\chapter{Ground state of \HgCl}
\label{sec:}

\section{Introduction}

organic conductors are the subject of intense study especially theoretically where a range of exotic ground states were proposed. Examples of new proposed ground states are pinball, three-fold charge metals and other states where charge frustration comes into play, \cite{HottaCrys2012}. Such materials have interesting phase diagrams with complex charge phases due to competitions between ground states. Such states are mostly described within the extended Hubbard model for quarter filled materials. In this chapter we observed a crossover to a new state as we lowered temperature of \HgCl.

\HgCl ~is a quasi-two dimensional material where there is minimal inter-layer interaction. The material is composed of alternating layers of cations and anions. The cation layer is made up of the ET molecule that defines the electronic and magnetic behavior of the material. Within the cation layer, the ET molecules are arranged in such a way to form $\kappa$-phase where every two molecules form a dimer and every dimer is considered a lattice site. Every two ET molecules give one electron to the Hg-Cl molecule. As a result, every lattice site has an unpaired hole on a dime $(BEDT-TTF)^{+1}_{2}$. In systems where there is strong intradimer bond, a dimer is considered a lattice site and thus the system is considered half filled. There are 2 dimers (4 molecules ) in a unit cell, where different parameters $t_d$, $t_i$, $t_j$, $V_d$, $V_i$ and $V_j$ control the behavior of the material. In such organic materials, electron-electron interaction is strong enough to prevent the double occupation of a state. Materials exhibiting such behavior are called Mott insulator. However, in cases of charge order where the charge doesn't distribute equally within a dimer, the system is treated like a quarter filled system. 

\HgCl~ exhibit MIT transition at 30 K that is driven by charge order. In the metallic state, the charge on $(BEDT-TTF)^{+1}_{2}$ is equally distributed on the two molecules within the dimer and is free to hop between dimers. As the material goes through phase transition, the charge on the molecules within a dimer deviate from 0.5e to 0.6e and 0.4e on charge rich and charge poor molecules in the insulating state. Such charge configuration give rise to electric dipoles that give rise to a ferro-electric state when these dipoles order. The configuration or long range order of the dipoles in theory could be either the degenerate chain stripe or the unique checkerboard stripe depending on the parameters $V$ and $V'$, see fig. \ref{fig:ChargeConfig}. Other configurations were proposed by \cite{Dayal2011} such as paired electron crystal.  However, it was proposed, Ref\cite{Drichko2014}, that the charge order is in the stripe configuration as shown in fig.\ref{fig:ChargeRaman2a},  

\begin{figure}[!ht]
\includegraphics[width=\textwidth]{ChargeConfig.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{Different Long range charge order configurations depending on V' and V, from Ref. \cite{HottaCrys2012}}
\label{fig:ChargeConfig}
\end{figure}

%the crystal axes changes \ref{} which lead to changes in the transfer integral(how it changes V?). such change leads to localization of charge within the dimer.The charge on the molecules within a dimer deviate from 0.5e on each molecule in the metallic state to 0.6e and 0.4e on charge rich and charge poor molecules in the insulating state. Such charge configuration give rise to electric dipoles that give rise to a ferro-electric state when these dipoles order. %Such picture of charge order deviates form that in reference ~cite{} where it is claimed that the dimers in the metallic also exhibit dipoles but that are randomly oriented and at 30K symmetry is broken through the orientation of dipoles in certain directions. 

%Mechanism of transition: Metal-insulator transition is driven by charge order; in the metallic state, the charge on $(BEDT-TTF)^{+1}_{2}$ is equally distributed on the two molecules within the dimer is free to hop between dimers (value of t?). As the material goes throug phase transition, the crystal axes changes  ref{} which lead to changes in the transfer integral(how it changes V?). such change leads to localization of charge within the dimer. 

%$\kappa$-ET$_{2}$-Hg-(SCN)$_{2}$-Cl go through phase transition at $=30K$ \cite{ref:HgCl1}, \cite{ref:HgCl2}. The phase transition is due charge disproportionation rather than structural or magnetic origin \cite{HgCl1}. charge disproportionation can't be explained by Hubbard model of dimerized Mott insulator. 

 
\section{Results}

High resolution Raman measurements were used to probe the charge degree of freedom for \HgCl. The sample was 1 mm by 0.5 mm single crystal.  The crystals used were grown as thin plates of typical size 1-2 mm wide and 0.05 mm thick using electrocrystallization method. The conductivity is low in the direction of layer stacking (a-axis ) and high in the layer plane and thus these systems are considered quasi-two dimensional materials. The probe was three-micron laser spot with 514 nm wavelength. The sample was cooled down to 4K in Janice cryostate.
%(order of 1.5^\time 10^{-7} ^S cm$^{-1}$ at RT) 

Raman spectroscopy was used to probe the charge degree of freedom for \HgCl. For every molecule there are 72 vibrational modes, the mode of interest is $\nu_{2}$ because it is sensitive to the charge density on the central C=C molecular bond. As the material is cooled down this charge ordering is manifested by the splitting of the $\nu_{2}$ band  at $1490$ cm$_{-1}$  into two modes,$\nu_{2(A)}$ and $\nu_{2(B)}$ at  1475  and  1507  $cm^{-1}$. The charge on each molecule within the dimer could be inferred from the location of the new bands using a linear relation, \cite{Yamamoto2005}. The charge distribution was calculated to be 0.6e on the charge rich molecule and 0.4e on the charge poor one creating an electric dipole. %The shape of the bands could be understood from the two-site jump model, \ref{}. 

High resolution measurements were taken of $\nu_{2}$ and $\nu_{3}$ vibrational modes and of low frequency modes. The material showed consistently MIT transition at $\sim 30$ K. We noticed that the transition took place at different temperatures, higher temperature on warming. However, such hysteresis was not observed in heat capacity measurements and that agrees with Ref. \cite{} where they didn't detect any hysteresis in resistivity measurements. Such hysteresis could be attributed to worse thermal conductivity in the insulating state and thus higher temperature of transition. During the transition we noticed mixed state where we could see the gradual decrease in the intensity of nu2 and increase in the intensity of $\nu_{2(A)}$ and $\nu_{2(B)}$. The frequency if $\nu_{2(B)}$ starts with smaller value and then as the temperature is lowered its intensity increases and its frequency stabilizes at 1498 cm, see fig. \ref{fig:ReentrantPhase}. From the frequency we calculated the charges to be 0.6e and 0.4e. %Fitting the data to two-site jump model?

\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{ReentrantPhase.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{ a) Temperature dependence of $\nu_{2}$, the measurement was taken on warming. b) Shows the three states of \HgCl: metallic state in red, charged ordered state in green and new re-entrant state in blue.}
\label{fig:ReentrantPhase}
\end{figure}

As the temperature is lowered below 10 K, charge order starts to disappear and a redistribution of spectral weight takes place, see fig. \ref{fig:ReentrantPhase}. This behavior is observed on warming and cooling. Also, different cooling rates, 10K/min, 1K/min and 0.2 K/min, produced the same result.  Also, we have noticed that some locations below 10K showed less spreading and spectral distribution closer to that of CO; this suggest that the ground state consists of domains. Also, the emv-coupled mode at $\sim 1400$ \cm was sensitive to such crossover; at MIT transition the intensity of corresponding band increases abruptly but as the temperature is lowered below 10 K its intensity decreases, see Fig. \ref{fig:LowFreqReentrantPhase} a). Similar behavior was detected at $\sim 450$ \cm, see Fig.\ref{fig:LowFreqReentrantPhase} b). The significance of these two bands and how they are sensitive to charge redistribution is still under investigation. 

\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{LowFreqReentrantPhase.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{ a) Temperature dependence of the emv-coupled mode at  1400 \cm. b) Temperature dependence of the low frequency mode at 450 \cm}
\label{fig:LowFreqReentrantPhase}
\end{figure}

\section{ Discussion}

Fitting charge sensitive bands shows that $\nu_{2(A)}$ frequency increases and that of $\nu_{2(B)}$ decreases, suggesting that the charge disproportionation starts to change from 0.6e and 0.4e to closer to 0.5e. Also, the mode $\nu_{2}$ starts to appear again where a significant spectral weight accumulates below 10K. There are few similarities and differences between this new state and the mixed state at the MIT transition; the frequency of  $\nu_{2(B)}$ decreases as the temperature is lowered below 10 K which is same behavior at the mixed state where the CO states starts to form. The spectral weight of $\nu_{2}$ starts to increase gradually but also with increasing width. Unlike in the mixed state , $\nu_{2}$, $\nu_{2(A)}$ and $\nu_{2(B)}$ merge together to form a very broad peak. In the mixed state the three bands stay separate. All this suggests that a crossover is taking place at very low temperature. 

\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{ReentrantPhaseFits.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a) Raman spectra with Lorentzian fits to the peaks. b) Temperature dependence of the frequency and widths of $\nu_{2}$, $\nu_{2(A)}$ and $\nu_{2(B)}$ peaks.}
\label{fig:ReentrantPhaseFits}
\end{figure}

Plots of the frequencies and widths of $\nu_{2}$, $\nu_{2(A)}$ and $\nu_{2(B)}$ vs temperature are shown in figure \ref{fig:ReentrantPhaseFits}. $\nu_{2(A)}$ and $\nu_{2(B)}$ bands represent equal number of charge rich and charge poor molecules. but have different spectral weights. To better understand the behavior and configuration of the charges we computed the percentage of the molecules contributing to the intensities of the bands. We fitted the bands to a Lorentzian peaks and then calculated the area of those peaks using the equation $1.59\times w\times h$ where $w$ is the full width at half maximum and $h$ is the intensity. We then normalized the areas of $\nu_{2}$, $\nu_{2(A)}$  and $\nu_{2(B)}$ using that of $\nu_{3}$. Taking the at 19 K the system is in pure charged ordered state and at 50K it is in pure metallic state; we used those spectra to calibrate the spectra and calculate the percentage of molecules with 0.5e, 0.4e and 0.6e at 4 K, see fig. \ref{fig:Yield}. We assumed that $100\%$ of molecules contribute to the spectral weight of $\nu_{2}$ at 50K, and $50\%$ of molecules are charge rich that contribute to the spectral weight of $\nu_{2(A)}$ at 19K. Similarly, $50\%$ of molecules are charge poor that contribute to the spectral weight of $\nu_{2(B)}$ at 19K. By adjusting for the difference in spectral weight we calculated the percentage of molecules contributing to the three bands at 4K. We used the fit in fig.\ref{fig:Yield} to calculate such percentages and found that the three charge densities, 0.6e, 0.4e and 0.5e are equally distributed among molecules in the material with each constituting one third of the total number of molecules. 
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{Yield.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{ a) Fitting the area under the $\nu_{2}$, $\nu_{2(A)}$ and $\nu_{2(B)}$ peaks to a polynomial function. b) Fitting the efficiency factors to  to calculate the ratio of molecules with certain charge.}
\label{fig:Yield}
\end{figure}

There are few theories that could explain the behavior that we are still investigating. First, it could be charge glass with the charge densities range from 0.4e to 0.6e randomly distributed among molecules. The redistribution of charges could suggest a re-entrant charge glass. The existence of domains also implies the coexistence of re-entrant charge glass and charge order. However, there was no evidence of Bose peak to imply glassy state. In addition, heat capacity measurements do not show any behavior at very low temperatures other than the lattice contribution $\sim T^{3}$. Fig. \ref{fig:HCandRT} shows that heat capacity at cooling rates 0.5K/min and 10K/min are identical which would not be the case if the ground state was a glassy state; i.e. no deviation from the $\sim T^{3}$ behavior at low temperature after fast cooling. We would expect glassiness or disorder affect Raman and heat capacity at low temperatures especially at very high cooling rates but that behavior was consistently absent. 
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{HCandRT.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{ a) RT plot of \HgCl ~from Ref. \cite{Gati2018}. b) Heat capacity measurement of \HgCl under different cooling rates, 0.5K/min (blue) and 10K/min (red)}
\label{fig:HCandRT}
\end{figure}

Another interesting explanation is that charge ordered state crosses over to an exotic metallic state, as suggest by ref. \cite{HottaCrys2012}, due to charge frustration in quarter filled organic compounds that destroys charge order leading to the the coexistence of solid and liquid components of charges. In such ground state, pinball liquid, only third of the charges contribute to the charge order (solid part) and the rest of the charges are mobile (liquid). Such states are ground states in the special case of $V'=V$ due to the the higher degeneracy than the charged ordered states. Such exotic metals due to charge frustrations are less conductive than normal metals. This could be suggested as the mechanism of the crossover in our material if we took the resistivity measurements in ref. \cite{Gati2018} into consideration where the resistivity showed a slight decrease at around 10K, see Fig. \ref{fig:HCandRT}. However, heat capacity at low temperatures still doesn't show any deviation from $T^{3}$ behavior. 


%into consideration where as accurate measurements  In \ref{}, 
%Two possible ground states, depending on the value of U, are the pinball liquid and three-fold charge metal.

A third possibility is that in the charge ordered state the dipoles are static without any fluctuations within the dimer. As we lower the temperature, the charge starts to fluctuate leading to the decreased difference between frequencies and the emergence of $\nu_{2}$. The broad peak could take place due to the large range of exchange frequency $\omega_{ex}$ from 0 to 40 \cm similar to \HgBr. That possibility is still under investigation. 

In conclusion, we took high resolution Raman measurement of \HgCl and detected the well understood MIT transition at $\sim 30$ K . However,  as we lower the temperature we detect a re-entrant state where the charge ordered state is destroyed. Our calculations suggest a rough estimate of equal charge distribution among molecules with each third of the molecules have charges 0.5e, 0.4e and 0.6e. Thermal expansion measurement from Ref. \cite{Gati2018}did not detect any change in the crystal constants at the crossover temperature 10K. Similarly, our heat capacity measurement didn't detect any sign of phase transition at 10 K.  Also, lattice contributions to heat capacity did not show any change under cooling rates. These measurements  suggest that the crossover is not due to change in lattice structure but rather due to frustration because of quantum effects. This, however, is under study; it could be argued that the entropy change is too small to be seen in heat capacity measurements  especially for these 2D charge transfer materials.   


%However, heat capacity measurements do not show any behavior at very low temperatures other than the lattice contribution \sim T^{3} . Also, comparing different cooling rates of (0.5K/min and 10K/min) didn't show any deviation from the \sim T^{3} behavior in low temperature heat capacity data. 
%heat capacity measurements didn't show any linear term due to glassy behavior at low temperatures. Heat capacity, however, doesn't show any peculiar behavior in the temperature range 2K to 20K, see fig. \ref{HC}. 


%Lattice contributions didnt aho any change , ref didn't see any change so frustration is most probabae
%There are two (complementary?) models to explain such behavior. 

%It was suggested that however the entropy change is too small to be dtetcted, 


