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陈浩南
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=== Phonons with Negligible Polarities
我们用 gamma 点的声子来近似弱极性的声子。
// 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
regardless of the wavevector of the incident and scattered light.
This approximation is widely adopted (cite) and justified by the fact that,
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.
#include "figure-discont.typ"
gamma 点处 18 个声子的表示。它们的拉曼张量的形状可以确定，但大小无法确定。
// 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 "table-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.
我们提出了一个新的办法来估计拉曼张量大小。
We propose a method to estimate the magnitudes of the Raman tensors of these phonons based on symmetry analysis.
This approach is founded on the assumption that the change in polarizability induced by atomic displacements in 4H-SiC
is primarily determined by the first- and second-nearest neighbors of the atom and the sign of the atomic charge,
while other factors (mass, bond length, etc.) only have small contributions.
Consequently, the Raman tensors of the calculated phonon modes can be estimated
before additional first-principles computations (see appendix for details),
and the results are summarized in @table-nopol.
The parameters $a_i$ exhibit significantly larger absolute values compared to $epsilon_i$, $eta_i$, and $zeta_i$,
indicating the E#sub[2] mode at 756.25 cm#super[-1] in simulation (mode 8)
possess a much higher Raman intensity than the others.
我们使用第一性原理计算得到了拉曼张量的大小，并与我们的结果进行了比较。
The Raman tensors and frequencies of the negligible-polar phonons were calculated using first-principles methods,
and the results are compared with both experimental data and theoretical predictions (@table-nopol).
The calculated phonon frequencies show good agreement with experimental data with a slight underestimation of 2-5%,
where the error may be attributed to the underestimation of interatomic forces by the PBE functional (cite).
The calculated Raman tensors are also consistent with experimental and theoretical results.
Among negligible-polar modes, the E#sub[2] mode at 776 cm#super[-1] in experiment (mode 8)
exhibits the highest Raman intensity,
followed by four modes with lower intensities that are also experimentally visible,
including the E#sub[2] modes at 195.5 cm#super[-1] (mode 1) and 203.3 cm#super[-1] (mode 2),
the E#sub[1] mode at 269.7 cm#super[-1] (mode 3), and the A#sub[1] mode at 609.5 cm#super[-1] (mode 6).
The E#sub[1] mode calculated at 746.91 cm#super[-1] (mode 7)
and the E#sub[2] mode calculated at 756.25 cm#super[-1] (mode 9)
are predicted to have much weaker Raman intensities and are located close to the most intense mode (mode 8),
making them indistinguishable in experimental spectra.
Additionally, the A#sub[1] mode calculated at 812.87 cm#super[-1] (mode 10)
exhibits a very weak Raman intensity in the basal-plane polarized configurations (xx and yy, only 0.01)
but shows an observable intensity when the polarization is along the z-axis (1.78).
Since most Raman experiments are performed in a back-scattering configuration with light incident along the z-direction
(i.e., with in-plane polarization)
and with photon energies much lower than the band gap,
this mode is typically not observed (cite).
However, it should become detectable if the incident light has a polarization component along the z-direction
(as in our experiment),
or when the excitation wavelength approaches resonance conditions (cite).
其它峰在其它章节中解释。
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 "table-nopol.typ"
#include "figure-raman.typ"
// TODO: 解释为什么 E1 可以看到

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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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#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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#page(flipped: true)[#figure({
set text(size: 9pt);
set par(justify: false);
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: 22, align: center + horizon, inset: (x: 3pt, y: 5pt),
m2[*Number of Mode*],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
m2[1], m2[2], m2[3], [4], [5], m2[6], m2[7], m2[8], m2[9], m2[10], [11], [12],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
m2[*Vibration Direction*], [x], [y], [x], [y], [x], [y], m(4)[z], [x], [y], [x], [y], [x], [y], m(4)[z],
table.cell(rowspan: 2)[*Representation*],
[C#sub[6v]], m2(E2), m2(E2), m2(E1), B1, B1, m2(A1), m2(E1), m2(E2), m2(E2), m2(A1), B1, B1,
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
[C#sub[2v]], A2, A1, A2, A1, B2, B1, B1, B1, m2(A1), B2, B1, A2, A1, A2, A1, m2(A1), B1, B1,
// TODO: 重新检查数据是否正确（主要是正负号）
table.cell(rowspan: 4)[*Raman Tensor*],
[Non-zero components],
// E2 E2 E1 2B1 A1
[xy], [xx, -yy], [xy], [xx, -yy], [xz], [yz], m2[None], [xx, yy], [zz],
// E1 E2 E2 A1 2B1
[xz], [yz], [xy], [xx, -yy], [xy], [xx, -yy], [xx, yy], [zz], m2[None],
[Analysis result (a.u.)],
// E2 E2 E1 2B1
m2[$2epsilon_2-2zeta_2-4eta_2$], m2[$2epsilon_2-2zeta_2$], m2[$-2epsilon_1-2zeta_1$], m2[-],
// A1 E1 E2
[$-2epsilon_5$], [$-2epsilon_6$], m2[$-2epsilon_1+2zeta_1$], m2[$8a_2+2epsilon_2-2zeta_2-4eta_2$],
// E2 A1 2B1
m2[$-2epsilon_2+2zeta_2$], [$-2zeta_5$], [$-2zeta_6$], m2[-],
[Calculation result (a.u.)],
// TODO: 改正正负号
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
m2[0.17], m2[1.13], m2[2.43], m2[-], [2.83], [1.79], m2[0.09], m2[88.54], m2[0.50], [0.01], [1.78], m2[-],
[Experiment result (a.u.)],
// TODO: 填充
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
m2[], m2[], m2[], m2[-], [], [], m2[Invisible], m2[], m3[Invisible], [], m2[-],
table.cell(rowspan: 3)[*Frequency*],
[Simulation (cm#super[-1])],
// E2 E2 E1 2B1 A1
m2[190.51], m2[197.84], m2[257.35], [389.96], [397.49], m2[591.90],
// E1 E2 E2 A1 2B1
m2[746.91], m2[756.25], m2[764.33], m2[812.87], [885.68], [894.13],
[Experiment (cm#super[-1])],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
m2[195.5], m2[203.3], m2[269.7], m2[-], m2[609.5], m2[Invisible], m2[776], m(3)[Invisible], [839], m2[-],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
[Error (%)], m2[2.6], m2[2.7], m2[4.6], m2[-], m2[2.9], m2[-], m2[2.5], m(3)[-], [3.1], m2[-],
table.cell(rowspan: 2)[*FWHM* (cm#super[-1])],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
[Simulation], m2[0.08], m2[0.09], m2[0.08], m2[-], m2[0.61], m2[3.97], m2[4.62], m2[4.01], m2[0.89], m2[-],
// TODO: 选取合适的实验并填充数据
[Experiment, zxxz],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
m2[2.61], m2[2.09], m2[1.98], m2[-], m2[2.64], m2[Invisible], m2[3.27], m3[Invisible], [], m2[-],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
m2[*Polarity*], m(4)[None], m2[Weak], m2[None], m(4)[Weak], m(4)[None], m2[Weak], m2[None],
)},
caption: [Negaligible-polarized Phonons at $Gamma$ Point.],
)<table-nopol>]

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#figure({
set text(size: 9pt);
set par(justify: false);
let m2(content) = table.cell(colspan: 2, 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: 7, align: center + horizon,
[*Representations in C#sub[6v]*], A1, B1, m2(E1), m2(E2),
[*Representations in C#sub[2v]*], A1, B1, B2, B1, A2, A1,
[*Vibration Direction*], [z], [z], [x], [y], [x], [y],
[*Raman Tensor*],
[$mat(a,,;,a,;,,b)$], [$0$], [$mat(,,a;,,;a,,;)$], [$mat(,,;,,a;,a,;)$], [$mat(,a,;a,,;,,;)$], [$mat(a,,;,-a,;,,;)$],
[*Raman scatter Intensity* #linebreak() (polarization of incident and scattered light)],
[xx/yy: $a^2$ #linebreak() zz: $b^2$ #linebreak() others: 0], [0],
m2[xz/yz: $a^2$ #linebreak() others: 0], m2[xx/xy/yy: $a^2$ #linebreak() others: 0],
)},
caption: [
Irreducible representations and raman tensors of phonons in 4H-SiC.
],
placement: none,
)<table-rep>