175 lines
12 KiB
XML
175 lines
12 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 was justified because
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the negligible-polarity phonon modes participated in Raman scattering
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were nearly identical across all incidence configurations,
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with their frequencies differing by only #sym.tilde 0.1 cm#super[-1]
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(see the intersection of gray solid lines and orange dashed lines in @figure-discont b and c).
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There were eight Raman-active negligible-polarity modes in 4H-SiC at the #sym.Gamma point,
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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 (see @figure-discont and @figure-raman).
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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 "figure-raman.typ"
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#include "table-rep.typ"
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// negligible-polarity modes (only modes visible in our experiments are labeled)
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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] 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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#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 $i in {1, 2, 5, 6}$)
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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.
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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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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-nopo-diff.
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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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=== Strong-polar Phonons
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沿着不同方向入射的话，强声子的模式是不同的。
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// 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。
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// 这导致不同方向的入射/散射光的声子模式不同。
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// 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。
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// 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）。
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// 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；
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// 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。
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// 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。
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// (没有在图上表示)
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半导体中的强极性声子模式强烈地依赖于入射光的方向，
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这是由于半导体中原子之间的长程库伦相互作用所致，
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表现在散射光谱中，Gamma点附近不连续（见@figure-discont）。
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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。
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这时会有一个 TO 模式（C6v 中的 E1）和一个 LO 模式（C6v 中的 A1）。
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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；
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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）。
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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。
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Strong-polarity phonon modes caused by different incident light directions are different,
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due to long-range Coulomb interactions between atoms in semiconductors,
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showing discontinuity in the scattering spectra near the #sym.Gamma point (see @figure-discont).
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For incident light propagating along the z direction (phonon modes on the A-#sym.Gamma line),
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symmetry of C#sub[6v] point group applies and leading to two modes (two peeks in Raman spectra),
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including an E#sub[1] mode (pink line in @figure-discont, vibration in-plane)
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and an A#sub[1] mode (green line in @figure-discont, vibration along z-direction).
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When the light is incident along other directions, symmetry in plane was broken and C#sub[6v] symmetry no longer holds,
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and there will be three phonon modes in theory.
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For example, when the light is incident along the y direction (phonon modes on the #{sym.Gamma}-K line),
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symmetry of C#sub[2v] applies and three modes exist in dispersion curves,
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including an A#sub[1] mode (green line in @figure-discont, vibration along z direction),
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a B#sub[2] mode (blue line in @figure-discont, vibration along x direction),
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and a B#sub[1] mode (red line in @figure-discont, vibration in y direction).
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When the light is incident along a direction between z and y,
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three phonon modes will exist, but vibration in the mixed direction.
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将理论/计算结果与实验对比。
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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。
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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。
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Many Raman experiments on 4H-SiC with incident light along the z direction have observed two peaks.
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However, no experiments have reported three peaks with incident light along other directions.
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In our experiment, we found the third, and it satisfied properties we expected.
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In our experiments, we found that the third peak only appears when focusing inside the sample.
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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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#include "table-pol.typ"
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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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