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@@ -6,16 +6,16 @@ Raman-active phonon modes were categorized into two groups,
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according to the distinct behaviors arising from different electrical polarities,
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including eight negligible-polarity modes (i.e., having zero or very weak electrical polarity),
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and three strong-polarity modes.
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The negligible-polarity modes exhibited minimal dependence on the wavevector direction,
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while the strong-polarity modes showed significant anisotropy.
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This distinction was clearly illustrated in @figure-discont,
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where the phonon dispersion relations of negligible-polarity modes (gray solid lines)
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were roughly continuous and flat at the #sym.Gamma point,
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leading to only #sym.tilde 0.1 cm#super[-1] frequency variations and similar vibration patterns
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across different incidence configurations.
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In contrast, the strong-polarity modes (colored solid lines) displayed discontinuities at the #sym.Gamma point,
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resulting in #sym.tilde 1 cm#super[-1] frequency shifts and distinct different vibration patterns
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under different incidence configurations.
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The negligible-polarity modes should exhibit minimal dependence on the wavevector direction,
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while the strong-polarity modes should show significant anisotropy.
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// This distinction was clearly illustrated in @figure-discont,
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// where the phonon dispersion relations of negligible-polarity modes (gray solid lines)
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// were roughly continuous and flat at the #sym.Gamma point,
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// leading to only #sym.tilde 0.1 cm#super[-1] frequency variations and similar vibration patterns
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// across different incidence configurations.
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// In contrast, the strong-polarity modes (colored solid lines) displayed discontinuities at the #sym.Gamma point,
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// resulting in #sym.tilde 10 cm#super[-1] frequency shifts and distinct different vibration patterns
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// under different incidence configurations.
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Consequently, the negligible-polarity modes were named
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according to their irreducible representations at the #sym.Gamma point and in order of increasing frequency,
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including A#sub[1]-1 to A#sub[1]-2, E#sub[1]-1 to E#sub[1]-2, and E#sub[2]-1 to E#sub[2]-4.
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@@ -23,29 +23,11 @@ In contrast, the strong-polarity modes were named according to their vibration d
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including TO-zOx (vibrating in the zOx plane and roughly perpendicular to the incidence direction),
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TO-y (vibrating along the y-axis), and LO (vibrating roughly parallel to the incidence direction).
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#include "figure-discont.typ"
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In order to further investigate the frequencies dependence on the wavevector,
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and to confirm the accuracy of both our experiments and calculations,
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we analyzed the negligible-polarity phonons off the #sym.Gamma point
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by comparing experimental and calculated results under various laser incidence directions.
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The E#sub[2]-3 peak served as a calibration reference under various experiments,
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// 第二段：我们具体计算了模式对极性的依赖。可以看到，弱极性声子的变化极小。与实验比较，也验证了我们实验的准确性。
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// 第三段：有两个弱极性模式还没有在实验上看到过。我们通过计算知道了它们的强度，并看到了其中一个。
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// 第四段：声子模式的强度可以从理论上得到解释。
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=== Negligible-polarity Phonons
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#include "figure-raman.typ"
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Peaks corresponding to seven Raman-active negligible-polarity phonons were observed in our experiments
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(only the E#sub[2]-4 mode was not observed),
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(only the E#sub[2]-4 mode was not observed, see @figure-raman),
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which is more than all previous experiments (where only five or six peaks were typically reported).
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To explain the discrepancy in experimental results, first-principles calculations were performed,
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To explain the discrepancy in experimental results,
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first-principles calculations on Raman tensors were performed,
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and the result was compared with experimental data and summarized in @table-nopol.
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Our calculation showed that the mode of E#sub[2]-1, E#sub[2]-2, E#sub[1]-1, A#sub[1]-1 and E#sub[2]-3
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had relatively high Raman intensities and well-separated frequencies,
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@@ -68,8 +50,17 @@ However, the E#sub[1]-2 mode was observable in our experiments of y(zx)#overline
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Our experiments reported the observation of the E#sub[1]-2 peak for the first time,
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and explained the discrepancy among previous experiments and ours with the help of our calculations.
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// TODO: 图中标注强调是拉曼活性的模式
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// TODO: b 图的标题歪了
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#include "figure-raman.typ"
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#include "table-nopol.typ"
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#include "table-pol.typ"
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// TODO: 合并两个表格到一页，删除 polar 中的多余信息
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It is noteworthy that the large variation in Raman tensor magnitudes among different modes
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was not yet theoretically understood.
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For example, the Raman tensor of the E#sub[2]-3 mode was substantially larger
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@@ -85,12 +76,59 @@ By analyzing the local environment of individual atoms,
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and several secondary components (reflecting environmental variations,
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denoted as $epsilon_i$, $eta_i$, and $zeta_i$，where $i in {1, 2, 5, 6}$,
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and $|epsilon_i| + |eta_i| + |zeta_i| << |a_i|$ was assumed).
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Detailed derivations were provided in @appd-predict, with results summarized in @table-nopol.
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Detailed derivations were provided in @appd-predict, with results summarized in @table-nopol and @table-pol.
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Notably, the E#sub[2]-3 mode was the only mode that retains the $a_i$ term,
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which indicating a constructive interference of contributions from the local environment of individual atoms.
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This stood in contrast to other negligible-polarity modes where such contributions tend to cancel out,
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explaining the exceptionally high Raman tensor magnitude observed for the E#sub[2]-3 mode.
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模式频率在倒空间中的依赖性被仔细研究并展示在@figure-rev 中。
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各个模式在 gamma 点附近的色散关系被计算出来并被展示在图 a 中。
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在弱极性模式中，频率偏移在 1cm -1 以内；在极性模式中，频率的变化达到了 10 cm -1 量级。其中，
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E2 模式没有极性，因此在 Gamma 点处无色散；离开 Gamma 点越远，频率对入射方向的依赖越大。
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这意味着相比于绿光，使用紫外光入射时，正入射与侧入射的频率差异会更大。
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A1 的极性不为零，它们的色散呈现瓣状，意味着无论使用什么激光波长，频率的变化都比较接近。
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E2-3 的频率几乎不变并且在任何角度的入射实验中都清晰可见 ，可以被选定为实验中的校准参考。
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图b给出了实验结果，其中 E2-1 由于较弱，导致样本标准差较大；E2-2 和 A1-1 的散射强度较强，因此标准差较小。
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图c对比了实验与计算的结果，可以看到两者吻合较好，只有 A1-1 的实验结果明显大于计算结果。
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这可以归结为一些原因。
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我们的研究对比了入射方向对极性较弱的声子模式的影响并在理论上进行了解释，以实现对拉曼光谱的更精确分析，并为掺杂和载流子对光谱影响的分析做了准备。
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#include "figure-rev.typ"
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#include "figure-discont.typ"
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The E#sub[2]-3 peak was calculated to be having virtually invariant frequency,
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and thus served as a calibration reference under various experiments.
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The peek of E#sub[2]-1, E#sub[2]-2, A#sub[1]-1 and TO-zOx modes were observable in both normal and edge incidence,
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thus was uesd to compare the frequency shifts between different incidence configurations.
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// TODO: 增加图例：各种入射激光
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// TODO: 增加 TO-zOx 的讨论
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// TODO: 条形图画成两个方向的
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// TODO: 换个方案拟合，考虑不对称，看能不能把这个误差填上
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// 第二段：我们具体计算了模式对极性的依赖。可以看到，弱极性声子的变化极小。与实验比较，也验证了我们实验的准确性。
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// 第三段：有两个弱极性模式还没有在实验上看到过。我们通过计算知道了它们的强度，并看到了其中一个。
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// 第四段：声子模式的强度可以从理论上得到解释。
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=== Negligible-polarity Phonons
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To achieve a more precise investigation of the Raman spectra
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and prepare for analyzing impurity and charge carrier effects,
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the analysis of negligible-polarity phonons off the #sym.Gamma point was conducted
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@@ -132,5 +170,5 @@ E1 的情况。
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// 我们预测，随着入射方向偏移，LO 峰会向着高频方向移动。此外，我们也注意到 LO 也会与载流子产生影响。
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// 在 n 型半导体中，LOPC 模式将代替 LO 模式；在 p 型半导体中，LO 模式仍然单独存在，但它的半高宽会受到载流子浓度的影响。
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#include "table-pol.typ"
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#include "figure-rev.typ"
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@@ -39,4 +39,4 @@
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)},
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caption: [Strong-polarized phonons near $Gamma$ point],
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)
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]
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<table-pol>]
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