Quantitatively Predicting Angle-Resolved Polarized Raman Intensity of Anisotropic Layered Materials

Abstract

Angle-resolved polarized Raman (ARPR) spectroscopy provides insights into optical anisotropy and symmetry-related electron-photon/electron-phonon couplings of anisotropic layered materials (ALMs). However, since their discovery over ten years ago, ARPR responses in ALM flakes has exhibited a puzzling dependence on flake thickness, excitation wavelength, and dielectric environment, complicating their understanding and prediction. By taking black phosphorus (BP) (⩾20 nm) flakes and four-layer Td-WTe₂ as examples, this study introduces intrinsic Raman tensors (R^int) and proposes strategies to predict the ARPR intensity profiles of thick and atomically-thin ALM flakes by considering birefringence, linear dichroism and multilayer interference inside multilayered structures with experimentally determined complex refractive indexes along in-plane axes and complex tensor elements of R^int for the corresponding phonon modes. The tensor elements of effective Raman tensors (R^eff), which are directly linked to the polarization vectors of incident and scattered light outside the ALM surface, are derived to quantitatively predict ARPR intensity for these ALM flakes, showing intricate dependence on ALM thickness, dielectric substrates, and excitation wavelengths. This framework can be extended to other ALM flakes from atomically-thin layers to bulk limit, facilitating comprehensive prediction of their ARPR intensity regardless of layer-dependent electronic properties.

Keywords: angle‐resolved polarized Raman intensity; anisotropic layered material; complex Raman tensor; complex refractive index; optical anisotropy.

Figure 1

a) Crystallographic structure of BP from the side and top views. b) Optical image of BP flake with $d_{\mathrm{BP}}$ = 83 nm measured by AFM (inset). c) Schematic diagram of ARPR spectroscopy setup with parallel polarization configuration. d) Raman spectra of BP flakes with $d_{\mathrm{BP}}$ = 35 and 83 nm on 90 nm‐SiO₂/Si substrate, and the corresponding ARPR intensity profiles of the e,g) $A_{\mathrm{g}}^1$ and f,h) $A_{\mathrm{g}}^2$ modes of BP flakes excited by λ_i = 633 nm, where filled circles and solid line represent the experimental and the fitted ARPR intensity profiles, respectively. The fitted $|c^{\mathrm{eff}}|/|a^{\mathrm{eff}}|$ and ${\varphi }_{\mathrm{ca}}^{\mathrm{eff}}$ are indicated under each polar plot.

Figure 2

a1) Schematic diagram of the multilayer interference of incident laser and scattered Raman signal inside multilayer structure, where the propagation paths of laser (blue lines) and scattered Raman signal (red lines) are presented separately. (Oblique incidence (scattering) for convenience). For clarity and simplicity, only a few representative reflection processes of the incident and scattered light are shown in the diagram. a2) Schematic illustration of the birefringence and linear dichroism in BP crystal. b) The setup for reflectance measurements. c) Experimental data (symbols) and fitted curves for the normalized reflectance of BP/90 nm‐SiO₂/Si relative to the 90 nm‐SiO₂/Si substrate as a function of $d_{\mathrm{BP}}$ along AC and ZZ axes at λ_i = 633 nm. The modulus square of the enhancement factors ($F_{\mathrm{i}}(y)$) for incident laser electric field inside BP/90 nm‐SiO₂/Si multilayer structures at λ_i = 633 nm along d) AC and e) ZZ axes of BP flake, respectively.

Figure 3

Experimental (filled circles) and fitted (solid lines) ARPR intensity profiles at λ_i = 633 nm of $A_{\mathrm{g}}^1$ and $A_{\mathrm{g}}^2$ modes in BP/90 nm‐SiO₂/Si with d _BP of a1,a2) 54 nm and b1,b2) ∼5000 nm (bulk), where the curves are fitted by Equation (5). Predicted c) $|c^{\mathrm{eff}}|/|a^{\mathrm{eff}}|$ and d) ${\varphi }_{\mathrm{ca}}^{\mathrm{eff}}$ for $A_{\mathrm{g}}^1$ and $A_{\mathrm{g}}^2$ modes of BP flakes on 90 nm‐SiO₂/Si substrate at λ_i = 633 nm.

Figure 4

Contour plots of $|c^{\mathrm{eff}}|/|a^{\mathrm{eff}}|$ and ${\varphi }_{\mathrm{ca}}^{\mathrm{eff}}$ for a,b) $A_{\mathrm{g}}^1$ and c,d) $A_{\mathrm{g}}^2$ modes with varied $d_{\mathrm{BP}}$ and $d_{{\mathrm{SiO}}_2}$ at λ_i = 633 nm. Experimental (filled circles) and predicted (solid lines) ARPR intensity profiles for $A_{\mathrm{g}}^1$ and $A_{\mathrm{g}}^2$ modes of BP flakes with e,f) $d_{\mathrm{BP}}$ = 166 nm on 225 nm‐SiO₂/Si substrate and g,h) $d_{\mathrm{BP}}$ = 107 nm on 300 nm‐SiO₂/Si substrate at λ_i = 633 nm.

Figure 5

Bar charts of the fitted averaged values of a) $|c^{\mathrm{int}}|/|a^{\mathrm{int}}|$ and b) ${\varphi }_{\mathrm{ca}}^{\mathrm{int}}$ for $A_{\mathrm{g}}^1$ and $A_{\mathrm{g}}^2$ modes at λ_i of 488 nm (blue), 532 nm (green) and 633 nm (red), respectively. The values are indicated above the corresponding bars. Contour plots of $|c^{\mathrm{eff}}|/|a^{\mathrm{eff}}|$ and ${\varphi }_{\mathrm{ca}}^{\mathrm{eff}}$ for $A_{\mathrm{g}}^2$ mode with varied $d_{\mathrm{BP}}$ and $d_{{\mathrm{SiO}}_2}$ at λ_i of c,d) 488 nm and e,f) 532 nm. Experimental (filled circles) and predicted (solid lines) ARPR intensity profiles for $A_{\mathrm{g}}^2$ mode of BP flakes with $d_{\mathrm{BP}}$ = 166 nm on 225 nm‐SiO₂/Si substrate and $d_{\mathrm{BP}}$ = 107 nm on 300 nm‐SiO₂/Si substrate at λ_i of g,h) 488 nm and i,j) 532 nm.

Figure 6

a) Crystallographic structure of Td‐WTe₂ from the side and top views. b) Polarized Raman spectra of 4L Td‐WTe₂ on 75 nm‐SiO₂/Si substrate at λ_i = 633 nm, when $\theta$ = 0° (e _i (e _s)∥X), 45° and 90° (e _i(e _s)∥Y). The A ₁ and A ₂ modes are indicated. c) Experimental (filled circles) and fitted (solid line) ARPR intensity profiles of the P2 mode in 4L Td‐WTe₂ on 75 nm‐SiO₂/Si substrate at λ_i = 633 nm. d) Experimental data (Symbols) and fitted curves for the normalized reflectance of 4L Td‐WTe₂/SiO₂/Si relative to the SiO₂/Si substrate as a function of $d_{{\mathrm{SiO}}_2}$ along X and Y axes at λ_i = 633 nm. The fitted $\stackrel{\sim }{n}$ for X and Y axes of Td‐WTe₂ are 3.11 + 1.02i and 3.31 + 0.95i, respectively (also see Table S1, Supporting Information). Predicted e) $|b^{\mathrm{eff}}|/|a^{\mathrm{eff}}|$ and f) ${\varphi }_{\mathrm{ba}}^{\mathrm{eff}}$ for the P2 mode in 4L Td‐WTe₂ on SiO₂/Si substrate versus $d_{{\mathrm{SiO}}_2}$ at λ_i = 633 nm. Experimental (filled circles) and the correspondingly predicted (solid lines) ARPR intensity for the P2 mode in 4L Td‐WTe₂ on g) 45 nm‐SiO₂/Si and h) 768 nm‐SiO₂/Si substrates at λ_i = 633 nm.