124 lines
8.2 KiB
XML
124 lines
8.2 KiB
XML
== Phonons in Perfect 4H-SiC
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These phonons were categorized into two groups and discussed separately, according to their electrical polarities:
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negligible-polarity phonons (i.e., zero or very weak electrical polarity), and strong-polarity phonons.
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=== Negligible-polarity Phonons
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The negligible-polarity phonons were initially analyzed at the #sym.Gamma point,
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disregarding the incidence configurations.
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This simplification is justified because the participating negligible-polarity phonon modes
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are nearly identical across all Raman experiments,
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with their frequencies differing by only #sym.tilde 0.1 cm#super[-1] difference in frequency
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(see gray lines in @figure-discont).
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There are eight Raman-active negligible-polar modes in 4H-SiC,
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corresponding to three irreducible representations of the C#sub[6v] group (A#sub[1], E#sub[1], and E#sub[2]).
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We named these modes as
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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,
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in order of increasing frequency.
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The Raman tensor forms of each mode were derived by further considering their representations in the C#sub[2v] group,
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and were summarized in @table-rep.
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#include "figure-discont.typ"
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#include "table-rep.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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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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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],
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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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#include "table-nopol.typ"
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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 $i in {1, 2, 3, 4}$)
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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 $i in {1, 2, 3, 4}$,
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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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Notably, the E#sub[2]-3 mode was the only mode that retains the $a_i$ term,
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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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thereby explaining the exceptionally high Raman tensor magnitude observed for the E#sub[2]-3 mode.
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To further investigate the Raman spectra,
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an 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 remained 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 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."
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#include "figure-e12.typ"
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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 figure and table.
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Our results further confirmed the accuracy of both our experiments and calculations.
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#include "figure-nopo-diff.typ"
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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。
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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。
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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。
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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。
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这可以被解释为频率移动的原因不同。
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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；
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而 A1 模式具有弱但非零的极性，它在 Gamma 点附近不连续，
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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。
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在 A1-1 模式中，阶跃大小占主导。
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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。
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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。
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在强极性声子中：
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在每一个入射方向的拉曼实验中，参与散射的声子都包含三个强极性声子模式：
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一个 LO，它的振动方向roughly平行于入射光方向；和两个 TO 模式，它们的振动方向roughly垂直于入射光方向。
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它们的性质（振动pattern，频率，拉曼张量）会dramatically改变accroding to 入射方向。
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例如，在正入射中，LO 模式属于 C6v 群中的 A1 表示（Named as n-LO），两个 TO 模式简并为 C6v 群中的 E1 表示（Named as n-TO）；
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在沿x方向的侧入射中，LO 模式属于 C2v 群中的 B1 表示（Named as e-LO），两个 TO 分别属于 C2v 群中的 B2 和 A1 表示（Named as e-TO-x 和 e-TO-y）。
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其中，正入射的 LO 会与电子耦合形成 LOPC 模式，从而导致频率的显著变化；侧入射的 LO 则由于对称性的原因，无法观测到。
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我们将它们的结果总结在表中。可以看到，我们的结果与实验和计算吻合较好。
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我们将声子被分为两类讨论：
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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；
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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。
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// #include "non-polar/default.typ"
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// #include "polar/default.typ"
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