= Method

== 4H-SiC wafer details

外延片的厚度、掺杂浓度、生长 C/Si 比，斜切角度。

5 个 6 寸的 p 型外延片被使用，我们将它们称为 W#sub[i]。
使用的衬底都是 n 型，前四个外延片的厚度为 1 微米，第五个外延片的厚度为 2 微米。
外延层的 Al 掺杂浓度分别为 0.1 3.8 5.1 6.4 10 E18 cm#super[-3]，使用 SIMS 测试。
生长时 Si/C 比分别为 0.7 1.2 1.6 2.4 2.0。、
所有外延片都有 4 度斜切。

Five 6-inch p-type epitaxial wafers (W#sub[1]–W#sub[5]) were fabricated on n-type substrates
  using the step-flow growth method with a 4° offcut angle.
The Al doping concentrations, determined by SIMS,
  were 0.1, 3.8, 5.1, 6.4, and 10 #sym.times 10 cm#super[-3] for W#sub[1]–W#sub[5], respectively.
W#sub[1]–W#sub[4] had an epitaxial layer thickness of 1 μm, while W#sub[5] had a thickness of 2 μm.
The Si/C ratios during growth were 0.7, 1.2, 1.6, 2.4, and 2.0 for the five wafers, respectively.

== Raman experiments setup

拉曼设备的型号。激光的波长，背散射。共焦针孔。

拉曼设备的型号是 LabRAM HR Evolution，使用背散射。
大部分实验中，我们使用 532 nm 的激光，少部分实验中使用 325 nm 的激光以观测紫外拉曼。
有三个不同的入射配置，包括正入射、掠入射、边入射。
考虑到 4 度斜切和 4H-SiC 几乎各向同性的折射率（2.73） @shaffer_refractive_1971 ，掠入射的入射角大约为 25 度。
在正散射过程中，我们使用 100 微米的共焦针孔，以尽可能提高 z 方向的分辨率 @song_depth_2020；其它情况使用常用的 200 微米针孔以提高信噪比。
此外，在正入射和边入射时，拉曼散射信号较强，因此我们使用较短的积分时间（约 60 秒），
  而在掠入射时，拉曼信号较弱，因此使用较长的积分时间（约 300 秒）。

All Raman experiments were conducted using a LabRAM HR Evolution system in a back-scattering configuration,
  where the scattered light was collected in the direction opposite to the incident laser.
A 532 nm laser was primarily used as the excitation source,
  while a 325 nm laser was employed for only ultraviolet Raman measurements.
Three distinct incidence configurations were utilized, as illustrated in @figure-incidence:
  (i) normal incidence, where the laser incident perpendicularly to the epitaxial surface;
  (ii) grazing incidence, where the laser incident nearly parallelly to the epitaxial surface;
  and (iii) edge incidence, where the laser is incident at the wafer edge and perpendicularly to the edge surface.
Considering the 4° offcut angle and the nearly isotropic refractive index of 2.73 for 4H-SiC @shaffer_refractive_1971,
  the refracted laser in grazing incidence forms an angle of approximately 25° with the c axis.
A 100 μm confocal pinhole was used for normal incidence to enhance axial (z-direction) resolution @song_depth_2020,
  while a 200 μm pinhole was employed for the other configurations to improve the signal-to-noise ratio.
The integration time was set to 60 seconds for normal and edge incidence,
  while it was extended to 300 seconds for grazing incidence due to the weaker Raman signal.

#include "figure-incidence.typ"

== Atomic model establishment

我们建立了三类模型：无缺陷、点缺陷和面缺陷。

Three types of models were established: defect-free models, point defect models, and surface defect models.

无缺陷和点缺陷的模型

无缺陷和点缺陷的模型尺寸约为 $12.4 angstrom times 10.7 #sym.angstrom times 10.1 #sym.angstrom$，包含了大约 128 个原子。
我们认为是足够大的，因为无缺陷模型的结果与实验差距在一定范围内，且继续扩大模型对准确程度没有提升。
对于点缺陷模型，我们考虑了 Si 空位、C 空位、N 替位、Al 替位。
分别记为 V#sub[Si]、V#sub[C]、N#sub[Si] 和 Al#sub[C]。
考虑到 SiC 的对称性 p63mc （引用），有两个不同的位点，记为 k 和 h，
  根据局部环境近似为立方（k）还是六角（h）。
此外，还有人提出，N 替换 C、C 替换 Si 的模型（引用），
  此结构除了 h 位与 k 位的区别以外，还需要考虑发生替换的两个原子位于面内还是面外（将会导致对称性的不同）。

The defect-free and point defect models were established using supercells
  with dimensions of $12.4 #sym.angstrom times 10.7 #sym.angstrom times 10.1 #sym.angstrom$,
  containing approximately 128 atoms.
This supercell size was found sufficient for accurately capturing the phonon properties
    of both defect-free and point-defect-containing 4H-SiC, 
  as the calculated phonon frequencies for the defect-free model deviated by less than 5% from experimental values,
  and further enlargement of the supercell yielded negligible changes.
Twelve point defect models were constructed,
  including Si vacancies (V#sub[Si]-h and V#sub[Si]-k), C vacancies (V#sub[C]-h and V#sub[C]-k),
  N substitutions at C sites (N#sub[C]-h and N#sub[C]-k),
  Al substitutions at Si sites (Al#sub[Si]-h and Al#sub[Si]-k),
  and complex defects involving N substitution at a C site followed by C substitution at a Si site
    @gerstmann_formation_2003
    (N#sub[C]C#sub[Si]-i-h, N#sub[C]C#sub[Si]-i-k, N#sub[C]C#sub[Si]-o-h and N#sub[C]C#sub[Si]-o-k).
Here, the suffixes -h and -k denote the quasi-hexagonal and quasi-cubic sites, respectively,
  while -i and -o denote the in-plane and out-of-plane configurations.

（图：a hk 位置 b 复杂缺陷 强调在复杂缺陷中，h k 根据 N 原子而定）

面缺陷的模型

对于面缺陷模型，主要考虑了三类 BPD（引用自己的文章），这些缺陷在室温下被认为是可以稳定存在的。
对于每类BPD，我们考虑两个模型，一个将两个 PD 包括在内，为了模拟 PBD 未分解或分解后边缘处的信号；
另一类则仅仅包含一个贯穿的层错，为了模拟 BPD 分解后在层错处的信号。

== Simulation details

计算工具和参数

第一性原理计算使用 VASP，使用 PBE PAW，平面波截断能设置为 800，自洽 1E-9，弛豫精度为 1E-8。
K 点网格根据模型大小不同，分别使用 555 和 ???。
涂抹使用 xxx 以统一比较点缺陷和无缺陷的模型。
弛豫的精度为 xxx。
声子计算使用 phonopy phono3py ufo，BEC 修正使用 xxx 的算法。（引用）
