158 lines
12 KiB
XML
158 lines
12 KiB
XML
=== Phonons with Negligible Polarities
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对弱极性声子的分析首先从理论出发，然后结合
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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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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
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should possess a much higher Raman scattering intensity than the others,
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which is consistent with experimental and first-principles calculation results in the following text.
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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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我们做了计算，估计哪些峰可以看到、哪些峰看不到。
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使用第一性原理计算得到了声子频率和拉曼张量的大小。
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我们的计算表明，E#sub[2]-1 E#sub[2]-2 E#sub[1]-1 A#sub[1]-1 E#sub[2]-3 模式的拉曼强度较高，并且被报道在绝大多数的拉曼实验中可以看到。
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A#sub[1]-2 被报道在一些实验中可以观测到而在另外一些实验中不可以观测到(cite)。
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我们的计算表明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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因此在正入射拉曼实验中可能难以看到，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 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%），在报道的实验中都不可见。
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E#sub[1]-2 与最强模式 E#sub[2]-3 表示不同，我们认为，通过选取合适的偏振配置，可能可以在实验中观察到。
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The Raman tensors and frequencies of the negligible-polar phonons were calculated using first-principles methods.
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The results showed that
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the E#sub[2]-1, E#sub[2]-2, E#sub[1]-1, A#sub[1]-1, and E#sub[2]-3 modes possess relatively high Raman intensities
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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 showed 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 had only 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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我们做了实验，比别人多看到了几个峰；和计算结果比对良好。
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拉曼散射实验在衬底上进行，并与第一性原理计算结果进行了比较。
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在我们的实验中，除了 E#sub[2]-4 以外的弱极性模式对应的拉曼散射峰都被观察到，多于其它实验；
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这包括 A#sub[1]-2 在 xzzx 下可以观察到，
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以及 E#sub[1]-2 在 xzyx 下通过额外延长积分时间而观察到。
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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。
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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 showed 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 had only 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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#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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