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= Introduction
// SiC 是很好的材料。
// 其中，4H-SiC 是SiC的一种多型，它的性质更好，近年来随着外延工艺的成熟而获得了更多的关注。
SiC is a promising wide-bandgap semiconductor material
with high critical electric field strength and high thermal conductivity.
It has been widely used in power electronic devices and has long attracted a lot of research
@casady_status_1996 @okumura_present_2006.
The 4H-SiC has a wider bandgap, higher critical electric field strength,
higher thermal conductivity, and higher electron mobility along the c-axis than other polytypes.
Currently, the 4H-SiC has gradually received more attention than other polytypes,
thanks to the development of epitaxy technology and the increasing application in the new energy industry
@tsuchida_recent_2018 @harada_suppression_2022 @sun_selection_2022. // TODO: 多引用一些近年来的文献，有很多
// 声子（量子化的原子振动）在理解晶体的原子结构以及热电性质方面起着重要作用。
// 声子可以通过多种实验技术来探测，包括 EELS、IR 吸收谱等。
// 拉曼光谱是最常用的方法，它提供了一种无损、非接触、快速和局部的声子测量方法，已被广泛用于确定晶体的原子结构（包括区分 SiC 的多型）。
Phonons (quantized atomic vibrations) play a fundamental role
in understanding the atomic structure
as well as the thermal and electrical properties
of crystals (including 4H-SiC).
They could be probed by various experimental techniques,
such as electron energy loss spectroscopy and infrared absorption spectroscopy.
Among these techniques,
Raman spectroscopy is the most commonly used method,
as it provides non-destructive, non-contact, rapid and spatially localized measurement of phonons
that near the #sym.Gamma point in reciprocal space.
Studies in Raman scattering of 4H-SiC have been conducted since as early as 1983
and have been widely employed to identification of different SiC polytypes.
// TODO: 增加引用文献
// 近年来，更多信息被从拉曼光谱中挖掘出来。
// LOPC 已经被用于快速估计 n 型 SiC 的掺杂浓度。
// 层错的拉曼光谱也已经被研究，可以被用于检测特定结构层错的存在和位置。
// 掺杂对拉曼光谱的潜在影响也已经被研究。
// 然而，拉曼光谱上仍有一些不知来源的峰；同时，一些也缺少一些理论上预测应该存在的峰。
// 此外，预测掺杂导致的新峰也没有说明原因。
Increasingly rich information has been extracted from Raman spectra of 4H-SiC.
Longitudinal optical phonon–plasmon coupling (LOPC) peek
has been utilized to rapidly estimate the doping concentration in n-type SiC.
Peeks associated with some stacking faults have also been investigated
and used to detect the presence and location of specific structural faults.
Moreover, the potential effects of doping on Raman spectra have been explored.
However, some unidentified peaks still appear in the Raman spectra,
while certain phonon modes predicted by theory remain unobserved.
In addition, the origins of newly emerged peaks induced by doping are often unclear or unexplained.
// TODO: 多举例，增加引用文献
In this paper, we do some things. Especially we do something for the first time.
// TODO: 完善
#include "section/introduction.typ"
= Method
@@ -137,314 +90,11 @@ experiment
== Phonons in Perfect 4H-SiC
#par()[#text()[#h(0.0em)]]
(There are 21 phonons in total.
We classified them into two categories: 18 negligible-polar phonons and 3 strong-polar phonons.)
// 拉曼活性的声子模式对应于 Gamma 点附近的声子模式。
// 根据这些声子模式的极性，我们将这些声子分成两类。
The phonons involved in Raman scattering are located in reciprocal space around the #sym.Gamma point,
at the exact positions are determined by the wavevectors of the incident and scattered light.
At each such position, there are 21 phonon modes (degenerate modes are counted as their multiplicity).
We classify these 21 phonons into two categories based on their polarities.
The 18 of 21 phonons are classified into negligible-polar phonons (i.e., phonons with zero or very weak polarity),
for which the effect of polarity can be ignored in the Raman scattering process;
and the other three phonons are strong-polar phonons,
where the polarity gives rise to observable effects in the Raman spectra.
(This classification make sense.)
This classification is based on the fact that
the four Si atoms in the primitive cell of 4H-SiC carry similar positive Born effective charges (BECs),
and the four C atoms carry similar negative BECs (see @table-bec).
In the 18 negligible-polar phonons,
the vibrations of two Si atoms are approximately opposite to those of the other two Si atoms,
and the same holds for the C atoms,
leading to cancellations of macroscopic polarity.
In contrast, in the three strong-polar phonons,
all Si atoms vibrate in the same direction, and all the C atoms vibrate in the opposite direction,
resulting in a strong dipole moment.
#figure({
set text(size: 9pt);
table(columns: 4, align: center + horizon,
table.cell(colspan: 2)[], table.cell(colspan: 2)[*BEC* (unit: |e|)],
table.cell(colspan: 2)[], [x / y direction], [z direction],
table.cell(rowspan: 2)[Si atom], [A/C layer], [2.667], [2.626],
[B layer], [2.674], [2.903],
table.cell(rowspan: 2)[C atom], [A/C layer], [-2.693], [-2.730],
[B layer], [-2.648], [-2.800],
)},
caption: [
Born effective charges of Si and C atoms in A/B/C/B layers of 4H-SiC, calculated using first principle method.
],
placement: none,
)<table-bec>
#include "section/perfect/default.typ"
=== Phonons with Negligible Polarities
#par()[#text()[#h(0.0em)]]
(We investigate phonons at Gamma instead of the exact location near Gamma.)
Phonons at the #sym.Gamma point were used
to approximate negligible-polar phonons that participating in Raman processes of any incident/scattered light.
This approximation is widely adopted and justified by the fact that, // TODO: cite
although the phonons participating in Raman processes are not these strictly located at the #sym.Gamma point,
they are very close to the #sym.Gamma point in reciprocal space
(about 0.01 nm#super[-1] in back-scattering configurations with 532 nm laser light,
which corresponds to only 1% of the smallest reciprocal lattice vector of 4H-SiC,
see orange dotted line in @figure-discont),
and their dispersion at #sym.Gamma point is continuous with vanishing derivatives.
Therefore, negligible-polar phonons involved in Raman processes
have nearly indistinguishable properties from those at the #sym.Gamma point,
and the phonon participating in Raman processes of different incident/scattered light directions
are all nearly identical to the phonons at the #sym.Gamma point.
#figure(
image("/画图/声子不连续/embed.svg"),
caption: [
(a) Phonon dispersion of 4H-SiC along the A–#sym.Gamma–K high-symmetry path.
Gray lines represent negligible-polar phonon modes,
while colored lines indicate strong-polar phonon modes.
The green, red and blue lines indicate the mode along the z-direction, y-direction and x-direction, respectively.
Along A-#sym.Gamma path, strong-polar modes along x- and y-directions are degenerated,
showing as a single purple line.
(b) Magnified view of the boxed region in (a).
The orange dashed lines mark the phonon wavevectors involved in Raman scattering
with incident light along the z- and y-directions.
],
placement: none,
)<figure-discont>
#par()[#text()[#h(0.0em)]]
(Representation of these 18 phonons, and the shape of their Raman tensors could be determined in advance.)
Phonons at the #sym.Gamma point satisfy the C#sub[6v] point group symmetry,
and the 18 negligible-polar phonons correspond to 12 irreducible representations of the C#sub[6v] point group:
2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2].
Phonons belonging to the A#sub[1] and B#sub[1] representations vibrate along the z-axis and are non-degenerate,
while those belonging to the E#sub[1] and E#sub[2] representations vibrate in-plane and are doubly degenerate.
Phonons of the B#sub[1] representation are Raman-inactive, as their Raman tensors vanish.
In contrast, phonons of the other representations are Raman-active,
and the non-zero components of their Raman tensor
can be determined by further considering their representation in the C#sub[2v] point group (see @table-rep).
These Raman-active phonons are potentially be visible in Raman experiment under appropriate polarization configurations.
However, whether a mode is sufficiently strong to be experimentally visible
depends on the magnitudes of its Raman tensor components,
which cannot be determined solely from symmetry analysis.
#figure({
let m2(content) = table.cell(colspan: 2, content);
set text(size: 9pt);
table(columns: 6, align: center + horizon, inset: (x: 3pt, y: 5pt),
[*Representations in C#sub[6v]*], [A#sub[1]], m2[E#sub[1]], m2[E#sub[2]],
[*Representations in C#sub[2v]*], [A#sub[1]], [B#sub[2]], [B#sub[1]], [A#sub[2]], [A#sub[1]],
[*Vibration Direction*], [z], [x], [y], [x], [y],
[*Raman Tensor of #linebreak() Individual Phonons*],
[$mat(a,,;,a,;,,b)$], [$mat(,,a;,,;a,,;)$], [$mat(,,;,,a;,a,;)$], [$mat(,a,;a,,;,,;)$], [$mat(a,,;,-a,;,,;)$],
[*Raman Intensity with Different #linebreak() Polarization Configurations*],
[xx/yy: $a^2$ #linebreak() zz: $b^2$ #linebreak() others: 0],
m2[xz/yz: $a^2$ #linebreak() others: 0], m2[xx/xy/yy: $a^2$ #linebreak() others: 0],
)},
caption: [
Raman-active representations of C#sub[6v] and C#sub[2v] point groups.
],
placement: none,
)<table-rep>
#par()[#text()[#h(0.0em)]]
(We propose a method to estimate the magnitudes of the Raman tensors of these phonons,
without first-principle calculations.
Here we only write out results, details are in appendix.)
// TODO: maybe it is better to assign Raman tensor to each bond, instead of atom
We propose a method to estimate the magnitudes of the Raman tensors of these phonons by symmetry analysis.
The method only takes the vibration directions of each atom in each phonon mode,
leaving the amplitudes unconsidered (see appendix for details),
and the result was summarized in @table-predmode.
In the Raman tensors in @table-predmode,
$a_i$ corresponding to the change of polarizability caused by movement of the Si atoms in A and C layers,
$epsilon_i$, $eta_i$ and $eta_i$ corresponding to the difference between different bilayers and different atoms.
Due to the similarity of environment in different bilayers and around different atoms,
the absolute values of $epsilon_i$, $eta_i$ and $zeta_i$ are expected to be much smaller than that of $a_i$,
thus the Raman tensors containing $a_i$ are expected to be much larger than those not containing $a_i$.
// Raman Tensor for A1: line1 xx/yy; line2 zz
// Raman Tensor for E1: x-dirc xz or y-dirc yx
// Raman Tensor for E2: x-dirc xy or y-dirc xx or y-dirc -yy
// TODO: remove LO TO or not?
#page(flipped: true)[#figure({
let m(n, content) = table.cell(colspan: n, content);
let m2(content) = table.cell(colspan: 2, content);
let m3(content) = table.cell(colspan: 3, content);
let m4(content) = table.cell(colspan: 4, content);
set text(size: 9pt);
set par(justify: false);
table(columns: 11, align: center + horizon, inset: (x: 3pt, y: 5pt),
[*Representation in C#sub[6v]*], m3[A#sub[1]], m3[E#sub[1]], m4[E#sub[2]],
[*Relative Vibration Direction*],
[Si: $+-+-$ #linebreak() C: $0000$], [Si: $0000$ #linebreak() C: $+-+-$], [Si: $++++$ #linebreak() C: $----$],
[Si: $+-+-$ #linebreak() C: $-+-+$], [Si: $+-+-$ #linebreak() C: $+-+-$], [Si: $++++$ #linebreak() C: $----$],
[Si: $++--$ #linebreak() C: $-++-$], [Si: $+--+$ #linebreak() C: $++--$],
[Si: $++--$ #linebreak() C: $+--+$], [Si: $+--+$ #linebreak() C: $--++$],
[*Vibration Direction*], m3[z], m3[x/y], m4[x/y],
[*Raman Tensor Predicted*], [xx/yy: $-2A_#text[Si] epsilon_5$ #linebreak() zz: $-2A_#text[Si]epsilon_6$],
[xx/yy: $-2A_#text[C]zeta_5$ #linebreak() zz: $-A_#text[C]zeta_6$],
[xx/yy: $2A_#text[Si] (2a_5+epsilon_5) + 2A_#text[C] (2a_5+eta_5+zeta_5)$ #linebreak() zz: $2A_#text[Si] (2a_6+epsilon_6) + 2A_#text[C] (2a_6+eta_6+zeta_6)$],
[xz/yz: $-2A_#text[Si]epsilon_1-2A_#text[C]zeta_1$],
[xz/yz: $-2A_#text[Si]epsilon_1+2A_#text[C]zeta_1$],
[xz/yz: $2A_#text[Si] (2a_1+epsilon_1) +2A_#text[C] (2a_1+2eta_1+zeta_1))$],
[xx/-yy/xy: $2A_#text[Si] (2a_2+epsilon_2) -2A_#text[C] (2a_2+2eta_2+zeta_2))$],
[xx/-yy/xy: $-2A_#text[Si]epsilon_2-2A_#text[C]zeta_2$],
[xx/-yy/xy: $2A_#text[Si] (2a_2+epsilon_2) +2A_#text[C] (2a_2+2eta_2+zeta_2))$],
[xx/-yy/xy: $-2A_#text[Si]epsilon_2+2A_#text[C]zeta_2$],
[*Raman Intensity Predicted*], m2[weak], [strong], m2[weak], [strong], m2[weak], [strong], [weak],
[*Raman Tensor Calculated*],
[-1.68 #linebreak() 1.34], [0.10 #linebreak() -1.33], [-7.68 #linebreak() 21.65],
[-1.56], [-0.30], [7.32], [-0.41], [1.06], [9.41], [-0.71],
// [*x*], [1 axial acoustic], [0 axial optical], [1 axial optical],
// [0 axial acoustic], [1 axial optical], [1 axial optical],
// m2[0.5 acoustic], m2[0.5 optical],
[*Type*], [axial acoustic], [axial optical], [longitudinal optical],
[planer acoustic], [planer optical], [transverse optical],
m2[planer acoustic], m2[planer optical],
[*Move-towards Atom-pairs* (In-plane/Out-plane)], [4/0], [0/4], [4/4], [0/4], [4/0], [4/4], [0/2], [2/0], m2[4/2],
// [*Predicted Frequency*], [low], [medium], [high], [medium], [low], [high], [low], [medium], m2[high],
[*Calculated Frequency*],
[591.90], [812.87], [933.80], [257.35], [746.91], [776.57], [190.51], [197.84], [756.25], [764.33]
)},
caption: [Predicted modes and their "Raman tensor"],
placement: none,
)<table-predmode>]
The Raman tensors and frequencies of the negligible-polar phonons were calculated using first-principles methods,
and the results are compared with experiment and theory (@table-nopol).
Calculated frequencies of these phonons are consistent with the experimental results
with a low-estimated error of about 2% to 5%, which might be due to the PBE functional used in the calculation (cite).
The Raman tensors of these phonons are also consistent with the experimental and theoretical results,
where E#sub[2] mode experimentally at 776 is the most intense phonon mode,
followed by four modes with lesser intensities
(E#sub[2] modes at 195.5 and 203.3, E#sub[1] mode at 269.7, A#sub[1] mode at 609.5).
The Raman scatter of the E#sub[1] mode calculately at 746.91 and E#sub[2] mode calculately at 756.25
are much weaker than the E#sub[2] mode calculated at 756.25 but located near it, according to our calculation,
thus it could not be distinguished from E#sub[2] mode calculated at 756.25,
which explains why they are not observed in experiments.
Moveever, the A#sub[1] mode calculated at 812.87
have a very weak Raman intensity in the basal plane (xx and yy, only 0.01)
but an observable intensity in the zz configuration (1.78).
Thus, this mode could not be observed in most Raman experiments (cite),
but could be observable when incident light propagate not along the z-direction (our experiment),
or the incident light wavelength is near the resonance condition (cite).
Besides, there are other peeks in the experiment.
The peek at 796 and 980 are caused by strong-polar phonons which will be discussed later.
Besides, there are small peeks at xxx,
which could not be explained in perfect 4H-SiC and will be discussed in the next section.
// TODO: 将一部分 phonons 改为 phonon modes
// 在论文中我们这样来称呼：phonon 对应某一个特征向量，而 modes 对应于一个子空间。
// 也就是说，简并的里面有两个或者无数个 phonon，但只有一个 mode
#page(flipped: true)[#figure({
let m(n, content) = table.cell(colspan: n, content);
let m2(content) = table.cell(colspan: 2, content);
let m3(content) = table.cell(colspan: 3, content);
let A1 = [A#sub[1]];
// let A2 = [A#sub[2]];
let B1 = [B#sub[1]];
// let B2 = [B#sub[2]];
let E1 = [E#sub[1]];
let E2 = [E#sub[2]];
set text(size: 9pt);
set par(justify: false);
table(columns: 27, align: center + horizon, inset: (x: 3pt, y: 5pt),
// [*Direction of Incident & Scattered Light*],
// m(26)[Any direction (not depend on direction of incident & scattered light)],
// TODO: 整理表格，使用 m2 m3 来代替
[*Number of Phonon*],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
[1], m2[2], [3], m2[4], [5], [6], [7], [8], m3[9], [10], [11], [12], m2[13], [14], m2[15], m3[16], [17], [18],
[*Vibration Direction*],
// E2 E2 E1 2B1 A1
[x], m2[y], [x], m(2)[y], [x], [y], m(2)[z], m(3)[z],
// E1 E2 E2 A1 2B1
[x], [y], [x], m(2)[y], [x], m(2)[y], m(3)[z], m(2)[z],
[*Representation #linebreak() in Group C#sub[6v]*],
m(3, E2), m(3, E2), m(2, E1), B1, B1, m(3, A1), m(2, E1), m(3, E2), m(3, E2), m(3, A1), B1, B1,
[*Raman-active or Not*],
m(8)[Raman-active], m(2)[Raman-inactive], m(14)[Raman-active], m(2)[Raman-inactive],
// [*Representation in Group C#sub[2v]*],
// // E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
// A2, m(2, A1), A2, m(2, A1), B2, B1, B1, B1, m(3, A1), B2, B1, A2, m(2, A1), A2, m(2, A1), m(3, A1), B1, B1,
[*Scattering in Polarization #linebreak() (non-zero Raman #linebreak() tenser components)*],
// E2 E2 E1 2B1 A1
[xy], [xx], [yy], [xy], [xx], [yy], [xz], [yz], m(2)[-], [xx], [yy], [zz],
// E1 E2 E2 A1 2B1
[xz], [yz], [xy], [xx], [yy], [xy], [xx], [yy], [xx], [yy], [zz], m(2)[-],
[*Raman Intensity (a.u.)*],
// E2 E2 E1 2B1 A1
m(3)[0.17], m(3)[1.13], m(2)[2.43], m(2)[0], m(2)[2.83], [1.79],
// E1 E2 E2 A1 2B1
m(2)[0.09], m(3)[88.54], m(3)[0.50], m(2)[0.01], [1.78], m(2)[0],
[*Visible in Common #linebreak() Raman Experiment or Not*],
// E2 E2 E1 2B1 A1
m(8)[Visible], m(2)[-], m(3)[Visible],
// E1 E2 E2 A1 2B1
m(2)[Invisible], m(3)[Visible], m(5)[Invisible], [Visible], m(2)[-],
[*Wavenumber #linebreak() (Simulation) (cm#super[-1])*],
// E2 E2 E1 2B1 A1
m(3)[190.51], m(3)[197.84], m(2)[257.35], [389.96], [397.49], m(3)[591.90],
// E1 E2 E2 A1 2B1
m(2)[746.91], m(3)[756.25], m(3)[764.33], m(3)[812.87], [885.68], [894.13],
[*Wavenumber #linebreak() (Experiment) (cm#super[-1])*],
// E2 E2 E1 2B1 A1
m(3)[195.5], m(3)[203.3], m(2)[269.7], m(2)[-], m(3)[609.5],
// E1 E2 E2 A1 2B1
m(2)[-], m(3)[776], m(5)[-], [839], m(2)[-],
[*FWHM #linebreak() (Simulation) (cm#super[-1])*],
// E2 E2 E1 2B1 A1
m(3)[0.08], m(3)[0.09], m(2)[0.08], m(2)[-], m(3)[0.61],
// E1 E2 E2 A1 2B1
m(2)[3.97], m(3)[4.62], m(3)[4.01], m(3)[0.89], m(2)[-],
// TODO: 怎么选取用于比较的合适的实验？
[*FWHM #linebreak() (Experiment, zxxz) (cm#super[-1])*],
// E2 E2 E1 2B1 A1
m(3)[2.61], m(3)[2.09], m(2)[1.98], m(2)[-], m(3)[2.64],
// E1 E2 E2 A1 2B1
m(2)[-], m(3)[3.27], m(3)[-], m(3)[-], m(2)[-],
[*Electrical Polarity*],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
m(6)[None], m(2)[Weak], m(2)[None], m(5)[Weak], m(6)[None], m(3)[Weak], m(2)[None],
)},
caption: [Negaligible-polarized Phonons at $Gamma$ Point],
)<table-nopol>]
#figure(
image("/画图/拉曼整体图/main.svg"),
caption: [
(a) Phonon dispersion of 4H-SiC along the A–#sym.Gamma–K high-symmetry path.
Gray lines represent negligible-polar phonon modes,
while colored lines indicate strong-polar phonon modes.
(b) Magnified view of the boxed region in (a).
The orange dashed lines mark the phonon wavevectors involved in Raman scattering
with incident light along the z- and y-directions.
]
)<raman>
// TODO: 画一个模拟的图，与实验图对比。
// 实验与计算基本相符。对于声子频率，计算总是低估大约 3%。
// 此外，一些较强的模式在预测无法看到的偏振中也可以看到。例如，一些在 xy 偏振中不应该看到的模式可以被看到了。
// 这个现象可以由 4度的斜切所解释：我们将材料略微踮起一些角度，就可以使得该模式减小。
// 这个现象也可以由材料或偏振片的微小角度来解释。
// 例如，我们将偏振方向转动 5 度，就可以得到这个模拟结果。
// 此外，由于使用的材料是沿着 c 轴切片的，所以我们在测量 y 入射时不得不将片子以略小于 90 度（约 75 度）的角度放置。这也导致实验与计算的偏差。
// TODO: 翻译成英文
#include "section/perfect/non-polar/default.typ"
=== Strong-polar Phonons

48
test-typst/section/introduction.typ Normal file
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@@ -0,0 +1,48 @@
// SiC 是很好的材料。
// 其中，4H-SiC 是SiC的一种多型，它的性质更好，近年来随着外延工艺的成熟而获得了更多的关注。
SiC is a promising wide-bandgap semiconductor material
with high critical electric field strength and high thermal conductivity.
It has been widely used in power electronic devices and has long attracted a lot of research
@casady_status_1996 @okumura_present_2006.
The 4H-SiC has a wider bandgap, higher critical electric field strength,
higher thermal conductivity, and higher electron mobility along the c-axis than other polytypes.
Currently, the 4H-SiC has gradually received more attention than other polytypes,
thanks to the development of epitaxy technology and the increasing application in the new energy industry
@tsuchida_recent_2018 @harada_suppression_2022 @sun_selection_2022. // TODO: 多引用一些近年来的文献，有很多
// 声子（量子化的原子振动）在理解晶体的原子结构以及热电性质方面起着重要作用。
// 声子可以通过多种实验技术来探测，包括 EELS、IR 吸收谱等。
// 拉曼光谱是最常用的方法，它提供了一种无损、非接触、快速和局部的声子测量方法，已被广泛用于确定晶体的原子结构（包括区分 SiC 的多型）。
Phonons (quantized atomic vibrations) play a fundamental role
in understanding the atomic structure
as well as the thermal and electrical properties
of crystals (including 4H-SiC).
They could be probed by various experimental techniques,
such as electron energy loss spectroscopy and infrared absorption spectroscopy.
Among these techniques,
Raman spectroscopy is the most commonly used method,
as it provides non-destructive, non-contact, rapid and spatially localized measurement of phonons
that near the #sym.Gamma point in reciprocal space.
Studies in Raman scattering of 4H-SiC have been conducted since as early as 1983
and have been widely employed to identification of different SiC polytypes.
// TODO: 增加引用文献
// 近年来，更多信息被从拉曼光谱中挖掘出来。
// LOPC 已经被用于快速估计 n 型 SiC 的掺杂浓度。
// 层错的拉曼光谱也已经被研究，可以被用于检测特定结构层错的存在和位置。
// 掺杂对拉曼光谱的潜在影响也已经被研究。
// 然而，拉曼光谱上仍有一些不知来源的峰；同时，一些也缺少一些理论上预测应该存在的峰。
// 此外，预测掺杂导致的新峰也没有说明原因。
Increasingly rich information has been extracted from Raman spectra of 4H-SiC.
Longitudinal optical phonon–plasmon coupling (LOPC) peek
has been utilized to rapidly estimate the doping concentration in n-type SiC.
Peeks associated with some stacking faults have also been investigated
and used to detect the presence and location of specific structural faults.
Moreover, the potential effects of doping on Raman spectra have been explored.
However, some unidentified peaks still appear in the Raman spectra,
while certain phonon modes predicted by theory remain unobserved.
In addition, the origins of newly emerged peaks induced by doping are often unclear or unexplained.
// TODO: 多举例，增加引用文献
In this paper, we do some things. Especially we do something for the first time.
// TODO: 完善

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#figure({
set text(size: 9pt);
table(columns: 4, align: center + horizon,
table.cell(colspan: 2)[], table.cell(colspan: 2)[*BEC* (unit: |e|)],
table.cell(colspan: 2)[], [x / y direction], [z direction],
table.cell(rowspan: 2)[Si atom], [A/C layer], [2.667], [2.626],
[B layer], [2.674], [2.903],
table.cell(rowspan: 2)[C atom], [A/C layer], [-2.693], [-2.730],
[B layer], [-2.648], [-2.800],
)},
caption: [
Born effective charges of Si and C atoms in A/B/C/B layers of 4H-SiC, calculated using first principle method.
],
placement: none,
)<table-bec>

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(There are 21 phonons in total.
We classified them into two categories: 18 negligible-polar phonons and 3 strong-polar phonons.)
// 拉曼活性的声子模式对应于 Gamma 点附近的声子模式。
// 根据这些声子模式的极性，我们将这些声子分成两类。
The phonons involved in Raman scattering are located in reciprocal space around the #sym.Gamma point,
at the exact positions are determined by the wavevectors of the incident and scattered light.
At each such position, there are 21 phonon modes (degenerate modes are counted as their multiplicity).
We classify these 21 phonons into two categories based on their polarities.
The 18 of 21 phonons are classified into negligible-polar phonons (i.e., phonons with zero or very weak polarity),
for which the effect of polarity can be ignored in the Raman scattering process;
and the other three phonons are strong-polar phonons,
where the polarity gives rise to observable effects in the Raman spectra.
(This classification make sense.)
This classification is based on the fact that
the four Si atoms in the primitive cell of 4H-SiC carry similar positive Born effective charges (BECs),
and the four C atoms carry similar negative BECs (see @table-bec).
In the 18 negligible-polar phonons,
the vibrations of two Si atoms are approximately opposite to those of the other two Si atoms,
and the same holds for the C atoms,
leading to cancellations of macroscopic polarity.
In contrast, in the three strong-polar phonons,
all Si atoms vibrate in the same direction, and all the C atoms vibrate in the opposite direction,
resulting in a strong dipole moment.
#include "bec.typ"

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// We investigate phonons at Gamma instead of the exact location near Gamma.
Phonons at the #sym.Gamma point were used
to approximate negligible-polar phonons that participating in Raman processes of any incident/scattered light.
This approximation is widely adopted and justified by the fact that, // TODO: cite
although the phonons participating in Raman processes are not these strictly located at the #sym.Gamma point,
they are very close to the #sym.Gamma point in reciprocal space
(about 0.01 nm#super[-1] in back-scattering configurations with 532 nm laser light,
which corresponds to only 1% of the smallest reciprocal lattice vector of 4H-SiC,
see orange dotted line in @figure-discont),
and their dispersion at #sym.Gamma point is continuous with vanishing derivatives.
Therefore, negligible-polar phonons involved in Raman processes
have nearly indistinguishable properties from those at the #sym.Gamma point,
and the phonon participating in Raman processes of different incident/scattered light directions
are all nearly identical to the phonons at the #sym.Gamma point.
#include "discont.typ"
// Representation of these 18 phonons, and the shape of their Raman tensors could be determined in advance.)
Phonons at the #sym.Gamma point satisfy the C#sub[6v] point group symmetry,
and the 18 negligible-polar phonons correspond to 12 irreducible representations of the C#sub[6v] point group:
2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2].
Phonons belonging to the A#sub[1] and B#sub[1] representations vibrate along the z-axis and are non-degenerate,
while those belonging to the E#sub[1] and E#sub[2] representations vibrate in-plane and are doubly degenerate.
Phonons of the B#sub[1] representation are Raman-inactive, as their Raman tensors vanish.
In contrast, phonons of the other representations are Raman-active,
and the non-zero components of their Raman tensor
can be determined by further considering their representation in the C#sub[2v] point group (see @table-rep).
These Raman-active phonons are potentially be visible in Raman experiment under appropriate polarization configurations.
However, whether a mode is sufficiently strong to be experimentally visible
depends on the magnitudes of its Raman tensor components,
which cannot be determined solely from symmetry analysis.
#include "rep.typ"
// We propose a method to estimate the magnitudes of the Raman tensors of these phonons,
// without first-principle calculations.
// Here we only write out results, details are in appendix.
// TODO: maybe it is better to assign Raman tensor to each bond, instead of atom
We propose a method to estimate the magnitudes of the Raman tensors of these phonons by symmetry analysis.
The method only takes the vibration directions of each atom in each phonon mode,
leaving the amplitudes unconsidered (see appendix for details),
and the result was summarized in @table-predmode.
In the Raman tensors in @table-predmode,
$a_i$ corresponding to the change of polarizability caused by movement of the Si atoms in A and C layers,
$epsilon_i$, $eta_i$ and $eta_i$ corresponding to the difference between different bilayers and different atoms.
Due to the similarity of environment in different bilayers and around different atoms,
the absolute values of $epsilon_i$, $eta_i$ and $zeta_i$ are expected to be much smaller than that of $a_i$,
thus the Raman tensors containing $a_i$ are expected to be much larger than those not containing $a_i$.
#include "predmode.typ"
The Raman tensors and frequencies of the negligible-polar phonons were calculated using first-principles methods,
and the results are compared with experiment and theory (@table-nopol).
Calculated frequencies of these phonons are consistent with the experimental results
with a low-estimated error of about 2% to 5%, which might be due to the PBE functional used in the calculation (cite).
The Raman tensors of these phonons are also consistent with the experimental and theoretical results,
where E#sub[2] mode experimentally at 776 is the most intense phonon mode,
followed by four modes with lesser intensities
(E#sub[2] modes at 195.5 and 203.3, E#sub[1] mode at 269.7, A#sub[1] mode at 609.5).
The Raman scatter of the E#sub[1] mode calculately at 746.91 and E#sub[2] mode calculately at 756.25
are much weaker than the E#sub[2] mode calculated at 756.25 but located near it, according to our calculation,
thus it could not be distinguished from E#sub[2] mode calculated at 756.25,
which explains why they are not observed in experiments.
Moreover, the A#sub[1] mode calculated at 812.87
have a very weak Raman intensity in the basal plane (xx and yy, only 0.01)
but an observable intensity in the zz configuration (1.78).
Thus, this mode could not be observed in most Raman experiments (cite),
but could be observable when incident light propagate not along the z-direction (our experiment),
or the incident light wavelength is near the resonance condition (cite).
Besides, there are other peeks in the experiment.
The peek at 796 and 980 are caused by strong-polar phonons which will be discussed later.
Besides, there are small peeks at xxx,
which could not be explained in perfect 4H-SiC and will be discussed in the next section.
// TODO: 将一部分 phonons 改为 phonon modes
// 在论文中我们这样来称呼：phonon 对应某一个特征向量，而 modes 对应于一个子空间。
// 也就是说，简并的里面有两个或者无数个 phonon，但只有一个 mode
#include "nopol.typ"
#include "raman.typ"
// TODO: 画一个模拟的图，与实验图对比。
// 实验与计算基本相符。对于声子频率，计算总是低估大约 3%。
// 此外，一些较强的模式在预测无法看到的偏振中也可以看到。例如，一些在 xy 偏振中不应该看到的模式可以被看到了。
// 这个现象可以由 4度的斜切所解释：我们将材料略微踮起一些角度，就可以使得该模式减小。
// 这个现象也可以由材料或偏振片的微小角度来解释。
// 例如，我们将偏振方向转动 5 度，就可以得到这个模拟结果。
// 此外，由于使用的材料是沿着 c 轴切片的，所以我们在测量 y 入射时不得不将片子以略小于 90 度（约 75 度）的角度放置。这也导致实验与计算的偏差。
// TODO: 翻译成英文

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#figure(
image("/画图/声子不连续/embed.svg"),
caption: [
(a) Phonon dispersion of 4H-SiC along the A–#sym.Gamma–K high-symmetry path.
Gray lines represent negligible-polar phonon modes,
while colored lines indicate strong-polar phonon modes.
The green, red and blue lines indicate the mode along the z-direction, y-direction and x-direction, respectively.
Along A-#sym.Gamma path, strong-polar modes along x- and y-directions are degenerated,
showing as a single purple line.
(b) Magnified view of the boxed region in (a).
The orange dashed lines mark the phonon wavevectors involved in Raman scattering
with incident light along the z- and y-directions.
],
placement: none,
)<figure-discont>

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#page(flipped: true)[#figure({
let m(n, content) = table.cell(colspan: n, content);
let m2(content) = table.cell(colspan: 2, content);
let m3(content) = table.cell(colspan: 3, content);
let A1 = [A#sub[1]];
// let A2 = [A#sub[2]];
let B1 = [B#sub[1]];
// let B2 = [B#sub[2]];
let E1 = [E#sub[1]];
let E2 = [E#sub[2]];
set text(size: 9pt);
set par(justify: false);
table(columns: 27, align: center + horizon, inset: (x: 3pt, y: 5pt),
// [*Direction of Incident & Scattered Light*],
// m(26)[Any direction (not depend on direction of incident & scattered light)],
// TODO: 整理表格，使用 m2 m3 来代替
[*Number of Phonon*],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
[1], m2[2], [3], m2[4], [5], [6], [7], [8], m3[9], [10], [11], [12], m2[13], [14], m2[15], m3[16], [17], [18],
[*Vibration Direction*],
// E2 E2 E1 2B1 A1
[x], m2[y], [x], m(2)[y], [x], [y], m(2)[z], m(3)[z],
// E1 E2 E2 A1 2B1
[x], [y], [x], m(2)[y], [x], m(2)[y], m(3)[z], m(2)[z],
[*Representation #linebreak() in Group C#sub[6v]*],
m(3, E2), m(3, E2), m(2, E1), B1, B1, m(3, A1), m(2, E1), m(3, E2), m(3, E2), m(3, A1), B1, B1,
[*Raman-active or Not*],
m(8)[Raman-active], m(2)[Raman-inactive], m(14)[Raman-active], m(2)[Raman-inactive],
// [*Representation in Group C#sub[2v]*],
// // E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
// A2, m(2, A1), A2, m(2, A1), B2, B1, B1, B1, m(3, A1), B2, B1, A2, m(2, A1), A2, m(2, A1), m(3, A1), B1, B1,
[*Scattering in Polarization #linebreak() (non-zero Raman #linebreak() tenser components)*],
// E2 E2 E1 2B1 A1
[xy], [xx], [yy], [xy], [xx], [yy], [xz], [yz], m(2)[-], [xx], [yy], [zz],
// E1 E2 E2 A1 2B1
[xz], [yz], [xy], [xx], [yy], [xy], [xx], [yy], [xx], [yy], [zz], m(2)[-],
[*Raman Intensity (a.u.)*],
// E2 E2 E1 2B1 A1
m(3)[0.17], m(3)[1.13], m(2)[2.43], m(2)[0], m(2)[2.83], [1.79],
// E1 E2 E2 A1 2B1
m(2)[0.09], m(3)[88.54], m(3)[0.50], m(2)[0.01], [1.78], m(2)[0],
[*Visible in Common #linebreak() Raman Experiment or Not*],
// E2 E2 E1 2B1 A1
m(8)[Visible], m(2)[-], m(3)[Visible],
// E1 E2 E2 A1 2B1
m(2)[Invisible], m(3)[Visible], m(5)[Invisible], [Visible], m(2)[-],
[*Wavenumber #linebreak() (Simulation) (cm#super[-1])*],
// E2 E2 E1 2B1 A1
m(3)[190.51], m(3)[197.84], m(2)[257.35], [389.96], [397.49], m(3)[591.90],
// E1 E2 E2 A1 2B1
m(2)[746.91], m(3)[756.25], m(3)[764.33], m(3)[812.87], [885.68], [894.13],
[*Wavenumber #linebreak() (Experiment) (cm#super[-1])*],
// E2 E2 E1 2B1 A1
m(3)[195.5], m(3)[203.3], m(2)[269.7], m(2)[-], m(3)[609.5],
// E1 E2 E2 A1 2B1
m(2)[-], m(3)[776], m(5)[-], [839], m(2)[-],
[*FWHM #linebreak() (Simulation) (cm#super[-1])*],
// E2 E2 E1 2B1 A1
m(3)[0.08], m(3)[0.09], m(2)[0.08], m(2)[-], m(3)[0.61],
// E1 E2 E2 A1 2B1
m(2)[3.97], m(3)[4.62], m(3)[4.01], m(3)[0.89], m(2)[-],
// TODO: 怎么选取用于比较的合适的实验？
[*FWHM #linebreak() (Experiment, zxxz) (cm#super[-1])*],
// E2 E2 E1 2B1 A1
m(3)[2.61], m(3)[2.09], m(2)[1.98], m(2)[-], m(3)[2.64],
// E1 E2 E2 A1 2B1
m(2)[-], m(3)[3.27], m(3)[-], m(3)[-], m(2)[-],
[*Electrical Polarity*],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
m(6)[None], m(2)[Weak], m(2)[None], m(5)[Weak], m(6)[None], m(3)[Weak], m(2)[None],
)},
caption: [Negaligible-polarized Phonons at $Gamma$ Point],
)<table-nopol>]

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// Raman Tensor for A1: line1 xx/yy; line2 zz
// Raman Tensor for E1: x-dirc xz or y-dirc yx
// Raman Tensor for E2: x-dirc xy or y-dirc xx or y-dirc -yy
// TODO: remove LO TO or not?
#page(flipped: true)[#figure({
let m(n, content) = table.cell(colspan: n, content);
let m2(content) = table.cell(colspan: 2, content);
let m3(content) = table.cell(colspan: 3, content);
let m4(content) = table.cell(colspan: 4, content);
set text(size: 9pt);
set par(justify: false);
table(columns: 11, align: center + horizon, inset: (x: 3pt, y: 5pt),
[*Representation in C#sub[6v]*], m3[A#sub[1]], m3[E#sub[1]], m4[E#sub[2]],
[*Relative Vibration Direction*],
[Si: $+-+-$ #linebreak() C: $0000$], [Si: $0000$ #linebreak() C: $+-+-$], [Si: $++++$ #linebreak() C: $----$],
[Si: $+-+-$ #linebreak() C: $-+-+$], [Si: $+-+-$ #linebreak() C: $+-+-$], [Si: $++++$ #linebreak() C: $----$],
[Si: $++--$ #linebreak() C: $-++-$], [Si: $+--+$ #linebreak() C: $++--$],
[Si: $++--$ #linebreak() C: $+--+$], [Si: $+--+$ #linebreak() C: $--++$],
[*Vibration Direction*], m3[z], m3[x/y], m4[x/y],
[*Raman Tensor Predicted*], [xx/yy: $-2A_#text[Si] epsilon_5$ #linebreak() zz: $-2A_#text[Si]epsilon_6$],
[xx/yy: $-2A_#text[C]zeta_5$ #linebreak() zz: $-A_#text[C]zeta_6$],
[xx/yy: $2A_#text[Si] (2a_5+epsilon_5) + 2A_#text[C] (2a_5+eta_5+zeta_5)$ #linebreak() zz: $2A_#text[Si] (2a_6+epsilon_6) + 2A_#text[C] (2a_6+eta_6+zeta_6)$],
[xz/yz: $-2A_#text[Si]epsilon_1-2A_#text[C]zeta_1$],
[xz/yz: $-2A_#text[Si]epsilon_1+2A_#text[C]zeta_1$],
[xz/yz: $2A_#text[Si] (2a_1+epsilon_1) +2A_#text[C] (2a_1+2eta_1+zeta_1))$],
[xx/-yy/xy: $2A_#text[Si] (2a_2+epsilon_2) -2A_#text[C] (2a_2+2eta_2+zeta_2))$],
[xx/-yy/xy: $-2A_#text[Si]epsilon_2-2A_#text[C]zeta_2$],
[xx/-yy/xy: $2A_#text[Si] (2a_2+epsilon_2) +2A_#text[C] (2a_2+2eta_2+zeta_2))$],
[xx/-yy/xy: $-2A_#text[Si]epsilon_2+2A_#text[C]zeta_2$],
[*Raman Intensity Predicted*], m2[weak], [strong], m2[weak], [strong], m2[weak], [strong], [weak],
[*Raman Tensor Calculated*],
[-1.68 #linebreak() 1.34], [0.10 #linebreak() -1.33], [-7.68 #linebreak() 21.65],
[-1.56], [-0.30], [7.32], [-0.41], [1.06], [9.41], [-0.71],
// [*x*], [1 axial acoustic], [0 axial optical], [1 axial optical],
// [0 axial acoustic], [1 axial optical], [1 axial optical],
// m2[0.5 acoustic], m2[0.5 optical],
[*Type*], [axial acoustic], [axial optical], [longitudinal optical],
[planer acoustic], [planer optical], [transverse optical],
m2[planer acoustic], m2[planer optical],
[*Move-towards Atom-pairs* (In-plane/Out-plane)], [4/0], [0/4], [4/4], [0/4], [4/0], [4/4], [0/2], [2/0], m2[4/2],
// [*Predicted Frequency*], [low], [medium], [high], [medium], [low], [high], [low], [medium], m2[high],
[*Calculated Frequency*],
[591.90], [812.87], [933.80], [257.35], [746.91], [776.57], [190.51], [197.84], [756.25], [764.33]
)},
caption: [Predicted modes and their "Raman tensor"],
placement: none,
)<table-predmode>]

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#figure(
image("/画图/拉曼整体图/main.svg"),
caption: [
(a) Phonon dispersion of 4H-SiC along the A–#sym.Gamma–K high-symmetry path.
Gray lines represent negligible-polar phonon modes,
while colored lines indicate strong-polar phonon modes.
(b) Magnified view of the boxed region in (a).
The orange dashed lines mark the phonon wavevectors involved in Raman scattering
with incident light along the z- and y-directions.
]
)<raman>

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#figure({
let m2(content) = table.cell(colspan: 2, content);
set text(size: 9pt);
table(columns: 6, align: center + horizon, inset: (x: 3pt, y: 5pt),
[*Representations in C#sub[6v]*], [A#sub[1]], m2[E#sub[1]], m2[E#sub[2]],
[*Representations in C#sub[2v]*], [A#sub[1]], [B#sub[2]], [B#sub[1]], [A#sub[2]], [A#sub[1]],
[*Vibration Direction*], [z], [x], [y], [x], [y],
[*Raman Tensor of #linebreak() Individual Phonons*],
[$mat(a,,;,a,;,,b)$], [$mat(,,a;,,;a,,;)$], [$mat(,,;,,a;,a,;)$], [$mat(,a,;a,,;,,;)$], [$mat(a,,;,-a,;,,;)$],
[*Raman Intensity with Different #linebreak() Polarization Configurations*],
[xx/yy: $a^2$ #linebreak() zz: $b^2$ #linebreak() others: 0],
m2[xz/yz: $a^2$ #linebreak() others: 0], m2[xx/xy/yy: $a^2$ #linebreak() others: 0],
)},
caption: [
Raman-active representations of C#sub[6v] and C#sub[2v] point groups.
],
placement: none,
)<table-rep>