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    pdftitle={Microscopic contributions to the deviation from Amontons friction law}
    }




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\title{Microscopic contributions to the deviation from Amontons friction law}
%%%\sititle{} % when using manuscript=suppinfo
\author*{Suresh Ravisankar}{ravissur@fel.cvut.cz}
\affiliation{Department of Control Engineering, Faculty of Electrical Engineering, Czech Technical University in Prague, Technicka 2, 16627 Prague 6, Czech Republic}
\author*[1]{Ravikant Kumar}{kumarra2@fel.cvut.cz}
\author[1]{Antonio Cammarata}
\author{Thilo Glatzel}
\affiliation{Department of Physics, University of Basel, 4056 Basel, Switzerland}
\author[1]{Tomas Polcar}
\maketitle
\tableofcontents





\section{Structural Information}
\label{sec:struct_info}
%
The geometry of each hetrostructure used in the molecular dynamics simulations with a total number of 360 atoms (144 TM, 36 M, 72 X, 108 Si) is provided in what follows in \textsc{poscar} format.

\verbatiminput{Au_MoS2.vasp}\label{struct:Au_MoS2_POSCAR}\addcontentsline{toc}{subsection}{Au/MoS$_2$/Si}

\verbatiminput{Au_MoSe2.vasp}\label{struct:Au_MoSe2_POSCAR}\addcontentsline{toc}{subsection}{Au/MoSe$_2$/Si}

\verbatiminput{Au_WS2.vasp}\label{struct:Au_WS2_POSCAR}\addcontentsline{toc}{subsection}{Au/WS2/Si}

\verbatiminput{Au_WSe2.vasp}\label{struct:Au_WSe2_POSCAR}\addcontentsline{toc}{subsection}{Au/WSe2/Si}

\verbatiminput{Ag_MoS2.vasp}\label{struct:Ag_MoS2_POSCAR}\addcontentsline{toc}{subsection}{Ag/MoS2/Si}

\verbatiminput{Ag_MoSe2.vasp}\label{struct:Ag_MoSe2_POSCAR}\addcontentsline{toc}{subsection}{Ag/MoSe2/Si}

\verbatiminput{Ag_WS2.vasp}\label{struct:Ag_WS2_POSCAR}\addcontentsline{toc}{subsection}{Ag/WS2/Si}

\verbatiminput{Ag_WSe2.vasp}\label{struct:Ag_WSe2_POSCAR}\addcontentsline{toc}{subsection}{Ag/WSe2/Si}
%

\section{Frictional Properties}
\begin{figure}[t]
  \centering
  \includegraphics[width=0.8\textwidth]{COF_3ms_SDError}
  \caption{$\mu$ using linear fit function on average friction force as a function of load for all the systems with a tip velocity of 3 m/s.}
  \label{fig:COF_Au_1}
\end{figure}
%

\begin{figure}
  \centering
  \includegraphics[width=0.8\textwidth]{COF_4ms_SDError}
  \caption{$\mu$ using linear fit function on average friction force as a function of load for all the systems with a tip velocity of 4 m/s.}
  \label{fig:COF_Au_2}
\end{figure}
%
\begin{figure}
  \centering
  \includegraphics[width=0.8\textwidth]{COF_5ms_SDError}
  \caption{$\mu$ using linear fit function on average friction force as a function of load for all the systems with a tip velocity of 5 m/s.}
  \label{fig:COF_Au_3}
\end{figure}
%
The \autoref{fig:COF_Au_1}, \autoref{fig:COF_Au_2} and \autoref{fig:COF_Au_3} shows friction force as a function of normal load for four Au/MX$_2$/Si systems expect Au/MoS$_2$/Si which is presented in the main manuscript, with linear fits used to extract the coefficient of friction $\mu$.
% 
Most of the systems exhibhit nonmonotonic nature with loads and extraction of $\mu$ from a linear fit results in large errors.
% 
This same trend is observed in Ag/MX$_2$/Si systems as shown in \autoref{fig:COF_Ag_1}, \autoref{fig:COF_Ag_2}, \autoref{fig:COF_Ag_3} and \autoref{fig:COF_Ag_4}.

%
\begin{figure}
  \centering
  \includegraphics[width=0.99\textwidth]{MoSe_Favg}
  \caption{Friction force profile of Au/MoSe$_2$/Si where tip moving with velocity 2 m/s for load of 0 nN.}
  \label{fig:MoSe_Favg}
\end{figure}
The friction force profile for the Au/MoSe$_2$/Si system is shown in \autoref{fig:MoSe_Favg}.
% 
In this case, both lateral slip and zigzag behaviour are suppressed and the tip moves along $\bm{y}$-direction.
% 
This is also observed in the Fourier transform (FT) spectra of Au/MoSe$_2$/Si where the second and third peaks, representing the lateral and ziz-zag motions, are absent (\autoref{fig:MoSe_Ftrans}).

%
\begin{figure}
  \centering
  \includegraphics[width=0.8\textwidth]{MoSe_Ftran}
  \caption{Spatial Fourier transform of Au/MoSe$_2$/Si with varying loads at velocity 2 m/s.}
  \label{fig:MoSe_Ftrans}
\end{figure}
%
\begin{figure}
  \centering
  \includegraphics[width=0.8\textwidth]{COF_2ms_Ag_SDError}
  \caption{$\mu$ extracted from a linear fit on the average friction force as a function of load with a tip velocity of 2 m/s.}
  \label{fig:COF_Ag_1}
\end{figure}
%
\begin{figure}
  \centering
  \includegraphics[width=0.8\textwidth]{COF_3ms_Ag_SDError}
  \caption{$\mu$ extracted from a linear fit function on the average friction force as a function of load with a tip velocity of 3 m/s.}
  \label{fig:COF_Ag_2}
\end{figure}
%
\begin{figure}
  \centering
  \includegraphics[width=0.8\textwidth]{COF_4ms_Ag_SDError}
  \caption{$\mu$ extracted from a linear fit function on the average friction force as a function of load with a tip velocity of 4 m/s.}
  \label{fig:COF_Ag_3}
\end{figure}
%
\begin{figure}
  \centering
  \includegraphics[width=0.8\textwidth]{COF_5ms_Ag_SDError}
  \caption{$\mu$ extracted from a linear fit function on the average friction force as a function of load with a tip velocity of 5 m/s.}
  \label{fig:COF_Ag_4}
\end{figure}
\begin{figure}
  \centering
  \includegraphics[width=0.8\textwidth]{MX_Ftrans_all}
  \caption{Spatial Fourier transform of friction force with monolayer a) MoS$_2$, b) MoSe$_2$, c) WS$_2$ and d) WSe$_2$ at different substrates.}
  \label{fig:All_Ftrans}
\end{figure}
% 
\autoref{fig:All_Ftrans} shows the spatial Fourier transform of the frictional force in the Ag/MX$_2$/Si and Au/MX$_2$/Si systems.
% 
In the Au/WSe$_2$/Si case, the first peak dominates than the Ag/WSe$_2$/Si system which leads to higher frictional force.
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\end{document}
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