diff --git a/paper/result/default.typ b/paper/result/default.typ index 0b46482..9171643 100644 --- a/paper/result/default.typ +++ b/paper/result/default.typ @@ -1,11 +1,8 @@ = Results and Discussion -我们先讨论无缺陷的情况，再讨论有缺陷的情况。 -无缺陷的情况可以解释绝大多数拉曼信号，其它情况从拉曼中的小峰和峰的改变中看出。 - Phonon modes in defect-free 4H-SiC were first analyzed, which account for the majority of the observed Raman signals. -Subsequently, the effects of defects and charge carriers were addressed, +Subsequently, the effects of impurities and charge carriers were addressed, which manifested as additional minor peaks and modifications to the primary peaks in the Raman spectra. #include "perfect/default.typ" diff --git a/paper/result/perfect/default.typ b/paper/result/perfect/default.typ index c12c05f..667d2a5 100644 --- a/paper/result/perfect/default.typ +++ b/paper/result/perfect/default.typ @@ -7,22 +7,26 @@ These phonons were categorized into two groups and discussed separately, accordi 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). +This simplification was justified because + the negligible-polarity phonon modes participated in Raman scattering + were nearly identical across all incidence configurations, + with their frequencies differing by only #sym.tilde 0.1 cm#super[-1] + (see the intersection of gray solid lines and orange dashed lines in @figure-discont b and c). -There are eight Raman-active negligible-polar modes in 4H-SiC, +There were eight Raman-active negligible-polarity modes in 4H-SiC at the #sym.Gamma point, 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. + in order of increasing frequency (see @figure-discont and @figure-raman). 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 "figure-raman.typ" #include "table-rep.typ" +// negligible-polarity modes (only modes visible in our experiments are labeled) + 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). @@ -43,7 +47,7 @@ The E#sub[2]-4 modes was calculated 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], +However, the E#sub[1]-2 mode was observable in our experiments of y(zx)#overline[y] with extended acquisition time, 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, @@ -58,51 +62,98 @@ For example, the Raman tensor of the E#sub[2]-3 mode was substantially larger 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 $i in {1, 2, 3, 4}$) + a dominant component (invariant across similar environments, denoted as $a_i$，where $i in {1, 2, 5, 6}$) and several secondary components (reflecting environmental variations, - denoted as $epsilon_i$, $eta_i$, and $zeta_i$，where $i in {1, 2, 3, 4}$, + denoted as $epsilon_i$, $eta_i$, and $zeta_i$，where $i in {1, 2, 5, 6}$, 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. + which 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. + 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 +To achieve a more precise investigation of the Raman spectra + and prepare for analyzing impurity and charge carrier effects, + the 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 + since its position was calculated to be 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." - -#include "figure-e12.typ" - 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. + and the experimental results were in good agreement with our calculations, as shown in @fig-nopo-diff. 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。 +=== Strong-polar Phonons + +沿着不同方向入射的话，强声子的模式是不同的。 + +// 在半导体的极性声子模式中，原子间存在长距离的库伦相互作用，导致散射谱在 Gamma 附近不再连续（引用），如图中的彩色线所示。 +// 这导致不同方向的入射/散射光的声子模式不同。 +// 具体来说，当入射光/散射光沿着 z 方向时，起作用的是 A-Gamma 线上的声子模式（图中的左半边的橘线），它们适用于群 C6v。 +// 这时会有一个 E1 模式（TO，振动方向在面内）和一个 A1 模式（LO，沿 z 振动）。 +// 而当沿着 y 方向入射时，起作用的是 Gamma-K 线上的声子模式（图中的右半边的橘线），它们不再适用于群 C6v，而只适用于群 C2v； +// 它会分裂成沿x、y、z 方向的三个声子模式（图中的右半边的蓝线），它们分别对应于群 C2v 的 A1、B1 和 B2 表示 TODO: 确认这个几个表示的名字。 +// 若考虑到到入射光不是严格沿着 z 方向，而是有一个小的角度（例如 10 度），则此时有一个声子模式沿着 x 方向，另外两个声子模式则为 y-z 两个方向的混合。 +// (没有在图上表示) + +半导体中的强极性声子模式强烈地依赖于入射光的方向， + 这是由于半导体中原子之间的长程库伦相互作用所致， + 表现在散射光谱中，Gamma点附近不连续（见@figure-discont）。 +具体来说，当入射光沿着 z 方向时，起作用的是 A-Gamma 线上的声子模式（图中的上半部分的橘线），它们适用于群 C6v。 +这时会有一个 TO 模式（C6v 中的 E1）和一个 LO 模式（C6v 中的 A1）。 +而当沿着 y 方向入射时，起作用的是 Gamma-K 线上的声子模式（图中下半部分的橘线），它们不再适用于群 C6v，而只适用于群 C2v； +这时将会出现三个模式，包括一个LO（沿y方向振动，对应C2v中的A1）和两个TO模式（根据振动方向，命名为B2-x和B2-y，对应C2v中的B1和B2）。 +当入射光不是严格沿着坐标轴方向，而是在xOy面内呈现一定的夹角时（掠入射，以及考虑了斜切的正入射），则此时有一个声子模式沿着 x 方向，另外两个声子模式则为 y-z 两个方向的混合。 + +Strong-polarity phonon modes caused by different incident light directions are different, + due to long-range Coulomb interactions between atoms in semiconductors, + showing discontinuity in the scattering spectra near the #sym.Gamma point (see @figure-discont). +For incident light propagating along the z direction (phonon modes on the A-#sym.Gamma line), + symmetry of C#sub[6v] point group applies and leading to two modes (two peeks in Raman spectra), + including an E#sub[1] mode (pink line in @figure-discont, vibration in-plane) + and an A#sub[1] mode (green line in @figure-discont, vibration along z-direction). +When the light is incident along other directions, symmetry in plane was broken and C#sub[6v] symmetry no longer holds, + and there will be three phonon modes in theory. +For example, when the light is incident along the y direction (phonon modes on the #{sym.Gamma}-K line), + symmetry of C#sub[2v] applies and three modes exist in dispersion curves, + including an A#sub[1] mode (green line in @figure-discont, vibration along z direction), + a B#sub[2] mode (blue line in @figure-discont, vibration along x direction), + and a B#sub[1] mode (red line in @figure-discont, vibration in y direction). +When the light is incident along a direction between z and y, + three phonon modes will exist, but vibration in the mixed direction. + +将理论/计算结果与实验对比。 + +我们将计算与实验结果进行了对比。衬底中的LO峰与plasmon耦合形成LOPC峰，因此与计算结果大不相同。 +对于TO峰，在正入射中，它与E2-3模式的距离为xxx；在掠入射中，它与E2-3模式的距离为xxx。 + +Many Raman experiments on 4H-SiC with incident light along the z direction have observed two peaks. +However, no experiments have reported three peaks with incident light along other directions. +In our experiment, we found the third, and it satisfied properties we expected. +In our experiments, we found that the third peak only appears when focusing inside the sample. + +E1 的情况。 + +注意到在正入射中，理论上不能被观察到的E#sub[1]-1模式也被观察到了。 +与弱极性的 E1-1 模式类似，我们也认为这是由于入射光并非完全沿 z 轴入射所致。 +但与弱极性 E1-1 模式不同的是，强极性 E1-1 模式在 xy 的偏振下并没有更强反而更弱。 +这是因为E1这时不再是严格的E1模式，而是分裂成了两个相近的模式。 +我们的计算表明，在2度的入射角下，E1分裂的两个模式非常接近。 +其中某个模式会怎样怎样，另一个会怎样怎样。 + +// 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。 +// 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。 + +#include "table-pol.typ" + -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。 -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。 -这可以被解释为频率移动的原因不同。 -对于 E2 模式，其极性为零，色散曲线在 Gamma 点附近连续，因此频率取决于色散曲线在Gamma附近的曲率； -而 A1 模式具有弱但非零的极性，它在 Gamma 点附近不连续， - 计算频率偏移需要同时考虑色散曲线曲率和阶跃大小的影响。 -在 A1-1 模式中，阶跃大小占主导。 -而在重掺杂n型衬底中，阶跃大小被载流子屏蔽显著减小，因此导致理想模型的计算结果与实验结果不符。 -通过将载流子屏蔽效应纳入计算，计算结果与实验结果吻合较好，如图所示。 在强极性声子中： diff --git a/paper/result/perfect/non-polar/figure-raman.typ b/paper/result/perfect/figure-raman.typ similarity index 100% rename from paper/result/perfect/non-polar/figure-raman.typ rename to paper/result/perfect/figure-raman.typ diff --git a/paper/result/perfect/non-polar/default.typ b/paper/result/perfect/non-polar/default.typ deleted file mode 100644 index 2db1699..0000000 --- a/paper/result/perfect/non-polar/default.typ +++ /dev/null @@ -1,24 +0,0 @@ - - -我们提出了一个新的办法来估计拉曼张量大小。 - -我们提出了一个办法来快速估计拉曼张量的大小。 -这个办法基于对称性分析，并加入了以下假设： - 每个原子对拉曼张量的贡献主要取决于第一近邻原子（它们的贡献记为 $a_i$），更远的原子则归结为小量（记为 $epsilon_i$ $eta_i$ $zeta_i$)。 - 此外，我们忽略了同一个振动模式中，同种原子振幅的绝对值的差异，只考虑它们振动方向的不同。 -因此，拉曼张量的大小可以在进一步的第一性原理计算之前给出，结果总结在表中。 -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）。 - - - - - -#include "figure-raman.typ" - -不同入射方向下有微小移动。 - -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，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。 - diff --git a/paper/result/perfect/polar/default.typ b/paper/result/perfect/polar/default.typ deleted file mode 100644 index a0d4887..0000000 --- a/paper/result/perfect/polar/default.typ +++ /dev/null @@ -1,62 +0,0 @@ -=== Strong-polar Phonons - -沿着不同方向入射的话，强声子的模式是不同的。 - -// 在半导体的极性声子模式中，原子间存在长距离的库伦相互作用，导致散射谱在 Gamma 附近不再连续（引用），如图中的彩色线所示。 -// 这导致不同方向的入射/散射光的声子模式不同。 -// 具体来说，当入射光/散射光沿着 z 方向时，起作用的是 A-Gamma 线上的声子模式（图中的左半边的橘线），它们适用于群 C6v。 -// 这时会有一个 E1 模式（TO，振动方向在面内）和一个 A1 模式（LO，沿 z 振动）。 -// 而当沿着 y 方向入射时，起作用的是 Gamma-K 线上的声子模式（图中的右半边的橘线），它们不再适用于群 C6v，而只适用于群 C2v； -// 它会分裂成沿x、y、z 方向的三个声子模式（图中的右半边的蓝线），它们分别对应于群 C2v 的 A1、B1 和 B2 表示 TODO: 确认这个几个表示的名字。 -// 若考虑到到入射光不是严格沿着 z 方向，而是有一个小的角度（例如 10 度），则此时有一个声子模式沿着 x 方向，另外两个声子模式则为 y-z 两个方向的混合。 -// (没有在图上表示) - -半导体中的强极性声子模式强烈地依赖于入射光的方向， - 这是由于半导体中原子之间的长程库伦相互作用所致， - 表现在散射光谱中，Gamma点附近不连续（见@figure-discont）。 -具体来说，当入射光沿着 z 方向时，起作用的是 A-Gamma 线上的声子模式（图中的上半部分的橘线），它们适用于群 C6v。 -这时会有一个 TO 模式（C6v 中的 E1）和一个 LO 模式（C6v 中的 A1）。 -而当沿着 y 方向入射时，起作用的是 Gamma-K 线上的声子模式（图中下半部分的橘线），它们不再适用于群 C6v，而只适用于群 C2v； -这时将会出现三个模式，包括一个LO（沿y方向振动，对应C2v中的A1）和两个TO模式（根据振动方向，命名为B2-x和B2-y，对应C2v中的B1和B2）。 -当入射光不是严格沿着坐标轴方向，而是在xOy面内呈现一定的夹角时（掠入射，以及考虑了斜切的正入射），则此时有一个声子模式沿着 x 方向，另外两个声子模式则为 y-z 两个方向的混合。 - -Strong-polar phonon modes caused by different incident light directions are different, - due to long-range Coulomb interactions between atoms in semiconductors, - showing discontinuity in the scattering spectra near the #sym.Gamma point (see @figure-discont). -For incident light propagating along the z direction (phonon modes on the A-#sym.Gamma line), - symmetry of C#sub[6v] point group applies and leading to two modes (two peeks in Raman spectra), - including an E#sub[1] mode (pink line in @figure-discont, vibration in-plane) - and an A#sub[1] mode (green line in @figure-discont, vibration along z-direction). -When the light is incident along other directions, symmetry in plane was broken and C#sub[6v] symmetry no longer holds, - and there will be three phonon modes in theory. -For example, when the light is incident along the y direction (phonon modes on the #{sym.Gamma}-K line), - symmetry of C#sub[2v] applies and three modes exist in dispersion curves, - including an A#sub[1] mode (green line in @figure-discont, vibration along z direction), - a B#sub[2] mode (blue line in @figure-discont, vibration along x direction), - and a B#sub[1] mode (red line in @figure-discont, vibration in y direction). -When the light is incident along a direction between z and y, - three phonon modes will exist, but vibration in the mixed direction. - -将理论/计算结果与实验对比。 - -我们将计算与实验结果进行了对比。衬底中的LO峰与plasmon耦合形成LOPC峰，因此与计算结果大不相同。 -对于TO峰，在正入射中，它与E2-3模式的距离为xxx；在掠入射中，它与E2-3模式的距离为xxx。 - -Many Raman experiments on 4H-SiC with incident light along the z direction have observed two peaks. -However, no experiments have reported three peaks with incident light along other directions. -In our experiment, we found the third, and it satisfied properties we expected. -In our experiments, we found that the third peak only appears when focusing inside the sample. - -E1 的情况。 - -注意到在正入射中，理论上不能被观察到的E#sub[1]-1模式也被观察到了。 -与弱极性的 E1-1 模式类似，我们也认为这是由于入射光并非完全沿 z 轴入射所致。 -但与弱极性 E1-1 模式不同的是，强极性 E1-1 模式在 xy 的偏振下并没有更强反而更弱。 -这是因为E1这时不再是严格的E1模式，而是分裂成了两个相近的模式。 -我们的计算表明，在2度的入射角下，E1分裂的两个模式非常接近。 -其中某个模式会怎样怎样，另一个会怎样怎样。 - -// 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。 -// 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。 - -#include "table-pol.typ" diff --git a/paper/result/perfect/polar/table-pol.typ b/paper/result/perfect/table-pol.typ similarity index 100% rename from paper/result/perfect/polar/table-pol.typ rename to paper/result/perfect/table-pol.typ