\chapter{Heat Capacity Measurements}
\label{sec:HeatCap}
%\chaptermark{Optional running chapter heading}


Heat capacity of three different materials was measured to compare similarities in phonon contribution and detect any differences. Heat capacity of iso-structures $\kappa$-Hg-Cl and $\kappa$-Hg-Br were taken at different cooling rates and magnetic fields. Also, measurements of spin liquid compound $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$ were taken to compare to literature values of phonon and spinon contributions. The main result is that all three compounds share almost same phonon contribution but have very different low temperature behavior. $\kappa$-Hg-Cl had $\sim T^{3}$ behavior at at very low temperatures while $\kappa$-Hg-Br and $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$ have comparable values of Sommerfeld coefficient. However, the linear term of $\kappa$-Hg-Br changes a little with magnetic field and cooling rates. Below are the results in more detail. 

\section{Results and Discussion}
\label{sec:HCresults}

\subsection { $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$}
We conducted heat capacity measurements on the spin liquid compound $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$ to verify that the results agree with literature and to compare to heat capacity of our new compounds. The measurements were done using Quantum Design setup based on the relaxation method.  We took the first measurement in zero and 5 T field on warming. The sample (0.38 mg) was cooled down at $0.5 K/min$ in zero field. Results for $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$, are shown in fig.\ref{fig:ET-Cu-CN3_general3}, fig.\ref{fig:ET-Cu-CN3_field} and fig. \ref{fig:k-ET-Cu-CN3_CoolingRate}. Fig.\ref{fig:ET-Cu-CN3_general3} a) is a plot of heat capacity vs temperature; the figure shows the same kink detected by other experiments at 6 K \cite{Yamashita2008}. The explanation of that kink is highly speculative and possible interpretations include the onset of spin liquid state. Fig.\ref{fig:ET-Cu-CN3_general3} b) is a plot of $C_{p}/T$ vs $T^2$ for temperature in the range 1.8 K to 3.8 K to show the linear contribution. The red line shows a linear fit to the low temperature portion of the data.  The fit to the data is a line with slope that represents cubic contribution due to lattice phonons, and intercept that represent linear contribution to the heat capacity. The coefficient of the linear term (Sommerfeld coefficient) is of the same order of that detected in literature \cite{Yamashita2008} with a value around $8.5 mJ K^{-2} mol^{-1}$ compared to $12 mJ K^{-4} mol^{-1}$. The lattice contribution $\beta=21 mJ K^{-4} mol^{-1}$ is measured by fitting the data to $\beta T^3$ and agrees perfectly with values found in literature. %Fig.\ref{fig:ET-Cu-CN3_general3} c) is a plot $C_{p}/T^{3}$ vs $T$ to show anomalies or contributions other than cubic and linear function of temperatures. The figure shows that anomaly at around 6K that corresponds to the kink observed.  

\begin{figure}[!ht]
\includegraphics[width=\textwidth]{ET-Cu-CN3_general3.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a) Plot of $C_p vs~ T$ of $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$. There is kink at 6 K that is detected in Ref.~\cite{Yamashita2008}. b) Plot of  $C_p/T vs~ T^{2}$, the linear fit is shown in red.}
\label{fig:ET-Cu-CN3_general3} 
\end{figure}
Fig.\ref{fig:ET-Cu-CN3_field} shows the effect of magnetic field on the heat capacity. Fig. \ref{fig:ET-Cu-CN3_field} a) is log-log plot of heat capacity vs temperature in zero filed (blue curve) and 5T (red curve). The data doesn't show any change in the kink shape or location due to field. In fig. \ref{fig:ET-Cu-CN3_field} b) The black line shows a linear fit to the low temperature portion of the data shown in red (5 T) and blue (0 T) curves. The Sommerfeld coefficient was not suppressed by applying magnetic field which agrees with previous experiments. 


\begin{figure}[!ht]
\includegraphics[width=\textwidth]{ET-Cu-CN3_field3.pdf} 
\makeatletter
\let\@currsize\normalsize
\caption{a) Plot of $C_p  vs ~T$ of $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$ at two fields. b)  Plot of $C_p T^{-1} vs ~T^2$  at low temperatures. The data was fitted to a line with the slope equals to the lattice contribution, $\beta$, and the intercept is the Sommerfeld coefficient, $\gamma$}
\label{fig:ET-Cu-CN3_field}
\end{figure}

Fig.\ref{fig:k-ET-Cu-CN3_CoolingRate} shows the effect of cooling rates on the different contributions to heat capacity. Heat capacity was measured on warming after cooling samples at rates 0.1 K/min, 0.5K/min and 2 K/min. Measurements at cooling rates 0.1 K/min and 2 K/min were measured using a 0.2 mg sample from 2K to 60 K. A larger sample, 0.38 mg, was used in the cooling cycle at 0.5 K/min. All measurements were taken at 0 Oe. Fig \ref{fig:k-ET-Cu-CN3_CoolingRate} a) is a plot of $C_{p}/T$ vs $T$ to show the effect of cooling rates on the general shape of the heat capacity curve. The plot doesn't show any variation in the shape or location of the kink. Also, no variation in the curve shape means no change in the lattice contribution. The inset is a plot of $C_{p}/T^{3}$ vs $T$ to magnify the kink at 6K; there is a slight change in the 0.5K/min cooling rate. However, because the is no change between the data taken at cooling rates 0.1 K/min and 2K/min, such change could be considered within error bars. Fig. \ref{fig:k-ET-Cu-CN3_CoolingRate} b) shows the low temperature contribution for different cooling rates as $C_{p}/T$ vs $T^2$. The low temperature data was fitted to a line with slope equal to the cubic term (lattice contribtion), and intercept equal to the linear term corresponding to spinon contribution. The lattice contribution in all three cooling rates is equal to $\sim 25 mJ/mole.K^4$. The linear term due to the spinon contribution agree within the error at cooling rates 0.1K/min and 2K/min. The value however, is lower than that reported in literature \cite{Yamashita2008}. However, at 0.5K/min, the linear term is larger than  other measurements. However, since there is no pattern whether increasing or decreasing values for the linear term with cooling rates, no conclusion could be derived. DR measurements at different cooling rates could help draw conclusive results. 

\begin{figure}[!ht]
\includegraphics[width=\textwidth]{k-ET-Cu-CN3_CoolingRateNew.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a) Plot of $C_{p}/T$ vs $T$ of of $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$ at cooling rates: 0.1 K/min (blue), 0.5 K/min (green) and 2 K/min (red). Inset is a plot of $C_{p}/T^{3}$ vs $T$.b) Plot of $C_{p}/T$ vs $T^{2}$ at cooling rates 0.1 K/min (blue), 0.5 K/min (green) and 2 K/min (red) . No significant difference is detected. }
\label{fig:k-ET-Cu-CN3_CoolingRate} 
\end{figure}

%The measurement taken at 0.5 K/min shows very good agreement with other cooling rates if multiplied by a small factor (1.17). Fig.\ref{fig:k-ET-Cu-CN3_CoolingRate} a) is a plot of $C_{p}/T$ vs $T$ that shows no significant change in the behavior of heat capacity between different cooling rates; the kink doesn't change position or shape. However, such change might only be seen in much higher cooling rates like was seen in charge glasses. Fitting low temperature  data (from 2K to 3K) for all cooling rates in fig. \ref{fig:k-ET-Cu-CN3_CoolingRate} b) to a line shows that the linear contribution drops in the data taken at cooling rate 0.1 K/min to $3.8 \pm 0.6 mJ K^{-2} mol^{-1}$ and at cooling rate 2K/min to $2.6 \pm 0.7 mJ K^{-2} mol^{-1}$  to when compared to the value of the first measurement at cooling rate of 0.5K/min. The measurement taken at 0.5 K/min cooling rate started at lower temperature (1.8 K instead of 2 K), but that corresponds to only one data point. The mass of the sample was larger which would make the data more reliable. However, to determine whether the cooling rate has a definite effect on the linear term, DR measurements of the compound need to be taken at at least three different cooling rates. 

%when we considered very low temperatures (below 4K), the contribution fitted the $T^{3}$ similar to literature \cite{}. However, we found difficulty fitting until around $30K$

\subsection{$\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Br}

We took heat capacity measurements of three samples of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Br. The three samples had masses 0.8 mg, 0.6 mg and 0.92 mg. Measurements were taken on warming at fields 0 T and 5T. Also different cooling rates were tested. Fig. \ref{fig:k-ET-Hg-Br_general} a) is a plot of $C_{p}/T$ vs $T$ at 0 Oe of samples cooled at 0.5K/min in zero field. The inset is a magnification of the feature (anomaly) at 80 K where MIT takes place. This anomaly is expected to be due to the phase transition. Further testing of the phase transition was taken using a technique on ppms where much longer heat pule was used. Fig. \ref{fig:k-ET-Hg-Br_general} b) is a plot of  $C_{p}/T$ vs $T^2$ at 0 T for a sample cooled at 0.5 K/min. The coefficient of the linear term (Sommerfeld coefficient) is around $7.6 mJ K^{-2} mol^{-1}$ compared to $9 mJ K^{-4} mol^{-1}$ in  $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$. The lattice contribution $\beta=18.4 mJ K^{-4} mol^{-1}$ was measured by fitting the data to $\beta T^3$ term. 

%Fig. \ref{fig:k-ET-Hg-Br_general} a) and b) show the fit for different contributions to heat capacity measured at 0 T of samples cooled at 0.5 K/min. Fig. \ref{fig:k-ET-Hg-Br_general} a) is a plot of $C_{p}$ vs $T$ that shows the change in the slope of the curve in different temperature ranges.  Fig. \ref{fig:k-ET-Hg-Br_general} b) is a plot of $C_{p}/T^3$ vs $T$ that shows acoustic phonons (Einstein model) at around 6K. Below 2.5 K there is an up-rise in the signal that is fitted to linearly in Fig. \ref{fig:k-ET-Hg-Br_general} b).

\begin{figure}[!ht]
\includegraphics[width=\textwidth]{k-ET-Hg-Br_general3.pdf} 
\makeatletter
\let\@currsize\normalsize
\caption{a) Plot of $C_p/T$  vs $T$ of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Br. Inset is a magnification of the kink at $\sim 80$ K.  b)  Plot of $C_p T^{-1} vs ~T^2$  at low temperatures. The data was fitted to a line with the slope equals to the lattice contribution, $\beta$, and the intercept is the Sommerfeld coefficient, $\gamma$}
\label{fig:k-ET-Hg-Br_general}
\end{figure}

Fig. \ref{fig:k-ET-Hg-Br_field} is plot of the heat capacity of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Br under 0T and 5T to compare effect of field on different contributions. Fig. \ref{fig:k-ET-Hg-Br_field} a) is a log-log plot of $C_{p}$ vs $T$ and no significant change is taking place.  Fig. \ref{fig:k-ET-Hg-Br_field} b) is a plot of $C_{p}/T$ vs $T^2$ at 0T and 5T. Because few data points were taken, no conclusion could be derived. DR measurements however, shows the effect of field on the linear terms that correspond to spinon excitations. %Fig. \ref{fig:k-ET-Hg-Br_field} c) shows no effect due to field within the error. magneto-elastic contributions? 

\begin{figure}[!ht]
\includegraphics[width=\textwidth]{k-ET-Hg-Br_field3.pdf} 
\makeatletter
\let\@currsize\normalsize
\caption{a) Log-Log plot of $C_p  vs ~T$ of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Br at two fields. b)  Plot of $C_p T^{-1} vs ~T^2$  at low temperatures. The data was fitted to a line with the slope equals to the lattice contribution, $\beta$, and the intercept is the Sommerfeld coefficient, $\gamma$}
\label{fig:k-ET-Hg-Br_field}
\end{figure}

Fig. \ref{fig:k-ET-Hg-Br_CoolingRate3} shows the effect of cooling rate on the lattice contribution, spinon contribution and phase transition. Fig. \ref{fig:k-ET-Hg-Br_CoolingRate3} a) is plot of $C_{p}/T$ vs $T$ at cooling rates 0.1K/min (blue), 0.5K/min (green) and 2K/min (red). The phase transition at around 80 K doesn't show any change in shape or temperature at either 0.1K/min or 2K/min.  Fig. \ref{fig:k-ET-Hg-Br_CoolingRate3} b) is a log-log plot of $C_{p}$ vs $T$ that shows a change in the low temperature data between 2K and 5K where at higher cooling rates, there is an additional contribution that is fitted in Fig. \ref{fig:k-ET-Hg-Br_CoolingRate3} c). The inset of  Fig. \ref{fig:k-ET-Hg-Br_CoolingRate3} b) is  a plot of $C_{p}/T^{3}$ vs $T$ that also confirms the same result; slight change in the low temperature behavior. A plot of low temperature data from T=2K to T=4.5 K is shown in Fig. \ref{fig:k-ET-Hg-Br_CoolingRate3} c). The plot of $C_{p}/T$ vs $T^2$ is fitted linearly to extract the lattice contribution (slope) and spinon excitation (intercept). The intercept value does get larger with increasing the cooling rate from $1 mJ/mole.K^2$ at 0.1K/min to $13.4 mJ/mole.K^2$ at 2K/min. The lattice term doesn't show any change and has value of $\sim 19 mJ K^{-4}mol^{-1}$.  

% Fig.\ref{fig:fig04} shows the both standard heat capacity and DR measurements. Both give very consistent results of the linear term with different samples. The linear term was about around $14 mJ K^{-2}mol^{-1}$ at a field of 5T in the DR measurements, however, it is double that value at zero field which means that the field has slight effect that needs to be investigated more carefully. To investigate the origin of that linear temperature dependence more carefully we need to consider possible sources, to name few, glassiness, charge degree of freedom, ferromagnetism in 2D or as mentioned before conduction electrons. The last two are ruled out, but charge degrees of freedom are investigated using Raman spectroscopy. 

\begin{figure}[!ht]
\includegraphics[width=\textwidth]{k-ET-Hg-Br_CoolingRate3.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a) Plot of $C_p/T$  vs ~$T$ of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Br at cooling rates 2K/min  (red) and 0.1K/min (blue). b)  Log-Log plot of $C_p/T$  vs ~$T$ for cooling rates 0.1K/min (blue), 0.5K/min (green) and 2K/min (red). The inset is a plot of $C_{p}/T^{3}$ vs $T$. c) Low temperature plot of $C_{p}/T$vs $T^2$. The data was fitted to a line with the slope equals to the lattice contribution, $\beta$, and the intercept is the Sommerfeld coefficient, $\gamma$.}
\label{fig:k-ET-Hg-Br_CoolingRate3} 
\end{figure}

\subsection{$\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl}

We measured heat capacities of few samples of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl with masses 0.56 mg, 1.11 mg and 0.93 mg at different cooling rates and under few values of magnetic field. The results of $C_p/T~ vs ~T$ are shown in fig.\ref{fig:k-ET-Hg-Cl_general3}a). There is an anomaly at 30 K (circled) that is a sign of a MIT phase transition. Interestingly the same feature at 80K found in the heat capacity of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Br also appears in  $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl and this could be attributed to CH$_{2}$ ordering. The inset magnifies both features. Fig.\ref{fig:k-ET-Hg-Cl_general3}b) is a plot of $C_{p}/T$ vs $T^{2}$ at low temperate that was fitted linearly. The fit showed consistently that $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Cl has no linear term and the low temperature behavior is dominated by lattice contributions $\sim T^{3}$. 


\begin{figure}[!ht]
\includegraphics[width=\textwidth]{k-ET-Hg-Cl_general3.pdf} 
\makeatletter
\let\@currsize\normalsize
\caption{a) Plot of $C_p/T$  vs $T$ of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Cl. Inset is a magnification of the anomaly at 30K and $\sim 80$ K.  b)  Plot of $C_p T^{-1} vs ~T^2$  at low temperatures. The data was fitted to a line with the slope equals to the lattice contribution, $\beta$, and the intercept is the Sommerfeld coefficient, $\gamma$}
\label{fig:k-ET-Hg-Cl_general3} 
\end{figure}


Fig.\ref{fig:k-ET-Hg-Cl_Hyst} a) and b) are plots of the anomaly due to phase transition at 30K. The measurements were done on warming (red) and cooling (blue) to check for hysteresis. No hysteresis was detected whether the sample was cooled down at 10 K/min (plot a) or at 0.1 K/min (plot b).
\begin{figure}[!ht]
\includegraphics[width=\textwidth]{k-ET-Hg-Cl_Hyst.pdf} 
\makeatletter
\let\@currsize\normalsize
\caption{a) Plot of $C_p$  vs $T$ of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Cl at cooling rate 10K/min. Measurements were done on cooling (blue) and warming (red). b)  Plot of $C_p$  vs $T$ of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Cl at cooling rate 0.1K/min.}
\label{fig:k-ET-Hg-Cl_Hyst}
\end{figure}
Fig. \ref{fig:k-ET-Hg-Cl_field} is a plot of the heat capacity of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Cl under 0T (blue) , 5T (green) and 9T (red) to study effect of magnetic field on different heat capacity components. Fig. \ref{fig:k-ET-Hg-Cl_field} a) is a log-log plot of $C_{p}$ vs $T$ and no significant change is taking place.  Fig. \ref{fig:k-ET-Hg-Cl_field} b) is a plot of $C_{p}/T$ vs $T^2$ at 0T, 5T and 9T. No changes was detected due to field at low temperature, with $\sim T^{3}$ behavior dominating the contribution to heat capacity.   
\begin{figure}[!ht]
\includegraphics[width=\textwidth]{k-ET-Hg-Cl_field3.pdf} 
\makeatletter
\let\@currsize\normalsize
\caption{a) Log-Log plot of $C_p  vs ~T$ of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Cl at two fields. b)  Plot of $C_p T^{-1} vs ~T^2$  at low temperatures. The data was fitted to a line with the slope equals to the lattice contribution, $\beta$, and the intercept is the Sommerfeld coefficient, $\gamma$}
\label{fig:k-ET-Hg-Cl_field}
\end{figure}

%\subsection{Kanoda compound}
%\subsection{Cooling Rate}

Fig. \ref{fig:k-ET-Hg-Cl_CoolingRate} shows the effect of cooling rate on the lattice contribution and phase transition. Fig. \ref{fig:k-ET-Hg-Cl_CoolingRate} a) is plot of $C_{p}/T$ vs $T$ at cooling rates 10K/min (blue) and 0.5K/min (red). The phase transition at around 30 K shows slight change in shape. The effect is magnified in inset of Fig. \ref{fig:k-ET-Hg-Cl_CoolingRate}a). Fig. \ref{fig:k-ET-Hg-Cl_CoolingRate}b) is a log-log plot of $C_{p}$ vs $T$ that shows no change in the low temperature behavior. The inset of  Fig. \ref{fig:k-ET-Hg-Cl_CoolingRate} b) is  a plot of $C_{p}/T^{3}$ vs $T$ that also confirms the same result with no change in phonon contribution. A plot of low temperature data from T=2K to T=4.5 K is shown in Fig. \ref{fig:k-ET-Hg-Cl_CoolingRate} c). The plot of $C_{p}/T$ vs $T^2$ is fitted linearly to extract the lattice contribution (slope) and any linear contrition if existed. The intercept value is zero at all cooling rates that might imply that no disorder or glassiness takes place. The lattice term doesn't show any change and has value of $\sim 20 mJ K^{-4}mol^{-1}$.  
\begin{figure}[!ht]
\includegraphics[width=\textwidth]{k-ET-Hg-Cl_CoolingRate3.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a) Plot of $C_p/T$  vs ~$T$ of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl at cooling rates 0.5K/min (red) and 10 K/min (blue). The inset is a close up of the anomaly due to phase transition at 30 K. b)  Log-Log plot of $C_p/T$  vs ~$T$ for cooling rates 0.5K/min (red) and 10 K/min (blue). Inset is a plot of $C_p/T^{3}$  vs ~$T$ for cooling rates 0.5K/min (red) and 10K/min (blue). c) Low temperature plot of $C_{p}/T$vs $T^2$. The data was fitted to a line with the slope equals to the lattice contribution, $\beta$, and the intercept is the Sommerfeld coefficient, $\gamma$.}
\label{fig:k-ET-Hg-Cl_CoolingRate} 
\end{figure}

Fig.\ref{fig:k-ET-Hg-Cl_Hyst_CoolingRate}a) is a plot of $C_p$ vs $T$ to compare the shape of the anomaly at 30 K at different cooling rates;10K/min (red curve) and 0.1 K/min (black curve). There is a slight shift in the temperature and shape of the anomaly that is still under investigation. Fig.\ref{fig:k-ET-Hg-Cl_Hyst_CoolingRate}b) is a plot where one of the curves was shifted in temperature to superimpose the two curves; there is slight change at temperatures higher than the MIT transition. Also, it is obvious that the anomaly has a complex shape with what looks like a double peak. The implication of which is still under study.
\begin{figure}[!ht]
\includegraphics[width=\textwidth]{k-ET-Hg-Cl_Hyst_CoolingRate3.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a) Plot of $C_{p}$ vs$T$ of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl of the anomaly at 30 K at cooling rates 10K/min (red) and 0.1K/min (back).b) Same plot as a) with the two curves superimposed.}
\label{fig:k-ET-Hg-Cl_Hyst_CoolingRate} 
\end{figure}

\section{Conclusion}
\label{sec:HCdiscussion}

In this section I will compare the results of heat capacity measurements of the three materials: $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Br (blue), $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Cl(green) and $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$ (red). Heat capacity of these materials generally have $\sim T^{3}$ dependence at low temperature. It was shown that all three materials have similar lattice contribution in the range of $\beta=18.4-24 mJ K^{-4} mol^{-1}$. The heat capacity curve was fit using two kinds of Debye phonons at two Debye temperatures: $T_{D1}\sim 440 K$ and$T_{D2}\sim 101 K$. In fig. \ref{fig:All_Lattice_Contribution} a) the anomaly at 6K distinguishes $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$ from the isostructural materials $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Br and $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Cl. It is hypothesized that the kink at 6K signifies the onset of spin liquid state. Heat capacoity of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Cl shows the anomaly at 30K due to the transition from metallic state  to charge ordered insulating transition.  Both $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Br and $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Cl  have an anomaly at $\sim 80K$ that could be attributed to $CH_{2}$ ordering. Fig. \ref{fig:All_Lattice_Contribution}b) shows a feature at around 10 K that is still under investigation. 


%$T_{D}$: 439.856930367, $T_{D2}$: 101.011178828, $T{E}$: 22.4533232149, $N_{e}$: 0.321160058413, $N_{D1}$: 51.67746168, $N_{D2}$: 7.0013793793, $N$: 59.0000011177

 Fig. \ref{fig:All_Lattice_Contribution} c) compares the linear term of all three materials; $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Br and $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$ show pronounced linear term at very low temperatures. The coefficient of the linear term (Sommerfeld coefficient) is around $7.6 mJ K^{-2} mol^{-1}$ compared to $9 mJ K^{-4} mol^{-1}$ in $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$. Generally, conduction electrons are the major source of a linear term in heat capacity measurements, however, because these compounds are Mott insulators at such low temperatures, conduction electron contribution is ruled out. Another reason for this linear term is believed to be due to spinons. The linear term doesn't  change under the effect of magnetic field in both materials. However, $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Br  shows more pronounced decrease of the Sommerfeld coefficient as the cooling rate is decreased, see fig.\ref{fig:k-ET-Cu-CN3_ET-Hg-Br_compare2}b) and c). That could indicate more disorder or glassiness at low temperatures.


\begin{figure}[!ht]
\includegraphics[width=\textwidth]{All_Contribution.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a) Plot of $C_p/T vs~ T$ of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Br (blue), $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Cl(green) and $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$ (red). b)  Plot of $C_p/T^{3} vs~ T$ of all three materials. c) Log-Log plot of $C_p$ vs $T$ of all three materials; the inset is a plot of $C_{p}/T$ vs $T^{2}$ at very low temperatures.}
\label{fig:All_Lattice_Contribution} 
\end{figure}

\begin{figure}[!ht]
\includegraphics[width=\textwidth]{k-ET-Cu-CN3_ET-Hg-Br_compare2.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a) Plot of $C_p/T$ vs $T$ of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$Br (blue) and $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$ (red). b)  plot of $C_{p}/T$ vs $T^{2}$ at very low temperatures of both materials cooled at 0.1K/min (left panel) and at 2K/min(right panel). c) Plot of $C_p/T^{3} vs~ T$ of both materials cooled at 0.1K/min (left panel) and at 2K/min(right panel)} 
\label{fig:k-ET-Cu-CN3_ET-Hg-Br_compare2}
\end{figure}