175 lines
10 KiB
XML
175 lines
10 KiB
XML
== Phonons in Perfect 4H-SiC
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// 第一段：弱极性与强极性声子模式表现截然不同。如图所示，弱极性声子模式几乎不依赖于波矢方向，而强极性声子模式则表现出显著的各向异性。
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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 (possessing zero or very weak polarity),
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and three strong-polarity modes.
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The negligible-polarity modes should exhibit minimal dependence on the wavevector direction,
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whereas the strong-polarity modes should show significant anisotropy.
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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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In contrast, the strong-polarity modes were designated based on their polarization relative to the wavevector.
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Since the laser incidence direction was restricted to the x-z plane in this work, these modes were labeled as follows:
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TO-xz (polarized in the x-z plane and roughly perpendicular to the wavevector),
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TO-y (polarized along the y-axis), and LO (roughly parallel to the wavevector).
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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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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, 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,
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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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making them observed in our experiments as well as most previous experiments.
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The A#sub[1]-2 mode was calculated to have very weak (0.01) and relatively strong (1.78) Raman intensity
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under in-plane polarization and z polarization, respectively,
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which is compatible with our experimental result
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that it could be observed clearly in the y(zz)#overline[y] configuration but hardly seen in our other experiments.
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This peek was reported to be observable in some experiments (cite) but not in others (cite),
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our calculation provided an explanation for this discrepancy.
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The E#sub[2]-4 modes was calculated
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to be located close to the most intense E#sub[2]-3 mode (< 10 cm#super[-1] away)
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and exhibit very weak Raman intensities (only 0.6% of the E#sub[2]-3 mode),
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making it not visible in our and all previous experiments.
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The E#sub[1]-2 mode was also located close to the E#sub[2]-3 mode and has weak Raman intensity,
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making it also unobservable in previous experiments.
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However, the E#sub[1]-2 mode was observable in our experiments of y(zx)#overline[y] with extended acquisition time,
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where the scattering of the E#sub[2]-3 mode was suppressed while that of the E#sub[1]-2 mode was enhanced,
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thanks to the different representations of these two modes.
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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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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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than those of other negligible-polarity modes (over 30 times larger than that of the second-strongest).
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This highlighted a significant gap in established theory
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that rigorous symmetry analysis could only predict the non-zero components of the Raman tensors,
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but not their magnitudes.
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In order to address the limitations of existing theories,
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a method for estimating the magnitudes of Raman tensors was proposed.
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By analyzing the local environment of individual atoms,
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this approach decomposed their contributions to the Raman tensor into two parts:
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a dominant component (invariant across similar environments, denoted as $a_i$，where $i in {1, 2, 5, 6}$)
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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 and @table-pol.
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Notably, the E#sub[2]-3 mode was the only mode that retains the $a_i$ term,determine
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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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The mode frequency dependence on the wavevector were thoroughly investigated,
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including both thoretical calculations (@figure-rev a), experimental measurements (@figure-rev b)
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and their comparisons (@figure-rev c).
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The E#sub[2]-3 mode frequency was calculated to remaine distinct in all incidence geometries (@figure-rev a),
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making it an ideal calibration reference for experiments.
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Meanwhile,
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the E#sub[2]-1, E#sub[2]-2 and A#sub[1]-1 mode showed a relatively larger dependence on the incidence direction,
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which is in good agreement with our experimental observations (@figure-rev c).
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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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#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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by comparing experimental and calculated results under various lazer incidence directions.
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The E#sub[2]-3 peak searved as a calibration reference under various experiments,
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since its position was calculated to be virtually invariant between normal and edge incidence
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(with a shift of only #sym.tilde 0.004 cm#super[-1]).
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The E#sub[2]-1, E#sub[2]-2, and A#sub[1]-1 modes exhibited observable shifts,
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and the experimental results were in good agreement with our calculations, as shown in fig.
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Our results further confirmed the accuracy of both our experiments and calculations.
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=== Strong-polarity Phonons
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The Strong-polarity phonon modes participated in Raman scattering
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exhibited significant variations depending on the incidence configurations
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(see the intersection of colored solid lines and orange dashed lines in @figure-discont b and c).
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For incident light propagating along the z direction,
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the C#sub[6v] point group applied and two modes were present,
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marked as normal-TO and normal-LO,
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and they were corresponding to the E#sub[1] and A#sub[1] representations
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and vibrations along the basle plane and z direction, respectively.
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The normal-LO would subsiquently couple with plasmons to form LOPC modes in n-type 4H-SiC.
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For incident light propagating along other directions,
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the C#sub[6v] group no longer held and three modes were present.
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Specifically, for incident light propagating along x direction, the C#sub[2v] group applied,
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and the three modes was named as edge-TO-z, edge-TO-y and edge-LO,
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which were corresponding to A#sub[1], B#sub[2] and B#sub[1] representations
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and vibrations along z, y and x directions, respectively.
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E1 的情况。
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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。
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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。
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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。
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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。
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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。
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其中某个模式会怎样怎样，另一个会怎样怎样。
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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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