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陈浩南
2025-05-09 12:47:29 +08:00
parent cd606acf6d
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@@ -157,15 +157,13 @@ In these 18 modes, the vibrations of two Si atoms are approximately opposite to
// 这18个声子对应于 $\mathrm{C_{6v}}$ 点群的 14 个表示：2A1 + 4B1 + 2E_1 + 4E2
// 其中，B1 表示没有拉曼活性，它的拉曼张量为零；其它表示的拉曼张量不为零，但张量的大小是否足够大到可以在实验上看到，则还需要第一性原理计算，不能直接通过表示来判断。
// 我们的计算结果如表所示。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。
The 18 negligible-polar phonons correspond to 14 irreducible representations of the C#sub[6v] point group:
2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2].
Phonons belonging to A#sub[1] and B#sub[1] representations are non-degenerate,
while phonons belonging to E#sub[1] and E#sub[2] representations are doubly degenerate.
Phonons belonging to B#sub[1] representation are Raman inactive, as their Raman tensors vanish.
In contrast, the Raman tensors of phonons belonging to other representations have non-zero components,
Phonons belonging to B#sub[1] representation are Raman-inactive, as their Raman tensors vanish.
In contrast, phonons belonging to other representations are Raman-active,
the Raman tensors of them have non-zero components,
indicating that these phonons might be visible in Raman experiment under appropriate polarization configurations.
However, the actual visibility of each phonon depends on the magnitudes of its Raman tensor components,
which cannot be inferred solely from symmetry analysis.
@@ -187,59 +185,34 @@ However, the actual visibility of each phonon depends on the magnitudes of its R
* 写出各个模式的拉曼张量（上面的线性组合）。即可以直接看到结果。
*/
// 我们计算了这些声子的拉曼张量与线宽，并与实验对比，结果如表所示。
// 不同方向入射/散射光的拉曼实验对应于 Gamma 附近、偏向于不同方向的声子。
// 其中 $\mathrm{A_1}$、$\mathrm{B_1}$ 为一维表示，对应于无简并的声子；$\mathrm{E_1}$、$\mathrm{E_2}$ 为二维表示，对应于二重简并的声子。
// 在拉曼实验中，起作用的声子并不严格在 Gamma 点；但大多数声子的色散谱在 Gamma 点连续且导数（斜率）为零，因此大多情况下可以沿用这个分类，少数情况我们稍后会专门讨论。
// 我们计算了拉曼活性声子的拉曼张量。
// 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 @nopol.
Some Raman-active phonons are not visible in experiments,
including E#sub[1] at ~746.91 cm#super[-1] and E#sub[2] at ~764.33 cm#super[-1],
causing their Raman intensity are relatively low and located close to strong modes.
The A#sub[1] phonon at ~812.87 cm#super[-1] is only visible
when both the incident and scattered light propagate along the z-direction,
since its Raman intensity in basal plane is too week to be recognized from the background.
We also calculated the linewidthes of these phonons at 300 K and compared them with experimental results,
as summarized in the table.
The atomic vibration amplitudes are listed separately in the Appendix.
// 4H-SiC 在 Gamma 点的声子共有 21 个模式，这些模式对应于点群 $\mathrm{C_{6v}}$ 的 14 个表示（$\mathrm{3A_1+4B_1+3E_1+4E_2}$，其中 $\mathrm{A_1}$、$\mathrm{B_1}$ 为一维表示，对应于无简并的声子；$\mathrm{E_1}$、$\mathrm{E_2}$ 为二维表示，对应于二重简并的声子）。在拉曼实验中，起作用的声子并不严格在 Gamma 点；但大多数声子的色散谱在 Gamma 点连续且导数（斜率）为零，因此大多情况下可以沿用这个分类，少数情况我们稍后会专门讨论。
// 我们计算了 4H-SiC 在 A-Gamma 和 Gamma-M 上的声子频率，如图和附录1所示。
// 在拉曼散射中，起作用的模式都是那些非常接近于 Gamma 的模式
// （如图中的点所示，分为位于 1/50 和 1/100 处，这两条线分别对应于拉曼散射在 z 方向入射/散射和 y 方向入射/散射）。
// 大多数声子模式在 Gamma 附近都是连续的，这使得它们的频率对入射光的方向不敏感；
// 然而，少数声子具有较强的极性，这使得声子之间存在长程的库伦相互作用（引用文献），并导致 gamma 附近的频率不同，如图中的某两条线所示。
// 据此，我们将无缺陷的 4H-SiC 的声子分成三类：
// 无拉曼活性或拉曼散射强度太弱的模式，它们在拉曼散射谱上不可见；
// 拉曼散射强度足够大且极性不强的模式，它们在拉曼散射谱上可以看到，且频率与拉曼入射光方向无关；
// 极性声子，它们在拉曼散射谱上可以看到，不仅频率与入射光方向有关，而且可与载流子发生一些相互作用。
// Phonons in defect-free 4H-SiC are calculated at A-$Gamma$ and $Gamma$-M,
// as shown in Figure \ref{fig:phonon} and Table \ref{tab:phonon}.
// Raman active phonons are very close to $Gamma$,
// as indicated by the points in the figure.
// Because of the consistency of the most phonon modes near $Gamma$,
// most of the phonon frequencies are insensitive to the direction of the incident light.
// However, some phonons have strong polarities,
// which leads to long-range Coulomb interactions between phonons,
// and results in different frequencies near $Gamma$,
// as shown by the two lines in the figure.
// Thus, we divide the phonons of defect-free 4H-SiC into three categories:
// (1) Raman inactive or too weak Raman intensity,
// which are invisible in the Raman scattering spectrum;
// (2) Raman active phonons with strong polarities,
// which are visible in the Raman scattering spectrum,
// and their frequencies are independent of the direction of the incident light;
// (3) Polar phonons,
// which are visible in the Raman scattering spectrum,
// and their frequencies depend on the direction of the incident light,
// and can interact with carriers.
#page(flipped: true)[
#let m(n, content) = table.cell(colspan: n, content);
#let mCell(n, content) = m(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]];
#figure(
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)],
#page(flipped: true)[#figure({
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]];
table(columns: 27, align: center + horizon, inset: (x: 2pt, y: 5pt),
// [*Direction of Incident & Scattered Light*],
// m(26)[Any direction (not depend on direction of incident & scattered light)],
[*Number of Phonon*],
// E2 E2 E1 2B1 A1
[1], m(2)[2], [3], m(2)[4], [5], [6], [7], [8], m(3)[9],
@@ -252,10 +225,10 @@ However, the actual visibility of each phonon depends on the magnitudes of its R
[x], [y], [x], m(2)[y], [x], m(2)[y], m(3)[z], [z], [z],
[*Representation 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,
[*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*],
// [*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 tenser components)*],
// E2 E2 E1 2B1 A1
[xy], [xx], [yy], [xy], [xx], [yy], [xz], [yz], [-], [-], [xx], [yy], [zz],
// E1 E2 E2 A1 2B1
@@ -280,15 +253,34 @@ However, the actual visibility of each phonon depends on the magnitudes of its R
m(3)[195.5], m(3)[203.3], m(2)[269.7], [-], [-], m(3)[609.5],
// E1 E2 E2 A1 2B1
m(2)[-], m(3)[776], m(3)[-], m(2)[-], [839], [-], [-],
[*FWHM (Simulation) (cm#super[-1])*],
// E2 E2 E1 2B1 A1
m(3)[1.11], m(3)[1.11], m(2)[1.11], [-], [-], m(3)[591.90],
// E1 E2 E2 A1 2B1
m(2)[1.11], m(3)[1.11], m(3)[1.11], m(3)[1.11], [-], [-],
[*FWHM (Experiment) (cm#super[-1])*],
// E2 E2 E1 2B1 A1
m(3)[1.11], m(3)[1.11], m(2)[1.11], [-], [-], m(3)[591.90],
// E1 E2 E2 A1 2B1
m(2)[-], m(3)[1.11], m(3)[-], m(3)[1.11], [-], [-],
[*Electrical Polarity*],
// E2 E2 E1 2B1 A1
m(3)[None], m(3)[None], m(2)[Weak], [None], [None], m(3)[Weak],
// E1 E2 E2 A1 2B1
m(2)[Weak], m(3)[None], m(3)[None], m(3)[Weak], [None], [None],
),
)},
caption: [Weak- and None-polarized phonons near $Gamma$ point],
)
)<nopol>]
#page(flipped: true)[
#figure({
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)];