144 lines
9.3 KiB
XML
144 lines
9.3 KiB
XML
== Phonons in Perfect 4H-SiC
|
||
|
||
// 第一段：弱极性与强极性声子模式表现截然不同。如图所示，弱极性声子模式几乎不依赖于波矢方向，而强极性声子模式则表现出显著的各向异性。
|
||
|
||
Owing to the distinct behaviors arising from their different electrical polarities,
|
||
Raman-active phonons were categorized into two groups:
|
||
negligible-polarity phonons (i.e., having zero or very weak electrical polarity),
|
||
and strong-polarity phonons.
|
||
The negligible-polarity phonons exhibited minimal dependence on the wavevector direction
|
||
(see the intersection of gray solid lines and orange dashed lines in @figure-discont b and c),
|
||
and were therefore usually analyzed at the #sym.Gamma point regardless of the incidence configurations;
|
||
specifically, there were eight such modes
|
||
labeled A#sub[1]-1 to A#sub[1]-2, E#sub[1]-1 to E#sub[1]-2, and E#sub[2]-1 to E#sub[2]-4,
|
||
according to their irreducible representations in the C#sub[6v] group and in order of increasing frequency.
|
||
In contrast, the strong-polarity phonons displayed significant anisotropy,
|
||
and required analysis under specific incidence configurations.
|
||
In this paper, we focused on incidence within the zOx plane,
|
||
where three modes were present and named according to their vibration directions:
|
||
TO-zOx (vibrating in the zOx plane and roughly perpendicular to the incidence direction),
|
||
TO-y (vibrating along the y-axis), and LO (vibrating roughly parallel to the incidence direction).
|
||
|
||
// 第二段：我们具体计算了模式对极性的依赖。可以看到，弱极性声子的变化极小。与实验比较，也验证了我们实验的准确性。
|
||
|
||
// 第三段：有两个弱极性模式还没有在实验上看到过。我们通过计算知道了它们的强度，并看到了其中一个。
|
||
|
||
// 第四段：声子模式的强度可以从理论上得到解释。
|
||
|
||
=== Negligible-polarity Phonons
|
||
|
||
The negligible-polarity phonons were initially analyzed at the #sym.Gamma point,
|
||
disregarding the incidence configurations.
|
||
This simplification was justified because
|
||
the negligible-polarity phonon modes participated in Raman scattering
|
||
were nearly identical across all incidence configurations,
|
||
with their frequencies differing by only #sym.tilde 0.1 cm#super[-1]
|
||
|
||
|
||
There were eight Raman-active negligible-polarity modes in 4H-SiC at the #sym.Gamma point,
|
||
corresponding to three irreducible representations of the C#sub[6v] group (A#sub[1], E#sub[1], and E#sub[2]).
|
||
We named these modes as
|
||
E#sub[2]-1, E#sub[2]-2, E#sub[1]-1, A#sub[1]-1, E#sub[1]-2, E#sub[2]-3, E#sub[2]-4, and A#sub[1]-2,
|
||
in order of increasing frequency (see @figure-discont and @figure-raman).
|
||
The Raman tensor forms of each mode were derived by further considering their representations in the C#sub[2v] group,
|
||
and were summarized in @table-rep.
|
||
|
||
#include "figure-discont.typ"
|
||
#include "figure-raman.typ"
|
||
#include "table-rep.typ"
|
||
|
||
Peaks corresponding to seven Raman-active negligible-polarity phonons were observed in our experiments
|
||
(only the E#sub[2]-4 mode was not observed),
|
||
which is more than all previous experiments (where only five or six peaks were typically reported).
|
||
To explain the discrepancy in experimental results, first-principles calculations were performed,
|
||
and the result was compared with experimental data and summarized in @table-nopol.
|
||
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 was calculated to have very weak (0.01) and relatively strong (1.78) Raman intensity
|
||
under in-plane polarization and z polarization, respectively,
|
||
which is compatible with our experimental result
|
||
that it could be observed clearly in the y(zz)#overline[y] configuration but hardly seen in our other experiments.
|
||
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[2]-4 modes was 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 0.6% of the E#sub[2]-3 mode),
|
||
making it not visible in our and all previous experiments.
|
||
The E#sub[1]-2 mode was also located close to the E#sub[2]-3 mode and has weak Raman intensity,
|
||
making it also unobservable in previous experiments.
|
||
However, the E#sub[1]-2 mode was observable in our experiments of y(zx)#overline[y] with extended acquisition time,
|
||
where the scattering of the E#sub[2]-3 mode was suppressed while that of the E#sub[1]-2 mode was enhanced,
|
||
thanks to the different representations of these two modes.
|
||
Our experiments reported the observation of the E#sub[1]-2 peak for the first time,
|
||
and explained the discrepancy among previous experiments and ours with the help of our calculations.
|
||
|
||
#include "table-nopol.typ"
|
||
|
||
It is noteworthy that the large variation in Raman tensor magnitudes among different modes
|
||
was not yet theoretically understood.
|
||
For example, the Raman tensor of the E#sub[2]-3 mode was substantially larger
|
||
than those of other negligible-polarity modes (over 30 times larger than that of the second-strongest).
|
||
This highlighted a significant gap in established theory
|
||
that rigorous symmetry analysis could only predict the non-zero components of the Raman tensors,
|
||
but not their magnitudes.
|
||
In order to address the limitations of existing theories,
|
||
a method for estimating the magnitudes of Raman tensors was proposed.
|
||
By analyzing the local environment of individual atoms,
|
||
this approach decomposed their contributions to the Raman tensor into two parts:
|
||
a dominant component (invariant across similar environments, denoted as $a_i$，where $i in {1, 2, 5, 6}$)
|
||
and several secondary components (reflecting environmental variations,
|
||
denoted as $epsilon_i$, $eta_i$, and $zeta_i$，where $i in {1, 2, 5, 6}$,
|
||
and $|epsilon_i| + |eta_i| + |zeta_i| << |a_i|$ was assumed).
|
||
Detailed derivations were provided in @appd-predict, with results summarized in @table-nopol.
|
||
Notably, the E#sub[2]-3 mode was the only mode that retains the $a_i$ term,
|
||
which indicating a constructive interference of contributions from the local environment of individual atoms.
|
||
This stood in contrast to other negligible-polarity modes where such contributions tend to cancel out,
|
||
explaining the exceptionally high Raman tensor magnitude observed for the E#sub[2]-3 mode.
|
||
|
||
To achieve a more precise investigation of the Raman spectra
|
||
and prepare for analyzing impurity and charge carrier effects,
|
||
the analysis of negligible-polarity phonons off the #sym.Gamma point was conducted
|
||
by comparing experimental and calculated results under various lazer incidence directions.
|
||
The E#sub[2]-3 peak searved as a calibration reference under various experiments,
|
||
since its position was calculated to be virtually invariant between normal and edge incidence
|
||
(with a shift of only #sym.tilde 0.004 cm#super[-1]).
|
||
The E#sub[2]-1, E#sub[2]-2, and A#sub[1]-1 modes exhibited observable shifts,
|
||
and the experimental results were in good agreement with our calculations, as shown in @fig-nopo-diff.
|
||
Our results further confirmed the accuracy of both our experiments and calculations.
|
||
|
||
#include "figure-nopo-diff.typ"
|
||
|
||
=== Strong-polarity Phonons
|
||
|
||
The Strong-polarity phonon modes participated in Raman scattering
|
||
exhibited significant variations depending on the incidence configurations
|
||
(see the intersection of colored solid lines and orange dashed lines in @figure-discont b and c).
|
||
For incident light propagating along the z direction,
|
||
the C#sub[6v] point group applied and two modes were present,
|
||
marked as normal-TO and normal-LO,
|
||
and they were corresponding to the E#sub[1] and A#sub[1] representations
|
||
and vibrations along the basle plane and z direction, respectively.
|
||
The normal-LO would subsiquently couple with plasmons to form LOPC modes in n-type 4H-SiC.
|
||
For incident light propagating along other directions,
|
||
the C#sub[6v] group no longer held and three modes were present.
|
||
Specifically, for incident light propagating along x direction, the C#sub[2v] group applied,
|
||
and the three modes was named as edge-TO-z, edge-TO-y and edge-LO,
|
||
which were corresponding to A#sub[1], B#sub[2] and B#sub[1] representations
|
||
and vibrations along z, y and x directions, respectively.
|
||
|
||
E1 的情况。
|
||
|
||
注意到在正入射中，理论上不能被观察到的E#sub[1]-1模式也被观察到了。
|
||
与弱极性的 E1-1 模式类似，我们也认为这是由于入射光并非完全沿 z 轴入射所致。
|
||
但与弱极性 E1-1 模式不同的是，强极性 E1-1 模式在 xy 的偏振下并没有更强反而更弱。
|
||
这是因为E1这时不再是严格的E1模式，而是分裂成了两个相近的模式。
|
||
我们的计算表明，在2度的入射角下，E1分裂的两个模式非常接近。
|
||
其中某个模式会怎样怎样，另一个会怎样怎样。
|
||
|
||
// 我们预测，随着入射方向偏移，LO 峰会向着高频方向移动。此外，我们也注意到 LO 也会与载流子产生影响。
|
||
// 在 n 型半导体中，LOPC 模式将代替 LO 模式；在 p 型半导体中，LO 模式仍然单独存在，但它的半高宽会受到载流子浓度的影响。
|
||
|
||
#include "table-pol.typ"
|
||
#include "figure-rev.typ"
|