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@@ -75,8 +75,6 @@ E#sub[1]-2 和 E#sub[2]-4 模式位于最强模式 E#sub[2]-3 附近，且具有
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使得它们在实验光谱中难以区分。
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此外，A#sub[1]-2 模式在基面极化配置（xx 和 yy，仅为 0.01）中具有非常弱的拉曼强度，这导致它在正入射的拉曼实验中通常不可见；
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但当偏振沿 z 轴时（1.78）则显示出可观测的强度。
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// E#sub[1]-1 模式在理论上无法在正入射中观察到，但在实验中仍然可以看到一个小峰。
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// 这被认为是因为入射光并非完全沿z轴入射，由于衬底斜切和共聚焦汇聚角。通过向不同方向倾斜衬底，我们可以使这个峰变高或变低，如附图所示，这印证了我们的猜想。
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The Raman tensors and frequencies of the negligible-polar phonons were calculated using first-principles methods,
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and the results were compared with both experimental data and theoretical predictions (@table-nopol).
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@@ -96,54 +94,29 @@ Additionally,
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but showed an observable intensity when the polarization is along the z-axis (1.78).
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Thus, it was not typically observed in normal incidence Raman experiments (cite),
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but could be clearly detected in edge incidence configurations in our experiments.
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// TODO: 这里缺两个引用
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不同方向的入射配置会导致微小但可观测的峰位差异。
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理想情况下不允许存在的效应被观察到。
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这包括不同入射方向下，弱极性峰位的微小移动，以及 E#sub[1]-1 模式在正入射中的可见性。
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在我们的计算中，从正入射到肩入射，E2-3的峰位几乎不变，E2-1与A1-1会有可观测的蓝移，同时E2-2会有可观测的红移。
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实验结果与计算结果基本一致，如图如表所示。
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E1模式在正入射中可以看到，尽管理论上不应该。
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这被认为是因为入射光并非完全沿z轴入射，由于衬底斜切和共聚焦汇聚角。
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不同方向偏振的结果印证了我们的猜想，如图所示。同时，略微倾斜衬底可以使这个峰变高或变低，如附图所示，这印证了我们的猜想。
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微小但可观测的峰位移动应该存在于不同入射配置的峰位中，因为非零的波矢长度。
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在我们的计算中，相比于正入射，肩入射的E2-1与E2-2的间距会减小0.49，同时A1-1会红移0.19。
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在实验中，E2-1与E2-2的间距减少0.36，A1-1红移了xxxx。
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Minor but observable peak shifts of negligible-polar phonons were expected between different incidence configurations,
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due to non-zero wavevector lengths.
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Some effects absent in the ideal model were observed experimentally,
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including minor but observable peak shifts of negligible-polar phonons between different incidence configurations,
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and the observability of the E#sub[1]-1 mode in normal incidence Raman experiments.
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The peak shifts were considered caused by the non-zero wavevector lengths of phonons.
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In our calculations, compared to normal incidence,
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the spacing between the E#sub[2]-1 mode and E#sub[2]-2 mode decreased by 0.49 cm#super[-1] under edge incidence,
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while the A#sub[1]-1 mode redshifted by 0.19 cm#super[-1].
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In our experiments, the spacing between the E#sub[2]-1 mode and E#sub[2]-2 mode decreased by 0.36 cm#super[-1],
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while the A#sub[1]-1 mode redshifted by xxxx cm#super[-1].
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the E#sub[2]-3 mode's frequency remained nearly unchanged,
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while the E#sub[2]-1, E#sub[2]-2, A#sub[1]-1 modes exhibited minor but observable shifts.
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The experimental results were in good agreement with our calculations, as shown in figure and table.
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The visibility of the E#sub[1]-1 mode in normal incidence Raman experiments were attributed to the fact that
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the incident light was not perfectly aligned along the z-axis,
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due to the substrate's slight tilt and the converging angle of the confocal setup.
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This was confirmed by the results under different polarization configurations and different tilt angles,
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as shown in figure.
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#include "figure-nopo-diff.typ"
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E1为什么可以被看到。
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E1在正入射中可以看到，尽管理论上不应该。
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这被认为是因为入射光并非完全沿z轴入射，由于衬底斜切和共聚焦汇聚角。
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通过向不同方向倾斜衬底，我们可以使这个峰变高或变低，如附图所示。
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此外，考虑到斜切方向（x方向），E1模式应该更容易在zxxz和zyxz中看到，而在zyyz中更难看到，这与实验一致。
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The E#sub[1]-1 mode was observed in normal incidence Raman experiments, despite being theoretically forbidden.
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This was attributed to the fact that the incident light was not perfectly aligned along the z-axis,
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due to the substrate's slight tilt and the converging angle of the confocal setup.
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This was confirmed by tilting the substrate in different directions,
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which caused the peak to become more or less pronounced, as shown in the inset figure.
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Furthermore, considering the tilt direction (x-direction),
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the E#sub[1]-1 mode should be more easily observed in the zxxz and zyxz configurations,
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while being more difficult to detect in the zyyz configuration,
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which is consistent with our experimental observations.
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#include "figure-e1.typ"
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// 其它峰在其它章节中解释。
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//
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// Besides, there are other peeks in the experiment.
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// The peek at 796 and 980 are caused by strong-polar phonons which will be discussed later.
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// Besides, there are small peeks at xxx,
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// which could not be explained in perfect 4H-SiC and will be discussed in the next section.
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// TODO: 将一部分 phonons 改为 phonon modes
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// 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。
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// 也就是说，简并的里面有两个或者无数个 phonon，但只有一个 mode
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// #include "figure-raman.typ"
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// TODO: 解释为什么 E1 可以看到
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@@ -1,5 +1,5 @@
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#figure(
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image("/画图/弱极性不同方向偏移/main.svg"),
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image("/画图/弱极性不同方向偏移/embed.svg"),
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caption: [
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(a) Phonon dispersion of 4H-SiC along the A–#sym.Gamma–K high-symmetry path.
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Gray lines represent negligible-polar phonon modes,
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@@ -5,6 +5,9 @@
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let m(n, content) = table.cell(colspan: n, content);
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let m2(content) = table.cell(colspan: 2, content);
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let m3(content) = table.cell(colspan: 3, content);
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let pm = sym.plus.minus;
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let m21(content) = table.cell(rowspan: 2, colspan: 1, content);
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let m22(content) = table.cell(rowspan: 2, colspan: 2, content);
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table(columns: 12, align: center + horizon,
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m2[*Notation of Mode*],
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[E#sub[2]-1], [E#sub[2]-2], [E#sub[1]-1], m2[A#sub[1]-1],
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@@ -19,22 +22,31 @@
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// E2 E2 E1 A1 E1 E2 E2 A1
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[Error (%)], [2.6], [2.7], [4.6], m2[2.9], [-], [2.5], [-], m2[3.1],
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table.cell(rowspan: 4)[*Raman Tensor*],
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[Non-zero components],
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[Non-zero components],
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// E2 E2 E1 A1
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[xy, xx, -yy], [xy, xx, -yy], [xz, yz], [xx, yy], [zz],
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// E1 E2 E2 A1
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[xz, yz], [xy, xx, -yy], [xy, xx, -yy], [xx, yy], [zz],
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[Analysis result (a.u.)],
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// E2 E2 E1 A1
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[$2epsilon_2-2zeta_2-4eta_2$], [$2epsilon_2-2zeta_2$], [$-2epsilon_1-2zeta_1$], [$-2epsilon_5$], [$-2epsilon_6$],
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// E1 E2 E2 A1
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[$-2epsilon_1+2zeta_1$], [$bold(8a_2)+2epsilon_2-2zeta_2-4eta_2$], [$-2epsilon_2+2zeta_2$], [$-2zeta_5$], [$-2zeta_6$],
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[Calculation result (a.u.)],
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// E2 E2 E1 A1 E1 E2 E2 A1
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[0.17], [1.13], [2.43], [2.83], [1.79], [0.09], [88.54], [0.50], [0.01], [1.78],
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[Experiment result #linebreak() (a.u.)],
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// E2 E2 E1 A1 E1 E2 E2 A1
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[0.15], [1.11], [], [1.64], [], [Invisible], [88.54], [Invisible], [Invisible], [],
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[Analysis result (a.u.)],
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// E2 E2 E1
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[$2epsilon_2-2zeta_2-4eta_2$], [$2epsilon_2-2zeta_2$], [$-2epsilon_1-2zeta_1$],
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// A1 E1 E2
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[$-2epsilon_5$], [$-2epsilon_6$], [$-2epsilon_1+2zeta_1$], [$bold(8a_2)+2epsilon_2-2zeta_2-4eta_2$],
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// E2 A1
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[$-2epsilon_2+2zeta_2$], [$-2zeta_5$], [$-2zeta_6$],
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[Calculation result (a.u.)],
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// E2 E2 E1 A1 E1 E2 E2 A1
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[0.17], [1.13], [2.43], [2.83], [1.79], [0.09], [88.54], [0.50], [0.01], [1.78],
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[Experiment result #linebreak() (a.u.)],
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// E2 E2 E1 A1 E1 E2 E2 A1
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[0.15], [1.11], [], [1.64], [], [Invisible], [88.54], [Invisible], [Invisible], [],
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table.cell(rowspan: 2)[*Raman Shift of Different Incidence Directions*],
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[Calculation result (cm#super[-1])],
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// E2 E2 E1 A1 E1 E2 E2 A1
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[0.263], [-0.221], m21[-], m2[-0.188], m21[-], [0.004], m21[-], m22[-],
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[Experiment result (cm#super[-1], average #pm standard deviation)],
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// E2 E2 A1 E2
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[0.158 #pm 0.060], [-0.245 #pm 0.026], m2[-0.279 #pm 0.016], [0 #linebreak() (used for calibration)],
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)},
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caption: [Negaligible-polarized Phonons at $Gamma$ Point.],
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)<table-nopol>]
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画图/弱极性不同方向偏移/embed.svg
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画图/弱极性不同方向偏移/拉曼.svg
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@@ -1,2 +1,2 @@
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1.954075029530483505e+02 1.953478857341484627e+02 1.953769267166200905e+02 1.954027970339571993e+02 1.953347589514454796e+02 1.953585750606797262e+02 1.953331279341388722e+02 1.954060267425776658e+02 1.952899705473047050e+02 1.953494452516382864e+02 1.953889540387470731e+02
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1.955143254061939047e+02 1.955314568194222034e+02 1.955552683333574748e+02 1.955481750180620395e+02 1.955953344290883251e+02 1.955747288170350373e+02 1.955405035622480341e+02 1.954405635948803877e+02 1.953860875562799890e+02 1.955043177012716740e+02 1.955407037484820592e+02
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1.955848052701602739e+02 1.955719549975438838e+02 1.955929051638321710e+02 1.956348447216788884e+02 1.954968231689395282e+02 1.955432332739998742e+02 1.955365842761710553e+02 1.956495763564502113e+02 1.955159160392721844e+02 1.956011491541673593e+02 1.956275997529293704e+02
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1.958012278006910094e+02 1.958162272575077907e+02 1.957831178516572948e+02 1.958249159186141810e+02 1.958364298063405897e+02 1.957933101182395603e+02 1.958021374449715495e+02 1.958134637560016813e+02 1.957881271460675521e+02 1.958187264806561529e+02 1.957841759645648096e+02
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@@ -1,2 +1,2 @@
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2.033596745420836385e+02 2.033350514440899133e+02 2.033520034103685248e+02 2.033359289059363277e+02 2.033325794937803437e+02 2.033390901492543037e+02 2.033161571980983240e+02 2.033605399415457100e+02 2.033257530283401309e+02 2.033291258845766265e+02 2.033518148664678051e+02
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2.030969863890181841e+02 2.031161714659073425e+02 2.031116343498714230e+02 2.031200227998062076e+02 2.030973971095251045e+02 2.031188254080149420e+02 2.031087887206043661e+02 2.030768375231341452e+02 2.030640772958760749e+02 2.030605849463428854e+02 2.030720189180035788e+02
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2.034024141958410041e+02 2.033769616315146891e+02 2.033951351274238277e+02 2.033767144659590258e+02 2.033742078065036765e+02 2.033807464342712876e+02 2.033596912050411163e+02 2.034018552200704733e+02 2.033676570593588053e+02 2.033706539863660225e+02 2.033939947741981484e+02
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2.031269701456583903e+02 2.031410180043692435e+02 2.031354010942689854e+02 2.031408982509935299e+02 2.031211049500984416e+02 2.031478830793148802e+02 2.031381933050760722e+02 2.031074067250477526e+02 2.030943949028673501e+02 2.030889933216531062e+02 2.030979516775117872e+02
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