== Phonons in Perfect 4H-SiC

These phonons were categorized into two groups and discussed separately, according to their electrical polarities:
  negligible-polarity phonons (i.e., zero or very weak electrical polarity), and strong-polarity phonons.

=== Negligible-polarity Phonons

The negligible-polarity phonons were initially analyzed at the #sym.Gamma point,
  disregarding the incidence configurations.
This simplification is justified because the participating negligible-polarity phonon modes
  are nearly identical across all Raman experiments,
  with their frequencies differing by only #sym.tilde 0.1 cm#super[-1] difference in frequency
  (see gray lines in @figure-discont).

There are eight Raman-active negligible-polar modes in 4H-SiC,
  corresponding to three irreducible representations of the C#sub[6v] group (A#sub[1], E#sub[1], and E#sub[2]).
We named these modes as
  E#sub[2]-1, E#sub[2]-2, E#sub[1]-1, A#sub[1]-1, E#sub[1]-2, E#sub[2]-3, E#sub[2]-4, and A#sub[1]-2,
  in order of increasing frequency.
The Raman tensor forms of each mode were derived by further considering their representations in the C#sub[2v] group,
  and were summarized in @table-rep.

#include "figure-discont.typ"
#include "table-rep.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],
  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 $i in {1, 2, 3, 4}$)
  and several secondary components (reflecting environmental variations, 
    denoted as $epsilon_i$, $eta_i$, and $zeta_i$，where $i in {1, 2, 3, 4}$,
    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,
  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,
  thereby explaining the exceptionally high Raman tensor magnitude observed for the E#sub[2]-3 mode.

To further investigate the Raman spectra,
  an 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 remained virtually invariant between normal and edge incidence
  (with a shift of only #sym.tilde 0.004 cm#super[-1]).

The E2​-1, E2​-2, and A1​-1 modes exhibited observable frequency shifts. These experimental results are in good agreement with our calculations, as shown in Figure X and Table Y."


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 figure and table.
Our results further confirmed the accuracy of both our experiments and calculations.

#include "figure-nopo-diff.typ"

为了进一步探索拉曼光谱，离开Gamma点的弱极性声子的分析也被进行，通过比对不同方向入射下的实验和计算结果。


为了检验我们实验和计算的准确性，我们测量和计算了不同入射方向下弱极性峰位的微小移动。
在我们的计算中，对比于正入射，肩入射中，E2-1与A1-1会有可观测的蓝移，同时E2-2会有可观测的红移。
E2-1 E2-2 的计算结果与实验吻合较好，A1-1与实验结果略有出入。
这可以被解释为频率移动的原因不同。
对于 E2 模式，其极性为零，色散曲线在 Gamma 点附近连续，因此频率取决于色散曲线在Gamma附近的曲率；
而 A1 模式具有弱但非零的极性，它在 Gamma 点附近不连续，
  计算频率偏移需要同时考虑色散曲线曲率和阶跃大小的影响。
在 A1-1 模式中，阶跃大小占主导。
而在重掺杂n型衬底中，阶跃大小被载流子屏蔽显著减小，因此导致理想模型的计算结果与实验结果不符。
通过将载流子屏蔽效应纳入计算，计算结果与实验结果吻合较好，如图所示。

在强极性声子中：

在每一个入射方向的拉曼实验中，参与散射的声子都包含三个强极性声子模式：
  一个 LO，它的振动方向roughly平行于入射光方向；和两个 TO 模式，它们的振动方向roughly垂直于入射光方向。
它们的性质（振动pattern，频率，拉曼张量）会dramatically改变accroding to 入射方向。
例如，在正入射中，LO 模式属于 C6v 群中的 A1 表示（Named as n-LO），两个 TO 模式简并为 C6v 群中的 E1 表示（Named as n-TO）；
  在沿x方向的侧入射中，LO 模式属于 C2v 群中的 B1 表示（Named as e-LO），两个 TO 分别属于 C2v 群中的 B2 和 A1 表示（Named as e-TO-x 和 e-TO-y）。
其中，正入射的 LO 会与电子耦合形成 LOPC 模式，从而导致频率的显著变化；侧入射的 LO 则由于对称性的原因，无法观测到。
我们将它们的结果总结在表中。可以看到，我们的结果与实验和计算吻合较好。

我们将声子被分为两类讨论：
  极性可以忽略的声子模式（即极性为零或非常弱的声子），在拉曼散射过程中它们的电极性造成的效应可以忽略；
  强极性声子模式，在拉曼光谱中电极性效应是可观测的，不可以被忽略。


// #include "non-polar/default.typ"
// #include "polar/default.typ"