\chapter{Organic Conductors}
\label{sec:OrgConduct}
%\chaptermark{Optional running chapter heading}

Organic materials were mainly electrical insulators and with the discovery of the first organic 1D superconductor $ (TMTSF)_{2} PF_{6} $, organic conductors started to attract a lot of attention \cite{book1}. Quasi-two dimensional organic conductors caught much attention recently because they were among the first strong candidates of QSL \cite{Kurosaki05,Pratt2011,Shimizu2003,Yamashita2008}. The crystal structure is arranged in alternating layers of cations and anions with the cation layers are formed of the organic BEDT-TTF (ET) molecules, see fig.\ref{fig:OrganCond0} b). The conductivity is low in the direction of layer stacking and high in the layer plane and thus, these systems are considered quasi-two dimensional materials, fig.\ref{fig:OrganCond0}a). Each BEDT-TTF molecule consists of 10 carbon,8 hydrogen and 8 sulfur atoms linked by covalent bonds, see Fig \ref{fig:OrganCond0} b). There are two terminal ethylene groups on both ends of the BEDT-TTF molecule. These ethylene groups are twisted out of the molecular plane which makes the molecule in either an eclipsed or staggered conformation. The charge transfer salts generally exhibit disorder at room temperature due to the random occurrence of both configurations while at low temperatures the staggered conformation dominates \cite{Jeschke2010}.

The ET molecules within the cation layers could have different arrangements and one of the most interesting arrangement is the $\kappa$-phase, fig.\ref{fig:OrganCond1} c). In that arrangement, every two molecules form a dimer and every dimer donates one electron to an anion giving rise to a half filled band. Due to this half filling (one hole per lattice site), these materials are expected to be conductors. However, due to the electron-electron interaction, there is an on-site repulsion between electrons, $U$, that opens a gap (equal to $U$) at the Fermi level forming a full lower Hubbard band and an empty upper Hubbard band, giving rise to Mott insulating state. Because these materials are insulators with half-filled band, they could act as a model system of 2D triangular or square lattice of localized electrons with spin $1/2$ on each lattice site. That model system is interesting because they could give rise to magnetic frustration with specific values of parameters. Fig. \ref{fig:OrganCond2} shows the equivalent 2D lattice with different parameters that control the behavior of the material. Parameter $t_{d}$ is the transfer integral between two molecules within a dimer, the larger the value of $t_{d}$ the larger the dimerization. Parameters $t_{p}$, $t_{q}$ and $t_{B}$  represent transfer integral between nearest neighbor and next nearest neighbor dimers. The effective transfer integrals are calculated as follows:
\begin{equation}
t=t_{p}+t_{q} 
\end{equation}
\begin{equation}
t^{'}=t_{b}/2
\end{equation}

Figure \ref{fig:OrganCond2} shows the effective transfer integrals on the 2D lattice and how they affect the geometry of the lattice; if $t'/t<<1$, the lattice is square with little frustration. However; if $t'/t\sim 1$, the lattice is isotropic triangular lattice. Values of $t'/t$ in between 0 and 1 give rise to anisotropic triangular lattice with varying degrees of geometric frustration. Magnetic frustration could be achieved through the interplay of geometric frustration and additional interactions between spins such as ring exchange. In the simplest case where geometric frustration in only included, the value of exchange interactions $J$ and $J'$ are calculated as follows:
\begin{equation}
J\approx 4t/U
\end{equation}
\begin{equation}
J'\approx 4t'/U
\end{equation}

Figure \ref{fig:OrganCond3} is a phase diagram that shows the effect of transfer integrals, and electron-electron interactions $U$ and $V$ on the behavior of the organic charge transfer materials \cite{HottaCrys2012}. The right axis in green shows the effect of dimerization, $t_{d}$, on the the ground state; as the dimerization increase, the ground state moves towards anti-ferromagnetic Mott insulator. As dimerization become less effective or in other words, $t_{d}$ become smaller, the state is pushed towards charge ordering and ferro-electricity. The top axis in red shows the effect of frustration; in less dimerized materials, frustration is expected to push the system form charge ordered state to exotic metal state. In more dimerized systems, frustration drives Mott anti-ferromagnetic state to Spin liquid state. Few materials are given as examples on the diagram in fig. \ref{fig:OrganCond3} for each of these ground states. Such phase diagrams show the richness of these materials where different phases or competition between ground states could be probed by applying hydro-static or chemical pressure (the substitution of one atom with another that has same chemical properties). Table 2.1 shows calculated transfers integrals for different materials using Density functional theory calculations (DFT) and the corresponding predicted ground state. 


\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{OrganCond0.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a)The structure of quasi-two dimensional organic conductors, an array of ET molecules form the cation layer along which the  conductivity is high. The cation layer alternate with the anion layer and the conductivity along the layer stacking is very low. Ref.~ \cite{layers}, b) BEDT-TTF molecule. }
\label{fig:OrganCond0}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{OrganCond1.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a) $ab$ plane of $\kappa$-phase organic conductor, adapted from reference \cite{Drichko2014}; it shows the alternating anion and cation layers. b) $ac$ place of $\kappa$-phase organic conductor showing that BEDT-TTF molecules are inclined with respect to lattice direction $a$, adapted from reference \cite{Gati2018};grey lines correspond to the anion layer, green circles correspond to the atoms of the anion layer and orange solid lines illustrate the interaction path between the S atoms in the BEDT-TTF layer and the anion layer. c) $bc$ plane of the $\kappa$ phase showing dimers where every two molecules form one dimers (circled) \cite{Drichko2014}. d) $bc$ plane of anion layer, adapted from reference  \cite{Drichko2014}. }
\label{fig:OrganCond1}
\end{figure}

\begin{figure}[htbp]
\includegraphics[width=\textwidth]{OrganCond2.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a) An illustration of bonding and transfer integrals between different molecules in the $\kappa$-phase, adopted from reference \cite{HottaCrys2012}. b) Adopted from \cite{Powell2006}, is an illustration of effective transfer integrals and corresponding exchange interaction between lattice sites}
\label{fig:OrganCond2}
\end{figure}

\begin{figure}[htbp]
\includegraphics[width=\textwidth]{OrganCond3.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{ Adopted from reference \cite{HottaCrys2012}. Phase diagram of charge transfer materials and the effect of frustration and dimerization on the ground states.}
\label{fig:OrganCond3}
\end{figure}


\makeatletter
\let\@currsize\normalsize
\makeatother
\ssp


\begin{longtable}{ c c c c }
\caption{Parameters of the Hubbard model adapted from references \cite{Drichko2014,Kandpal2009}}\\
\hline
Compound & $t^{'}/t$  & $U/t$ &Ground state \\  
\hline
\endfirsthead
\multicolumn{2}{@{}l}{\textbf{Table \thetable} \ldots continued} \\
\hline
Compound & $t^{'}/t$  & $U/t$ & Ground state \\   
\hline
\endhead
$\kappa$-(BEDT-TTF)$_{2}$Cu[N(CN)$_{2}$]Br & 0.42  &5.1  &  Superconductor \\
$\kappa$-(BEDT-TTF)$_{2}$Cu[N(CN)$_{2}$]Cl & 0.44  & 5.5   & Antiferromagnet  \\
$\kappa$-(BEDT-TTF)$_{2}$Cu$_{2}$(CN)$_{3}$& 0.83  & 7.3 & Spin Liquid  \\
$\kappa$-(BEDT-TTF)$_{2}$Cu (SCN)$_{2}$& 0.58 &6.0 & Superconductor\\
$\kappa$-(BEDT-TTF)$_{2}$Hg(SCN)$_{2}$Cl (300 K) & 0.80   &5.1  & Metal    \\
$\kappa$-(BEDT-TTF)$_{2}$Hg(SCN)$_{2}$Cl (50 K)& 0.84   &  4.37  & Quantum dipole solid?  \\
$\kappa$-(BEDT-TTF)$_{2}$Hg(SCN)$_{2}$Br &  & & Spin Liquid  \\

\hline
\end{longtable}
\dsp
\clearpage

\section*{Materials under Study:  $\kappa$-(ET)$_2$ Hg(SCN)$_2$ Br and  $\kappa$-(ET)$_2$ Hg(SCN)$_2$ Cl}

In this work I focused on the study of two isostructural Mott insulators that could be accurately modeled by a 2D triangular lattice of dimers. Crystal structures of the studied materials, \HgBr\ and \HgCl\, were published in references ~\cite{Aldoshina1993,Drichko2014}. In Table 2.2  information about crystal structure of both structures is presented.  It shows that the volume of the unit cell for \HgBr\ compound is somewhat larger than \HgCl. The effect of replacing Br atom with the smaller atom Cl, called chemical substitution/pressure, is equivalent to applying hydrostatic pressure on \HgBr. Therefore, studying both materials could give us clues on the effect of pressure on transfer integrals and subsequent change of the ground state. Subsequently, helping us understand mechanism of exotic new ground states. Theoretical calculations of the electronic structure are necessary to complement experimental results and help devise models to better understand the mechanism of novel ground states. %how the change of parameters affects define the ground state.

The organic conductor \HgBr~ shows quite different behavior from its iso-structure \HgCl ~even though they have almost identical crystal structure. Figure \ref{fig:RT1} shows the RT plot of both materials. \HgBr~ goes through metal to insulator phase transition (MIT) at 80 K while \HgCl~ transitions at 30K. Figure \ref{fig:RT2} shows the effect of hydrostatic pressure on the RT curve of \HgCl. It proves that applying pressure suppresses the MIT transition and at a certain critical pressure the material stays metallic. Such behavior could explain why the MIT temperature of \HgBr~ is higher than that of \HgCl. However, there are still a lot of  unknowns about theses materials. First, magnetic properties are still under investigation due to some contradicting results; \HgCl~ was claimed to go through anti-ferromagnetic transition at around 27 K \cite{Yasin2012} but subsequent measurements failed to detect such transition \cite{Brown2019}. It was speculated that it could form a spin singlet ground state but no experimental proof until now. Reference \cite{Hemmida2018} claimed that \HgBr~ showed weak ferromagnetism below 20K arising from disorder. Mechanism and order of phase transition of both materials are still under debate. Even though it is agreed that \HgCl~ becomes insulator due to charge order, the order of transition is under debate. Reference \cite{Gati2018} claims that  \HgCl~ goes through first order phase transition based on thermal expansion measurements, see fig. \ref{fig:HgCl-ThermalExpansion}, and the existence of hysteresis in RT plots. However, we didn't find any evidence on 1st order phase transition from long pulse heat capacity measurements neither does reference \cite{Yasin2012} see such hysteresis in RT measurements. Similarly \HgBr~ is still under investigations to understand the the mechanism and order of the phase transition. References \cite{Hemmida2018} claims that the phase transition is 1st order based on their heat capacity measurements while our measurements failed to show such 1st order transition. The mechanism by which \HgBr~ transitions from metallic to insulating state is still under investigation too; no Charge order or pronounced structural change could account for the phase transition and thus, it is not a simple Mott transition. It is suggested that charge and magnetic fluctuations could account for MIT transition \cite{Ivek2017}. All these experiments prove that both these materials are very interesting and more experimental and theoretical studies are needed.

\HgCl~ is unusual that it is a charge ordered $\kappa$ phase material. Also, according to theoretical calculations, \HgBr~ is expected to show high frustration with $j^{'}/j=0.83$ and this is a potentially good starting point for obtaining a spin liquid behavior. Thus, it is imperative to study the charge degree of freedom of these materials generally not probed or well understood with other non-spectroscopic measurements. Also, a lot of theoretical models are being developed that study the effect of charge order and its coupling with spin to understand exotic phases of matter. Our objective is to characterize \HgBr~ compound and understand its ground state behavior. We have performed heat capacity measurements to probe excitations at very low temperatures. Also, Raman measurements were taken to study if the compound shows a charge degree of freedom similar to that of \HgCl. Magnetic excitations were also probed by taking low-frequency Raman measurement of the compound. 
The aims for studying these two systems can be summarized as follows:
\begin{itemize}
\item -\HgCl~ is a known charge ordered material. We wanted to study charge degree of freedom for a $\kappa$-phase.
\item -\HgBr ~was not studied, and was interesting as a potential spin liquid.
\end{itemize}


\begin{figure}[htbp]
\includegraphics[width=\textwidth]{RT1.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{Adapted from reference ~\cite{Ivek2017}. Shows the RT plot of bothe \HgCl~ and \HgBr.}
\label{fig:RT1}
\end{figure}

\begin{figure}[htbp]
\includegraphics[width=\textwidth]{RT2.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{Adapted from reference ~\cite{Gati2018}. RT plot of \HgCl~ under pressure. At a critical pressure the MIT is suppressed. }
\label{fig:RT2}
\end{figure}

\begin{figure}[htbp]
\includegraphics[width=\textwidth]{HgCl-ThermalExpansion.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{Adapted from reference ~\cite{Gati2018}. Thermal expansion measurements of \HgCl. }
\label{fig:HgCl-ThermalExpansion}
\end{figure}


\makeatletter
\let\@currsize\normalsize
\makeatother
\ssp


\begin{longtable}{ c c c }
\caption{Crystal structure data for \HgBr\ and \HgCl}\\
\hline
Formula     & $\kappa$-(ET)$_2$Hg(SCN)$_2$Br (\HgBr)~\cite{Aldoshina1993}  & $\kappa$-(ET)$_2$Hg(SCN)$_2$Cl (\HgCl)~\cite{Drichko2014} \\  
\hline
\endfirsthead
\multicolumn{2}{@{}l}{\textbf{Table \thetable} \ldots continued} \\
\hline
Formula     & $\kappa$-(ET)$_2$Hg(SCN)$_2$Br (\HgBr)~\cite{Aldoshina1993}  & $\kappa$-(ET)$_2$Hg(SCN)$_2$Cl (\HgCl)~\cite{Drichko2014} \\  
\hline
\endhead
Space group &   $C2/c$            & $C2/c$    \\
$a$ (\AA)   &    37.09(1)       &   36.9564    \\
$b$ (\AA)   &    8.338(3)       &  8.2887(2)     \\
$c$ (\AA)   &    11.738(5)      &  11.7503(3)     \\
$\alpha$ (deg) &  90          &    90           \\
$\beta$  (deg) &  89.71       &    90.067       \\
$\gamma$ (deg) &   90         &    90           \\
$V$ ($A^3$)    &   3441 (2)     &    3564.29   \\
Z              &    4    &     4      \\ 
\hline
\end{longtable}
\dsp
\clearpage