== Phonons in Perfect 4H-SiC

// 第一段：弱极性与强极性声子模式表现截然不同。如图所示，弱极性声子模式几乎不依赖于波矢方向，而强极性声子模式则表现出显著的各向异性。

Raman-active phonon modes were categorized into two groups,
  according to the distinct behaviors arising from different electrical polarities,
  including eight negligible-polarity modes (i.e., having zero or very weak electrical polarity),
  and three strong-polarity modes.
The negligible-polarity modes exhibited minimal dependence on the wavevector direction,
  while the strong-polarity modes showed significant anisotropy.
This distinction was clearly illustrated in @figure-discont,
  where the phonon dispersion relations of negligible-polarity modes (gray solid lines)
    were roughly continuous and flat at the #sym.Gamma point,
  leading to only #sym.tilde 0.1 cm#super[-1] frequency variations and similar vibration patterns
    across different incidence configurations.
In contrast, the strong-polarity modes (colored solid lines) displayed discontinuities at the #sym.Gamma point,
  resulting in #sym.tilde 1 cm#super[-1] frequency shifts and distinct different vibration patterns
  under different incidence configurations.
Consequently, the negligible-polarity modes were named
  according to their irreducible representations at the #sym.Gamma point and in order of increasing frequency,
  including A#sub[1]-1 to A#sub[1]-2, E#sub[1]-1 to E#sub[1]-2, and E#sub[2]-1 to E#sub[2]-4.
In contrast, the strong-polarity modes were named according to their vibration directions and incidence configurations,
  including TO-zOx (vibrating in the zOx plane and roughly perpendicular to the incidence direction),
  TO-y (vibrating along the y-axis), and LO (vibrating roughly parallel to the incidence direction).

#include "figure-discont.typ"

In order to further investigate the frequencies dependence on the wavevector,
  and to confirm the accuracy of both our experiments and calculations,
  we analyzed the negligible-polarity phonons off the #sym.Gamma point
  by comparing experimental and calculated results under various laser incidence directions.
The E#sub[2]-3 peak served as a calibration reference under various experiments,

// 第二段：我们具体计算了模式对极性的依赖。可以看到，弱极性声子的变化极小。与实验比较，也验证了我们实验的准确性。

// 第三段：有两个弱极性模式还没有在实验上看到过。我们通过计算知道了它们的强度，并看到了其中一个。

// 第四段：声子模式的强度可以从理论上得到解释。

=== Negligible-polarity Phonons


#include "figure-raman.typ"

Peaks corresponding to seven Raman-active negligible-polarity phonons were observed in our experiments
  (only the E#sub[2]-4 mode was not observed),
  which is more than all previous experiments (where only five or six peaks were typically reported).
To explain the discrepancy in experimental results, first-principles calculations were performed,
  and the result was compared with experimental data and summarized in @table-nopol.
Our calculation showed that the mode of E#sub[2]-1, E#sub[2]-2, E#sub[1]-1, A#sub[1]-1 and E#sub[2]-3
  had relatively high Raman intensities and well-separated frequencies,
  making them observed in our experiments as well as most previous experiments.
The A#sub[1]-2 mode was calculated to have very weak (0.01) and relatively strong (1.78) Raman intensity 
  under in-plane polarization and z polarization, respectively,
  which is compatible with our experimental result 
  that it could be observed clearly in the y(zz)#overline[y] configuration but hardly seen in our other experiments.
This peek was reported to be observable in some experiments (cite) but not in others (cite),
  our calculation provided an explanation for this discrepancy.
The E#sub[2]-4 modes was calculated
  to be located close to the most intense E#sub[2]-3 mode (< 10 cm#super[-1] away)
  and exhibit very weak Raman intensities (only 0.6% of the E#sub[2]-3 mode),
  making it not visible in our and all previous experiments.
The E#sub[1]-2 mode was also located close to the E#sub[2]-3 mode and has weak Raman intensity,
  making it also unobservable in previous experiments.
However, the E#sub[1]-2 mode was observable in our experiments of y(zx)#overline[y] with extended acquisition time,
  where the scattering of the E#sub[2]-3 mode was suppressed while that of the E#sub[1]-2 mode was enhanced,
  thanks to the different representations of these two modes.
Our experiments reported the observation of the E#sub[1]-2 peak for the first time,
  and explained the discrepancy among previous experiments and ours with the help of our calculations.

#include "table-nopol.typ"

It is noteworthy that the large variation in Raman tensor magnitudes among different modes
  was not yet theoretically understood.
For example, the Raman tensor of the E#sub[2]-3 mode was substantially larger
  than those of other negligible-polarity modes (over 30 times larger than that of the second-strongest).
This highlighted a significant gap in established theory
  that rigorous symmetry analysis could only predict the non-zero components of the Raman tensors,
  but not their magnitudes.
In order to address the limitations of existing theories,
  a method for estimating the magnitudes of Raman tensors was proposed.
By analyzing the local environment of individual atoms,
  this approach decomposed their contributions to the Raman tensor into two parts:
  a dominant component (invariant across similar environments, denoted as $a_i$，where $i in {1, 2, 5, 6}$)
  and several secondary components (reflecting environmental variations, 
    denoted as $epsilon_i$, $eta_i$, and $zeta_i$，where $i in {1, 2, 5, 6}$,
    and $|epsilon_i| + |eta_i| + |zeta_i| << |a_i|$ was assumed).
Detailed derivations were provided in @appd-predict, with results summarized in @table-nopol.
Notably, the E#sub[2]-3 mode was the only mode that retains the $a_i$ term,
  which indicating a constructive interference of contributions from the local environment of individual atoms.
This stood in contrast to other negligible-polarity modes where such contributions tend to cancel out,
  explaining the exceptionally high Raman tensor magnitude observed for the E#sub[2]-3 mode.

To achieve a more precise investigation of the Raman spectra
    and prepare for analyzing impurity and charge carrier effects,
  the analysis of negligible-polarity phonons off the #sym.Gamma point was conducted
  by comparing experimental and calculated results under various lazer incidence directions.
The E#sub[2]-3 peak searved as a calibration reference under various experiments,
  since its position was calculated to be virtually invariant between normal and edge incidence
  (with a shift of only #sym.tilde 0.004 cm#super[-1]).
The E#sub[2]-1, E#sub[2]-2, and A#sub[1]-1 modes exhibited observable shifts,
  and the experimental results were in good agreement with our calculations, as shown in fig.
Our results further confirmed the accuracy of both our experiments and calculations.

=== Strong-polarity Phonons

The Strong-polarity phonon modes participated in Raman scattering
  exhibited significant variations depending on the incidence configurations
  (see the intersection of colored solid lines and orange dashed lines in @figure-discont b and c).
For incident light propagating along the z direction,
  the C#sub[6v] point group applied and two modes were present,
  marked as normal-TO and normal-LO,
  and they were corresponding to the E#sub[1] and A#sub[1] representations
    and vibrations along the basle plane and z direction, respectively.
The normal-LO would subsiquently couple with plasmons to form LOPC modes in n-type 4H-SiC.
For incident light propagating along other directions,
  the C#sub[6v] group no longer held and three modes were present.
Specifically, for incident light propagating along x direction, the C#sub[2v] group applied,
  and the three modes was named as edge-TO-z, edge-TO-y and edge-LO,
  which were corresponding to A#sub[1], B#sub[2] and B#sub[1] representations
  and vibrations along z, y and x directions, respectively.

E1 的情况。

注意到在正入射中，理论上不能被观察到的E#sub[1]-1模式也被观察到了。
与弱极性的 E1-1 模式类似，我们也认为这是由于入射光并非完全沿 z 轴入射所致。
但与弱极性 E1-1 模式不同的是，强极性 E1-1 模式在 xy 的偏振下并没有更强反而更弱。
这是因为E1这时不再是严格的E1模式，而是分裂成了两个相近的模式。
我们的计算表明，在2度的入射角下，E1分裂的两个模式非常接近。
其中某个模式会怎样怎样，另一个会怎样怎样。

// 我们预测，随着入射方向偏移，LO 峰会向着高频方向移动。此外，我们也注意到 LO 也会与载流子产生影响。
// 在 n 型半导体中，LOPC 模式将代替 LO 模式；在 p 型半导体中，LO 模式仍然单独存在，但它的半高宽会受到载流子浓度的影响。

#include "table-pol.typ"
#include "figure-rev.typ"