100 lines
6.2 KiB
XML
100 lines
6.2 KiB
XML
=== Phonons with Negligible Polarities
|
||
|
||
我们用 gamma 点的声子来近似弱极性的声子。
|
||
|
||
// We investigate phonons at Gamma instead of the exact location near Gamma.
|
||
Phonons at the #sym.Gamma point were used
|
||
to approximate negligible-polar phonons that participating in Raman processes
|
||
regardless of the wavevector of the incident and scattered light.
|
||
This approximation is widely adopted (cite) and justified by the fact that,
|
||
although the phonons participating in Raman processes are not these strictly located at the #sym.Gamma point,
|
||
they are very close to the #sym.Gamma point in reciprocal space
|
||
(about 0.01 nm#super[-1] in back-scattering configurations with 532 nm laser light,
|
||
which corresponds to only 1% of the smallest reciprocal lattice vector of 4H-SiC,
|
||
see orange dotted line in @figure-discont),
|
||
and their dispersion at #sym.Gamma point is continuous with vanishing derivatives.
|
||
Therefore, negligible-polar phonons involved in Raman processes
|
||
have nearly indistinguishable properties from those at the #sym.Gamma point.
|
||
|
||
#include "figure-discont.typ"
|
||
|
||
gamma 点处 18 个声子的表示。它们的拉曼张量的形状可以确定，但大小无法确定。
|
||
|
||
// Representation of these 18 phonons, and the shape of their Raman tensors could be determined in advance.)
|
||
Phonons at the #sym.Gamma point satisfy the C#sub[6v] point group symmetry,
|
||
and the 18 negligible-polar phonons correspond to 12 irreducible representations of the C#sub[6v] point group:
|
||
2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2].
|
||
Phonons belonging to the A#sub[1] and B#sub[1] representations vibrate along the z-axis and are non-degenerate,
|
||
while those belonging to the E#sub[1] and E#sub[2] representations vibrate in-plane and are doubly degenerate.
|
||
Phonons of the B#sub[1] representation are Raman-inactive, as their Raman tensors vanish.
|
||
In contrast, phonons of the other representations are Raman-active,
|
||
and the non-zero components of their Raman tensor
|
||
can be determined by further considering their representation in the C#sub[2v] point group (see @table-rep).
|
||
These Raman-active phonons are potentially be visible in Raman experiment under appropriate polarization configurations.
|
||
However, whether a mode is sufficiently strong to be experimentally visible
|
||
depends on the magnitudes of its Raman tensor components,
|
||
which cannot be determined solely from symmetry analysis.
|
||
|
||
#include "table-rep.typ"
|
||
|
||
// We propose a method to estimate the magnitudes of the Raman tensors of these phonons,
|
||
// without first-principle calculations.
|
||
// Here we only write out results, details are in appendix.
|
||
|
||
我们提出了一个新的办法来估计拉曼张量大小。
|
||
|
||
We propose a method to estimate the magnitudes of the Raman tensors of these phonons based on symmetry analysis.
|
||
This approach is founded on the assumption that the change in polarizability induced by atomic displacements in 4H-SiC
|
||
is primarily determined by the first- and second-nearest neighbors of the atom and the sign of the atomic charge,
|
||
while other factors (mass, bond length, etc.) only have small contributions.
|
||
Consequently, the Raman tensors of the calculated phonon modes can be estimated
|
||
before additional first-principles computations (see appendix for details),
|
||
and the results are summarized in @table-nopol.
|
||
The parameters $a_i$ exhibit significantly larger absolute values compared to $epsilon_i$, $eta_i$, and $zeta_i$,
|
||
indicating the E#sub[2] mode at 756.25 cm#super[-1] in simulation (mode 8)
|
||
possess a much higher Raman intensity than the others.
|
||
|
||
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。
|
||
|
||
The Raman tensors and frequencies of the negligible-polar phonons were calculated using first-principles methods,
|
||
and the results are compared with both experimental data and theoretical predictions (@table-nopol).
|
||
The calculated phonon frequencies show good agreement with experimental data with a slight underestimation of 2-5%,
|
||
where the error may be attributed to the underestimation of interatomic forces by the PBE functional (cite).
|
||
The calculated Raman tensors are also consistent with experimental and theoretical results.
|
||
Among negligible-polar modes, the E#sub[2] mode at 776 cm#super[-1] in experiment (mode 8)
|
||
exhibits the highest Raman intensity,
|
||
followed by four modes with lower intensities that are also experimentally visible,
|
||
including the E#sub[2] modes at 195.5 cm#super[-1] (mode 1) and 203.3 cm#super[-1] (mode 2),
|
||
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).
|
||
The E#sub[1] mode calculated at 746.91 cm#super[-1] (mode 7)
|
||
and the E#sub[2] mode calculated at 756.25 cm#super[-1] (mode 9)
|
||
are predicted to have much weaker Raman intensities and are located close to the most intense mode (mode 8),
|
||
making them indistinguishable in experimental spectra.
|
||
Additionally, the A#sub[1] mode calculated at 812.87 cm#super[-1] (mode 10)
|
||
exhibits a very weak Raman intensity in the basal-plane polarized configurations (xx and yy, only 0.01)
|
||
but shows an observable intensity when the polarization is along the z-axis (1.78).
|
||
Since most Raman experiments are performed in a back-scattering configuration with light incident along the z-direction
|
||
(i.e., with in-plane polarization)
|
||
and with photon energies much lower than the band gap,
|
||
this mode is typically not observed (cite).
|
||
However, it should become detectable if the incident light has a polarization component along the z-direction
|
||
(as in our experiment),
|
||
or when the excitation wavelength approaches resonance conditions (cite).
|
||
|
||
其它峰在其它章节中解释。
|
||
|
||
Besides, there are other peeks in the experiment.
|
||
The peek at 796 and 980 are caused by strong-polar phonons which will be discussed later.
|
||
Besides, there are small peeks at xxx,
|
||
which could not be explained in perfect 4H-SiC and will be discussed in the next section.
|
||
|
||
// TODO: 将一部分 phonons 改为 phonon modes
|
||
// 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。
|
||
// 也就是说，简并的里面有两个或者无数个 phonon，但只有一个 mode
|
||
|
||
#include "table-nopol.typ"
|
||
|
||
#include "figure-raman.typ"
|
||
|
||
// TODO: 解释为什么 E1 可以看到
|