105 lines
7.2 KiB
XML
105 lines
7.2 KiB
XML
// We investigate phonons at Gamma instead of the exact location near Gamma.
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Phonons at the #sym.Gamma point were used
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to approximate negligible-polar phonons that participating in Raman processes
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regardless of the wavevector of the incident and scattered light.
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This approximation is widely adopted (cite) and justified by the fact that,
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although the phonons participating in Raman processes are not these strictly located at the #sym.Gamma point,
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they are very close to the #sym.Gamma point in reciprocal space
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(about 0.01 nm#super[-1] in back-scattering configurations with 532 nm laser light,
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which corresponds to only 1% of the smallest reciprocal lattice vector of 4H-SiC,
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see orange dotted line in @figure-discont),
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and their dispersion at #sym.Gamma point is continuous with vanishing derivatives.
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Therefore, negligible-polar phonons involved in Raman processes
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have nearly indistinguishable properties from those at the #sym.Gamma point.
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#include "figure-discont.typ"
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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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// TODO: maybe it is better to assign Raman tensor to each bond, instead of atom
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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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As a result,
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the phonon modes with the strongest Raman intensities can be predicted
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prior to first-principles calculations and experiments (appendix),
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and the Raman tensors of the calculated phonon modes can be estimated
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before additional first-principles computations.
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Further details are provided in the appendix.
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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 very weak Raman intensity in the configuration polarized in the basal-plane (xx and yy, only 0.01)
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but an observable intensity in the configuration polarized along 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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(polarized in-plane)
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with photon energies much lower than the band gap,
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this mode is generally not observed (cite).
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However, it should become detectable when the incident light have z-direction polarization part (as in our experiment)
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or when the incident light wavelength is near resonance conditions (cite).
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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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The method only takes the vibration directions of each atom in each phonon mode,
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leaving the amplitudes unconsidered (see appendix for details),
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and the result was summarized in @table-predmode.
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In the Raman tensors in @table-predmode,
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$a_i$ corresponding to the change of polarizability caused by movement of the Si atoms in A and C layers,
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$epsilon_i$, $eta_i$ and $eta_i$ corresponding to the difference between different bilayers and different atoms.
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Due to the similarity of environment in different bilayers and around different atoms,
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the absolute values of $epsilon_i$, $eta_i$ and $zeta_i$ are expected to be much smaller than that of $a_i$,
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thus the Raman tensors containing $a_i$ are expected to be much larger than those not containing $a_i$.
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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 "raman.typ"
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// 实验与计算基本相符。对于声子频率，计算总是低估大约 3%。
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// 此外，一些较强的模式在预测无法看到的偏振中也可以看到。例如，一些在 xy 偏振中不应该看到的模式可以被看到了。
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// 这个现象可以由 4度的斜切所解释：我们将材料略微踮起一些角度，就可以使得该模式减小。
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// 这个现象也可以由材料或偏振片的微小角度来解释。
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// 例如，我们将偏振方向转动 5 度，就可以得到这个模拟结果。
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// 此外，由于使用的材料是沿着 c 轴切片的，所以我们在测量 y 入射时不得不将片子以略小于 90 度（约 75 度）的角度放置。这也导致实验与计算的偏差。
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// TODO: 翻译成英文 |