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\begin{document}

\preprint{APS/123-QED}

\title{Title Title Title}

\author{Haonan Chen}\altaffiliation{Physics Department, XYZ University.}

\begin{abstract}
An article usually includes an abstract, a concise summary of the work
covered at length in the main body of the article. 
\end{abstract}

\maketitle

\section{Introduction}\label{sec_introduction}

% SiC 是很好的材料。
% 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。
% 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。

SiC is a promising wide-bandgap semiconductor material
  with high critical electric field strength and high thermal conductivity.
  It has been widely used in power electronic devices and has long attracted a lot of research
  \cite{casady_status_1996, okumura_present_2006}.
% The 4H-SiC has a wider bandgap, higher critical electric field strength,
%   higher thermal conductivity, and higher electron mobility along the c-axis than other polytypes.
% Currently, the 4H-SiC has gradually received more attention than other polytypes,
%   thanks to the development of epitaxy technology and the increasing application in the new energy industry
%   \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
% 
% Currently, the 4H-SiC has gradually received more attention than other polytypes,
%   thanks to the development of epitaxy technology and the increasing application in the new energy industry
%   \cite{tsuchida_recent_2018, harada_suppression_2022, sun_selection_2022}. 


% 某某人做了什么



% 4H-SiC is a promising wide-bandgap semiconductor material
%    with high critical electric field strength and high thermal conductivity.
%    It has been widely used in power electronic devices and has long attracted a lot of research
%    \cite{casady_status_1996, okumura_present_2006}.
%  SiC has more than 250 polytypes \cite{cheung_silicon_2006}, the 3C-SiC has been widely studied in the past decades
% 139       \cite{dompoint_kinetics_2011, izhevskyi_review_2000, kimoto_bulk_2016, tang_atomic_2007, blumenau_effect_2005,
% 140         bernardini_interaction_2005, rodney_ab_2017, blumenau_straight_2002, blumenau_structure_2003,
% 141         vashishta_interaction_2007}.
% 
% 147     It shows an exclusive application in power electronic devices,
% 148       such as the Schottky barrier diodes and the power metal-oxide-semiconductor field-effect transistors
% 149       \cite{bhatnagar_comparison_1993, kimoto_material_2015}.
% 150     However, these applications are negatively affected by Shockley stacking faults (SSFs),
% 151       which are a class of common defects in 4H-SiC.
% 152
% 153     The structure of SiC could be considered as a stack of Si and C atomic layers,
% 154       and only the stacking sequence of exactly repeating A-B-C-B leads to the formation of 4H-SiC without defects.
% 155     The differences in formation energies between different stacking positions are only about
% 156         \qty{1}{\meV/atom} \cite{kimoto_bulk_2016},


SiC has many excellent properties and wide applications.

Phonons in SiC are important. They can influence the properties of SiC and can be used to characterize the materials.

There are many existing studies on phonons in SiC, but they have some shortcomings.

In this paper, we do some things. We do something for the first time.

\section{Methods}\label{sec_methods}

\section{Results}\label{sec_results}

\subsection{Phonons in Perfect 4H-SiC}

% 拉曼活性的声子模式对应于 Gamma 点附近的声子模式。
% 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。

Raman scattering peeks correspond to phonons located near $\Gamma$ point in reciprocal space.
We classified these phonons into three categories according to their behavior in Raman scattering:
  (1) phonons could not be observed in Raman scattering spectrum,
    either because they are Raman inactive or their scattering intensity is too weak;
  (2) phonons could be observed in Raman scattering spectrum and with weak or no polarities,
    their frequencies were independent of the direction of the incident light;
  (3) strong polar phonons,
    which were visible in Raman scattering spectrum,
    and their frequencies depend on the direction of the incident light.

% 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。
% 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
%   （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）。
% 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；
%   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。
% 据此，我们将无缺陷的 4H-SiC 的声子分成三类：
%   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；
%   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；
%   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。

Phonons in defect-free 4H-SiC are calculated at A-$\Gamma$ and $\Gamma$-M,
  as shown in Figure \ref{fig:phonon} and Table \ref{tab:phonon}.
Raman active phonons are very close to $\Gamma$,
  as indicated by the points in the figure.
Because of the consistency of the most phonon modes near $\Gamma$,
  most of the phonon frequencies are insensitive to the direction of the incident light.
However, some phonons have strong polarities,
  which leads to long-range Coulomb interactions between phonons,
  and results in different frequencies near $\Gamma$,
  as shown by the two lines in the figure.
Thus, we divide the phonons of defect-free 4H-SiC into three categories:
  (1) Raman inactive or too weak Raman intensity,
    which are invisible in the Raman scattering spectrum;
  (2) Raman active phonons with strong polarities,
    which are visible in the Raman scattering spectrum,
    and their frequencies are independent of the direction of the incident light;
  (3) Polar phonons,
    which are visible in the Raman scattering spectrum,
    and their frequencies depend on the direction of the incident light,
    and can interact with carriers.

% insert fig1.svg
\begin{figure}[h]
  \centering
  \includegraphics{../画图/声子不连续/整体图.pdf}
  \caption{Phonon dispersion of defect-free 4H-SiC.}
  \label{fig:phonon}
\end{figure}


\subsubsection{}

\appendix

\section{A little more on appendixes}

\begin{sidewaystable}
  \centering
  {
    \caption{Weak- and None-polarized phonons near $\Gamma$ point}
    \begin{tblr}{
      hlines,vlines,colsep=2pt,width=\textwidth,
      colspec={
        X[10,c,m]
        *{3}{X[c,m]}          % E2
        *{3}{X[c,m]}          % E2
        *{2}{X[c,m]}          % E1
        *{2}{X[4,c,m]}          % 2B1
        *{2}{X[c,m]} X[2,c,m] % A1
        *{2}{X[c,m]}          % E1
        *{3}{X[c,m]}          % E2
        *{3}{X[c,m]}          % E2
        *{2}{X[c,m]} X[2,c,m] % A1
        *{2}{X[4,c,m]}          % 2B1
      }
    }
      \textbf{Direction of Incident \& Scattered Light}
        & \SetCell[c=26]{}{Any direction \\ (not depend on direction of incident \& scattered light)}
        & & & & & & & & & & & & & & & & & & & & & & & & &
      \\
      \textbf{Number of Phonon}
        & 1 & \SetCell[c=2]{} 2 &   % E2
        & 3 & \SetCell[c=2]{} 4 &   % E2
        & 5 & 6                     % E1
        & 7 & 8                     % 2B1
        & \SetCell[c=3]{} 9 & &     % A1
        & 10 & 11                   % E1
        & 12 & \SetCell[c=2]{} 13 & % E2
        & 14 & \SetCell[c=2]{} 15 & % E2
        & \SetCell[c=3]{} 16 & &    % A1
        & 17 & 18                   % 2B1
      \\
      \textbf{Vibration Direction}
        & x & \SetCell[c=2]{} y & % E2
        & x & \SetCell[c=2]{} y & % E2
        & x & y                   % E1
        & z & z                   % 2B1
        & \SetCell[c=3]{} z & &   % A1
        & x & y                   % E1
        & x & \SetCell[c=2]{} y & % E2
        & x & \SetCell[c=2]{} y & % E2
        & \SetCell[c=3]{} z & &   % A1
        & z & z                   % 2B1
      \\
      \textbf{Representation in Group $\mathrm{C_{6v}}$}
        & \SetCell[c=3]{} $\mathrm{E_2}$ & &  % E2
        & \SetCell[c=3]{} $\mathrm{E_2}$ & &  % E2
        & \SetCell[c=2]{} $\mathrm{E_1}$ &    % E1
        & $\mathrm{B_1}$ & $\mathrm{B_1}$     % 2B1
        & \SetCell[c=3]{} $\mathrm{A_1}$ & &  % A1
        & \SetCell[c=2]{} $\mathrm{E_1}$ &    % E1
        & \SetCell[c=3]{} $\mathrm{E_2}$ & &  % E2
        & \SetCell[c=3]{} $\mathrm{E_2}$ & &  % E2
        & \SetCell[c=3]{} $\mathrm{A_1}$ & &  % A1
        & $\mathrm{B_1}$ & $\mathrm{B_1}$     % 2B1
      \\
      \textbf{Representation in Group $\mathrm{C_{2v}}$}
        & $\mathrm{A_2}$ & \SetCell[c=2]{} $\mathrm{A_1}$ & % E2
        & $\mathrm{A_2}$ & \SetCell[c=2]{} $\mathrm{A_1}$ & % E2
        & $\mathrm{B_2}$ & $\mathrm{B_1}$                   % E1
        & $\mathrm{B_1}$ & $\mathrm{B_1}$                   % 2B1
        & \SetCell[c=3]{} $\mathrm{A_1}$ & &                % A1
        & $\mathrm{B_2}$ & $\mathrm{B_1}$                   % E1
        & $\mathrm{A_2}$ & \SetCell[c=2]{} $\mathrm{A_1}$ & % E2
        & $\mathrm{A_2}$ & \SetCell[c=2]{} $\mathrm{A_1}$ & % E2
        & \SetCell[c=3]{} $\mathrm{A_1}$ & &                % A1
        & $\mathrm{B_1}$ & $\mathrm{B_1}$                   % 2B1
      \\
      \textbf{Scattering in Polarization}
        & xy & xx & yy  % E2
        & xy & xx & yy  % E2
        & xz & yz       % E1
        & - & -         % 2B1
        & xx & yy & zz  % A1
        & xz & yz       % E1
        & xy & xx & yy  % E2
        & xy & xx & yy  % E2
        & xx & yy & zz  % A1
        & - & -         % 2B1
      \\
      \textbf{Raman Intensity (a.u.)}
        & \SetCell[c=3]{} $0.17$ & &        % E2
        & \SetCell[c=3]{} $1.13$ & &        % E2
        & \SetCell[c=2]{} $2.43$ &          % E1
        & $0$ & $0$                         % 2B1
        & \SetCell[c=2]{} $2.83$ & & $1.79$ % A1
        & \SetCell[c=2]{} $0.09$ &          % E1
        & \SetCell[c=3]{} $88.54$ & &       % E2
        & \SetCell[c=3]{} $0.50$ & &        % E2
        & \SetCell[c=2]{} $0.01$ & & $1.78$ % A1
        & $0$ & $0$                         % 2B1
      \\
      \textbf{Visible in Common Raman Experiment}
        & \SetCell[c=3]{} Yes & &     % E2
        & \SetCell[c=3]{} Yes & &     % E2
        & \SetCell[c=2]{} Yes &       % E1
        & No & No                     % 2B1
        & \SetCell[c=3]{} Yes & &     % A1
        & \SetCell[c=2]{} No  &       % E1
        & \SetCell[c=3]{} Yes & &     % E2
        & \SetCell[c=3]{} No  & &     % E2
        & \SetCell[c=2]{} No  & & Yes % A1
        & No & No                     % 2B1
      \\
      \textbf{Wavenumber (Simulation) ($\mathrm{cm^{-1}}$)}
        & \SetCell[c=3]{} $190.51$ & &  % E2
        & \SetCell[c=3]{} $197.84$ & &  % E2
        & \SetCell[c=2]{} $257.35$ &    % E1
        & $389.96$ & $397.49$           % 2B1
        & \SetCell[c=3]{} $591.90$ & &  % A1
        & \SetCell[c=2]{} $746.91$ &    % E1
        & \SetCell[c=3]{} $756.25$ & &  % E2
        & \SetCell[c=3]{} $764.33$ & &  % E2
        & \SetCell[c=3]{} $812.87$ & &  % A1
        & $885.68$ & $894.13$           % 2B1
      \\
      \textbf{Wavenumber (Experiment) ($\mathrm{cm^{-1}}$)}
        & \SetCell[c=3]{} $195.5$ & & % E2
        & \SetCell[c=3]{} $203.3$ & & % E2
        & \SetCell[c=2]{} $269.7$ &   % E1
        & - & -                       % 2B1
        & \SetCell[c=3]{} $609.5$ & & % A1
        & \SetCell[c=2]{} - &         % E1
        & \SetCell[c=3]{} $776$ & &   % E2
        & \SetCell[c=3]{} - & &       % E2
        & \SetCell[c=2]{} - & $839$   % A1
        & - & -                       % 2B1
      \\
      \textbf{Electrical Polarity}
        & \SetCell[c=3]{} None & &  % E2
        & \SetCell[c=3]{} None & &  % E2
        & \SetCell[c=2]{} Weak &    % E1
        & None & None               % 2B1
        & \SetCell[c=3]{} Weak & &  % A1
        & \SetCell[c=2]{} Weak &    % E1
        & \SetCell[c=3]{} None & &  % E2
        & \SetCell[c=3]{} None & &  % E2
        & \SetCell[c=3]{} Weak & &  % A1
        & None & None               % 2B1
      \\
    \end{tblr}
  }
  {
    \caption{Strong-polarized phonons near $\Gamma$ point}
    \begin{tblr}{
      hlines,vlines,colsep=2pt,width=\textwidth,hspan=even,
      colspec={
        X[5,c,m]
        X[c,m]                  % z x
        X[c,m]                  % z y
        *{2}{X[c,m]}X[c,m,1.5]  % z z
        X[c,m,2]X[c,m]X[c,m,2]  % y z
        X[c,m,1.5]              % y x
        X[c,m,2]                % y y
        *{3}{X[c,m]}X[c,m,1.5]  % 45 y&z mainly z
        X[c,m,1.5]              % 45 x
        *{4}{X[c,m]}            % 45 y&z mainly y
      }
    }
      \textbf{Direction of Incident \& Scattered Light}
        & \SetCell[c=5]{} z & & & &
        & \SetCell[c=5]{} y & & & &
        & \SetCell[c=9]{} between z and y, 10\textdegree{} to z & & & & &  & & &
      \\
      \textbf{Number of Phonon}
        & 1 & 2                   % z E1
        & \SetCell[c=3]{} 3 & &   % z A1
        & \SetCell[c=3]{} 1 & &   % y z
        & 2                       % y x
        & 3                       % y y
        & \SetCell[c=4]{} 1 & & & % 45 y&z mainly z
        & 2                       % 45 x
        & \SetCell[c=4]{} 3 & & & % 45 y&z mainly y
      \\
      \textbf{Vibration Direction}
        & {x \\ (TO)} & {y \\ (TO)}                           % z E1
        & \SetCell[c=3]{} z (LO) & &                          % z A1
        & \SetCell[c=3]{} z (TO) & &                          % y z
        & {x \\ (TO)}                                         % y x
        & {y \\ (LO)}                                         % y y
        & \SetCell[c=4]{} {y-z mixed \\ (LO-TO mixed)} & & &  % 45 y&z mainly z
        & x (TO)                                              % 45 x
        & \SetCell[c=4]{} {y-z mixed \\ (LO-TO mixed)} & & &  % 45 y&z mainly y
      \\
      \textbf{Representation in Group $\mathrm{C_{6v}}$}
        & \SetCell[c=2]{} $\mathrm{E_1}$ &
        & \SetCell[c=3]{} $\mathrm{A_1}$ & &
        & \SetCell[c=14]{} Not applicable & & & & &  & & & & &  & & &
      \\
      \textbf{Representation in Group $\mathrm{C_{2v}}$}
        & $\mathrm{B_2}$ & $\mathrm{B_1}$       % z E1
        & \SetCell[c=3]{} $\mathrm{A_1}$ & &    % z A1
        & \SetCell[c=3]{} $\mathrm{A_1}$ & &    % y z
        & $\mathrm{B_2}$                        % y x
        & $\mathrm{B_1}$                        % y y
        & \SetCell[c=4]{} Not Applicable & & &  % 45 y&z mainly z
        & $\mathrm{B_2}$                        % 45 x
        & \SetCell[c=4]{} Not Applicable & & &  % 45 y&z mainly y
      \\
      \textbf{Scattering in Polarization}
        & xz & yz           % z E1
        & xx & yy & zz      % z A1
        & xx & yy & zz      % y z
        & xz                % y x
        & yz                % y y
        & xx & yy & yz & zz % 45 y&z mainly z
        & xz                % 45 x
        & xx & yy & yz & zz % 45 y&z mainly y
      \\
      \textbf{Raman Intensity (a.u.)}
        & \SetCell[c=2]{} $53.52$ &                     % z E1
        & \SetCell[c=2]{} $58.26$ & & $464.69$          % 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}
