668 lines
34 KiB
Typst
668 lines
34 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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#import "@preview/physica:0.9.4": pdv, super-T-as-transpose
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#show: super-T-as-transpose
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#set par.line(numbering: "1")
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#set par(justify: true)
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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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#set heading(numbering: "1.")
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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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(There are 21 phonons in total.
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We classified them into two categories: 18 negligible-polar phonons and 3 strong-polar phonons.)
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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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The phonons involved in Raman scattering are located in reciprocal space
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at positions determined by the difference between the wavevectors of the incident and scattered light.
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At each such position, there are 21 phonon modes (excluding translational modes).
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We classify these 21 phonons into two categories based on their polarities.
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The 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 make sense.)
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This classification is based on the fact that
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the four Si atoms in the primitive cell of 4H-SiC 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: [
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Born effective charges of Si and C atoms in A/B/C/B layers of 4H-SiC, calculated using first principle method.
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],
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placement: none,
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)<table-bec>
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=== Phonons with Negligible Polarities
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(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 of any incident/scattered light.
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This approximation is widely adopted and justified by the fact that, // TODO: cite
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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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dispersion of negligible-polar phonons 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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(Representation of these 18 phonons, and the shape of their Raman tensors could be determined in advance.)
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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 tensors
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can be determined by further considering the C#sub[2v] point group (see @table-rep).
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These Raman-active phonons may appear in Raman spectra under appropriate polarization configurations.
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However, the actual visibility of each mode depends on the magnitude of its Raman tensor components,
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which cannot be determined solely from symmetry analysis.
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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 might be visible in Raman experiment under appropriate polarization configurations.
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However, whethear 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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#figure({
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let m2(content) = table.cell(colspan: 2, content);
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table(columns: 6, align: center + horizon, inset: (x: 3pt, y: 5pt),
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[*Representations in C6v*], [A#sub[1]], m2[E#sub[1]], m2[E#sub[2]],
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[*Representations in C2v*], [A#sub[1]], [B#sub[2]], [B#sub[1]], [A#sub[2]], [A#sub[1]],
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[*Vibration Direction*], [z], [x], [y], [x], [y],
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[*Raman Tensor of #linebreak() Individual Phonons*],
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[$mat(a,,;,a,;,,b)$], [$mat(,,a;,,;a,,;)$], [$mat(,,;,,a;,a,;)$], [$mat(,a,;a,,;,,;)$], [$mat(a,,;,-a,;,,;)$],
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[*Raman Intensity with Different #linebreak() Polarization Configurations*],
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[xx/yy: $a^2$ #linebreak() zz: $b^2$ #linebreak() others: 0],
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m2[xz/yz: $a^2$ #linebreak() others: 0], m2[xx/xy/yy: $a^2$ #linebreak() others: 0],
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)},
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caption: [
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Raman-active representations of C#sub[6v] and C#sub[2v] point groups.
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],
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placement: none,
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)<table-rep>
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(We propose a method to estimate the magnitudes of the Raman tensors of these phonons.
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Here we write out its main steps, 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 by symmetry analysis (see appendix for details).
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The center principle is to assign the Raman tensor (i.e., change of polarizability caused by atomic displacement)
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to each atom in the unit cell.
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This including the following steps:
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- Write out the change of polarizability caused by displacement of Si atom in A and C layer,
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Where unknown non-zero components are denoted by $a_1$, $a_2$, $a_5$, $a_6$.
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For example, when we move the Si atom in A layer slightly towards the x+ direction in $d$ distance,
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the change of polarizability should be $mat(,a_2,a_1;a_2,,;a_1,,)d$.
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This could be done by conclusion above.
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- The Si atom in B layer have similar local environment as the A and C layer, with only a little difference.
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We denote these difference by $epsilon_1$, $epsilon_2$, $epsilon_5$, $epsilon_6$,
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and the absolute value of $epsilon_i$ should be much smaller than $a_i$.
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For example, when we move the Si atom in B layer slightly towards the x+ direction in $d$ distance,
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the change of polarizability should be $mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,)d$.
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- The local environment of C atom in A layer is similar to the Si atom in A layer with charge reversed and
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the system reversed along xy plane.
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We denote these difference by $eta_1$, $eta_2$, $eta_5$, $eta_6$,
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and the absolute value of $epsilon_i$ should be much smaller than $a_i$.
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For example, when we move the C atom in A layer slightly towards the x+ direction in $d$ distance,
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the change of polarizability should be $mat(,a_2+eta_2,-a_1-eta_1;a_2+eta_2,,;-a_1-eta_1,,)d$.
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- Similar to the case in Si atoms, we derive the change of polarizability
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caused by moving C atom in B layer slightly towards the x+ direction in $d$ distance,
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which should be $mat(,-a_2-eta_2-zeta_2,-a_1-eta_1-zeta_1;-a_2-eta_2-zeta_2,,;-a_1-eta_1-zeta_1,,)d$.
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Lets assign Raman tensor onto each atom.
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That is, Raman tensor is derivative of the polarizability with respect to the atomic displacement:
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$
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alpha = pdv(chi, u)
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$
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where $u$ should be the displacement of the atom corresponding to a phonon mode.
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But, even when $u$ is *NOT* the displacement of a phonon
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(for example, lets only slightly move Si atom in A layer, keeping other atoms fixed),
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the (high-frequency) polarizability is still well-defined,
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and the will still cause a change in the polarizability.
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Even more, the group representation theory is still applicable in this condition:
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the only thing that matters is, when applying $g$ to the system,
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the tensor transformed into $g^(-1) alpha g$ or $g alpha g^(-1)$,
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no matter $alpha$ is Raman tensor or something else, or it is related to a phonon or not.
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Thus, we can, in principle, "assign" Raman tensor of a phonon, to each atom.
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This "assign" is unique since both the atom movement and all phonons have 24 dimensions.
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Next, we consider what these single-atom-caused "Raman tensors" looks like.
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For example, what happens if we move the Si atom in A layer slightly along the x+ direction?
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Consider also move the Si atom in C layer slightly, along x+ or x- direction.
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How about the Raman tensor caused by the both two atoms?
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In first case, this is B2 representation in E1 representation. Thus the Raman tensor should be something like:
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$
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mat(,,2a_1;,,;2a_1,,;)
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$
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In the second case, it is A2 in E2. It turns out:
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$
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mat(,2a_2,;2a_2,,;,,;)
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$
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The average of these two tensors should be the s"Raman tensor" cause by move only the Si atom in A layer,
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slightly towards x+ direction.
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$
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mat(,a_2,a_1;a_2,,;a_1,,;)
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$
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The difference should be the "Raman tensor" of the second atom.
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$
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mat(,-a_2,a_1;-a_2,,;a_1,,;)
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$
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// This approach applied relied on the fact that, all Si atom in 4H-SiC is "distinguishable" by the symmetry operations.
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// I mean, what will happen if we have two Si atoms in A layer?
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// Apparently, we could not extract the "Raman tensor" of only one of the two atoms.
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// This is the case for the 6H-SiC.
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// Hence, we will provide a more general approach to estimate the "Raman tensor" of a single atom.
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Consider the Si atom in the B1 layer.
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It lives in an environment quite similar to the A layer.
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Thus, the "Raman tensor" caused by it should be similar to the one caused by the A layer:
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$
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mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,;)
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$
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Similar to the Si atom in B2 layer:
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$
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mat(,-a_2-epsilon_2,a_1+epsilon_1;-a_2-epsilon_2,,;a_1+epsilon_1,,;)
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$
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Same approach applied for Si atom vibrate in y direction.
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When we move both Si atoms in A and C layer in y+ direction,
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it is B1 in E1, thus the "Raman tensor" should be:
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$
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mat(,,;,,2a_3;,2a_3,;)
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$
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And if we move Si in A layer towards y+ but Si in C layer towards y-,
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it is A2 in E2:
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$
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mat(2a_4,,;,-2a_4,;,,;)
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$
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Thus we get the "Raman tensor" of Si atom in A layer sololy move towards y+ direction:
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$
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mat(a_4,,;,-a_4,a_3;,a_3,;)
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$
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and the "Raman tensor" of Si atom in C layer towards y+ direction:
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$
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mat(-a_4,,;,a_4,a_3;,a_3,;)
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$
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Same applied for the Si atom in B layer:
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$
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mat(a_4+epsilon_4,,;,-a_4-epsilon_4,a_3+epsilon_3;,a_3+epsilon_3,;)
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$
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$
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mat(-a_4-epsilon_4,,;,a_4+epsilon_4,a_3+epsilon_3;,a_3+epsilon_3,;)
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$
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Before consider z-direction, it is important to note that, $a_1$ $a_2$ $a_3$ $a_4$ are not independent.
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Consider vibration along x+ direction (lets say the distance is $d$).
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System energy caused by external electric field and vibration is:
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$
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E^T (mat(,,2a_1;,,;2a_1,,) d) E
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$
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Apply C#sub[3] to atom vibration and external field, energy should not change. We got:
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$
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(mat(-1/2,-sqrt(3)/2,;sqrt(3)/2,-1/2,;,,1)E)^T ( mat(,,2a_1;,,;2a_1,,)(-1/2 d) + mat(,,;,,2a_3;,2a_3,)(sqrt(3)/2 d) )
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(mat(-1/2,-sqrt(3)/2,;sqrt(3)/2,-1/2,;,,1)E)
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$
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It is equal to:
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$
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E^T (mat(,,1/2 a_1 + 3/2 a_3;,,sqrt(3)/2 a_1 - sqrt(3)/2 a_3;1/2 a_1 + 3/2 a_3,sqrt(3)/2 a_1 - sqrt(3)/2 a_3,) d) E
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$
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Thus:
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$
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1/2 a_1 + 3/2 a_3 = 2a_1 #linebreak()
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sqrt(3)/2 a_1 - sqrt(3)/2 a_3 = 0
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$
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Thus $a_1 = a_3$.
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Apply the same method, we get $abs(a_2) = abs(a_4)$.
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Since we have not define the sign of $a_4$, we could take $a_2 = a_4$.
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Same for $epsilon$.
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Now consider what if we move the Si atom in A layer along z+ direction.
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If we move the Si atom in C layer along z+ direction, it is A1:
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$
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mat(2a_5,,;,2a_5,;,,2a_6;)
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$
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If we move the Si atom in C layer along z- direction, it is B1:
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$
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0
|
||
$
|
||
Thus we get the "Raman tensor" of Si atom in A or C layer towards z+ direction:
|
||
$
|
||
mat(a_5,,;,a_5,;,,a_6;)
|
||
$
|
||
|
||
Lets consider the C atom in A layer.
|
||
It should be somehow similar to the Si atom in A layer, but with a negative sign in some places,
|
||
and then add or subtract some little value.
|
||
Actually, the "transformation" of Si atom in A layer to C atom in A layer applied in the following steps:
|
||
- reverse charge.
|
||
- reverse system along xy plane.
|
||
First we consider the first step.
|
||
Taking the define of electricity tenser:
|
||
$
|
||
P = chi E
|
||
$
|
||
Lets reverse charge of the system, say we now have electricity tensor $chi'$. We get:
|
||
$
|
||
-P = chi'(-E)
|
||
$
|
||
Thus we get $chi' = chi$, the first step does not change the electricity tensor, nor the "Raman tensor".
|
||
|
||
Now we consider the second step.
|
||
For electricity tensor, it will become:
|
||
$
|
||
mat(1,,;,1,;,,-1) chi mat(1,,;,1,;,,-1)
|
||
$
|
||
For $u$, when it is along x or y direction, it will not change. When it is along z direction, it will become $-u$.
|
||
|
||
So in conclusion, Raman tensor of C atom in A layer could be estimated from the Raman tensor of Si atom in A layer, by:
|
||
- for movement alone x and y direction, xz yz should be applied a negative sign.
|
||
- for movement alone z direction, xx xy yy zz should be applied a negative sign.
|
||
Export "Raman tensor" of C atom in C layer from C atom in A layer, in the same way.
|
||
|
||
Now consider the C atom in B1 layer.
|
||
Is it similar to the C atom in A layer, just like that for Si atom?
|
||
No. It turns out to be similar to the C atom in C layer.
|
||
|
||
We summarize these stuff into @table-singleatom.
|
||
Furthermore, we list predicted modes and their Raman tensors, in @table-predmode.
|
||
|
||
- $a$: Raman tensor of Si atom in A layer, large value.
|
||
- $epsilon$: Difference of Raman tensors of Si atom in A and B1 layer, small value.
|
||
- $eta$: Difference of Raman tensors of C and Si atom in A layer, small value.
|
||
- $zeta$: Difference of Raman tensors of C atoms in A and B layer, small value.
|
||
|
||
Frequency could be estimated by, how many atoms are moving towards its neighbor.
|
||
|
||
#page(flipped: true)[#figure({
|
||
table(columns: 4, align: center + horizon, inset: (x: 3pt, y: 5pt),
|
||
[*Move Direction*], [x], [y], [z],
|
||
[C A], [$mat(,a_2+eta_2,-a_1-eta_1;a_2+eta_2,,;-a_1-eta_1,,;)$],
|
||
[$mat(a_2+eta_2,,;,-a_2-eta_2,-a_1-eta_1;,-a_1-eta_1,;)$], [$mat(-a_5-eta_5,,;,-a_5-eta_5,;,,-a_6-eta_6;)$],
|
||
[Si A], [$mat(,a_2,a_1;a_2,,;a_1,,;)$], [$mat(a_2,,;,-a_2,a_1;,a_1,;)$], [$mat(a_5,,;,a_5,;,,a_6;)$],
|
||
[C, B1], [$mat(,-a_2-eta_2-zeta_2,-a_1-eta_1-zeta_1;-a_2-eta_2-zeta_2,,;-a_1-eta_1-zeta_1,,;)$],
|
||
[$mat(-a_2-eta_2-zeta_2,,;,a_2+eta_2+zeta_2,-a_1-eta_1-zeta_1;,-a_1-eta_1-zeta_1,;)$],
|
||
[$mat(-a_5-eta_5-zeta_5,,;,-a_5-eta_5-zeta_5,;,,-a_6-eta_6-zeta_6;)$],
|
||
[Si B1], [$mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,;)$],
|
||
[$mat(a_2+epsilon_2,,;,-a_2-epsilon_2,a_1+epsilon_1;,a_1+epsilon_1,;)$],
|
||
[$mat(a_5+epsilon_5,,;,a_5+epsilon_5,;,,a_6+epsilon_6;)$],
|
||
[C, C], [$mat(,-a_2-eta_2,-a_1-eta_1;-a_2-eta_2,,;-a_1-eta_1,,;)$],
|
||
[$mat(-a_2-eta_2,,;,a_2+eta_2,-a_1-eta_1;,-a_1-eta_1,;)$], [$mat(-a_5-eta_5,,;,-a_5-eta_5,;,,-a_6-eta_6;)$],
|
||
[Si C], [$mat(,-a_2,a_1;-a_2,,;a_1,,;)$], [$mat(-a_2,,;,a_2,a_1;,a_1,;)$], [$mat(a_5,,;,a_5,;,,a_6;)$],
|
||
[C, B2], [$mat(,a_2+eta_2+zeta_2,-a_1-eta_1-zeta_1;a_2+eta_2+zeta_2,,;-a_1-eta_1-zeta_1,,;)$],
|
||
[$mat(a_2+eta_2+zeta_2,,;,-a_2-eta_2-zeta_2,-a_1-eta_1-zeta_1;,-a_1-eta_1-zeta_1,;)$],
|
||
[$mat(-a_5-eta_5-zeta_5,,;,-a_5-eta_5-zeta_5,;,,-a_6-eta_6-zeta_6;)$],
|
||
[Si B2], [$mat(,-a_2-epsilon_2,a_1+epsilon_1;-a_2-epsilon_2,,;a_1+epsilon_1,,;)$],
|
||
[$mat(-a_2-epsilon_2,,;,a_2+epsilon_2,a_1+epsilon_1;,a_1+epsilon_1,;)$],
|
||
[$mat(a_5+epsilon_5,,;,a_5+epsilon_5,;,,a_6+epsilon_6;)$],
|
||
)},
|
||
caption: ["Raman tensor" caused by single atom],
|
||
placement: none,
|
||
)<table-singleatom>]
|
||
|
||
// Raman Tensor for A1: line1 xz/yz; line2 zz
|
||
// Raman Tensor for E1: x-dirc xz or y-dirc yx
|
||
// Raman Tensor for E2: x-dirc xy or y-dirc xx or y-dirc -yy
|
||
// Relative Vibration Direction: col1 C ABCB col2 Si ABCB
|
||
#page(flipped: true)[#figure({
|
||
let m(n, content) = table.cell(colspan: n, content);
|
||
let m2(content) = table.cell(colspan: 2, content);
|
||
let m3(content) = table.cell(colspan: 3, content);
|
||
let m4(content) = table.cell(colspan: 4, content);
|
||
table(columns: 11, align: center + horizon, inset: (x: 3pt, y: 5pt),
|
||
[*Representation in C#sub[6v]*], m3[A#sub[1]], m3[E#sub[1]], m4[E#sub[2]],
|
||
[*x*], m2[0.5], [1], m2[0.5], [1], m2[0.25], m2[0.75],
|
||
[*Relative Vibration Direction*],
|
||
[$++\ --\ ++\ --$], [$+-\ -+\ +-\ -+$], [$+-\ +-\ +-\ +-$],
|
||
[$++\ --\ ++\ --$], [$+-\ -+\ +-\ -+$], [$+-\ +-\ +-\ +-$],
|
||
[$++\ +-\ --\ -+$], [$++\ --\ --\ ++$], [$++\ -+\ --\ +-$], [$+-\ ++\ -+\ --$],
|
||
[*Vibration Direction*], m3[z], m3[x/y], m4[x/y],
|
||
[*Raman Tensor Predicted*], [$2(zeta_5-epsilon_5)$ #linebreak() $2(zeta_6-epsilon_6)$],
|
||
[$2(epsilon_5+zeta_5)$ #linebreak() $2(epsilon_6+zeta_6)$],
|
||
[$-4(2a_5+eta_5+epsilon_5+zeta_5)$ #linebreak() $-4(2a_6+eta_6+epsilon_6+zeta_6)$],
|
||
[$2(zeta_1-epsilon_1)$], [$2(epsilon_1+zeta_1)$], [$-4(2a_1+eta_1+epsilon_1+zeta_1)$],
|
||
[$-2(zeta_2+epsilon_2)$], [$2(2eta_2+zeta_2-epsilon_2)$], [$2(4a_2+2eta_2+zeta_2+epsilon_2)$],
|
||
[$2(epsilon_2-zeta_2)$],
|
||
[*Raman Intensity Predicted*], m2[weak], [strong], m2[weak], [strong], m2[weak], [strong], [weak],
|
||
[*Raman Tensor Calculated*],
|
||
[0.10 #linebreak() -1.33], [-1.68 #linebreak() 1.34], [7.68 #linebreak() -21.65],
|
||
[-1.56], [-0.30], [7.32], [1.06], [0.41], [9.41], [0.17],
|
||
[*Atom-pair that Move Relatively In-plane*], [4], [0], [4], [4], [0], [4], [0], [2], [4], [4],
|
||
[*Atom-pair that Move Relatively Out-plane*], [0], [4], [4], [0], [4], [4], [2], [0], [2], [2],
|
||
[*Predicted Frequency*], [low], [medium], [high], [medium], [low], [high], [low], [medium], m2[high],
|
||
[*Calculated Frequency*],
|
||
[591.90], [812.87], [933.80], [746.91], [257.35], [776.57], [197.84], [190.51], [756.25], [764.33]
|
||
)},
|
||
caption: [Predicted modes and their "Raman tensor"],
|
||
placement: none,
|
||
)<table-predmode>]
|
||
|
||
// 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。
|
||
// 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。
|
||
// 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。
|
||
// 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。
|
||
The Raman tensors of these Raman-active phonons were calculated using first-principles methods,
|
||
and the results are summarized and compared with experimental results in @table-nopol.
|
||
Two Raman-active modes are not observed in our experiments,
|
||
including the E#sub[1] mode at 746.91 cm#super[-1] and the E#sub[2] mode at 764.33 cm#super[-1],
|
||
due to their relatively low Raman intensities, broad FWHM values, and their proximity to stronger modes.
|
||
The A#sub[1] phonon at 812.87 cm#super[-1] is Raman-active
|
||
in both in-plane (xx and xy) and out-of-plane (zz) polarization configurations,
|
||
but it is only visible when both the incident and scattered light propagate along the z-direction (zz),
|
||
as its Raman intensity in basal plane is too week to be distinguished from the noise.
|
||
We also calculated the linewidths of these phonons at 300 K and compared them with experimental results,
|
||
as summarized in the @table-nopol.
|
||
The atomic vibration amplitudes are listed separately in the Appendix.
|
||
|
||
// 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
|
||
|
||
#page(flipped: true)[#figure({
|
||
let m(n, content) = table.cell(colspan: n, content);
|
||
let m2(content) = table.cell(colspan: 2, content);
|
||
let m3(content) = table.cell(colspan: 3, 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]];
|
||
table(columns: 27, align: center + horizon, inset: (x: 3pt, y: 5pt),
|
||
// [*Direction of Incident & Scattered Light*],
|
||
// m(26)[Any direction (not depend on direction of incident & scattered light)],
|
||
// TODO: 整理表格，使用 m2 m3 来代替
|
||
[*Number of Phonon*],
|
||
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
|
||
[1], m2[2], [3], m2[4], [5], [6], [7], [8], m3[9], [10], [11], [12], m2[13], [14], m2[15], m3[16], [17], [18],
|
||
[*Vibration Direction*],
|
||
// E2 E2 E1 2B1 A1
|
||
[x], m2[y], [x], m(2)[y], [x], [y], m(2)[z], m(3)[z],
|
||
// E1 E2 E2 A1 2B1
|
||
[x], [y], [x], m(2)[y], [x], m(2)[y], m(3)[z], m(2)[z],
|
||
[*Representation #linebreak() in Group C#sub[6v]*],
|
||
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,
|
||
[*Raman-active or Not*],
|
||
m(8)[Raman-active], m(2)[Raman-inactive], m(14)[Raman-active], m(2)[Raman-inactive],
|
||
// [*Representation in Group C#sub[2v]*],
|
||
// // E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
|
||
// 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,
|
||
[*Scattering in Polarization #linebreak() (non-zero Raman #linebreak() tenser components)*],
|
||
// E2 E2 E1 2B1 A1
|
||
[xy], [xx], [yy], [xy], [xx], [yy], [xz], [yz], m(2)[-], [xx], [yy], [zz],
|
||
// E1 E2 E2 A1 2B1
|
||
[xz], [yz], [xy], [xx], [yy], [xy], [xx], [yy], [xx], [yy], [zz], m(2)[-],
|
||
[*Raman Intensity (a.u.)*],
|
||
// E2 E2 E1 2B1 A1
|
||
m(3)[0.17], m(3)[1.13], m(2)[2.43], m(2)[0], m(2)[2.83], [1.79],
|
||
// E1 E2 E2 A1 2B1
|
||
m(2)[0.09], m(3)[88.54], m(3)[0.50], m(2)[0.01], [1.78], m(2)[0],
|
||
[*Visible in Common #linebreak() Raman Experiment or Not*],
|
||
// E2 E2 E1 2B1 A1
|
||
m(8)[Visible], m(2)[-], m(3)[Visible],
|
||
// E1 E2 E2 A1 2B1
|
||
m(2)[Invisible], m(3)[Visible], m(5)[Invisible], [Visible], m(2)[-],
|
||
[*Wavenumber #linebreak() (Simulation) (cm#super[-1])*],
|
||
// E2 E2 E1 2B1 A1
|
||
m(3)[190.51], m(3)[197.84], m(2)[257.35], [389.96], [397.49], m(3)[591.90],
|
||
// E1 E2 E2 A1 2B1
|
||
m(2)[746.91], m(3)[756.25], m(3)[764.33], m(3)[812.87], [885.68], [894.13],
|
||
[*Wavenumber #linebreak() (Experiment) (cm#super[-1])*],
|
||
// E2 E2 E1 2B1 A1
|
||
m(3)[195.5], m(3)[203.3], m(2)[269.7], m(2)[-], m(3)[609.5],
|
||
// E1 E2 E2 A1 2B1
|
||
m(2)[-], m(3)[776], m(5)[-], [839], m(2)[-],
|
||
[*FWHM #linebreak() (Simulation) (cm#super[-1])*],
|
||
// E2 E2 E1 2B1 A1
|
||
m(3)[0.08], m(3)[0.09], m(2)[0.08], m(2)[-], m(3)[0.61],
|
||
// E1 E2 E2 A1 2B1
|
||
m(2)[3.97], m(3)[4.62], m(3)[4.01], m(3)[0.89], m(2)[-],
|
||
[*FWHM #linebreak() (Experiment) (cm#super[-1])*],
|
||
// E2 E2 E1 2B1 A1
|
||
m(3)[1.11], m(3)[1.11], m(2)[1.11], m(2)[-], m(3)[591.90],
|
||
// E1 E2 E2 A1 2B1
|
||
m(2)[-], m(3)[1.11], m(3)[-], m(3)[1.11], m(2)[-],
|
||
[*Electrical Polarity*],
|
||
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
|
||
m(6)[None], m(2)[Weak], m(2)[None], m(5)[Weak], m(6)[None], m(3)[Weak], m(2)[None],
|
||
)},
|
||
caption: [Negaligible-polarized Phonons at $Gamma$ Point],
|
||
)<table-nopol>]
|
||
|
||
#figure(
|
||
image("/画图/拉曼整体图/main.svg"),
|
||
caption: [
|
||
(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。
|
||
// (没有在图上表示)
|
||
|
||
#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")
|