From 5cef649cebc0301c65199093e6515cbf8e9a8a3b Mon Sep 17 00:00:00 2001 From: chn Date: Wed, 19 Nov 2025 13:28:40 +0800 Subject: [PATCH] =?UTF-8?q?=E5=B0=86=E6=97=A0=E7=BC=BA=E9=99=B7=E5=86=85?= =?UTF-8?q?=E5=AE=B9=E5=90=88=E5=B9=B6=E5=88=B0=E4=B8=80=E4=B8=AA=E6=96=87?= =?UTF-8?q?=E4=BB=B6=E4=B8=AD?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- paper/appendix/predict/default.typ | 2 +- paper/appendix/representation/default.typ | 2 +- paper/method/simulation.typ | 14 ++ paper/result/perfect/default.typ | 123 +++++++++++++----- paper/result/perfect/non-polar/default.typ | 82 ------------ .../perfect/{non-polar => }/table-nopol.typ | 0 .../perfect}/table-rep.typ | 0 7 files changed, 105 insertions(+), 118 deletions(-) rename paper/result/perfect/{non-polar => }/table-nopol.typ (100%) rename paper/{appendix/representation => result/perfect}/table-rep.typ (100%) diff --git a/paper/appendix/predict/default.typ b/paper/appendix/predict/default.typ index 4316b68..cecd59f 100644 --- a/paper/appendix/predict/default.typ +++ b/paper/appendix/predict/default.typ @@ -1,7 +1,7 @@ #import "@preview/physica:0.9.5": pdv, super-T-as-transpose #show: super-T-as-transpose -== Approximation of Raman tensor of 4H-SiC +== Approximation of Raman tensor of 4H-SiC 近似的核心思路。 diff --git a/paper/appendix/representation/default.typ b/paper/appendix/representation/default.typ index 022d378..d5a493b 100644 --- a/paper/appendix/representation/default.typ +++ b/paper/appendix/representation/default.typ @@ -7,7 +7,7 @@ The phonons at the #sym.Gamma point in 4H-SiC (space group P6#sub[3]mc) and negligible-polar modes (2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2]). The form of the Raman tensors for each representation is well known and summarized in @table-rep. -#include "table-rep.typ" +// #include "table-rep.typ" By further decomposing the doubly degenerate modes (E#sub[1] and E#sub[2] of C#sub[6v] group) into the C#sub[2v] point group, diff --git a/paper/method/simulation.typ b/paper/method/simulation.typ index 58d0183..cfcb5e7 100644 --- a/paper/method/simulation.typ +++ b/paper/method/simulation.typ @@ -7,3 +7,17 @@ K 点网格根据模型大小不同，分别使用 555 和 ???。 涂抹使用 xxx 以统一比较点缺陷和无缺陷的模型。 弛豫的精度为 xxx。 声子计算使用 phonopy phono3py ufo，BEC 修正使用 xxx 的算法。（引用） + +// while 强极性声子that参与一个方向的拉曼散射 are totally different from those that参与其它方向的拉曼散射。 + +// that +// where atoms of the same species vibrated in opposite directions, +// resulting in effective cancellation of electrical polarization during the Raman process +// (gray lines in @figure-discont); +// +// where atoms of the same species vibrated in phase, +// resulting in strong electrical polarization and observable effects in Raman spectra +// (colored lines in @figure-discont). + + // #include "figure-discont.typ" + diff --git a/paper/result/perfect/default.typ b/paper/result/perfect/default.typ index 3a2f5e5..cb7365b 100644 --- a/paper/result/perfect/default.typ +++ b/paper/result/perfect/default.typ @@ -1,31 +1,96 @@ == Phonons in Perfect 4H-SiC -我们将声子分为两类讨论：弱极性的声子，和强极性的声子。 -这是因为， - 弱极性声子that参与一个方向的拉曼散射 are nearly the same as those that参与其它方向的拉曼散射 （only ~ 0.1 cm-1 difference in frequency)， - while 强极性声子that参与一个方向的拉曼散射 are totally different from those that参与其它方向的拉曼散射。 +These phonons were categorized into two groups and discussed separately, according to their electrical polarities: + negligible-polarity phonons (i.e., zero or very weak electrical polarity), and strong-polarity phonons. -在弱极性声子中： +=== Negligible-polarity Phonons -4H-SiC 拥有8个拉曼活性的弱极性模式，对应于 C6v 点群的三个不可约表示 of E2, E1, and A1。 -按频率从低到高排序，将它们命名为 E2-1, E2-2, E1-1, A1-1, E1-2, E2-3, E2-4, A1-2。 -通过进一步在 C2v 群中考虑，可以给出每个pattern 的拉曼张量形式（总结于表中）。 +The negligible-polarity phonons were initially analyzed at the #sym.Gamma point, + disregarding the incidence configurations. +This simplification is justified because the participating negligible-polarity phonon modes + are nearly identical across all Raman experiments, + with their frequencies differing by only #sym.tilde 0.1 cm#super[-1] difference in frequency + (see gray lines in @figure-discont). -在我们的实验中，它们中的七个峰被看见（只有 E2-4 没有被看见），which is more than previous experiments（只有五或六个峰被看见）。 -为了解释实验结果，第一性原理计算被进行，并且结果与实验数据被比较。 -E#sub[2]-1 E#sub[2]-2 E#sub[1]-1 A#sub[1]-1 E#sub[2]-3 有较高的拉曼强度且位置比较分散，这导致它们在大多数实验中都可以看到。 -A#sub[1]-2 在面内偏振下拉曼散射强度非常弱，但在 z 轴偏振下有可观测的强度，which is compatible with our experimental result that 在正入射拉曼实验中，它很难被看到，但在侧入射时通过选取合适的偏振配置应该可以观察到。因此，它在一些实验中可以观察到而在另外一些实验中不可以观察到(cite)。 -E#sub[2]-4 距离最强模式 E#sub[2]-3 很近，且具有很弱的拉曼强度，因此在我们的实验和报道的实验中都不可见。 -E#sub[1]-2 与 E#sub[2]-3 也有短的距离from E#sub[2]-3和弱的拉曼散射强度，但它的表示与 E#sub[2]-3 不同。 -在我们的实验中，通过选定某个偏振，E#sub[2]-3 的散射强度被抑制、E#sub[1]-2 的散射强度被增强，从而使得 E#sub[1]-2 可以被观察到(cite)。 -我们的实验第一次观察到了 E#sub[1]-2 峰的位置，并且与计算和理论吻合较好。 +There are eight Raman-active negligible-polar modes in 4H-SiC, + corresponding to three irreducible representations of the C#sub[6v] group (A#sub[1], E#sub[1], and E#sub[2]). +We named these modes as + E#sub[2]-1, E#sub[2]-2, E#sub[1]-1, A#sub[1]-1, E#sub[1]-2, E#sub[2]-3, E#sub[2]-4, and A#sub[1]-2, + in order of increasing frequency. +The Raman tensor forms of each mode were derived by further considering their representations in the C#sub[2v] group, + and were summarized in @table-rep. + +#include "figure-discont.typ" +#include "table-rep.typ" + +Peaks corresponding to seven Raman-active negligible-polarity phonons were observed in our experiments + (only the E#sub[2]-4 mode was not observed), + which is more than all previous experiments (where only five or six peaks were typically reported). +To explain the discrepancy in experimental results, first-principles calculations were performed, + and the result was compared with experimental data and summarized in @table-nopol. +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 + had relatively high Raman intensities and well-separated frequencies, + making them observed in our experiments as well as most previous experiments. +The A#sub[1]-2 mode was calculated to have very weak (0.01) and relatively strong (1.78) Raman intensity + under in-plane polarization and z polarization, respectively, + which is compatible with our experimental result + that it could be observed clearly in the y(zz)#overline[y] configuration but hardly seen in our other experiments. +This peek was reported to be observable in some experiments (cite) but not in others (cite), + our calculation provided an explanation for this discrepancy. +The E#sub[2]-4 modes was calculated + to be located close to the most intense E#sub[2]-3 mode (< 10 cm#super[-1] away) + and exhibit very weak Raman intensities (only 0.6% of the E#sub[2]-3 mode), + making it not visible in our and all previous experiments. +The E#sub[1]-2 mode was also located close to the E#sub[2]-3 mode and has weak Raman intensity, + making it also unobservable in previous experiments. +However, the E#sub[1]-2 mode was observable in our experiments of y(zx)#overline[y], + where the scattering of the E#sub[2]-3 mode was suppressed while that of the E#sub[1]-2 mode was enhanced, + thanks to the different representations of these two modes. +Our experiments reported the observation of the E#sub[1]-2 peak for the first time, + and explained the discrepancy among previous experiments and ours with the help of our calculations. + +#include "table-nopol.typ" + +It is noteworthy that the large variation in Raman tensor magnitudes among different modes + was not yet theoretically understood. +For example, the Raman tensor of the E#sub[2]-3 mode was substantially larger + than those of other negligible-polarity modes (over 30 times larger than that of the second-strongest). +This highlighted a significant gap in established theory + that rigorous symmetry analysis could only predict the non-zero components of the Raman tensors, + but not their magnitudes. + +In order to address the limitations of existing theories, + a method for estimating the magnitudes of Raman tensors was proposed. +By analyzing the local environment of individual atoms, + this approach decomposed their contributions to the Raman tensor into two parts: + a dominant component (invariant across similar environments, denoted as $a_i$，where $i in {1, 2, 3, 4}$) + and several secondary components (reflecting environmental variations, + denoted as $epsilon_i$, $eta_i$, and $zeta_i$，where $i in {1, 2, 3, 4}$, + and $|epsilon_i| + |eta_i| + |zeta_i| << |a_i|$ was assumed). +Detailed derivations were provided in @appd-predict, with results summarized in @table-nopol. +Notably, the E#sub[2]-3 mode was the only mode that retains the $a_i$ term, + indicating a constructive interference of contributions from the local environment of individual atoms. +This stood in contrast to other negligible-polarity modes where such contributions tend to cancel out, + thereby explaining the exceptionally high Raman tensor magnitude observed for the E#sub[2]-3 mode. + +To further investigate the Raman spectra, + an analysis of negligible-polarity phonons off the #sym.Gamma point was conducted + by comparing experimental and calculated results under various lazer incidence directions. +The E#sub[2]-3 peak searved as a calibration reference under various experiments, + since its position remained virtually invariant between normal and edge incidence + (with a shift of only #sym.tilde 0.004 cm#super[-1]). + +The E2​-1, E2​-2, and A1​-1 modes exhibited observable frequency shifts. These experimental results are in good agreement with our calculations, as shown in Figure X and Table Y." + + +The E#sub[2]-1, E#sub[2]-2, and A#sub[1]-1 modes exhibited observable shifts, + and the experimental results were in good agreement with our calculations, as shown in figure and table. +Our results further confirmed the accuracy of both our experiments and calculations. + +// #include "figure-nopo-diff.typ" + +为了进一步探索拉曼光谱，离开Gamma点的弱极性声子的分析也被进行，通过比对不同方向入射下的实验和计算结果。 -注意到 E2-3 的拉曼张量远远大于其它所有弱极性模式（是第二强模式的 80 倍以上），which 缺少理论上的解释。 -经典的严格的对称性分析只能给出拉曼张量的非零分量，但无法对非零分量的大小做出预测。 -为了解释这一现象，a theoratic approach based on 经典的对称性分析 was proposed by introduce 一些近似。 -详细的方法见附录，and 结果总结在表中，其中 $a_i$ 被期待远大于其它量 $epsilon_i$ $eta_i$ $zeta_i$，thus 拉曼张量的大小 could be estimated subsequently。 -我们的结果表明，E2-3 模式的拉曼散射强度远高于其它振动模式可以被解释为所有原子对拉曼张量贡献的相长干涉，这与其他弱极性模式不同（他们的贡献相互抵消）。 -这个方法可以被潜在用于快速估计其它结构中弱极性模式的拉曼强度，从而在昂贵的第一性原理计算计算拉曼张量之前，预测哪些模式可能在实验中被观察到。 为了检验我们实验和计算的准确性，我们测量和计算了不同入射方向下弱极性峰位的微小移动。 在我们的计算中，对比于正入射，肩入射中，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。 @@ -52,16 +117,6 @@ E2-1 E2-2 的计算结果与实验吻合较好，A1-1与实验结果略有出入 极性可以忽略的声子模式（即极性为零或非常弱的声子），在拉曼散射过程中它们的电极性造成的效应可以忽略； 强极性声子模式，在拉曼光谱中电极性效应是可观测的，不可以被忽略。 -These phonons were categorized into two groups and discussed separately, according to their electrical polarities: - phonons with negligible polarities (i.e., zero or very weak electrical polarity), - where atoms of the same species vibrated in opposite directions, - resulting in effective cancellation of electrical polarization during the Raman process - (gray lines in @figure-discont); - and phonons with strong polarities - where atoms of the same species vibrated in phase, - resulting in strong electrical polarization and observable effects in Raman spectra - (colored lines in @figure-discont). -#include "figure-discont.typ" -#include "non-polar/default.typ" -#include "polar/default.typ" +// #include "non-polar/default.typ" +// #include "polar/default.typ" diff --git a/paper/result/perfect/non-polar/default.typ b/paper/result/perfect/non-polar/default.typ index 87cba1b..2db1699 100644 --- a/paper/result/perfect/non-polar/default.typ +++ b/paper/result/perfect/non-polar/default.typ @@ -1,25 +1,4 @@ -=== Phonons with Negligible Polarities -对弱极性声子的分析的传统理论分析。（Gamma 近似，C6v表示） - -对弱极性声子的理论分析，使用 Gamma 点的声子来近似。 -这基于这样的事实：这些声子的色散曲线在 Gamma 点附近连续且非常接近 Gamma 点，并且已经被广泛使用 @_n-sic_2008。 -对称性分析从理论上指出了这些声子属于 C6v 的 12 个表示。 -其中，B1 无活性，其它则有可能在拉曼实验中观察到。 -然而，模式的拉曼散射强弱无法仅从对称性分析中推断出来。 - -The negligible-polar phonons were theoretically analyzed by approximating their properties at the #sym.Gamma point. -This approximation is based on the fact that - the dispersion of these phonons is continuous and very close to the #sym.Gamma point, - and has been widely adopted in the literature @_n-sic_2008. -Symmetry analysis pointed out that - these phonons correspond to twelve irreducible representations of the C#sub[6v] point group and were named accrodingly - (A#sub[1]-1 to A#sub[1]-2, B#sub[1]-1 to B#sub[1]-4, E#sub[1]-1 to E#sub[1]-2, and E#sub[2]-1 to E#sub[2]-4, - see appendix for details). -Among which the B#sub[1] representation is Raman inactive, - while other eight modes are potentially observable in Raman experiments. -However, it should be noted that the intensity of Raman scattering from each mode - cannot be inferred from symmetry considerations alone. 我们提出了一个新的办法来估计拉曼张量大小。 @@ -31,59 +10,9 @@ However, it should be noted that the intensity of Raman scattering from each mod 我们的结果表明，E2-3 模式的拉曼散射强度远高于其它振动模式，这与实验和第一性原理计算结果一致。 我们的研究表明，这个峰的高拉曼强度来自于所有键的贡献的相长干涉，这与其他弱极性模式不同（他们的贡献相互抵消）。 -To overcome this limitation, - a method to estimate the magnitudes of the Raman tensors of each mode was proposed (see appendix for details). -This approach was founded on the symmetry analysis and incorporates the assumption - that the primary contribution from each atom to the Raman tensor arises from its nearest neighbors (denoted as $a_i$), - while contributions from more distant atoms are much smaller (denoted as $epsilon_i$, $eta_i$, and $zeta_i$). -Furthermore, the absolute amplitude differences among atoms of the same type within a phonon mode was neglected, - and only their vibrational directions were considered. -This enables a preliminary estimation of the Raman tensor magnitudes prior to expensive first-principles calculations, - with the results summarized in @table-nopol. -Our analysis gave the result that the E#sub[2]-3 mode possess a much higher Raman intensity - and should be clearly observable in Raman experiments, - while the observability of other modes depends on their frequencies were well-separated or not - and could only be confirmed by further first-principles calculations or experiments. -Our result showed that - the high Raman intensity of this mode arises from the constructive interference of contributions from all bonds, - in contrast to other negligible-polar modes where contributions tend to cancel each other out. -#include "table-nopol.typ" -实验看到了7个峰，比别人多。 -我们做了实验，看到了其中 7 个峰，比别人的实验要多。 -为了解释实验结果，我们进行了第一性原理计算。 -E#sub[2]-1 E#sub[2]-2 E#sub[1]-1 A#sub[1]-1 E#sub[2]-3 这几个模式在大多数实验中都可以看到，这是因为它们有较高的拉曼强度且位置比较分散。 -A#sub[1]-2 在一些实验中可以观察到而在另外一些实验中不可以观察到(cite)，这是因为它在面内偏振下拉曼散射强度非常弱，但在 z 轴偏振下有可观测的强度； - 因此在我们的正入射拉曼实验中它很难被看到，但在侧入射时通过选取合适的偏振配置应该可以观察到。 -E#sub[1]-2 和 E#sub[2]-4 距离最强模式 E#sub[2]-3 很近，且具有很弱的拉曼强度，在报道的实验中都不可见。 -在我们的实验中，通过选取合适的偏振配置和延长积分时间，E#sub[2]-3 被压制后，E#sub[1]-2 也被观察到了。 -我们的实验第一次指出了 E#sub[1]-2 峰，并且与计算和理论吻合较好。 - -Raman experiments were established and seven negligible-polar phonon peaks were observed, - which is more than all previous experiments (where only five or six peaks were typically reported). -To further understand our experimental results, - first-principles calculations were performed, - and the result was compared with experimental data. -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 - had relatively high Raman intensities and well-separated frequencies, - making them observed in our experiments as well as most previous experiments. -The A#sub[1]-2 mode had only a very weak Raman intensity with in-plane polarization (e.g., zyy#overline[z]), - while showed an observable intensity when the polarization is along the z-axis (e.g., xzzx). -Thus, it could be observed clearly in our edge incidence experiments (xzzx), - but hardly seen in normal incidence configurations. -This peek was reported to be observable in some experiments (cite) but not in others (cite), - our calculation provided an explanation for this discrepancy. -The E#sub[1]-2 and E#sub[2]-4 modes are calculated - to be located close to the most intense E#sub[2]-3 mode (< 10 cm#super[-1] away) - and exhibit very weak Raman intensities (only 0.1% and 0.6% of the E#sub[2]-3 mode, respectively), - making them hard to observe in reported experiments. -However, in our experiments, - with edge incidence configuration (zxy#overline[z]) and extended integration time, - the E#sub[2]-3 peek was suppressed and the E#sub[1]-2 mode was also observed. -Our experiments reported the E#sub[1]-2 peak for the first time, - and the experimental results were in good agreement with both calculations and theoretical analysis. #include "figure-raman.typ" @@ -93,14 +22,3 @@ Our experiments reported the E#sub[1]-2 peak for the first time, 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。 实验结果与计算结果基本一致，如图如表所示。 -To further validate the accuracy of our experiments and calculations, - we measured and calculated the peak shifts of negligible-polar phonons - under different incidence configurations. -In our calculations, comparing with normal incidence, - the E#sub[2]-3 peek remained nearly unchanged under edge incidence (only shifted about 0.004 cm#super[-1]), - thus we use it as a reference to calibrate other peak positions. -The E#sub[2]-1, E#sub[2]-2, and A#sub[1]-1 modes exhibited observable shifts, - and the experimental results were in good agreement with our calculations, as shown in figure and table. -Our results further confirmed the accuracy of both our experiments and calculations. - -#include "figure-nopo-diff.typ" diff --git a/paper/result/perfect/non-polar/table-nopol.typ b/paper/result/perfect/table-nopol.typ similarity index 100% rename from paper/result/perfect/non-polar/table-nopol.typ rename to paper/result/perfect/table-nopol.typ diff --git a/paper/appendix/representation/table-rep.typ b/paper/result/perfect/table-rep.typ similarity index 100% rename from paper/appendix/representation/table-rep.typ rename to paper/result/perfect/table-rep.typ