According to the outline of the definition of Raman intensity, the Raman intensities of A 1 g , A 2 g , B 2g and B 1 3g modes satisfy the variation of Eq. (4) and Eq. (6) with the change of rotation angle. By fitting the APR spectra of A 1 g , A 2 g , B 2g and B 1 3g modes, the ratio relationship of Raman tensor elements can be observed, as shown in Table 1. For A 1 g mode, the ratio of Raman tensor elements amplitudes a 1 to c 1 is 3.33, and that of c 1 to b 1 is 4.8. Consequently, Raman tensor elements amplitudes a 1 , b 1 and c 1 satisfy the relationship of a 1 > c 1 > b 1 . Similarly, for A 2 g mode, its diagonal Raman tensor elements amplitudes also satisfy the relationship of a 2 > c 2 > b 2 .
Raman tensor element R q ij is also defined as the derivative of susceptibility χ ij with regard to atom position [12], thus where M μ and V prim represent the atomic mass of μ th atom and volume of unit cell respectively; r l (μ) is the position of μ th atom along l direction and e q l ðμÞ is the eigenvector of qth phonon of μth atom. Since polarizability α ij is linear with susceptibility χ ij (α ij = ε 0 χ ij ), Raman tensor element R q ij can be written as where ∂α ij /∂r l (μ) can be defined as the differential polarizability with Raman intensity directly reflected. Take A 1 g mode as an example, the differential polarizability along aaxis is larger than that along c-axis, which determines that there will be a larger Raman intensity when the polarization direction of excitation light is parallel to a-axis.
In addition, the susceptibility χ ij and relative permittivity ε ij are interrelated (χ ij = ε ij − 1). Thus, Raman tensor element can also be written as More importantly, the relative permittivity ε ij can be calculated by ab-inito calculation. The Raman tensor elements of A 1 g and A 2 g modes can be calculated by the vari-ation of relative permittivity before and after corresponding vibration. Here, the Raman tensor elements of various vibration modes are calculated via Vienna Ab-initio Simulation Package (VASP) [29][30][31][32][33][34][35][36][37][38][39][40] and phonon calculation software 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 eV cutoff was adopted, and Perdew-Burke-Ernzerhof functional was used as pseudopotentials. The relaxation of electron cannot be stopped until the free energy change is smaller than 10 −8 eV and the force between two steps is smaller than 10 − 3 eV =Å. A high-density k-mesh of 50 × 1 × 50 was performed for self-consistent calculation to obtain inductance coefficient tensors, as shown in Table 1. For A 1 g mode, the calculated Raman tensor elements amplitudes ratio of a 1 to c 1 is 1.94, and that of c 1 to b 1 is 1.32. Therefore, a 1 , b 1 and c 1 satisfy the relationship of a 1 > c 1 > b 1 , which is in accordance with measured result. Similarly, the relationship among Raman tensor elements amplitudes a 2 , b 2 and c 2 also satisfy a 2 > c 2 > b 2 . Based on ab-initio calculation, it can be concluded that the strong Raman anisotropies of A 1 g and A 2 g modes come from different differential polarizabilities along different crystal axes.
Besides, the phase angle of Raman tensor element can also be calculated via ab-initio calculation. As we have claimed above, Raman tensor element is proportional to the derivative of the relative permittivity ε ij with respect to spatial position of atoms. However, relative permittivity ε ij is usually composed of a real part ε 0 ij and an imaginary part ε 0 0 ij , leading to a correction of the expression of Raman tensor element, that is Calculated results are shown in Table 1, where the Raman phase differences (ϕ a 1 c 1 , g are 0.23π, 0.03π, 0.01π, 0.26π respectively. By fitting the APR spectra of A 1 g and A 2 g modes based on Eq. (4) and Eq. (6), the phase angels can also be derived. The Raman phase differences ϕ a 1 c 1 and ϕ b 1 c 1 of A 1 g modes are 0.23π and 0, and those of A 2 g mode are 0.44π and 0.28π. Compared with the calculated ϕ a 2 c 2 , the experimental ϕ a 2 c 2 is much larger, which has puzzled us for a long time.
In the Raman scattering process, except for 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 ϕ a 2 c 2 .
Recently, a birefringence theory [41,42] has been proposed to elaborate the Raman phase difference in transparent crystal, which has been confirmed by the materials such as AlN, GaN [16,43,44]. It is generally believed that due to a nearly negligible penetration depth, the theory is invalid in narrow band-gap materials. However, this understanding may be incomplete. For opaque optical crystals, the birefringence effect may also have a significant modulation influence on Raman phase difference. According to the definition of Raman scattering intensity and the deduction in Supporting Information, a phase factor ϕ and a constant w could be introduced into the Raman scattering intensity expression of A 2 g mode in Eq. (4). Thus, when excitation light propagates along b-axis, Raman scattering intensity of A 2 g mode can be written as where ϕ and w are functions respect to refractivity and extinction coefficient, given by ϕ ¼ arctg n xr n zi − n xi n zr n xi n zr þ n xr n zi ; ð12Þ w ¼ n xr n zi − n xi n zr n 2 xi þ n 2 where n xr and n zr denote refractive index along x and z directions respectively, and n xi and n zi are extinction coefficients along x and z directions respectively. As shown in Fig. 3, considering the birefringence effect, the theoretical value of Raman phase difference ϕ a 2 c 2 of A 2 g mode is 83°, which is close to experimental Raman phase difference 79°. 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 the APR spectra of basal and cross planes of BP systematically with complete Raman tensors obtained. In addition, the strong Raman anisotropies of A 1 g and A 2 g modes were discovered in this research. Via first-principle calculation, the strong anisotropy has been confirmed as originating from the different differential polarizability along different crystal axes. Besides, it is found that when excitation light propagates along b-axis, the calculated Raman phase difference of A 2 g mode is different from the experimental value, which may be owing to the modulation of birefringence effect.