376 lines
15 KiB
TeX
376 lines
15 KiB
TeX
\documentclass[preprintnumbers]{revtex4-2}
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% \linespread{1.5}
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\usepackage{tikz}
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\usetikzlibrary{shapes, arrows.meta, positioning}
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\tikzstyle{box} = [rectangle, draw=black, fill=blue!10, rounded corners,
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text width=10cm, align=left, minimum height=1cm]
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\tikzstyle{arrow} = [->, thick, >=Stealth]
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\usepackage{lineno}
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\newcolumntype{C}[1]{>{\centering\arraybackslash}m{#1}}
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\usepackage{multirow}
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\usepackage{makecell}
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\usepackage{amsmath}
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\usepackage{tabularx}
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\usepackage{booktabs}
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\usepackage{tabularray}
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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{Ann Author}\altaffiliation[Also at ]{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 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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\newcommand{\twocol}[1]{\multicolumn{2}{C{1cm}|}{#1}}
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\newcommand{\threecol}[1]{\multicolumn{3}{C{1.5cm}|}{#1}}
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\newcommand{\wthreecol}[1]{\multicolumn{3}{C{2cm}|}{#1}}
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\newcommand{\allcol}[1]{\multicolumn{26}{C{16cm}|}{#1}}
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\setlength\tabcolsep{0pt} % 不加这句话的话，表格的竖线的宽度会被计入，导致合并后的单元格不居中
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\caption{Weak- and None-polarized phonons near $\Gamma$ point}
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\begin{tabular}{
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| C{4cm} % header
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| C{0.5cm} | C{0.5cm} | C{0.5cm} % E2
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| C{0.5cm} | C{0.5cm} | C{0.5cm} % E2
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| C{0.5cm} | C{0.5cm} % E1
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| C{1cm} | C{1cm} % 2B1
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| C{0.5cm} | C{0.5cm} | C{1cm} % A1
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| C{0.5cm} | C{0.5cm} % E1
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| C{0.5cm} | C{0.5cm} | C{0.5cm} % E2
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| C{0.5cm} | C{0.5cm} | C{0.5cm} % E2
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| C{0.5cm} | C{0.5cm} | C{1cm} % A1
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| C{1cm} | C{1cm} % 2B1
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| }
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\hline
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\textbf{\makecell{Direction of Incident \\ \& Scattered Light}}
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& \allcol{\makecell{Any direction \\ (not depend on direction of incident \& scattered light)}}
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\\ \hline
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\textbf{Number of Phonon}
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& 1 & \twocol{2} % E2
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& 3 & \twocol{4} % E2
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& 5 & 6 % E1
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& 7 & 8 % 2B1
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& \wthreecol{9} % A1
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& 10 & 11 % E1
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& 12 & \twocol{13} % E2
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& 14 & \twocol{15} % E2
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& \wthreecol{16} % A1
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& 17 & 18 % 2B1
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\\ \hline
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\textbf{Vibration Direction}
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& x & \twocol{y} % E2
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& x & \twocol{y} % E2
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& x & y % E1
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& z & z % 2B1
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& \wthreecol{z} % A1
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& x & y % E1
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& x & \twocol{y} % E2
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& x & \twocol{y} % E2
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& \wthreecol{z} % A1
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& z & z % 2B1
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\\ \hline
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\textbf{Representation in Group $\mathrm{C_{6v}}$}
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& \threecol{$\mathrm{E_2}$} % E2
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& \threecol{$\mathrm{E_2}$} % E2
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& \twocol{$\mathrm{E_1}$} % E1
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& $\mathrm{B_1}$ & $\mathrm{B_1}$ % 2B1
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& \wthreecol{$\mathrm{A_1}$} % A1
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& \twocol{$\mathrm{E_1}$} % E1
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& \threecol{$\mathrm{E_2}$} % E2
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& \threecol{$\mathrm{E_2}$} % E2
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& \wthreecol{$\mathrm{A_1}$} % A1
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& $\mathrm{B_1}$ & $\mathrm{B_1}$ % 2B1
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\\ \hline
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\textbf{Representation in Group $\mathrm{C_{2v}}$}
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& $\mathrm{A_2}$ & \twocol{$\mathrm{A_1}$} % E2
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& $\mathrm{A_2}$ & \twocol{$\mathrm{A_1}$} % 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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& \wthreecol{$\mathrm{A_1}$} % A1
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& $\mathrm{B_2}$ & $\mathrm{B_1}$ % E1
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& $\mathrm{A_2}$ & \twocol{$\mathrm{A_1}$} % E2
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& $\mathrm{A_2}$ & \twocol{$\mathrm{A_1}$} % E2
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& \wthreecol{$\mathrm{A_1}$} % A1
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& $\mathrm{B_1}$ & $\mathrm{B_1}$ % 2B1
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\\ \hline
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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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\\ \hline
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\textbf{Raman Intensity (a.u.)}
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& \threecol{$0.17$} % E2
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& \threecol{$1.13$} % E2
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& \twocol{$2.43$} % E1
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& $0$ & $0$ % 2B1
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& \twocol{$2.83$} & $1.79$ % A1
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& \twocol{$0.09$} % E1
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& \threecol{$88.54$} % E2
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& \threecol{$0.50$} % E2
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& \twocol{$0.01$} & $1.78$ % A1
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& $0$ & $0$ % 2B1
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\\ \hline
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\textbf{Visible in Common Raman Experiment}
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& \threecol{Yes} % E2
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& \threecol{Yes} % E2
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& \twocol{Yes} % E1
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& No & No % 2B1
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& \wthreecol{Yes} % A1
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& \twocol{No} % E1
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& \threecol{Yes} % E2
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& \threecol{No} % E2
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& \twocol{No} & Yes % A1
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& No & No % 2B1
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\\ \hline
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\textbf{Wavenumber (Simulation) ($\mathrm{cm^{-1}}$)}
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& \threecol{$190.51$} % E2
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& \threecol{$190.51$} % E2
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& \twocol{$257.35$} % E1
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& $389.96$ & $389.96$ % 2B1
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& \wthreecol{$591.90$} % A1
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& \twocol{$746.91$} % E1
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& \threecol{$756.25$} % E2
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& \threecol{$764.33$} % E2
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& \wthreecol{$812.87$} % A1
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& $885.68$ & $894.13$ % 2B1
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\\ \hline
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\textbf{Wavenumber (Experiment) ($\mathrm{cm^{-1}}$)}
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& \threecol{$195.5$} % E2
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& \threecol{$203.3$} % E2
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& \twocol{$269.7$} % E1
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& - & - % 2B1
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& \wthreecol{$609.5$} % A1
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& \twocol{-} % E1
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& \threecol{$776$} % E2
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& \threecol{-} % E2
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& \twocol{-} & $839$ % A1
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& - & - % 2B1
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\\ \hline
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\textbf{Electrical Polarity}
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& \threecol{None} % E2
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& \threecol{None} % E2
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& \twocol{Weak} % E1
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& None & None % 2B1
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& \wthreecol{Weak} % A1
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& \twocol{Weak} % E1
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& \threecol{None} % E2
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& \threecol{None} % E2
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& \wthreecol{Weak} % A1
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& None & None % 2B1
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\\ \hline
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\end{tabular}
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}
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{
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\newcommand{\tworow}[1]{\multirow{2}{*}{#1}}
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\newcommand{\twocol}[2]{\multicolumn{2}{C{#1}|}{#2}}
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\newcommand{\threecol}[2]{\multicolumn{3}{C{#1}|}{#2}}
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\newcommand{\fourcol}[1]{\multicolumn{4}{C{2cm}|}{#1}}
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\setlength\tabcolsep{0pt} % 不加这句话的话，表格的竖线的宽度会被计入，导致合并后的单元格不居中
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\caption{Strong-polarized phonons near $\Gamma$ point}
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\begin{tabular}{
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| C{4cm} % header
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| C{1cm} | C{1cm} % z E1
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| C{0.7cm} | C{0.7cm} | C{1cm} % z A1
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| C{0.5cm} | C{0.5cm} | C{0.5cm} % y z
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| C{1cm} % y x
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| C{1cm} % y y
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| C{0.5cm} | C{0.5cm} | C{0.5cm} | C{0.5cm} % 45 y&z mainly z
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| C{1cm} % 45 x
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| C{0.5cm} | C{0.5cm} | C{0.5cm} | C{0.5cm} % 45 y&z mainly y
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| }
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\hline
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\textbf{Direction of Incident \& Scattered Light}
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& \multicolumn{5}{C{4cm}|}{z}
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& \multicolumn{5}{C{3.9cm}|}{y}
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& \multicolumn{9}{C{5cm}|}{between z and y, 10\textdegree{} to z}
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\\ \hline
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\textbf{Number of Phonon}
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& 1 & 2 % z E1
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& \threecol{2.4cm}{3} % z A1
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& \threecol{1.5cm}{1} % y z
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& 2 % y x
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& 3 % y y
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& \fourcol{1} % 45 y&z mainly z
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& 2 % 45 x
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& \fourcol{3} % 45 y&z mainly y
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\\ \hline
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\textbf{Vibration Direction}
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& x (TO) & y (TO) % z E1
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& \threecol{2.4cm}{z (LO)} % z A1
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& \threecol{1.5cm}{z (TO)} % y z
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& x (TO) % y x
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& y (LO) % y y
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& \fourcol{y-z mixed (LO-TO mixed)} % 45 y&z mainly z
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& x (TO) % 45 x
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& \fourcol{y-z mixed (LO-TO mixed)} % 45 y&z mainly y
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\\ \hline
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\textbf{Representation in Group $\mathrm{C_{6v}}$}
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& \twocol{2cm}{$\mathrm{E_1}$}
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& \threecol{2.4cm}{$\mathrm{A_1}$}
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& \multicolumn{14}{C{8.5cm}|}{Not applicable}
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\\ \hline
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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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& \threecol{2.4cm}{$\mathrm{A_1}$} % z A1
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& \threecol{1.5cm}{$\mathrm{A_1}$} % y z
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& $\mathrm{B_2}$ % y x
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& $\mathrm{B_1}$ % y y
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& \fourcol{Not Applicable} % 45 y&z mainly z
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& $\mathrm{B_2}$ % 45 x
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& \fourcol{Not Applicable} % 45 y&z mainly y
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\\ \hline
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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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\\ \hline
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\textbf{Raman Intensity (a.u.)}
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& \twocol{2cm}{$53.52$} % z E1
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& \twocol{1.4cm}{$53.52$} & $464.69$ % z A1
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& \twocol{1cm}{$56.86$} & $454.09$ % y z
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& $53.52$ % y x
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& $53.55$ % y y
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& \twocol{1cm}{$53.71$} & $3.20$ & $425.98$ % 45 y&z mainly z
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& $53.56$ % 45 x
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& \twocol{1cm}{$3.60$} & $50.36$ & $27.99$ % 45 y&z mainly y
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\\ \hline
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\textbf{Visible in Common Raman Experiment}
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& \twocol{2cm}{Yes} % z E1
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& \twocol{1.4cm}{Yes (LOPC)} & No % z A1
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& Yes (overfocused) & No & Yes (overfocused) % y z
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& Yes % y x
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& Yes (LOPC) % y y
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& \fourcol{???} % 45 y&z mainly z
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& ??? % 45 x
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& \fourcol{???} % 45 y&z mainly y
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\\ \hline
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\textbf{Wavenumber (Simulation) ($\mathrm{cm^{-1}}$)}
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& \twocol{2cm}{$776.57$} % z E1
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& \threecol{2.4cm}{$933.80$} % z A1
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& \threecol{1.5cm}{$761.80$} % y z
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& $776.57$ % y x
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& $941.33$ % y y
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& \fourcol{$762.76$} % 45 y&z mainly z
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& $776.57$ % 45 x
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& \fourcol{$940.86$} % 45 y&z mainly y
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\\ \hline
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\textbf{Electrical Polarity} & \multicolumn{19}{C{12.9cm}|}{Strong}
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\\ \hline
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\end{tabular}
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}
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\label{tab:phonon}
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\end{sidewaystable}
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\end{document}
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