118 lines
8.5 KiB
XML
118 lines
8.5 KiB
XML
=== Phonons with Negligible Polarities
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对弱极性声子的分析的传统理论分析。（Gamma 近似，C6v表示）
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对弱极性声子的理论分析，使用 Gamma 点的声子来近似。
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这基于这样的事实：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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对称性分析从理论上指出了这些声子属于 C6v 的 12 个表示。
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其中，B1 无活性，其它则有可能在拉曼实验中观察到。
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然而，模式的拉曼散射强弱无法仅从对称性分析中推断出来。
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The negligible-polar phonons were theoretically analyzed by approximating their properties at the #sym.Gamma point.
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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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Symmetry analysis pointed out that
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these phonons correspond to twelve irreducible representations of the C#sub[6v] point group and were named accrodingly
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(A#sub[1]-1 to A#sub[1]-2, B#sub[1]-1 to B#sub[1]-4, E#sub[1]-1 to E#sub[1]-2, and E#sub[2]-1 to E#sub[2]-4,
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see appendix for details).
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Among which the B#sub[1] representation is Raman inactive,
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while other eight modes are potentially observable in Raman experiments.
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However, it should be noted that the intensity of Raman scattering from each mode
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cannot be inferred from symmetry considerations alone.
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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，这与实验和第一性原理计算结果一致。
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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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To overcome this limitation,
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a method to estimate the magnitudes of the Raman tensors of each mode 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 expensive 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 possess a much higher Raman intensity
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and should be clearly observable in Raman experiments,
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while the observability of other modes depends on their frequencies were well-separated or not
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and could only be confirmed by further first-principles calculations or experiments.
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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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实验看到了7个峰，比别人多。
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我们做了实验，看到了其中 7 个峰，比别人的实验要多。
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为了解释实验结果，我们进行了第一性原理计算。
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E#sub[2]-1 E#sub[2]-2 E#sub[1]-1 A#sub[1]-1 E#sub[2]-3 这几个模式在大多数实验中都可以看到，这是因为它们有较高的拉曼强度且位置比较分散。
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A#sub[1]-2 在一些实验中可以观察到而在另外一些实验中不可以观察到(cite)，这是因为它在面内偏振下拉曼散射强度非常弱，但在 z 轴偏振下有可观测的强度；
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因此在我们的正入射拉曼实验中它很难被看到，但在侧入射时通过选取合适的偏振配置应该可以观察到。
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E#sub[1]-2 和 E#sub[2]-4 距离最强模式 E#sub[2]-3 很近，且具有很弱的拉曼强度，在报道的实验中都不可见。
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在我们的实验中，通过选取合适的偏振配置和延长积分时间，E#sub[2]-3 被压制后，E#sub[1]-2 也被观察到了。
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我们的实验第一次指出了 E#sub[1]-2 峰，并且与计算和理论吻合较好。
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Raman experiments were established and seven negligible-polar phonon peaks were 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 further understand our experimental results,
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first-principles calculations were performed,
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and the result was compared with experimental data.
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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 had only a very weak Raman intensity with in-plane polarization (e.g., zyy#overline[z]),
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while showed an observable intensity when the polarization is along the z-axis (e.g., xzzx).
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Thus, it could be observed clearly in our edge incidence experiments (xzzx),
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but hardly seen in normal incidence configurations.
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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[1]-2 and E#sub[2]-4 modes are 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 0.1% and 0.6% of the E#sub[2]-3 mode, respectively),
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making them hard to observe in reported experiments.
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However, in our experiments,
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with edge incidence configuration (zxy#overline[z]) and extended integration time,
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the E#sub[2]-3 peek was suppressed and the E#sub[1]-2 mode was also observed.
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Our experiments reported the E#sub[1]-2 peak for the first time,
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and the experimental results were in good agreement with both calculations and theoretical analysis.
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#include "figure-raman.typ"
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E#sub[1]-1 模式在正入射中可见。
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E#sub[1]-1 模式在正入射拉曼实验中被观察到尽管在理论上是不允许的，这被认为是因为入射光并非完全沿c轴入射。
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如图所示，由于非零的数值孔径，激光汇聚到样品上时存在一个锥形角度（不完全平行于c轴），使得 E#sub[1]-1 模式在正入射中总是可见的。
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由于衬底斜切（沿着x方向），对于 zxxz 和 zxyz 配置，入射光或散射光的偏振将具有更多的 z 分量，从而使 E#sub[1]-1 模式的强度增加。
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通过略微倾斜衬底可以补偿斜切导致的效应，如附图所示。
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不同入射方向下弱极性峰位的微小移动被观察到，这是由于非零波矢长度的声子引起的。
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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。
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实验结果与计算结果基本一致，如图如表所示。
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Some effects absent in previous theory were observed experimentally,
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including minor but observable peak shifts of negligible-polar phonons between different incidence configurations,
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and the observability of the E#sub[1]-1 mode in normal incidence Raman experiments.
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The peak shifts were considered caused by the non-zero wavevector lengths of phonons.
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In our calculations, compared to normal incidence,
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the E#sub[2]-3 mode's frequency remained nearly unchanged,
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while the E#sub[2]-1, E#sub[2]-2, A#sub[1]-1 modes exhibited minor but observable shifts.
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The experimental results were in good agreement with our calculations, as shown in figure and table.
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The visibility of the E#sub[1]-1 mode in normal incidence Raman experiments were attributed to the fact that
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the incident light was not perfectly aligned along the z-axis,
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due to the substrate's slight tilt and the converging angle of the confocal setup.
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This was confirmed by the results under different polarization configurations and different tilt angles,
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as shown in figure.
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#include "figure-nopo-diff.typ"
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#include "figure-e1.typ"
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