=== Phonons with Negligible Polarities

对弱极性声子的分析的传统理论分析。（Gamma 近似，C6v表示）

对弱极性声子的理论分析，使用 Gamma 点的声子来近似。
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。
对称性分析从理论上指出了这些声子属于 C6v 的 12 个表示。
其中，B1 无活性，其它则有可能在拉曼实验中观察到。
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。

The negligible-polar phonons were theoretically analyzed by approximating their properties at the #sym.Gamma point.
This approximation is based on the fact that
    the dispersion of these phonons is continuous and very close to the #sym.Gamma point,
  and has been widely adopted in the literature @_n-sic_2008. 
Symmetry analysis pointed out that
  these phonons correspond to twelve irreducible representations of the C#sub[6v] point group and were named accrodingly
  (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,
  see appendix for details).
Among which the B#sub[1] representation is Raman inactive,
  while other eight modes are potentially observable in Raman experiments.
However, it should be noted that the intensity of Raman scattering from each mode
  cannot be inferred from symmetry considerations alone.

我们提出了一个新的办法来估计拉曼张量大小。

我们提出了一个办法来快速估计拉曼张量的大小。
这个办法基于对称性分析，并加入了以下假设：
  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 $epsilon_i$ $eta_i$ $zeta_i$)。
  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。
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。
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。
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）。

To overcome this limitation,
  a method to estimate the magnitudes of the Raman tensors of each mode was proposed (see appendix for details).
This approach was founded on the symmetry analysis and incorporates the assumption
  that the primary contribution from each atom to the Raman tensor arises from its nearest neighbors (denoted as $a_i$),
  while contributions from more distant atoms are much smaller (denoted as $epsilon_i$, $eta_i$, and $zeta_i$).
Furthermore, the absolute amplitude differences among atoms of the same type within a phonon mode was neglected,
  and only their vibrational directions were considered.
This enables a preliminary estimation of the Raman tensor magnitudes prior to expensive first-principles calculations,
  with the results summarized in @table-nopol.
Our analysis gave the result that the E#sub[2]-3 mode possess a much higher Raman intensity
    and should be clearly observable in Raman experiments,
  while the observability of other modes depends on their frequencies were well-separated or not
  and could only be confirmed by further first-principles calculations or experiments.
Our result showed that
  the high Raman intensity of this mode arises from the constructive interference of contributions from all bonds,
  in contrast to other negligible-polar modes where contributions tend to cancel each other out.

#include "table-nopol.typ"

实验看到了7个峰，比别人多。

我们做了实验，看到了其中 7 个峰，比别人的实验要多。
为了解释实验结果，我们进行了第一性原理计算。
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。
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；
  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。
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。
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。
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。

Raman experiments were established and seven negligible-polar phonon peaks were observed,
  which is more than all previous experiments (where only five or six peaks were typically reported).
To further understand our experimental results,
  first-principles calculations were performed,
  and the result was compared with experimental data.
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
  had relatively high Raman intensities and well-separated frequencies,
  making them observed in our experiments as well as most previous experiments.
The A#sub[1]-2 mode had only a very weak Raman intensity with in-plane polarization (e.g., zyy#overline[z]),
  while showed an observable intensity when the polarization is along the z-axis (e.g., xzzx).
Thus, it could be observed clearly in our edge incidence experiments (xzzx),
  but hardly seen in normal incidence configurations.
This peek was reported to be observable in some experiments (cite) but not in others (cite),
  our calculation provided an explanation for this discrepancy.
The E#sub[1]-2 and E#sub[2]-4 modes are calculated
  to be located close to the most intense E#sub[2]-3 mode (< 10 cm#super[-1] away)
  and exhibit very weak Raman intensities (only 0.1% and 0.6% of the E#sub[2]-3 mode, respectively),
  making them hard to observe in reported experiments.
However, in our experiments,
  with edge incidence configuration (zxy#overline[z]) and extended integration time,
  the E#sub[2]-3 peek was suppressed and the E#sub[1]-2 mode was also observed.
Our experiments reported the E#sub[1]-2 peak for the first time,
  and the experimental results were in good agreement with both calculations and theoretical analysis.

#include "figure-raman.typ"

E#sub[1]-1 模式在正入射中可见。

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。
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。
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。
通过略微倾斜衬底可以补偿斜切导致的效应，如附图所示。

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。
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。
实验结果与计算结果基本一致，如图如表所示。

Some effects absent in previous theory were observed experimentally,
  including minor but observable peak shifts of negligible-polar phonons between different incidence configurations,
  and the observability of the E#sub[1]-1 mode in normal incidence Raman experiments.
The peak shifts were considered caused by the non-zero wavevector lengths of phonons.
In our calculations, compared to normal incidence,
  the E#sub[2]-3 mode's frequency remained nearly unchanged,
  while the E#sub[2]-1, E#sub[2]-2, A#sub[1]-1 modes exhibited minor but observable shifts.
The experimental results were in good agreement with our calculations, as shown in figure and table.
The visibility of the E#sub[1]-1 mode in normal incidence Raman experiments were attributed to the fact that
  the incident light was not perfectly aligned along the z-axis,
  due to the substrate's slight tilt and the converging angle of the confocal setup.
This was confirmed by the results under different polarization configurations and different tilt angles,
  as shown in figure.

#include "figure-nopo-diff.typ"

#include "figure-e1.typ"
