Angle-Dependent Raman Scattering Studies on Anisotropic Properties of Crystalline Hexagonal 4H-SiC

Raman scattering spectroscopy (RSS) has the merits of non-destructiveness, fast analysis, and identification of SiC polytype materials. By way of angle-dependent Raman scattering (ADRS), the isotropic characteristics are confirmed for c-face 4H-SiC, while the anisotropic properties of a-face 4H-SiC are revealed and studied in detail via combined experiments and theoretical calculation. The variation functional relationship of the angle between the incident laser polarization direction and the parallel (perpendicular) polarization direction was well established. The selection rules of wurtzite 4H-SiC are deduced, and the intensity variations of the A1, E2, and E1 Raman phonon modes dependent on the incident angle are calculated, and well-matched with experimental data. Raman tensor elements of various modes are determined.

Since the late 1980s, high-quality and large-size wafers of both 6H-and 4H-SiC have come into industrial production. Gradually, the wafer size increased, up to 8 inches recently, and the wafer crystalline quality improved over time and is very widely used for device applications. Wafers of SiC are also promising substrates for nitride semiconductor growth due to their compatible lattice structure and similar thermal expansion coefficients, as well as for other materials.
Raman scattering spectroscopy (RSS) has the merits of non-destructiveness, fast analysis, and being free from special preparation for samples. It has been proven to be a useful and informative tool for the investigation and identification of SiC polytype materials. from a 2-inch wafer. Some c-face 4H-SiC experimental samples are pieces at the size of 5 × 5 mm or 10 × 10 mm, cut of commerce wafer materials (typically 1 3/8 to 2 inch).
Raman spectra were measured in backscattering geometry at room temperature (RT), by a Raman micro-spectrometer, connected with a Charge Coupled Device (CCD) for signal detection, named 'Finder One' from Zolix Company. It is equipped with four diode lasers possessing excitation wavelengths of 457, 532, 650, and 785 nm. In the current work, the 532 nm excitation laser was used, with incident light through the x50 object (with 0.55 numerical aperture and 4 mm focal length) focusing on the "a" surface or c-face of the 4H-SiC crystal sample, while the scattered light was collected via this same object. To make the scattered light parallel with or perpendicular to the incident light, a rotatable polarizer was inserted within the scattered light path. The sample stage could be rotated with 360 • , i.e., the incident light could be rotated with 360 • to obtain the angular dependent Raman spectra. Figure 1 shows (a) the hexagonal wurtzite crystal structure and (b) the experimental arrangement. The XYZ coordinates with the crystalline axis of the 4H-SiC crystal. The Z-axis is along the c-axis of the wurtzite 4H-SiC crystal, i.e., (0001) direction. The Y-axis is along the normal direction of the a-face of 4H-SiC. The rotation angle θ is the angle of inclusion between the ξ axis and the X-axis. The incident light travels along the Y-axis, the SiC sample is rotated around the Y-axis, and rotation angle θ lies between the incident light polarizing vector in the X-Z plane and X-axis.

Experimental
The experimental 4H-SiC samples used in this work were purchased from CREE Company. A piece of un-doped a-face (1120) 4H-SiC with the size of 5 × 5 mm, was cut from a 2-inch wafer. Some c-face 4H-SiC experimental samples are pieces at the size of 5 × 5 mm or 10 × 10 mm, cut of commerce wafer materials (typically 1 3/8 to 2 inch).
Raman spectra were measured in backscattering geometry at room temperature (RT), by a Raman micro-spectrometer, connected with a Charge Coupled Device (CCD) for signal detection, named 'Finder One' from Zolix Company. It is equipped with four diode lasers possessing excitation wavelengths of 457, 532, 650, and 785 nm. In the current work, the 532 nm excitation laser was used, with incident light through the x50 object (with 0.55 numerical aperture and 4 mm focal length) focusing on the "a" surface or c-face of the 4H-SiC crystal sample, while the scattered light was collected via this same object. To make the scattered light parallel with or perpendicular to the incident light, a rotatable polarizer was inserted within the scattered light path. The sample stage could be rotated with 360°, i.e., the incident light could be rotated with 360° to obtain the angular dependent Raman spectra. Figure 1 shows (a) the hexagonal wurtzite crystal structure and (b) the experimental arrangement. The XYZ coordinates with the crystalline axis of the 4H-SiC crystal. The Zaxis is along the c-axis of the wurtzite 4H-SiC crystal, i.e., (0001) direction. The Y-axis is along the normal direction of the a-face of 4H-SiC. The rotation angle θ is the angle of inclusion between the ξ axis and the X-axis. The incident light travels along the Y-axis, the SiC sample is rotated around the Y-axis, and rotation angle θ lies between the incident light polarizing vector in the X-Z plane and X-axis. In this paper, we present the RT backscattering Raman spectra of the semi-polar aface (1120) and c-face (0001) 4H-SiC crystal for the relationship between the lattice phonon vibration modes and the rotation angles of the wurtzite structure, which shows the crystal anisotropy or isotropy characteristics.  In this paper, we present the RT backscattering Raman spectra of the semi-polar a-face (1120) and c-face (0001) 4H-SiC crystal for the relationship between the lattice phonon vibration modes and the rotation angles of the wurtzite structure, which shows the crystal anisotropy or isotropy characteristics. Figure 2 presents the angle-dependent Raman spectra of the c-face (0001) 4H-SiC, measured at RT with the excitation of 532 nm. The hexagonal wurtzite 4H-SiC is a tetrahedrally coordinated semiconductor compound and belongs to the space group C 4 6v (P6 3 mc) in the primitive cell. All the atoms occupy sites of symmetry C 3v [7,11]. Main Raman active modes include E 2 (LA), 203.5 cm −1 ; A 1 (LA), 610.5 cm −1 ; E 2 (TO), 777.0 cm −1 ; E 1 (TO), 788.1 cm −1 ; and A 1 (LO) 967.0 cm −1 , as shown in Figure 2, in which all above Raman phonon modes are identical at different rotation angles of 0 • -330 • (0 • , i.e., 360 • ). This indicates the isotropic characteristics of the c-face 4H-SiC crystal [30]. C 4 6v(P63mc) in the primitive cell. All the atoms occupy sites of symmetry C3v [7,11]. Main Raman active modes include E2 (LA), 203.5 cm −1 ; A1 (LA), 610.5 cm −1 ; E2 (TO), 777.0 cm −1 ; E1 (TO), 788.1 cm −1 ; and A1 (LO) 967.0 cm −1 , as shown in Figure 2, in which all above Raman phonon modes are identical at different rotation angles of 0°-330° (0°, i.e., 360°). This indicates the isotropic characteristics of the c-face 4H-SiC crystal [30].  Figure 3 shows the rotation Raman spectra at the a-plane (1120) of 4H-SiC, in the wavenumber range of 580-820 cm −1 and with the angle rotated from 0° to 360° (step size of 5°), under (a) parallel and (b) perpendicular polarization, respectively. Within Figure 3, the A1 (LA), E2 (TO), and E1 (TO) modes are observed at 610.03 cm −1 , 776.49 cm −1, and 788.08 cm −1 , respectively. The intensity and phase variations of these Raman modes and their dependences on the rotation angle are clearly shown. Further, there exists a shoulder peak in the close-right of E2 (TO) at ~783 cm −1 , which is assigned as the A1 (TO) mode of 4H-SiC [29]. In addition, we provide a single point scan Raman spectrum of the a-face 4H-SiC, for reference, included in the supplementary, with an inset and showing E1 (TA) at 266 cm −1 , E1 (TA) at 610 cm −1 , E2 (TO) at 775 cm −1 , E1 (TO) at 788 cm −1 , A1 (LO) at 976 cm −1 and LOplasma coupling (LOPC) broadband in the right side of A1 (LO) [17][18][19].  Figure 3 shows the rotation Raman spectra at the a-plane (1120) of 4H-SiC, in the wavenumber range of 580-820 cm −1 and with the angle rotated from 0 • to 360 • (step size of 5 • ), under (a) parallel and (b) perpendicular polarization, respectively. Within Figure 3, the A 1 (LA), E 2 (TO), and E 1 (TO) modes are observed at 610.03 cm −1 , 776.49 cm −1, and 788.08 cm −1 , respectively. The intensity and phase variations of these Raman modes and their dependences on the rotation angle are clearly shown. Further, there exists a shoulder peak in the close-right of E 2 (TO) at~783 cm −1 , which is assigned as the A 1 (TO) mode of 4H-SiC [29]. In addition, we provide a single point scan Raman spectrum of the a-face 4H-SiC, for reference, included in the Supplementary, with an inset and showing E 1 (TA) at 266 cm −1 , E 1 (TA) at 610 cm −1 , E 2 (TO) at 775 cm −1 , E 1 (TO) at 788 cm −1 , A 1 (LO) at 976 cm −1 and LO-plasma coupling (LOPC) broadband in the right side of A 1 (LO) [17][18][19].

Raman Selection Rules
The 4H-SiC crystal is a compound semiconductor with a hexagonal wurtzite structure and belongs to the space group C 4 6v (P63mc) with eight atoms in the primitive cell [7,11]. In its Brillion zone, the lattice vibration produces nine optic branches and three acoustic branches, including the A 1 , B 1 , E 1, and E 2 vibration modes [7,11,17]. According to the group theory, A 1 and E 1 acoustic modes are Raman and infrared (IR) active. The A 1 phonon is polarized along the Y-axis direction, while the E 1 phonon is polarized within the XZ plane (see the Coordinate System in Figure 1). The E 2 mode is only Raman active, while B 1 belongs to the non-active mode. Due to the creation of polar bonds, there appear frequency shifts for A 1 and E 1 Raman modes with symmetries. The A 1 and E 1 Raman modes are split into the longitudinal phonon and transverse phonon modes, forming the A 1 (TO, LO) and E 1 (TO, LO) modes.

Raman Selection Rules
The 4H-SiC crystal is a compound semiconductor with a hexagonal wurtzite structure and belongs to the space group C 4 6v (P63mc) with eight atoms in the primitive cell [7,11]. In its Brillion zone, the lattice vibration produces nine optic branches and three acoustic branches, including the A1, B1, E1, and E2 vibration modes [7,11,17]. According to the group theory, A1 and E1 acoustic modes are Raman and infrared (IR) active. The A1 phonon is polarized along the Y-axis direction, while the E1 phonon is polarized within the XZ plane (see the Coordinate System in Figure 1). The E2 mode is only Raman active, while B1 belongs to the non-active mode. Due to the creation of polar bonds, there appear frequency shifts for A1 and E1 Raman modes with symmetries. The A1 and E1 Raman The Raman scattered light intensities can be expressed as [36][37][38][39][40][41]: where e i and e s present the polarization vector for the incident and scattered light, respectively. R is the 2nd-order Raman tensor with the form of a 3 × 3 vector matrix [36][37][38][39][40][41]. This vector matrix represents the characteristics of Raman scattering phonon modes. The A 1 , E 1, and E 2 Raman active modes in wurtzite structural materials are expressed below [36][37][38][39][40][41]: where a, b, c, and d represent Raman tensor elements. In Raman backscattering measurements on the a-plane of 4H-SiC single crystalline material, referring to the hexagonal axis structure, the polarized vectors for the incident and scattered light can be expressed as: and where θ (θ s ) represents the angle between the sample Y-axis and the incident (scattered) light vector, respectively. Here, we discuss the case of the parallel and perpendicular configuration, from which, according to the calculation based upon Equation (1), the Raman intensity at the surface of the a-plane from the 4H-SiC crystal can be expressed as: I // (A 1 ) ∼|a| 2 sin 4 θ + |b| 2 cos 4 θ + |a||b| 2 sin 2 2θ cos(ϕ a−b ), I ⊥ (E 2 ) ∼|d| 2 sin 2 θ cos 2 θ, I(E 2 ) ∼ I // (E 2 ) + I ⊥ (E 2 ).
From the above formalism, we can know that the phase difference ϕ a−b between the Raman tensor elements a and b possesses an influence on the Raman scattering intensity of A 1 mode. From Equation (7), the values of |a|, |b| and phase difference ϕ a−b in the scattered light intensity I // (A 1 ) are certain. They can be determined through the values of I // (A 1 ) at θ = 0 • and 90 • , respectively; however, the values of |a|, |b| and phase difference ϕ a−b in the scattered light intensity I ⊥ (A 1 ) are different from that of I // (A 1 ). As known from Equation (8), the values of |a|, |b| and phase difference ϕ a−b in the scattered light intensity I // (A 1 ) have also influenced the scattered light intensity I ⊥ (A 1 ).
Based on the above formalism, we can perform the theoretical calculation and experimental measurements on the 4H-SiC anisotropy Raman scattering. Based on the dynamic atomic theory, different elements of the vibration modes could display variations related to their space structures. In the following, we discuss the anisotropy characteristics and the effects on the Raman tensor of various vibration modes. In addition, to determine the Raman tensor for the A 1 mode, it can employ the Raman selection rules to perform the theoretical modeling. Both the A 1 and E 1 phonon modes are both Raman and IR active, as shown in Figure 4c-d, with similar intensity variation trends for both A 1 and E 1 modes; however, the E 2 mode, with only Raman active, is different from A 1 and E 1 modes.

Rotation Raman Spectral Intensities and Analyses
As shown in Figure 4e-f for the strongest E 2 mode, under the case of parallel polarization [y(xx)y-y(zz)y], with the rotation angle increased from 0 • to 90 • , the Raman intensity of E 2 mode increases continually; while with the rotation angle increased from 90 • to 180 • , the Raman intensity of E 2 mode decreases continually. At the rotation angles of 0 • , 90 • , 180 • , 270 • , and 360 • , the A 1 mode has the maxima intensity, under the parallel polarization; while at near 45 • , 140 • , 225 • , and 310 • , the A 1 mode is weakest. Under the case of perpendicular polarization [y(xz)y-y(zx)y], the Raman intensity variation of E 2 mode has the same variation trend as the A 1 mode, while the E 1 mode possesses a different variation trend from A 1 and E 2 .

Raman Tensor Element Analyses
Raman intensities of A 1 , E 1, and E 2 phonon modes under the polarization states of parallel [y(xx)y-y(zz)y] and perpendicular [y(xz)y-y(zx)y] are shown in Figure 4. The fitting results for the relationship of Raman intensity and ratio angle under two polarization cases are expressed in Figure 4a-f, with small errors. To determine the Raman tensor elements, the fitting procedure by the least square method and Formulas (7)-(15) with angle variation of function fitting is applied. The best-fitting curves are basically coincident with the experimental data, as shown in Figure 4. The relative values of Raman tensor elements for 4H-SiC crystal are calculated by way of the ratio calculation of fitting parameters are overviewed and shown in Table 1. As applying function fitting with Formulas (7)-(15) on data of Figure 4a-f plus elaboration, proper error bars were obtained and put into Table 1 also. It is found that different Raman modes are completely different. The phase difference and anisotropic ratio of each Raman mode characterize similar information, but there exist phase shifts among different elements. In addition, the values of phase shift and anisotropic ratio are determined from the corresponding Raman mode. Each element of Raman tensor represents its single directional vibration and corelated with another element; therefore, from experimental and theoretical results, the anisotropies of the 4H-SiC sample can be determined. (c,d) The intensity of the E1 signal from the a-plane surface versus the rotation angle for parallel and perpendicular polarization vectors, respectively. (e,f) The intensity of the E2 mode signal versus the rotation angle for parallel and perpendicular polarizations.

Raman Tensor Element Analyses
Raman intensities of A1, E1, and E2 phonon modes under the polarization states of parallel [y(xx)y-y(zz)y] and perpendicular [y(xz)y-y(zx)y] are shown in Figure 4. The fitting results for the relationship of Raman intensity and ratio angle under two polarization cases are expressed in Figure 4a-f, with small errors. To determine the Raman tensor elements, the fitting procedure by the least square method and Formulas (7)- (15) with angle variation of function fitting is applied. The best-fitting curves are basically coincident with the experimental data, as shown in Figure 4. The relative values of Raman tensor elements for 4H-SiC crystal are calculated by way of the ratio calculation of fitting The 4H-SiC material belonging to a wide bandgap semiconductor compound possesses the wurtzite structure with the hexagonal space group o C 4 6v (P6 3 mc). Based upon the atomic dynamics, by way of Raman scattering measurements, different vibration elements can display the sensitivities correlated to their own space group [42]. The A 1 , E 1 , and E 2 modes in wurtzite 4H-SiC are Raman active. Among these modes, the A 1 polarization direction coincides with the Y-axis. In a doped semiconductor, the A 1 mode, by way of the resonant coupling between polarized phonon and plasma (free carriers), exerts an influence on the electronic transport properties [43]. As expressed in Table 1, the relative values of Raman tensor elements for the 4H-SiC crystal are calculated with the help of the ratio calculation on fitting parameters. From Raman scattering spectra under the parallel and perpendicular polarization, the phase difference ϕ a−b of Raman tensor elements |a| and |b| (i.e., the parallel and perpendicular vectors) for the 4H-SiC crystal are determined as 117.37 • and 92.86 • , respectively. It can be seen from the article of Strach et al. [44] that the selection of parameters a, b and phase difference ϕ a−b is not unique for the perpendicular vectors of the incident and scattered light. So, the ratio of perpendicular polarized vectors in Table 1 is not well defined; however, this ratio is comparatively coincident with the result estimated from the incident and scattered polarized vectors in parallel alignment. For wurtzite material, re-collecting measurements on Raman spectra can confirm the relationship between the anisotropy ratio and the phase difference of Raman tensor elements in wurtzite crystal. W. Zheng et al. [39][40][41] have studied the elucidation of "phase difference" in Raman tensor formalism on wurtzite compounds, namely, AlN, GaN, ZnO, and SiC, and demonstrated the values of phase difference being unanimously confined to around π/2π/2, while the anisotropic ratios display obvious differences in the four compounds. It is reasonable to believe that the phase difference should not be treated as an intrinsic property such as the anisotropic ratio. In addition, it is noted that the anisotropy properties of 4H-SiC materials can also be investigated by way of nanoindentation and scratch experiments [45].

Conclusions
In summary, from this work for wurtzite 4H-SiC single crystalline material, the rotation, i.e., the angle-dependent Raman scattering (ADSS), has been studied by the combination of experiments and theories. A series of promising and significant results are achieved. The isotropy characteristics were confirmed from the c-face 4H-SiC, while the anisotropy characteristic properties were revealed from the a-face 4H-SiC. The variation functional relationship of the Raman phonon modes versus the angle between the incident laser polarization direction and the parallel (perpendicular) polarization direction was obtained. By way of the parametrization on incident light and scattered light polarization vectors, the selection rules of wurtzite SiC are calculated and well established. Corresponding Raman selection rules are derived according to measured scattering geometries to illustrate the angle dependence. Based upon the selection rules, the intensity variations of the A 1 , E 2, and E 1 modes dependent on the rotation angle are calculated, and the Raman tensor elements of various modes are well-deduced. These detailed theoretical calculation results on the Raman selection role, Raman tensor elements, and the variations of Raman spectral intensities, as well as the phonon anisotropy properties of wurtzite 4H-SiC, are significant and matched well with experimental data.
Our penetrating investigations are helpful to better understand the mechanisms and find ways to further improve the material design and growth of hexagonal SiC, as well as their wide range of applications.