127 lines
8.9 KiB
XML
127 lines
8.9 KiB
XML
=== Phonons with Negligible Polarities
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对弱极性声子的理论分析，首先使用 Gamma 点的声子来近似，然后再讨论不同入射方向导致的差异。
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使用Gamma 点的声子来近似，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。
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Negligible-polar phonons were theoretically analyzed,
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starting with the approximation using phonons at the #sym.Gamma point,
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followed by a discussion of modifications arising from non-zero wavevectors.
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This approximation is based on the fact that
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the dispersion of these phonons is continuous and very close to the #sym.Gamma point,
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and has been widely adopted in the literature @_n-sic_2008.
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18 个声子属于 12 个表示。拉曼张量的形状可以确定，大小不能。
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使用对称性分析来从理论上研究声子。
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18 个 Gamma 点的弱极性声子包含了 C#sub[6v] 点群的12 个不可约表示（2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2]）。
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通过进一步考虑 C#sub[6v] 中简并表示（E1 和 E2）在 C#sub[2v] 中的表示，所有声子的拉曼张量的非零分量可以确定，如表所示。
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其中，B#sub[1] 模式具有零拉曼张量，不参与拉曼散射；
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其它表示的模式具有非零拉曼张量分量，可能可以在适当的偏振配置下在拉曼实验中观察到。
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然而，模式是否足够强以在实验中可见取决于其拉曼张量分量的大小，而仅通过对称性分析无法确定这些大小。
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Symmetry analysis was utilized
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to theoretically investigate the properties of 18 negligible-polar phonons at the #sym.Gamma point.
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These phonons correspond to twelve irreducible representations of the C#sub[6v] point group
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(2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2]).
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By further decomposing the doubly degenerate modes (E#sub[1] and E#sub[2] of C#sub[6v] group)
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in the C#sub[2v] point group,
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the form of the 18 negligible-polar phonons' Raman tensors can be determined, as summarized in @table-rep.
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Phonons of the B#sub[1] representation in C#sub[6v] possess zero Raman tensors
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and thus do not contribute to Raman scattering,
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while other phonons have non-zero Raman tensor components,
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making them potentially observable in Raman experiments under appropriate incidence and polarization configurations.
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It should be noted, however, that the observability in Raman experiment depends not only on the form of Raman tensor,
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but also on the magnitude of its Raman tensor components,
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which cannot be inferred from symmetry considerations alone.
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#include "table-rep.typ"
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我们提出了一个新的办法来估计拉曼张量大小。
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我们提出了一个办法来快速估计拉曼张量的大小。
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这个办法基于对称性分析，并加入了以下假设：
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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 $epsilon_i$ $eta_i$ $zeta_i$)。
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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。
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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。
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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，这与实验一致。
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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）。
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A method to estimate the magnitudes of the Raman tensors of each mode from their vibration patterns (eigenvectors)
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was proposed (see appendix for details).
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This approach was founded on the symmetry analysis and incorporates the assumption
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that the primary contribution from each atom to the Raman tensor arises from its nearest neighbors (denoted as $a_i$),
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while contributions from more distant atoms are much smaller (denoted as $epsilon_i$, $eta_i$, and $zeta_i$).
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Furthermore, the absolute amplitude differences among atoms of the same type within a phonon mode was neglected,
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and only their vibrational directions were considered.
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This enables a preliminary estimation of the Raman tensor magnitudes prior to detailed first-principles calculations,
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with the results summarized in @table-nopol.
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Our analysis gave the result that the E#sub[2]-3 mode (at about 756.25 cm#super[-1] in simulation)
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should possess a much higher Raman scattering intensity than the others,
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which is consistent with experimental observations,
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where at about 776 cm#super[-1] a very strong E#sub[2] peak is observed.
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Our result showed that
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the high Raman intensity of this mode arises from the constructive interference of contributions from all bonds,
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in contrast to other negligible-polar modes where contributions tend to cancel each other out.
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#include "table-nopol.typ"
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我们使用第一性原理计算得到了频率和拉曼张量的大小，并与我们的结果进行了比较。
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声子频率和拉曼张量的大小被使用第一性原理计算，并与实验结果和理论预测进行了比较（@table-nopol）。
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计算的声子频率与实验数据有很好的吻合，误差在 2-5% 之间，这个误差可能是由于 PBE 泛函对原子间力的低估（cite）。
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计算的拉曼张量也与实验和理论结果基本一致，这包括强度最高的模式 E#sub[2]-3，
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其次是四个强度较低但在实验中清晰可见的模式，包括 E#sub[2]-1、E#sub[2]-2、E#sub[1]-1 和 A#sub[1]-1。
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E#sub[1]-2 和 E#sub[2]-4 模式位于最强模式 E#sub[2]-3 附近，且具有很弱的拉曼强度（分别为最强模式的0.1%和0.6%），
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使得它们在实验光谱中难以区分。
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此外，A#sub[1]-2 模式在基面极化配置（xx 和 yy，仅为 0.01）中具有非常弱的拉曼强度，这导致它在正入射的拉曼实验中通常不可见；
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但当偏振沿 z 轴时（1.78）则显示出可观测的强度。
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// E#sub[1]-1 模式在理论上无法在正入射中观察到，但在实验中仍然可以看到一个小峰。
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// 这被认为是因为入射光并非完全沿z轴入射，由于衬底斜切和共聚焦汇聚角。通过向不同方向倾斜衬底，我们可以使这个峰变高或变低，如附图所示，这印证了我们的猜想。
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The Raman tensors and frequencies of the negligible-polar phonons were calculated using first-principles methods,
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and the results were compared with both experimental data and theoretical predictions (@table-nopol).
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The calculated phonon frequencies were in good agreement with experimental values,
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with a slight underestimation of 2-5%,
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which might be attributed to the known tendency of the PBE functional underestimating interatomic forces (cite).
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The calculated Raman tensors were also consistent with both experimental and theoretical results.
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Among negligible-polar modes, the E#sub[2]-3 mode exhibited the highest Raman intensity,
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followed by four modes with lower intensities that were also visible in normal incidence Raman experiments,
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including the E#sub[2]-1 mode, the E#sub[2]-2 mode, the E#sub[1]-1 mode, and the A#sub[1]-1 mode.
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The E#sub[1]-2 mode and E#sub[2]-4 mode were not visible in our Raman experiments,
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as they were located close to the most intense E#sub[2]-3 mode (only about 10 cm#super[-1] away)
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with very weak Raman intensities (only 0.1% and 0.6% of the E#sub[2]-3 mode, respectively).
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Additionally,
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the A#sub[1]-2 mode exhibited a very weak Raman intensity under the incident light with in-plane polarization
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(only 0.01),
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but showed an observable intensity when the polarization is along the z-axis (1.78).
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Thus, it was not typically observed in normal incidence Raman experiments (cite),
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but could be clearly detected in edge incidence configurations in our experiments.
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// TODO: 这里缺两个引用
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不同方向的入射配置会导致微小但可观测的峰位差异。
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非零长度的波矢（i.e. 参与散射的声子不在 Gamma 点）导致不同入射配置的峰位具有微小但可观测的差异，如色散图所示。
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相比于正入射，肩入射时，E2-1与E2-2的间距会减小、E2-2会展宽；E2-3会展宽，同时略微蓝移动。
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我们的计算结果为xxx，实验结果为xxx。
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// 198.43184855558061
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// 199.03816532139587
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// 257.87910853505343
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// 258.4149982771832
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// 其它峰在其它章节中解释。
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//
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// Besides, there are other peeks in the experiment.
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// The peek at 796 and 980 are caused by strong-polar phonons which will be discussed later.
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// Besides, there are small peeks at xxx,
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// which could not be explained in perfect 4H-SiC and will be discussed in the next section.
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// TODO: 将一部分 phonons 改为 phonon modes
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// 在论文中我们这样来称呼：phonon 对应某一个特征向量，而 modes 对应于一个子空间。
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// 也就是说，简并的里面有两个或者无数个 phonon，但只有一个 mode
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// #include "figure-raman.typ"
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// TODO: 解释为什么 E1 可以看到
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