== 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 was justified because
  the negligible-polarity phonon modes participated in Raman scattering
  were nearly identical across all incidence configurations,
  with their frequencies differing by only #sym.tilde 0.1 cm#super[-1]
  (see the intersection of gray solid lines and orange dashed lines in @figure-discont b and c).

There were eight Raman-active negligible-polarity modes in 4H-SiC at the #sym.Gamma point,
  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 (see @figure-discont and @figure-raman).
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 "figure-raman.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] 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-nopo-diff.
Our results further confirmed the accuracy of both our experiments and calculations.

#include "figure-nopo-diff.typ"

=== Strong-polar Phonons

沿着不同方向入射的话，强声子的模式是不同的。

// 在半导体的极性声子模式中，原子间存在长距离的库伦相互作用，导致散射谱在 Gamma 附近不再连续（引用），如图中的彩色线所示。
// 这导致不同方向的入射/散射光的声子模式不同。
// 具体来说，当入射光/散射光沿着 z 方向时，起作用的是 A-Gamma 线上的声子模式（图中的左半边的橘线），它们适用于群 C6v。
// 这时会有一个 E1 模式（TO，振动方向在面内）和一个 A1 模式（LO，沿 z 振动）。
// 而当沿着 y 方向入射时，起作用的是 Gamma-K 线上的声子模式（图中的右半边的橘线），它们不再适用于群 C6v，而只适用于群 C2v；
// 它会分裂成沿x、y、z 方向的三个声子模式（图中的右半边的蓝线），它们分别对应于群 C2v 的 A1、B1 和 B2 表示 TODO: 确认这个几个表示的名字。
// 若考虑到到入射光不是严格沿着 z 方向，而是有一个小的角度（例如 10 度），则此时有一个声子模式沿着 x 方向，另外两个声子模式则为 y-z 两个方向的混合。
// (没有在图上表示)

半导体中的强极性声子模式强烈地依赖于入射光的方向，
  这是由于半导体中原子之间的长程库伦相互作用所致，
  表现在散射光谱中，Gamma点附近不连续（见@figure-discont）。
具体来说，当入射光沿着 z 方向时，起作用的是 A-Gamma 线上的声子模式（图中的上半部分的橘线），它们适用于群 C6v。
这时会有一个 TO 模式（C6v 中的 E1）和一个 LO 模式（C6v 中的 A1）。
而当沿着 y 方向入射时，起作用的是 Gamma-K 线上的声子模式（图中下半部分的橘线），它们不再适用于群 C6v，而只适用于群 C2v；
这时将会出现三个模式，包括一个LO（沿y方向振动，对应C2v中的A1）和两个TO模式（根据振动方向，命名为B2-x和B2-y，对应C2v中的B1和B2）。
当入射光不是严格沿着坐标轴方向，而是在xOy面内呈现一定的夹角时（掠入射，以及考虑了斜切的正入射），则此时有一个声子模式沿着 x 方向，另外两个声子模式则为 y-z 两个方向的混合。

Strong-polarity phonon modes caused by different incident light directions are different,
  due to long-range Coulomb interactions between atoms in semiconductors,
  showing discontinuity in the scattering spectra near the #sym.Gamma point (see @figure-discont).
For incident light propagating along the z direction (phonon modes on the A-#sym.Gamma line),
  symmetry of C#sub[6v] point group applies and leading to two modes (two peeks in Raman spectra),
  including an E#sub[1] mode (pink line in @figure-discont, vibration in-plane)
    and an A#sub[1] mode (green line in @figure-discont, vibration along z-direction).
When the light is incident along other directions, symmetry in plane was broken and C#sub[6v] symmetry no longer holds,
  and there will be three phonon modes in theory.
For example, when the light is incident along the y direction (phonon modes on the #{sym.Gamma}-K line),
  symmetry of C#sub[2v] applies and three modes exist in dispersion curves,
  including an A#sub[1] mode (green line in @figure-discont, vibration along z direction),
  a B#sub[2] mode (blue line in @figure-discont, vibration along x direction),
  and a B#sub[1] mode (red line in @figure-discont, vibration in y direction).
When the light is incident along a direction between z and y,
  three phonon modes will exist, but vibration in the mixed direction.

将理论/计算结果与实验对比。

我们将计算与实验结果进行了对比。衬底中的LO峰与plasmon耦合形成LOPC峰，因此与计算结果大不相同。
对于TO峰，在正入射中，它与E2-3模式的距离为xxx；在掠入射中，它与E2-3模式的距离为xxx。

Many Raman experiments on 4H-SiC with incident light along the z direction have observed two peaks.
However, no experiments have reported three peaks with incident light along other directions.
In our experiment, we found the third, and it satisfied properties we expected.
In our experiments, we found that the third peak only appears when focusing inside the sample.

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"




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"