395 lines
22 KiB
Typst
395 lines
22 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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// TODO: fix indent of first line
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#show figure.caption: it => {
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set text(10pt)
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// TODO: how to align correctly?
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align(center, box(align(left, it), width: 80%))
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}
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#set page(
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// paper: "us-letter",
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// header: align(right)[
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// A fluid dynamic model for
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// glacier flow
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// ],
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numbering: "1/1",
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)
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// TODO: why globally set placement not work?
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// #set figure(placement: none)
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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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= 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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// TODO: 增加引用文献
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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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// TODO: 多举例，增加引用文献
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In this paper, we do some things. Especially we do something for the first time.
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// TODO: 完善
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= Method
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// TODO
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calc
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experiment
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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 atom vibrations (phonons) located near the #sym.Gamma point in reciprocal space,
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and the exact location of these phonons is determined by the wavevectors of incident and scattered light.
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On each site of the Brillouin zone near the #sym.Gamma point,
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there are 21 phonon modes in 4H-SiC.
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We classified these phonons into two categories based on their polarities.
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18 of 21 phonons are classified into negligible-polar phonons (i.e., phonons with zero or very weak polarity),
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for which the effect of polarity can be ignored in the Raman scattering process;
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and the other three phonons are strong-polar phonons,
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where the polarity gives rise to observable effects in the Raman spectra.
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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 positive Born effective charges (BECs),
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and the four C atoms carry similar negative BECs (see @table-bec).
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In the 18 negligible-polar phonons,
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the vibrations of two Si atoms are approximately opposite to those of the other two Si atoms,
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so do C atoms,
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leading to cancellations of macroscopic polarity.
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While in the three strong-polar phonons,
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all Si atoms vibrate in the same direction, so do C atoms,
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leading to a net dipole moment.
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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* (unit: |e|)],
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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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placement: none,
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)<table-bec>
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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 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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as shown by the orange dotted line in @figure-discont.
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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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and the phonon participating in Raman processes of different incident/scattered light directions
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are all nearly identical to the phonons at the #sym.Gamma point.
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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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The green, red and blue lines indicate the mode along the z-direction, y-direction and x-direction, respectively.
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Along A-#sym.Gamma path, strong-polar modes along x- and y-directions are degenerated,
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showing as a single purple line.
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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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placement: none,
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)<figure-discont>
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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
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// 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 vibration along z-axis and are non-degenerate,
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while phonons belonging to E#sub[1] and E#sub[2] representations vibrate in plane and 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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// 各个模式的声子可以使用怎样的偏振光看到（即拉曼张量的非零分量）可以联合考虑 C6v 和 C2v 的表示来判断，如表所示。
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// TODO: 翻译
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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 computed solely from symmetry analysis.
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// TODO: 画个表
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Here we propose a method to estimate the magnitudes of the Raman tensors of these phonons.
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// TODO: 写出来这个方法，并验证。
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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频率及拉曼张量，并与实验对比，如表如图所示。
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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 @table-nopol.
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Two Raman-active modes are not observed in our experiments,
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including the E#sub[1] mode at 746.91 cm#super[-1] and the E#sub[2] mode at 764.33 cm#super[-1],
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due to their relatively low Raman intensities, broad FWHM values, and their proximity to stronger modes.
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The A#sub[1] phonon at 812.87 cm#super[-1] is Raman-active
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in both in-plane (xx and xy) and out-of-plane (zz) polarization configurations,
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but it is only visible when both the incident and scattered light propagate along the z-direction (zz),
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as its Raman intensity in basal plane is too week to be distinguished from the noise.
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We also calculated the linewidths of these phonons at 300 K and compared them with experimental results,
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as summarized in the @table-nopol.
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The atomic vibration amplitudes are listed separately in the Appendix.
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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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#page(flipped: true)[#figure({
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let m(n, content) = table.cell(colspan: n, content);
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let m2(content) = table.cell(colspan: 2, content);
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let m3(content) = table.cell(colspan: 3, 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: 3pt, 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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// TODO: 整理表格，使用 m2 m3 来代替
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[*Number of Phonon*],
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// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
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[1], m2[2], [3], m2[4], [5], [6], [7], [8], m3[9], [10], [11], [12], m2[13], [14], m2[15], m3[16], [17], [18],
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[*Vibration Direction*],
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// E2 E2 E1 2B1 A1
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[x], m2[y], [x], m(2)[y], [x], [y], m(2)[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], m(2)[z],
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[*Representation #linebreak() 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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[*Raman-active or Not*],
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m(8)[Raman-active], m(2)[Raman-inactive], m(14)[Raman-active], m(2)[Raman-inactive],
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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 #linebreak() tenser components)*],
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// E2 E2 E1 2B1 A1
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[xy], [xx], [yy], [xy], [xx], [yy], [xz], [yz], m(2)[-], [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], m(2)[-],
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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], m(2)[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], m(2)[0],
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[*Visible in Common #linebreak() Raman Experiment or Not*],
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// E2 E2 E1 2B1 A1
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m(8)[Visible], m(2)[-], m(3)[Visible],
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// E1 E2 E2 A1 2B1
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m(2)[Invisible], m(3)[Visible], m(5)[Invisible], [Visible], m(2)[-],
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[*Wavenumber #linebreak() (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 #linebreak() (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(2)[-], m(3)[609.5],
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// E1 E2 E2 A1 2B1
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m(2)[-], m(3)[776], m(5)[-], [839], m(2)[-],
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[*FWHM #linebreak() (Simulation) (cm#super[-1])*],
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// E2 E2 E1 2B1 A1
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m(3)[0.08], m(3)[0.09], m(2)[0.08], m(2)[-], m(3)[0.61],
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// E1 E2 E2 A1 2B1
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m(2)[3.97], m(3)[4.62], m(3)[4.01], m(3)[0.89], m(2)[-],
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[*FWHM #linebreak() (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(2)[-], 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], m(2)[-],
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[*Electrical Polarity*],
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// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
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m(6)[None], m(2)[Weak], m(2)[None], m(5)[Weak], m(6)[None], m(3)[Weak], m(2)[None],
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)},
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caption: [Negaligible-polarized Phonons at $Gamma$ Point],
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)<table-nopol>]
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#figure(
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image("/画图/拉曼整体图/main.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.
|
||
Gray lines represent negligible-polar phonon modes,
|
||
while colored lines indicate strong-polar phonon modes.
|
||
(b) Magnified view of the boxed region in (a).
|
||
The orange dashed lines mark the phonon wavevectors involved in Raman scattering
|
||
with incident light along the z- and y-directions.
|
||
]
|
||
)<raman>
|
||
|
||
// TODO: 画一个模拟的图，与实验图对比。
|
||
|
||
// 实验与计算基本相符。对于声子频率，计算总是低估大约 3%。
|
||
// 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。
|
||
// 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。
|
||
// 这个现象也可以由材料或偏振片的微小角度来解释。
|
||
// 例如，我们将偏振方向转动 5 度，就可以得到这个模拟结果。
|
||
// 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。
|
||
// TODO: 翻译成英文
|
||
|
||
=== Strong-polar Phonons
|
||
|
||
// 在半导体的极性声子模式中，原子间存在长距离的库伦相互作用，导致散射谱在 Gamma 附近不再连续（引用），如图中的彩色线所示。
|
||
// 这导致不同方向的入射/散射光的声子模式不同。
|
||
// 具体来说，当入射光/散射光沿着 z 方向时，起作用的是 A-Gamma 线上的声子模式（图中的左半边的橘线），它们适用于群 C6v。
|
||
// 这时会有一个 E1 模式（TO，振动方向在面内）和一个 A1 模式（LO，沿 z 振动）。
|
||
// 而当沿着 y 方向入射时，起作用的是 Gamma-K 线上的声子模式（图中的右半边的橘线），它们不再适用于群 C6v，而只适用于群 C2v；
|
||
// 它会分裂成沿x、y、z 方向的三个声子模式（图中的右半边的蓝线），它们分别对应于群 C2v 的 A1、B1 和 B2 表示 TODO: 确认这个几个表示的名字。
|
||
// 若考虑到到入射光不是严格沿着 z 方向，而是有一个小的角度（例如 10 度），则此时有一个声子模式沿着 x 方向，另外两个声子模式则为 y-z 两个方向的混合。
|
||
|
||
#page(flipped: true)[
|
||
#figure({
|
||
// 使用 m2 m3
|
||
let m(n, content) = table.cell(colspan: n, content);
|
||
let A1 = [A#sub[1]];
|
||
let A2 = [A#sub[2]];
|
||
let B1 = [B#sub[1]];
|
||
let B2 = [B#sub[2]];
|
||
let E1 = [E#sub[1]];
|
||
let E2 = [E#sub[2]];
|
||
let NA = [Not Applicable]
|
||
let yzmix = [y-z mixed#linebreak() (LO-TO mixed)];
|
||
let lopc = [Yes#linebreak() (LOPC)];
|
||
let overf = [Yes#linebreak() (overfocused)];
|
||
table(columns: 20, align: center + horizon, inset: (x: 3pt, y: 5pt),
|
||
[*Direction of Incident & Scattered Light*], m(5)[z], m(5)[y], m(9)[between z and y, 10#sym.degree to z],
|
||
// z y 45 y&z
|
||
[*Number of Phonon*], [1], [2], m(3)[3], m(3)[1], [2], [3], m(4)[1], [2], m(4)[3],
|
||
[*Vibration Direction*],
|
||
[x#linebreak() (TO)], [y#linebreak() (TO)], m(3)[z (LO)], // z
|
||
m(3)[z (TO)], [x#linebreak() (TO)], [y (LO)], // y
|
||
m(4, yzmix), [x#linebreak() (TO)], m(4, yzmix), // 45 y&z
|
||
[*Representation in Group C#sub[6v]*], m(2, E1), m(3, A1), m(14, NA),
|
||
// z y 45 y&z
|
||
[*Representation in Group C#sub[2v]*], B2, B1, m(3, A1), m(3, A1), B2, B1, m(4, NA), B2, m(4, NA),
|
||
[*Scattering in Polarization*],
|
||
[xz], [yz], [xx], [yy], [zz], // z
|
||
[xx], [yy], [zz], [xz], [yz], // y
|
||
[xx], [yy], [yz], [zz], [xz], [xx], [yy], [yz], [zz], // 45 y&z
|
||
[*Raman Intensity (a.u.)*],
|
||
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],
|
||
)
|
||
]
|
||
|
||
// TODO: 这句话放哪里？
|
||
// whose dispersion curves exhibit discontinuity near the #sym.Gamma point (also shown in @phonon),
|
||
|
||
|
||
#bibliography("./ref.bib", title: "Reference", style: "american-physics-society")
|