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陈浩南
2025-05-18 15:50:14 +08:00
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@@ -1,10 +1,12 @@
#import "@preview/starter-journal-article:0.4.0": article, author-meta
#import "@preview/tablem:0.2.0": tablem
#import "@preview/physica:0.9.4": pdv, super-T-as-transpose
#import "@preview/physica:0.9.5": pdv, super-T-as-transpose
#show: super-T-as-transpose
#set par.line(numbering: "1")
#set par(justify: true)
// 思源宋体，也算是宋体吧
#set text(font: ("Times New Roman", "Source Han Serif SC"))
// TODO: fix indent of first line
#show figure.caption: it => {
set text(10pt)
@@ -112,16 +114,40 @@ experiment
= Results and Discussion
- 无缺陷：
我们将声子分为两类，一类是极性比较弱的（18个），一类是比较强的（3个）。
- 弱极性的声子：
- 使用 Gamma 点的声子模式来近似。根据对称性可以明确预测它们的拉曼张量。
- 我们提出了一个方法，直接根据对称性来估计声子模式的拉曼张量，或者反过来，估计拉曼光谱中峰对应的原子振动模式。估计的结果大多数是正确的。、
- TODO: 可能换成使用原子对（键）来估计，要比使用原子来估计要更合理。
- TODO: 描述估计频率的方法，以及确认 x 的定义的正确性。
- 我们使用第一性原理计算了各种性质，它与实验、预测相符。
- TODO: 将峰宽列出来，将模拟图画出来。
- 某某峰 is reported 在某人的实验中可以看到而在某人的实验中看不到。我们 propose 它的确存在，但只能通过共振拉曼或者zz偏振才能看到。
- TODO: 引用文献。
- TODO: 确认一下最后一次实验中，峰偏移等是否与掺杂有明显关系，以及这个关系与之前是否相同。
- 强极性的声子：
- 强极性声子在 Gamma 附近散射谱不连续，它的声子模式由入射光的方向决定。在入射光不沿 z 轴的情况下，使用 C6v 群不再适用。
- TODO: 写文字
- 在接近 y 轴入射时，可以看到分裂。这个模式可能对表面敏感。
- TODO: 佐证它对表面敏感
- 对于 LO，可能形成 LOPC
- 有缺陷的情况：
- TODO: 描述缺陷原子的振动
- TODO: 计算拉曼张量，描述光谱的可能变化
== 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
at positions determined by the difference between the wavevectors of the incident and scattered light.
At each such position, there are 21 phonon modes (excluding translational modes).
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;
@@ -135,13 +161,14 @@ This classification is based on the fact that
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,
so do C atoms,
and the same holds for the C atoms,
leading to cancellations of macroscopic polarity.
While in the three strong-polar phonons,
all Si atoms vibrate in the same direction, so do C atoms,
leading to a net dipole moment.
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(
#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],
@@ -149,7 +176,7 @@ While in the three strong-polar phonons,
[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.
],
@@ -158,16 +185,19 @@ While in the three strong-polar phonons,
=== 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,
dispersion of negligible-polar phonons near the #sym.Gamma point is continuous with vanishing derivatives,
and their wavevector is very small (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),
as shown by the orange dotted line in @figure-discont.
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
@@ -189,17 +219,12 @@ Therefore, negligible-polar phonons involved in Raman processes
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 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 tensors
can be determined by further considering the C#sub[2v] point group (see @table-rep).
These Raman-active phonons may appear in Raman spectra under appropriate polarization configurations.
However, the actual visibility of each mode depends on the magnitude of its Raman tensor components,
which cannot be determined solely from symmetry analysis.
The 18 negligible-polar phonons correspond to 12 irreducible representations of the C#sub[6v] point group:
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.
@@ -207,16 +232,17 @@ Phonons of the B#sub[1] representation are Raman-inactive, as their Raman tensor
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 might be visible in Raman experiment under appropriate polarization configurations.
However, whethear a mode is sufficiently strong to be experimentally visible
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 C6v*], [A#sub[1]], m2[E#sub[1]], m2[E#sub[2]],
[*Representations in C2v*], [A#sub[1]], [B#sub[2]], [B#sub[1]], [A#sub[2]], [A#sub[1]],
[*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*],
[$mat(a,,;,a,;,,b)$], [$mat(,,a;,,;a,,;)$], [$mat(,,;,,a;,a,;)$], [$mat(,a,;a,,;,,;)$], [$mat(a,,;,-a,;,,;)$],
@@ -230,13 +256,244 @@ However, whethear a mode is sufficiently strong to be experimentally visible
placement: none,
)<table-rep>
(We propose a method to estimate the magnitudes of the Raman tensors of these phonons.
Here we write out its main steps, details are in appendix.)
#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 by symmetry analysis (see appendix for details).
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,
$epsilon_i$ and $eta_i$ corresponding to the difference between the A/C layers and B layers,
and $eta_i$ corresponding the difference between the Si and C atoms.
The absolute values of $a_i$ is expected to be much larger than that of $epsilon_i$, $eta_i$ and $zeta_i$,
thus the Raman tensors containing $a_i$ are expected to be much larger than those not containing $a_i$.
It could be seen that,
our prediction is mostly consistent with the first principle calculation and experiment.
// Raman Tensor for A1: line1 xz/yz; 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
// Relative Vibration Direction: col1 C ABCB col2 Si ABCB
#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);
let A1 = [A#sub[1]];
let E1 = [E#sub[1]];
let E2 = [E#sub[2]];
set text(size: 9pt);
set par(justify: false)
table(columns: 9, align: center + horizon, inset: (x: 2pt, y: 5pt),
// TODO: explain where x comes from
// TODO: 校验这个表的数据（确认没有标错列、没有标错正负号）
[*$x$*], m2[0.25], m4[0.5], m2[0.75],
[*Representation in C#sub[6v]*], m2(E2), E1, A1, E1, A1, m2(E2),
table.cell(rowspan: 2, [*Vibration Direction* (ABCB layer)]), m2[x/y], [x/y], [z], [x/y], [z], m2[x/y],
[Si: $+--+$ #linebreak() C: $++--$], [Si: $+--+$ #linebreak() C: $+--+$],
m2[Si: $+-+-$ #linebreak() C: $+-+-$], m2[Si: $-+-+$ #linebreak() C: $+-+-$],
[Si: $++--$ #linebreak() C: $+--+$], [Si: $-++-$ #linebreak() C: $++--$],
[*Raman Tensor Predicted*],
[$-2(zeta_2+epsilon_2)$], [$2(2eta_2+zeta_2-epsilon_2)$],
[$2(zeta_1-epsilon_1)$], [$2(zeta_5-epsilon_5)$ #linebreak() $2(zeta_6-epsilon_6)$],
[$2(epsilon_1+zeta_1)$], [$2(epsilon_5+zeta_5)$ #linebreak() $2(epsilon_6+zeta_6)$],
[$2(4a_2+2eta_2+zeta_2+epsilon_2)$], [$2(epsilon_2-zeta_2)$],
[*Raman Intensity Predicted*], m2[weak], m4[weak], [strong], [weak],
[*Raman Tensor Calculated*],
[1.06], [0.41], [-1.56], [0.10 #linebreak() -1.33], [-0.30], [-1.68 #linebreak() 1.34], [9.41], [0.17],
[*Move-towards Atom-pairs* (In-plane/Out-plane)], [0/2], [2/0], m2[4/0], m2[0/4], m2[4/2],
[*Predicted Frequency*], [low], [medium], [medium], [low], [low], [medium], m2[high],
[*Calculated Frequency*],
[197.84], [190.51], [746.91], [591.90], [257.35], [812.87], [756.25], [764.33]
)},
caption: [Predicted modes and their "Raman tensor"],
placement: none,
)<table-predmode>]
as well as the frequencies
,
except for the E#sub[2] mode experimentally at 200 cm#super[-1], which we expected to be at higher frequencies.
// 我们计算了拉曼活性声子的频率及拉曼张量，并与实验对比，如表如图所示。
// 其中有几个声子的拉曼活性较弱，有几个比较强。强的都可以在实验上看到；但弱的能否看到则取决于它是否恰好位于强模式的附近。
// 其中，xxx 和xxx 位于强模式的附近，它们在实验上无法看到；xxx 只在 z 方向入射/散射时可以看到；xxx 则在任意方向都能看到。
// 我们同样计算了这些声子在 300K 下的展宽，并与实验对比，结果如表所示。原子的振幅另外列于附录中。
The Raman tensors of these Raman-active phonons were calculated using first-principles methods,
and the results are summarized and compared with experimental results in @table-nopol.
Two Raman-active modes are not observed in our experiments,
including the E#sub[1] mode at 746.91 cm#super[-1] and the E#sub[2] mode at 764.33 cm#super[-1],
due to their relatively low Raman intensities, broad FWHM values, and their proximity to stronger modes.
The A#sub[1] phonon at 812.87 cm#super[-1] is Raman-active
in both in-plane (xx and xy) and out-of-plane (zz) polarization configurations,
but it is only visible when both the incident and scattered light propagate along the z-direction (zz),
as its Raman intensity in basal plane is too week to be distinguished from the noise.
We also calculated the linewidths of these phonons at 300 K and compared them with experimental results,
as summarized in the @table-nopol.
The atomic vibration amplitudes are listed separately in the Appendix.
// 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]];
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)[-],
[*FWHM #linebreak() (Experiment) (cm#super[-1])*],
// E2 E2 E1 2B1 A1
m(3)[1.11], m(3)[1.11], m(2)[1.11], m(2)[-], m(3)[591.90],
// E1 E2 E2 A1 2B1
m(2)[-], m(3)[1.11], m(3)[-], m(3)[1.11], 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
=== Strong-polar Phonons
// 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)
// 极性声子模式还会与载流子发生较强的相互作用。
#page(flipped: true)[
#figure({
// 使用 m2 m3
let m(n, content) = table.cell(colspan: n, 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]];
let NA = [Not Applicable]
let yzmix = [y-z mixed#linebreak() (LO-TO mixed)];
let lopc = [Yes#linebreak() (LOPC)];
let overf = [Yes#linebreak() (overfocused)];
table(columns: 20, align: center + horizon, inset: (x: 3pt, y: 5pt),
[*Direction of Incident & Scattered Light*], m(5)[z], m(5)[y], m(9)[between z and y, 10#sym.degree to z],
// z y 45 y&z
[*Number of Phonon*], [1], [2], m(3)[3], m(3)[1], [2], [3], m(4)[1], [2], m(4)[3],
[*Vibration Direction*],
[x#linebreak() (TO)], [y#linebreak() (TO)], m(3)[z (LO)], // z
m(3)[z (TO)], [x#linebreak() (TO)], [y (LO)], // y
m(4, yzmix), [x#linebreak() (TO)], m(4, yzmix), // 45 y&z
[*Representation in Group C#sub[6v]*], m(2, E1), m(3, A1), m(14, NA),
// z y 45 y&z
[*Representation in Group C#sub[2v]*], B2, B1, m(3, A1), m(3, A1), B2, B1, m(4, NA), B2, m(4, NA),
[*Scattering in Polarization*],
[xz], [yz], [xx], [yy], [zz], // z
[xx], [yy], [zz], [xz], [yz], // y
[xx], [yy], [yz], [zz], [xz], [xx], [yy], [yz], [zz], // 45 y&z
[*Raman Intensity (a.u.)*],
m(2)[53.52], m(2)[58.26], [464.69], // z
m(2)[58.26], [454.09], [53.52], [53.55], // y
m(2)[53.71], [3.20], [425.98], [53.56], m(2)[3.60], [50.36], [27.99], // 45 y&z
[*Visible in Common Raman Experiment*],
m(2)[Yes], m(2, lopc), [No], // z
overf, [No], overf, [Yes], lopc, // y
m(4)[???], [???], m(4)[???], // 45 y&z
[*Wavenumber (Simulation) (cm#super[-1])*],
// z y 45 y&z
m(2)[776.57], m(3)[933.80], m(3)[761.80], [776.57], [941.33], m(4)[762.76], [776.57], m(4)[940.86],
[*Electrical Polarity*], m(19)[Strong]
)},
caption: [Strong-polarized phonons near $Gamma$ point],
)
]
The center principle is to assign the Raman tensor (i.e., change of polarizability caused by atomic displacement)
to each atom in the unit cell.
@@ -425,7 +682,6 @@ Furthermore, we list predicted modes and their Raman tensors, in @table-predmode
- $eta$: Difference of Raman tensors of C and Si atom in A layer, small value.
- $zeta$: Difference of Raman tensors of C atoms in A and B layer, small value.
Frequency could be estimated by, how many atoms are moving towards its neighbor.
#page(flipped: true)[#figure({
table(columns: 4, align: center + horizon, inset: (x: 3pt, y: 5pt),
@@ -453,213 +709,6 @@ Frequency could be estimated by, how many atoms are moving towards its neighbor.
placement: none,
)<table-singleatom>]
// Raman Tensor for A1: line1 xz/yz; 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
// Relative Vibration Direction: col1 C ABCB col2 Si ABCB
#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);
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]],
[*x*], m2[0.5], [1], m2[0.5], [1], m2[0.25], m2[0.75],
[*Relative Vibration Direction*],
[$++\ --\ ++\ --$], [$+-\ -+\ +-\ -+$], [$+-\ +-\ +-\ +-$],
[$++\ --\ ++\ --$], [$+-\ -+\ +-\ -+$], [$+-\ +-\ +-\ +-$],
[$++\ +-\ --\ -+$], [$++\ --\ --\ ++$], [$++\ -+\ --\ +-$], [$+-\ ++\ -+\ --$],
[*Vibration Direction*], m3[z], m3[x/y], m4[x/y],
[*Raman Tensor Predicted*], [$2(zeta_5-epsilon_5)$ #linebreak() $2(zeta_6-epsilon_6)$],
[$2(epsilon_5+zeta_5)$ #linebreak() $2(epsilon_6+zeta_6)$],
[$-4(2a_5+eta_5+epsilon_5+zeta_5)$ #linebreak() $-4(2a_6+eta_6+epsilon_6+zeta_6)$],
[$2(zeta_1-epsilon_1)$], [$2(epsilon_1+zeta_1)$], [$-4(2a_1+eta_1+epsilon_1+zeta_1)$],
[$-2(zeta_2+epsilon_2)$], [$2(2eta_2+zeta_2-epsilon_2)$], [$2(4a_2+2eta_2+zeta_2+epsilon_2)$],
[$2(epsilon_2-zeta_2)$],
[*Raman Intensity Predicted*], m2[weak], [strong], m2[weak], [strong], m2[weak], [strong], [weak],
[*Raman Tensor Calculated*],
[0.10 #linebreak() -1.33], [-1.68 #linebreak() 1.34], [7.68 #linebreak() -21.65],
[-1.56], [-0.30], [7.32], [1.06], [0.41], [9.41], [0.17],
[*Atom-pair that Move Relatively In-plane*], [4], [0], [4], [4], [0], [4], [0], [2], [4], [4],
[*Atom-pair that Move Relatively Out-plane*], [0], [4], [4], [0], [4], [4], [2], [0], [2], [2],
[*Predicted Frequency*], [low], [medium], [high], [medium], [low], [high], [low], [medium], m2[high],
[*Calculated Frequency*],
[591.90], [812.87], [933.80], [746.91], [257.35], [776.57], [197.84], [190.51], [756.25], [764.33]
)},
caption: [Predicted modes and their "Raman tensor"],
placement: none,
)<table-predmode>]
// 我们计算了拉曼活性声子的频率及拉曼张量，并与实验对比，如表如图所示。
// 其中有几个声子的拉曼活性较弱，有几个比较强。强的都可以在实验上看到；但弱的能否看到则取决于它是否恰好位于强模式的附近。
// 其中，xxx 和xxx 位于强模式的附近，它们在实验上无法看到；xxx 只在 z 方向入射/散射时可以看到；xxx 则在任意方向都能看到。
// 我们同样计算了这些声子在 300K 下的展宽，并与实验对比，结果如表所示。原子的振幅另外列于附录中。
The Raman tensors of these Raman-active phonons were calculated using first-principles methods,
and the results are summarized and compared with experimental results in @table-nopol.
Two Raman-active modes are not observed in our experiments,
including the E#sub[1] mode at 746.91 cm#super[-1] and the E#sub[2] mode at 764.33 cm#super[-1],
due to their relatively low Raman intensities, broad FWHM values, and their proximity to stronger modes.
The A#sub[1] phonon at 812.87 cm#super[-1] is Raman-active
in both in-plane (xx and xy) and out-of-plane (zz) polarization configurations,
but it is only visible when both the incident and scattered light propagate along the z-direction (zz),
as its Raman intensity in basal plane is too week to be distinguished from the noise.
We also calculated the linewidths of these phonons at 300 K and compared them with experimental results,
as summarized in the @table-nopol.
The atomic vibration amplitudes are listed separately in the Appendix.
// 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]];
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)[-],
[*FWHM #linebreak() (Experiment) (cm#super[-1])*],
// E2 E2 E1 2B1 A1
m(3)[1.11], m(3)[1.11], m(2)[1.11], m(2)[-], m(3)[591.90],
// E1 E2 E2 A1 2B1
m(2)[-], m(3)[1.11], m(3)[-], m(3)[1.11], 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
=== Strong-polar Phonons
// 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)
#page(flipped: true)[
#figure({
// 使用 m2 m3
let m(n, content) = table.cell(colspan: n, 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]];
let NA = [Not Applicable]
let yzmix = [y-z mixed#linebreak() (LO-TO mixed)];
let lopc = [Yes#linebreak() (LOPC)];
let overf = [Yes#linebreak() (overfocused)];
table(columns: 20, align: center + horizon, inset: (x: 3pt, y: 5pt),
[*Direction of Incident & Scattered Light*], m(5)[z], m(5)[y], m(9)[between z and y, 10#sym.degree to z],
// z y 45 y&z
[*Number of Phonon*], [1], [2], m(3)[3], m(3)[1], [2], [3], m(4)[1], [2], m(4)[3],
[*Vibration Direction*],
[x#linebreak() (TO)], [y#linebreak() (TO)], m(3)[z (LO)], // z
m(3)[z (TO)], [x#linebreak() (TO)], [y (LO)], // y
m(4, yzmix), [x#linebreak() (TO)], m(4, yzmix), // 45 y&z
[*Representation in Group C#sub[6v]*], m(2, E1), m(3, A1), m(14, NA),
// z y 45 y&z
[*Representation in Group C#sub[2v]*], B2, B1, m(3, A1), m(3, A1), B2, B1, m(4, NA), B2, m(4, NA),
[*Scattering in Polarization*],
[xz], [yz], [xx], [yy], [zz], // z
[xx], [yy], [zz], [xz], [yz], // y
[xx], [yy], [yz], [zz], [xz], [xx], [yy], [yz], [zz], // 45 y&z
[*Raman Intensity (a.u.)*],
m(2)[53.52], m(2)[58.26], [464.69], // z
m(2)[58.26], [454.09], [53.52], [53.55], // y
m(2)[53.71], [3.20], [425.98], [53.56], m(2)[3.60], [50.36], [27.99], // 45 y&z
[*Visible in Common Raman Experiment*],
m(2)[Yes], m(2, lopc), [No], // z
overf, [No], overf, [Yes], lopc, // y
m(4)[???], [???], m(4)[???], // 45 y&z
[*Wavenumber (Simulation) (cm#super[-1])*],
// z y 45 y&z
m(2)[776.57], m(3)[933.80], m(3)[761.80], [776.57], [941.33], m(4)[762.76], [776.57], m(4)[940.86],
[*Electrical Polarity*], m(19)[Strong]
)},
caption: [Strong-polarized phonons near $Gamma$ point],
)
]
// TODO: 这句话放哪里？
// whose dispersion curves exhibit discontinuity near the #sym.Gamma point (also shown in @phonon),