417 lines
18 KiB
TeX
417 lines
18 KiB
TeX
\documentclass[preprintnumbers,aps,prl,nofootinbib]{revtex4-2}
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% \linespread{1.5}
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\usepackage{graphicx}
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\usepackage{dcolumn}
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\usepackage{bm}
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\usepackage[mathlines]{lineno}
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% \setCJKmainfont{Microsoft YaHei}
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% \setCJKsansfont{YouYuan}
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\linenumbers
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\renewcommand\linenumberfont{\normalfont}
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\usepackage{array}
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\usepackage{svg}
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\usepackage{rotating}
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\usepackage{amsmath}
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\usepackage{tabularray}
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\usepackage{siunitx}
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\DeclareSIUnit\angstrom{\textup{\AA}}
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\DeclareSIUnit\elementarycharge{\textup{e}}
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\sisetup{range-units=single,range-phrase=-}
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\usepackage{soul}
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\usepackage[svgnames]{xcolor}
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% \newcommand{\del}[1]{\textcolor{red}{\st{#1}}}
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% \DeclareRobustCommand{\add}[1]{{\sethlcolor{LightGreen}\hl{#1}}}
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% \usepackage[colorlinks]{hyperref}
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% \usepackage[style=cms]{citation-style-language}
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% \addbibresource{ref.bib}
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\begin{document}
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\preprint{APS/123-QED}
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\title{Title Title Title}
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\author{Haonan Chen}\altaffiliation{Physics Department, XYZ University.}
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\begin{abstract}
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An article usually includes an abstract, a concise summary of the work
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covered at length in the main body of the article.
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\end{abstract}
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\maketitle
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\section{Introduction}\label{sec_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 中的声子与材料的性质密切相关。通常来说，通过拉曼来区分多型。我们相信可以通过声子来挖掘更多的信息。
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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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\cite{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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% \cite{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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%
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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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% \cite{tsuchida_recent_2018, harada_suppression_2022, sun_selection_2022}.
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% 某某人做了什么
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% 4H-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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% \cite{casady_status_1996, okumura_present_2006}.
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% SiC has more than 250 polytypes \cite{cheung_silicon_2006}, the 3C-SiC has been widely studied in the past decades
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% 139 \cite{dompoint_kinetics_2011, izhevskyi_review_2000, kimoto_bulk_2016, tang_atomic_2007, blumenau_effect_2005,
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% 140 bernardini_interaction_2005, rodney_ab_2017, blumenau_straight_2002, blumenau_structure_2003,
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% 141 vashishta_interaction_2007}.
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%
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% 147 It shows an exclusive application in power electronic devices,
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% 148 such as the Schottky barrier diodes and the power metal-oxide-semiconductor field-effect transistors
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% 149 \cite{bhatnagar_comparison_1993, kimoto_material_2015}.
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% 150 However, these applications are negatively affected by Shockley stacking faults (SSFs),
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% 151 which are a class of common defects in 4H-SiC.
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% 152
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% 153 The structure of SiC could be considered as a stack of Si and C atomic layers,
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% 154 and only the stacking sequence of exactly repeating A-B-C-B leads to the formation of 4H-SiC without defects.
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% 155 The differences in formation energies between different stacking positions are only about
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% 156 \qty{1}{\meV/atom} \cite{kimoto_bulk_2016},
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SiC has many excellent properties and wide applications.
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Phonons in SiC are important. They can influence the properties of SiC and can be used to characterize the materials.
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There are many existing studies on phonons in SiC, but they have some shortcomings.
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In this paper, we do some things. We do something for the first time.
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\section{Methods}\label{sec_methods}
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\section{Results}\label{sec_results}
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\subsection{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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% 根据这些声子模式在拉曼实验中的表现，我们将这些声子分成三个部分。
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Raman scattering peeks correspond to phonons located near $\Gamma$ point in reciprocal space.
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We classified these phonons into three categories according to their behavior in Raman scattering:
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(1) phonons could not be observed in Raman scattering spectrum,
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either because they are Raman inactive or their scattering intensity is too weak;
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(2) phonons could be observed in Raman scattering spectrum and with weak or no polarities,
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their frequencies were independent of the direction of the incident light;
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(3) strong polar phonons,
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which were visible in Raman scattering spectrum,
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and their frequencies depend on the direction of the incident light.
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% 我们计算了 4H-SiC 在 A-Gamma 和 Gamma-M 上的声子频率，如图和附录1所示。
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% 在拉曼散射中，起作用的模式都是那些非常接近于 Gamma 的模式
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% （如图中的点所示，分为位于 1/50 和 1/100 处，这两条线分别对应于拉曼散射在 z 方向入射/散射和 y 方向入射/散射）。
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% 大多数声子模式在 Gamma 附近都是连续的，这使得它们的频率对入射光的方向不敏感；
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% 然而，少数声子具有较强的极性，这使得声子之间存在长程的库伦相互作用（引用文献），并导致 gamma 附近的频率不同，如图中的某两条线所示。
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% 据此，我们将无缺陷的 4H-SiC 的声子分成三类：
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% 无拉曼活性或拉曼散射强度太弱的模式，它们在拉曼散射谱上不可见；
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% 拉曼散射强度足够大且极性不强的模式，它们在拉曼散射谱上可以看到，且频率与拉曼入射光方向无关；
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% 极性声子，它们在拉曼散射谱上可以看到，不仅频率与入射光方向有关，而且可与载流子发生一些相互作用。
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Phonons in defect-free 4H-SiC are calculated at A-$\Gamma$ and $\Gamma$-M,
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as shown in Figure \ref{fig:phonon} and Table \ref{tab:phonon}.
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Raman active phonons are very close to $\Gamma$,
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as indicated by the points in the figure.
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Because of the consistency of the most phonon modes near $\Gamma$,
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most of the phonon frequencies are insensitive to the direction of the incident light.
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However, some phonons have strong polarities,
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which leads to long-range Coulomb interactions between phonons,
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and results in different frequencies near $\Gamma$,
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as shown by the two lines in the figure.
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Thus, we divide the phonons of defect-free 4H-SiC into three categories:
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(1) Raman inactive or too weak Raman intensity,
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which are invisible in the Raman scattering spectrum;
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(2) Raman active phonons with strong polarities,
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which are visible in the Raman scattering spectrum,
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and their frequencies are independent of the direction of the incident light;
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(3) Polar phonons,
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which are visible in the Raman scattering spectrum,
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and their frequencies depend on the direction of the incident light,
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and can interact with carriers.
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% insert fig1.svg
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\begin{figure}[h]
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\centering
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\includegraphics{../画图/声子不连续/整体图.pdf}
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\caption{Phonon dispersion of defect-free 4H-SiC.}
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\label{fig:phonon}
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\end{figure}
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\subsubsection{}
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\appendix
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\section{A little more on appendixes}
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\begin{sidewaystable}
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\centering
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{
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\caption{Weak- and None-polarized phonons near $\Gamma$ point}
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\begin{tblr}{
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hlines,vlines,colsep=2pt,width=\textwidth,
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colspec={
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X[10,c,m]
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*{3}{X[c,m]} % E2
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*{3}{X[c,m]} % E2
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*{2}{X[c,m]} % E1
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*{2}{X[4,c,m]} % 2B1
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*{2}{X[c,m]} X[2,c,m] % A1
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*{2}{X[c,m]} % E1
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*{3}{X[c,m]} % E2
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*{3}{X[c,m]} % E2
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*{2}{X[c,m]} X[2,c,m] % A1
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*{2}{X[4,c,m]} % 2B1
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}
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}
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\textbf{Direction of Incident \& Scattered Light}
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& \SetCell[c=26]{}{Any direction \\ (not depend on direction of incident \& scattered light)}
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& & & & & & & & & & & & & & & & & & & & & & & & &
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\\
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\textbf{Number of Phonon}
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& 1 & \SetCell[c=2]{} 2 & % E2
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& 3 & \SetCell[c=2]{} 4 & % E2
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& 5 & 6 % E1
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& 7 & 8 % 2B1
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& \SetCell[c=3]{} 9 & & % A1
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& 10 & 11 % E1
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& 12 & \SetCell[c=2]{} 13 & % E2
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& 14 & \SetCell[c=2]{} 15 & % E2
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& \SetCell[c=3]{} 16 & & % A1
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& 17 & 18 % 2B1
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\\
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\textbf{Vibration Direction}
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& x & \SetCell[c=2]{} y & % E2
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& x & \SetCell[c=2]{} y & % E2
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& x & y % E1
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& z & z % 2B1
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& \SetCell[c=3]{} z & & % A1
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& x & y % E1
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& x & \SetCell[c=2]{} y & % E2
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& x & \SetCell[c=2]{} y & % E2
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& \SetCell[c=3]{} z & & % A1
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& z & z % 2B1
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\\
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\textbf{Representation in Group $\mathrm{C_{6v}}$}
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& \SetCell[c=3]{} $\mathrm{E_2}$ & & % E2
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& \SetCell[c=3]{} $\mathrm{E_2}$ & & % E2
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& \SetCell[c=2]{} $\mathrm{E_1}$ & % E1
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& $\mathrm{B_1}$ & $\mathrm{B_1}$ % 2B1
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& \SetCell[c=3]{} $\mathrm{A_1}$ & & % A1
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& \SetCell[c=2]{} $\mathrm{E_1}$ & % E1
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& \SetCell[c=3]{} $\mathrm{E_2}$ & & % E2
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& \SetCell[c=3]{} $\mathrm{E_2}$ & & % E2
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& \SetCell[c=3]{} $\mathrm{A_1}$ & & % A1
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& $\mathrm{B_1}$ & $\mathrm{B_1}$ % 2B1
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\\
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\textbf{Representation in Group $\mathrm{C_{2v}}$}
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& $\mathrm{A_2}$ & \SetCell[c=2]{} $\mathrm{A_1}$ & % E2
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& $\mathrm{A_2}$ & \SetCell[c=2]{} $\mathrm{A_1}$ & % E2
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& $\mathrm{B_2}$ & $\mathrm{B_1}$ % E1
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& $\mathrm{B_1}$ & $\mathrm{B_1}$ % 2B1
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& \SetCell[c=3]{} $\mathrm{A_1}$ & & % A1
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& $\mathrm{B_2}$ & $\mathrm{B_1}$ % E1
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& $\mathrm{A_2}$ & \SetCell[c=2]{} $\mathrm{A_1}$ & % E2
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& $\mathrm{A_2}$ & \SetCell[c=2]{} $\mathrm{A_1}$ & % E2
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& \SetCell[c=3]{} $\mathrm{A_1}$ & & % A1
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& $\mathrm{B_1}$ & $\mathrm{B_1}$ % 2B1
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\\
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\textbf{Scattering in Polarization}
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& xy & xx & yy % E2
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& xy & xx & yy % E2
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& xz & yz % E1
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& - & - % 2B1
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& xx & yy & zz % A1
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& xz & yz % E1
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& xy & xx & yy % E2
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& xy & xx & yy % E2
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& xx & yy & zz % A1
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& - & - % 2B1
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\\
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\textbf{Raman Intensity (a.u.)}
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& \SetCell[c=3]{} $0.17$ & & % E2
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& \SetCell[c=3]{} $1.13$ & & % E2
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& \SetCell[c=2]{} $2.43$ & % E1
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& $0$ & $0$ % 2B1
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& \SetCell[c=2]{} $2.83$ & & $1.79$ % A1
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& \SetCell[c=2]{} $0.09$ & % E1
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& \SetCell[c=3]{} $88.54$ & & % E2
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& \SetCell[c=3]{} $0.50$ & & % E2
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& \SetCell[c=2]{} $0.01$ & & $1.78$ % A1
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& $0$ & $0$ % 2B1
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\\
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\textbf{Visible in Common Raman Experiment}
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& \SetCell[c=3]{} Yes & & % E2
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& \SetCell[c=3]{} Yes & & % E2
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& \SetCell[c=2]{} Yes & % E1
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& No & No % 2B1
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& \SetCell[c=3]{} Yes & & % A1
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& \SetCell[c=2]{} No & % E1
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& \SetCell[c=3]{} Yes & & % E2
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& \SetCell[c=3]{} No & & % E2
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& \SetCell[c=2]{} No & & Yes % A1
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& No & No % 2B1
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\\
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\textbf{Wavenumber (Simulation) ($\mathrm{cm^{-1}}$)}
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& \SetCell[c=3]{} $190.51$ & & % E2
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& \SetCell[c=3]{} $190.51$ & & % E2
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& \SetCell[c=2]{} $257.35$ & % E1
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& $389.96$ & $389.96$ % 2B1
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& \SetCell[c=3]{} $591.90$ & & % A1
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& \SetCell[c=2]{} $746.91$ & % E1
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& \SetCell[c=3]{} $756.25$ & & % E2
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& \SetCell[c=3]{} $764.33$ & & % E2
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& \SetCell[c=3]{} $812.87$ & & % A1
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& $885.68$ & $894.13$ % 2B1
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\\
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\textbf{Wavenumber (Experiment) ($\mathrm{cm^{-1}}$)}
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& \SetCell[c=3]{} $195.5$ & & % E2
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& \SetCell[c=3]{} $203.3$ & & % E2
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& \SetCell[c=2]{} $269.7$ & % E1
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& - & - % 2B1
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& \SetCell[c=3]{} $609.5$ & & % A1
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& \SetCell[c=2]{} - & % E1
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& \SetCell[c=3]{} $776$ & & % E2
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& \SetCell[c=3]{} - & & % E2
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& \SetCell[c=2]{} - & $839$ % A1
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& - & - % 2B1
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\\
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\textbf{Electrical Polarity}
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& \SetCell[c=3]{} None & & % E2
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& \SetCell[c=3]{} None & & % E2
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& \SetCell[c=2]{} Weak & % E1
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& None & None % 2B1
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& \SetCell[c=3]{} Weak & & % A1
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& \SetCell[c=2]{} Weak & % E1
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& \SetCell[c=3]{} None & & % E2
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& \SetCell[c=3]{} None & & % E2
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& \SetCell[c=3]{} Weak & & % A1
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& None & None % 2B1
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\\
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\end{tblr}
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}
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{
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\caption{Strong-polarized phonons near $\Gamma$ point}
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\begin{tblr}{
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hlines,vlines,colsep=2pt,width=\textwidth,hspan=even,
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colspec={
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X[5,c,m]
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X[c,m] % z x
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X[c,m] % z y
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*{2}{X[c,m]}X[c,m,1.5] % z z
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X[c,m,2]X[c,m]X[c,m,2] % y z
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X[c,m,1.5] % y x
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X[c,m,2] % y y
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*{3}{X[c,m]}X[c,m,1.5] % 45 y&z mainly z
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||
X[c,m,1.5] % 45 x
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*{4}{X[c,m]} % 45 y&z mainly y
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||
}
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||
}
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||
\textbf{Direction of Incident \& Scattered Light}
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||
& \SetCell[c=5]{} z & & & &
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||
& \SetCell[c=5]{} y & & & &
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||
& \SetCell[c=9]{} between z and y, 10\textdegree{} to z & & & & & & & &
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||
\\
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||
\textbf{Number of Phonon}
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||
& 1 & 2 % z E1
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||
& \SetCell[c=3]{} 3 & & % z A1
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||
& \SetCell[c=3]{} 1 & & % y z
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||
& 2 % y x
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||
& 3 % y y
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||
& \SetCell[c=4]{} 1 & & & % 45 y&z mainly z
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||
& 2 % 45 x
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||
& \SetCell[c=4]{} 3 & & & % 45 y&z mainly y
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||
\\
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\textbf{Vibration Direction}
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& {x \\ (TO)} & {y \\ (TO)} % z E1
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& \SetCell[c=3]{} z (LO) & & % z A1
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||
& \SetCell[c=3]{} z (TO) & & % y z
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||
& {x \\ (TO)} % y x
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& {y \\ (LO)} % y y
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||
& \SetCell[c=4]{} {y-z mixed \\ (LO-TO mixed)} & & & % 45 y&z mainly z
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& x (TO) % 45 x
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& \SetCell[c=4]{} {y-z mixed \\ (LO-TO mixed)} & & & % 45 y&z mainly y
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\\
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\textbf{Representation in Group $\mathrm{C_{6v}}$}
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||
& \SetCell[c=2]{} $\mathrm{E_1}$ &
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& \SetCell[c=3]{} $\mathrm{A_1}$ & &
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& \SetCell[c=14]{} Not applicable & & & & & & & & & & & & &
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\\
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\textbf{Representation in Group $\mathrm{C_{2v}}$}
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||
& $\mathrm{B_2}$ & $\mathrm{B_1}$ % z E1
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||
& \SetCell[c=3]{} $\mathrm{A_1}$ & & % z A1
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||
& \SetCell[c=3]{} $\mathrm{A_1}$ & & % y z
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||
& $\mathrm{B_2}$ % y x
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||
& $\mathrm{B_1}$ % y y
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||
& \SetCell[c=4]{} Not Applicable & & & % 45 y&z mainly z
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||
& $\mathrm{B_2}$ % 45 x
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||
& \SetCell[c=4]{} Not Applicable & & & % 45 y&z mainly y
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||
\\
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\textbf{Scattering in Polarization}
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& xz & yz % z E1
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||
& xx & yy & zz % z A1
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||
& xx & yy & zz % y z
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||
& xz % y x
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||
& yz % y y
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||
& xx & yy & yz & zz % 45 y&z mainly z
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||
& xz % 45 x
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& xx & yy & yz & zz % 45 y&z mainly y
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||
\\
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\textbf{Raman Intensity (a.u.)}
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||
& \SetCell[c=2]{} $53.52$ & % z E1
|
||
& \SetCell[c=2]{} $53.52$ & & $464.69$ % z A1
|
||
& \SetCell[c=2]{} $56.86$ & & $454.09$ % y z
|
||
& $53.52$ % y x
|
||
& $53.55$ % y y
|
||
& \SetCell[c=2]{} $53.71$ & & $3.20$ & $425.98$ % 45 y&z mainly z
|
||
& $53.56$ % 45 x
|
||
& \SetCell[c=2]{} $3.60$ & & $50.36$ & $27.99$ % 45 y&z mainly y
|
||
\\
|
||
\textbf{Visible in Common Raman Experiment}
|
||
& \SetCell[c=2]{} Yes & % z E1
|
||
& \SetCell[c=2]{} {Yes \\ (LOPC)} & & No % z A1
|
||
& Yes (overfocused) & No & Yes (overfocused) % y z
|
||
& Yes % y x
|
||
& {Yes \\ (LOPC)} % y y
|
||
& \SetCell[c=4]{} ??? & & & % 45 y&z mainly z
|
||
& ??? % 45 x
|
||
& \SetCell[c=4]{} ??? & & & % 45 y&z mainly y
|
||
\\
|
||
\textbf{Wavenumber (Simulation) ($\mathrm{cm^{-1}}$)}
|
||
& \SetCell[c=2]{} $776.57$ & % z E1
|
||
& \SetCell[c=3]{} $933.80$ & & % z A1
|
||
& \SetCell[c=3]{} $761.80$ & & % y z
|
||
& $776.57$ % y x
|
||
& $941.33$ % y y
|
||
& \SetCell[c=4]{} $762.76$ & & & % 45 y&z mainly z
|
||
& $776.57$ % 45 x
|
||
& \SetCell[c=4]{} $940.86$ & & & % 45 y&z mainly y
|
||
\\
|
||
\textbf{Electrical Polarity} & \SetCell[c=19]{} Strong & & & & & & & & & & & & & & & & & &
|
||
\end{tblr}
|
||
}
|
||
\label{tab:phonon}
|
||
\end{sidewaystable}
|
||
|
||
\bibliography{ref}
|
||
|
||
\end{document}
|