100 lines
6.4 KiB
XML
100 lines
6.4 KiB
XML
=== Phonons with Negligible Polarities
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用 gamma 点的声子来近似。
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Gamma 点的声子被用于近似参与拉曼散射的弱极性声子，无论入射配置。
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这个近似是因为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。
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Phonons at the #sym.Gamma point were used to approximate negligible-polar phonons,
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regardless of the wavevector of the incident and scattered light.
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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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这些声子的拉曼张量的形状可以通过对称性分析来确定。
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Gamma 点声子满足 C#sub[6v] 点群对称性，18 个模式包含了 2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2] 共 12 个表示；
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其中简并表示（E1 和 E2）的拉曼张量的形状可以通过进一步考虑在 C#sub[2v] 点群中的表示来确定，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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// Representation of these 18 phonons, and the shape of their Raman tensors could be determined in advance.)
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Phonons at the #sym.Gamma point satisfy the C#sub[6v] point group symmetry,
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and the 18 negligible-polar phonons correspond to 12 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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Phonons belonging to the A#sub[1] and B#sub[1] representations vibrate along the z-axis and are non-degenerate,
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while those belonging to the E#sub[1] and E#sub[2] representations vibrate in-plane and are doubly degenerate.
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Phonons of the B#sub[1] representation are Raman-inactive, as their Raman tensors vanish.
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In contrast, phonons of the other representations are Raman-active,
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and the non-zero components of their Raman tensor
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can be determined by further considering their representation in the C#sub[2v] point group (see @table-rep).
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These Raman-active phonons are potentially be visible in Raman experiment under appropriate polarization configurations.
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However, whether a mode is sufficiently strong to be experimentally visible
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depends on the magnitudes of its Raman tensor components,
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which cannot be determined solely from symmetry analysis.
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#include "table-rep.typ"
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// We propose a method to estimate the magnitudes of the Raman tensors of these phonons,
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// without first-principle calculations.
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// Here we only write out results, details are in appendix.
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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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We propose a method to estimate the magnitudes of the Raman tensors of these phonons based on symmetry analysis.
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This approach is founded on the assumption that the change in polarizability induced by atomic displacements in 4H-SiC
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is primarily determined by the first- and second-nearest neighbors of the atom and the sign of the atomic charge,
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while other factors (mass, bond length, etc.) only have small contributions.
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Consequently, the Raman tensors of the calculated phonon modes can be estimated
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before additional first-principles computations (see appendix for details),
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and the results are summarized in @table-nopol.
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The parameters $a_i$ exhibit significantly larger absolute values compared to $epsilon_i$, $eta_i$, and $zeta_i$,
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indicating the E#sub[2] mode at 756.25 cm#super[-1] in simulation (mode 8)
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possess a much higher Raman intensity than the others.
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我们使用第一性原理计算得到了拉曼张量的大小，并与我们的结果进行了比较。
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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 are compared with both experimental data and theoretical predictions (@table-nopol).
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The calculated phonon frequencies show good agreement with experimental data with a slight underestimation of 2-5%,
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where the error may be attributed to the underestimation of interatomic forces by the PBE functional (cite).
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The calculated Raman tensors are also consistent with experimental and theoretical results.
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Among negligible-polar modes, the E#sub[2] mode at 776 cm#super[-1] in experiment (mode 8)
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exhibits the highest Raman intensity,
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followed by four modes with lower intensities that are also experimentally visible,
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including the E#sub[2] modes at 195.5 cm#super[-1] (mode 1) and 203.3 cm#super[-1] (mode 2),
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the E#sub[1] mode at 269.7 cm#super[-1] (mode 3), and the A#sub[1] mode at 609.5 cm#super[-1] (mode 6).
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The E#sub[1] mode calculated at 746.91 cm#super[-1] (mode 7)
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and the E#sub[2] mode calculated at 756.25 cm#super[-1] (mode 9)
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are predicted to have much weaker Raman intensities and are located close to the most intense mode (mode 8),
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making them indistinguishable in experimental spectra.
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Additionally, the A#sub[1] mode calculated at 812.87 cm#super[-1] (mode 10)
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exhibits a very weak Raman intensity in the basal-plane polarized configurations (xx and yy, only 0.01)
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but shows an observable intensity when the polarization is along the z-axis (1.78).
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Since most Raman experiments are performed in a back-scattering configuration with light incident along the z-direction
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(i.e., with in-plane polarization)
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and with photon energies much lower than the band gap,
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this mode is typically not observed (cite).
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However, it should become detectable if the incident light has a polarization component along the z-direction
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(as in our experiment),
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or when the excitation wavelength approaches resonance conditions (cite).
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其它峰在其它章节中解释。
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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 "table-nopol.typ"
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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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