321 lines
18 KiB
Typst
321 lines
18 KiB
Typst
#import "@preview/starter-journal-article:0.4.0": article, author-meta
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#import "@preview/tablem:0.2.0": tablem
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#set par.line(numbering: "1")
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#show figure.caption: it => {
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set align(left)
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set text(10pt)
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show: box.with(width: 80%)
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it
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}
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#show: article.with(
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title: "Article Title",
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authors: (
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"Haonan Chen": author-meta(
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"xmu",
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// email: "chn@chn.moe",
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),
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"Junyong Kang": author-meta(
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"xmu",
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email: "jykang@xmu.edu.cn"
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)
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),
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affiliations: (
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"xmu": "Xiamen University",
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),
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abstract: [#lorem(100)],
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keywords: ("Typst", "Template", "Journal Article"),
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// template: (body: (body) => {
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// show heading.where(level: 1): it => block(above: 1.5em, below: 1.5em)[
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// #set pad(bottom: 2em, top: 1em)
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// #it.body
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// ]
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// set par(first-line-indent: (amount: 2em, all: true))
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// set footnote(numbering: "1")
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// body
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// })
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)
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#set figure(placement: none)
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= Introduction
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// SiC 是很好的材料。
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// 其中，4H-SiC 是SiC的一种多型，它的性质更好，近年来随着外延工艺的成熟而获得了更多的关注。
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SiC is a promising wide-bandgap semiconductor material
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with high critical electric field strength and high thermal conductivity.
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It has been widely used in power electronic devices and has long attracted a lot of research
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@casady_status_1996 @okumura_present_2006.
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The 4H-SiC has a wider bandgap, higher critical electric field strength,
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higher thermal conductivity, and higher electron mobility along the c-axis than other polytypes.
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Currently, the 4H-SiC has gradually received more attention than other polytypes,
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thanks to the development of epitaxy technology and the increasing application in the new energy industry
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@tsuchida_recent_2018 @harada_suppression_2022 @sun_selection_2022. // TODO: 多引用一些近年来的文献，有很多
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// 声子（量子化的原子振动）在理解晶体的原子结构以及热电性质方面起着重要作用。
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// 声子可以通过多种实验技术来探测，包括 EELS、IR 吸收谱等。
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// 拉曼光谱是最常用的方法，它提供了一种无损、非接触、快速和局部的声子测量方法，已被广泛用于确定晶体的原子结构（包括区分 SiC 的多型）。
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Phonons (quantized atomic vibrations) play a fundamental role
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in understanding the atomic structure
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as well as the thermal and electrical properties
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of crystals (including 4H-SiC).
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They could be probed by various experimental techniques,
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such as electron energy loss spectroscopy and infrared absorption spectroscopy.
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Among these techniques,
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Raman spectroscopy is the most commonly used method,
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as it provides non-destructive, non-contact, rapid and spatially localized measurement of phonons
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that near the #sym.Gamma point in reciprocal space.
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Studies in Raman scattering of 4H-SiC have been conducted since as early as 1983
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and have been widely employed to identification of different SiC polytypes.
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// 近年来，更多信息被从拉曼光谱中挖掘出来。
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// LOPC 已经被用于快速估计 n 型 SiC 的掺杂浓度。
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// 层错的拉曼光谱也已经被研究，可以被用于检测特定结构层错的存在和位置。
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// 掺杂对拉曼光谱的潜在影响也已经被研究。
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// 然而，拉曼光谱上仍有一些不知来源的峰；同时，一些也缺少一些理论上预测应该存在的峰。
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// 此外，预测掺杂导致的新峰也没有说明原因。
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Increasingly rich information has been extracted from Raman spectra of 4H-SiC.
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Longitudinal optical phonon–plasmon coupling (LOPC) peek
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has been utilized to rapidly estimate the doping concentration in n-type SiC.
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Peeks associated with some stacking faults have also been investigated
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and used to detect the presence and location of specific structural faults.
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Moreover, the potential effects of doping on Raman spectra have been explored.
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However, some unidentified peaks still appear in the Raman spectra,
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while certain phonon modes predicted by theory remain unobserved.
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In addition, the origins of newly emerged peaks induced by doping are often unclear or unexplained.
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In this paper, we do some things. We do something for the first time.
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= Method
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= Results and Discussion
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== Phonons in Perfect 4H-SiC
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// 拉曼活性的声子模式对应于 Gamma 点附近的声子模式。
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// 根据这些声子模式的极性，我们将这些声子分成两类。
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Raman scattering peeks correspond to phonons located near the #sym.Gamma point in reciprocal space.
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We classified these phonons into two categories based on their polarities:
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(i) negligible-polar phonons (i.e., phonons with no polarity or very weak polarity),
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whose dispersion curves are continuous near the #sym.Gamma point (as shown in @phonon),
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and for which the effect of polarity can be ignored in the Raman scattering process;
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and (ii) strong-polar phonons,
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whose dispersion curves exhibit discontinuity near the #sym.Gamma point (also shown in @phonon),
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where the polarity gives rise to observable effects in the Raman spectra.
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#figure(
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image("/画图/声子不连续/embed.svg"),
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caption: [
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(a) Phonon dispersion of 4H-SiC along the A–#sym.Gamma–K high-symmetry path.
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Gray lines represent negligible-polar phonon modes,
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while colored lines indicate strong-polar phonon modes.
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(b) Magnified view of the boxed region in (a).
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The orange dashed lines mark the phonon wavevectors involved in Raman scattering
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with incident light along the z- and y-directions.
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]
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)<phonon>
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=== Phonons with Negligible Polarities
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// 我们使用 Gamma 点的声子模式来近似拉曼过程中的非极性声子。
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// 这个近似被广泛使用，并且由于这个原因而被认为是可行的：
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// 尽管拉曼过程中起作用的声子并不是那些严格在 Gamma 点的，
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// 但这些声子模式的散射谱在 Gamma 附近连续且导数为零，且波矢很小（在本文中大约 0.01 A，只有c轴的大约2%）。
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// 因此，它们的性质与 Gamma 点的声子模式区别不大。
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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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This approximation is widely adopted 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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their dispersion near the #sym.Gamma point is continuous with vanishing derivatives,
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and their wavevector is very small (about 0.01 nm#super[-1] in this paper,
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which corresponds to only 1% of the smallest reciprocal lattice vector of 4H-SiC),
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as shown by the orange dotted line in @phonon.
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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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// 4H-SiC 在 Gamma 有 21 个distinct phonons。
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// 其中 18 个被归类为极性较弱的声子。这是因为，4H-SiC 的原胞中，4 个 Si 原子所带的有效电荷差别不大，四个 C 原子所带的有效电荷差别也不大。
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// 在这些声子模式中，原胞中的 2 个 Si 原子运动方向与另外 2 个 Si 原子相反，2 个 C 原子的运动方向与另外 2 个 C 原子相反，极性相互抵消。
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There are 21 distinct phonons at the #sym.Gamma point in 4H-SiC.
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Among them, 18 phonons are classified as negligible-polar phonons.
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This classification is based on the fact that
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the four Si atoms in the primitive cell carry similar Born effective charges (BEC),
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as do the four C atoms, as shown in @bec.
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In these 18 modes, the vibrations of two Si atoms are approximately opposite to those of the other two Si atoms,
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and similarly for the C atoms,
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leading to cancellations of macroscopic polarity.
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#figure(
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table(columns: 4, align: center + horizon,
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table.cell(colspan: 2)[], table.cell(colspan: 2)[*BEC*],
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table.cell(colspan: 2)[], [x / y direction], [z direction],
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table.cell(rowspan: 2)[Si atom], [A/C layer], [2.667], [2.626],
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[B layer], [2.674], [2.903],
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table.cell(rowspan: 2)[C atom], [A/C layer], [-2.693], [-2.730],
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[B layer], [-2.648], [-2.800],
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),
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caption: [Born effective charges of Si and C atoms in A/B/C/B layers of 4H-SiC.]
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)<bec>
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// 这18个声子对应于 $\mathrm{C_{6v}}$ 点群的 14 个表示：2A1 + 4B1 + 2E_1 + 4E2
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// 其中，B1 表示没有拉曼活性，它的拉曼张量为零；其它表示的拉曼张量不为零，但张量的大小是否足够大到可以在实验上看到，则还需要第一性原理计算，不能直接通过表示来判断。
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The 18 negligible-polar phonons correspond to 14 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 A#sub[1] and B#sub[1] representations are non-degenerate,
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while phonons belonging to E#sub[1] and E#sub[2] representations are doubly degenerate.
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Phonons belonging to B#sub[1] representation are Raman-inactive, as their Raman tensors vanish.
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In contrast, phonons belonging to other representations are Raman-active,
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the Raman tensors of them have non-zero components,
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indicating that these phonons might be visible in Raman experiment under appropriate polarization configurations.
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However, the actual visibility of each phonon depends on the magnitudes of its Raman tensor components,
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which cannot be inferred solely from symmetry analysis.
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/*
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这里应该有办法来估计。下面是我总结的规律：
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按照我们规定的 ABCB 层序，并将拉曼张量的大小归结为键长的变化的话：
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* 对于 E2 表示（AC层运动方向必须相反，B1/B2层运动方向必须相反，因此只讨论A和B1层）
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* A 层内部的那个竖的键，同向运动会导致比较大的拉曼张量
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* B1 层内部的那个竖的键，反向运动会导致比较大的拉曼张量
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* A 层和 B1 层之间的那个横的键，反向运动会导致比较大的拉曼张量
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我们或许可以通过这个路径来探索：
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* 首先，根据 C3v 点群的表示，写出每个键的拉曼张量。这包括：
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* 对于 A 内竖着的键，考虑连着的两个原子和第一近邻原子，对称性为 C3v。写出此时的拉曼张量。
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* 对于 B1 内竖着的键，它也是 C3v，它此时的拉曼张量是 h 下稍微变动的结果。写下这个结果。
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* 对于 A 到 B1 的横着的键，它是 C3v 。写下这个结果。
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* 对于 B1 到 C 的横着的键，它是 C3v 。写下这个结果为之前的结果的微微变动。
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* 对于其它键，根据对称性由上面的结果直接写出。
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* 写出各个模式的拉曼张量（上面的线性组合）。即可以直接看到结果。
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*/
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// 我们计算了拉曼活性声子的拉曼张量。
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// 其中有几个声子的拉曼活性较弱，有几个比较强。强的都可以在实验上看到；但弱的能否看到则取决于它是否恰好位于强模式的附近。
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// 其中，xxx 和xxx 位于强模式的附近，它们在实验上无法看到；xxx 只在 z 方向入射/散射时可以看到；xxx 则在任意方向都能看到。
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// 我们同样计算了这些声子在 300K 下的展宽，并与实验对比，结果如表所示。原子的振幅另外列于附录中。
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The Raman tensors of these Raman-active phonons were calculated using first-principles methods,
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and the results are summarized and compared with experimental results in @nopol.
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Some Raman-active phonons are not visible in experiments,
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including E#sub[1] at ~746.91 cm#super[-1] and E#sub[2] at ~764.33 cm#super[-1],
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causing their Raman intensity are relatively low and located close to strong modes.
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The A#sub[1] phonon at ~812.87 cm#super[-1] is only visible
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when both the incident and scattered light propagate along the z-direction,
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since its Raman intensity in basal plane is too week to be recognized from the background.
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We also calculated the linewidthes of these phonons at 300 K and compared them with experimental results,
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as summarized in the table.
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The atomic vibration amplitudes are listed separately in the Appendix.
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#page(flipped: true)[#figure({
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let m(n, content) = table.cell(colspan: n, content);
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let A1 = [A#sub[1]];
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// let A2 = [A#sub[2]];
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let B1 = [B#sub[1]];
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// let B2 = [B#sub[2]];
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let E1 = [E#sub[1]];
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let E2 = [E#sub[2]];
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table(columns: 27, align: center + horizon, inset: (x: 2pt, y: 5pt),
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// [*Direction of Incident & Scattered Light*],
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// m(26)[Any direction (not depend on direction of incident & scattered light)],
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[*Number of Phonon*],
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// E2 E2 E1 2B1 A1
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[1], m(2)[2], [3], m(2)[4], [5], [6], [7], [8], m(3)[9],
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// E1 E2 E2 A1 2B1
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[10], [11], [12], m(2)[13], [14], m(2)[15], m(3)[16], [17], [18],
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[*Vibration Direction*],
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// E2 E2 E1 2B1 A1
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[x], m(2)[y], [x], m(2)[y], [x], [y], [z], [z], m(3)[z],
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// E1 E2 E2 A1 2B1
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[x], [y], [x], m(2)[y], [x], m(2)[y], m(3)[z], [z], [z],
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[*Representation in Group C#sub[6v]*],
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m(3, E2), m(3, E2), m(2, E1), B1, B1, m(3, A1), m(2, E1), m(3, E2), m(3, E2), m(3, A1), B1, B1,
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// [*Representation in Group C#sub[2v]*],
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// // E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
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// A2, m(2, A1), A2, m(2, A1), B2, B1, B1, B1, m(3, A1), B2, B1, A2, m(2, A1), A2, m(2, A1), m(3, A1), B1, B1,
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[*Scattering in Polarization#linebreak() (non-zero Raman tenser components)*],
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// E2 E2 E1 2B1 A1
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[xy], [xx], [yy], [xy], [xx], [yy], [xz], [yz], [-], [-], [xx], [yy], [zz],
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// E1 E2 E2 A1 2B1
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[xz], [yz], [xy], [xx], [yy], [xy], [xx], [yy], [xx], [yy], [zz], [-], [-],
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[*Raman Intensity (a.u.)*],
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// E2 E2 E1 2B1 A1
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m(3)[0.17], m(3)[1.13], m(2)[2.43], [0], [0], m(2)[2.83], [1.79],
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// E1 E2 E2 A1 2B1
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m(2)[0.09], m(3)[88.54], m(3)[0.50], m(2)[0.01], [1.78], [0], [0],
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[*Visible in Common Raman Experiment*],
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// E2 E2 E1 2B1 A1
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m(3)[Yes], m(3)[Yes], m(2)[Yes], [No], [No], m(3)[Yes],
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// E1 E2 E2 A1 2B1
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m(2)[No], m(3)[Yes], m(3)[No], m(2)[No], [Yes], [No], [No],
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[*Wavenumber (Simulation) (cm#super[-1])*],
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// E2 E2 E1 2B1 A1
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m(3)[190.51], m(3)[197.84], m(2)[257.35], [389.96], [397.49], m(3)[591.90],
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// E1 E2 E2 A1 2B1
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m(2)[746.91], m(3)[756.25], m(3)[764.33], m(3)[812.87], [885.68], [894.13],
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[*Wavenumber (Experiment) (cm#super[-1])*],
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// E2 E2 E1 2B1 A1
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m(3)[195.5], m(3)[203.3], m(2)[269.7], [-], [-], m(3)[609.5],
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// E1 E2 E2 A1 2B1
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m(2)[-], m(3)[776], m(3)[-], m(2)[-], [839], [-], [-],
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[*FWHM (Simulation) (cm#super[-1])*],
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// E2 E2 E1 2B1 A1
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m(3)[1.11], m(3)[1.11], m(2)[1.11], [-], [-], m(3)[591.90],
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// E1 E2 E2 A1 2B1
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m(2)[1.11], m(3)[1.11], m(3)[1.11], m(3)[1.11], [-], [-],
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[*FWHM (Experiment) (cm#super[-1])*],
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// E2 E2 E1 2B1 A1
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m(3)[1.11], m(3)[1.11], m(2)[1.11], [-], [-], m(3)[591.90],
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// E1 E2 E2 A1 2B1
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m(2)[-], m(3)[1.11], m(3)[-], m(3)[1.11], [-], [-],
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[*Electrical Polarity*],
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// E2 E2 E1 2B1 A1
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m(3)[None], m(3)[None], m(2)[Weak], [None], [None], m(3)[Weak],
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// E1 E2 E2 A1 2B1
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m(2)[Weak], m(3)[None], m(3)[None], m(3)[Weak], [None], [None],
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)},
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caption: [Weak- and None-polarized phonons near $Gamma$ point],
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)<nopol>]
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#page(flipped: true)[
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#figure({
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let m(n, content) = table.cell(colspan: n, content);
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let A1 = [A#sub[1]];
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let A2 = [A#sub[2]];
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let B1 = [B#sub[1]];
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let B2 = [B#sub[2]];
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let E1 = [E#sub[1]];
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let E2 = [E#sub[2]];
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let NA = [Not Applicable];
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let yzmix = [y-z mixed#linebreak() (LO-TO mixed)];
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let lopc = [Yes#linebreak() (LOPC)];
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let overf = [Yes#linebreak() (overfocused)];
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table(columns: 20, align: center + horizon, inset: (x: 3pt, y: 5pt),
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[*Direction of Incident & Scattered Light*], m(5)[z], m(5)[y], m(9)[between z and y, 10#sym.degree to z],
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// z y 45 y&z
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[*Number of Phonon*], [1], [2], m(3)[3], m(3)[1], [2], [3], m(4)[1], [2], m(4)[3],
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[*Vibration Direction*],
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[x#linebreak() (TO)], [y#linebreak() (TO)], m(3)[z (LO)], // z
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m(3)[z (TO)], [x#linebreak() (TO)], [y (LO)], // y
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m(4, yzmix), [x#linebreak() (TO)], m(4, yzmix), // 45 y&z
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[*Representation in Group C#sub[6v]*], m(2, E1), m(3, A1), m(14, NA),
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// z y 45 y&z
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[*Representation in Group C#sub[2v]*], B2, B1, m(3, A1), m(3, A1), B2, B1, m(4, NA), B2, m(4, NA),
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[*Scattering in Polarization*],
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[xz], [yz], [xx], [yy], [zz], // z
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[xx], [yy], [zz], [xz], [yz], // y
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[xx], [yy], [yz], [zz], [xz], [xx], [yy], [yz], [zz], // 45 y&z
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[*Raman Intensity (a.u.)*],
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m(2)[53.52], m(2)[58.26], [464.69], // z
|
||
m(2)[58.26], [454.09], [53.52], [53.55], // y
|
||
m(2)[53.71], [3.20], [425.98], [53.56], m(2)[3.60], [50.36], [27.99], // 45 y&z
|
||
[*Visible in Common Raman Experiment*],
|
||
m(2)[Yes], m(2, lopc), [No], // z
|
||
overf, [No], overf, [Yes], lopc, // y
|
||
m(4)[???], [???], m(4)[???], // 45 y&z
|
||
[*Wavenumber (Simulation) (cm#super[-1])*],
|
||
// z y 45 y&z
|
||
m(2)[776.57], m(3)[933.80], m(3)[761.80], [776.57], [941.33], m(4)[762.76], [776.57], m(4)[940.86],
|
||
[*Electrical Polarity*], m(19)[Strong]
|
||
)},
|
||
caption: [Strong-polarized phonons near $Gamma$ point],
|
||
)
|
||
]
|
||
|
||
#bibliography("./ref.bib", title: "Reference", style: "american-physics-society")
|