\chapter{Experimental Methods}

\section{Raman Spectroscopy}
\label{sec:RamanSpect}
%\chaptermark{Optional running chapter heading}

One way of characterizing and understanding behavior of materials is through light-matter interaction. Different properties of materials could be probed by varying the frequency/energy and polarization of light and analyzing the scattered radiation. For example, elastic x-ray could be used to probe crystal structure while inelastic x-ray (Resonant inelastic X-ray scattering (RIXS)) is used to investigate the electronic structure of atoms, and in X-ray magnetic circular dichroism (XMCD), the polarization is used to probe spin and orbital magnetic moment of an atom. Raman relies on inelastic scattering of monochromatic light  during which energy is transferred between photons and the sample and as a result the scattered radiation has different frequency. Generally, electromagnetic waves in the visible, near infrared, or near ultraviolet range interacts and probes molecular vibrations, phonons or other excitations in the system. Therefore, results of Raman spectroscopy measurements are very rich; we can probe electronic excitations, vibrational modes of molecules and magnetic excitations. 

When using Raman spectroscopy, a monochromatic beam of electromagnetic radiation is focused on the sample and the intensity of the scattered radiation is measured as a function of its wavelength. Fig.\ref{fig:Picture06} shows a model for the dispersion of light by molecules; a photon is absorbed by the molecule to be excited to a virtual state and then decays to a new vibrational state emitting a photon of different energy. By measuring the frequency of the scattered photon, the energy absorbed by the excitation could be calculated and plotted as excitation spectrum.  Fig.\ref{fig:Picture07} shows Raman spectrum plotted as Intensity vs Raman shift or Raman wave number $\omega$, expressed in $cm^{-1}$, which is the difference in frequency between the scattered light and the incident electromagnetic radiation:

\begin{equation}
\omega=\omega_{S}/c-\omega_{I}/c
\end{equation}
 Peaks in the spectrum represent excitations. Location of bands give information about the energy scale of the excitations under study, and the polarization dependence provide information about the symmetry of molecular vibrations and magnetic excitations. Raman Spectrum consists of the Rayleigh line which is due to elastic scattering contribution, Stokes lines and anti-Stokes lines due to the inelastic scattering. Stokes lines are generally more intense because lower vibrational levels are more populated than higher levels and thus, are the ones that we probe in our experiment. 


\begin{figure}[htbp]
\includegraphics[width=\textwidth]{Picture06.jpg}
\makeatletter
\let\@currsize\normalsize
\caption{ A model of Raman scattering. A molecule absorbs a photon to a virtual state and then decays to a vibrational excited state by emitting a photon with lower frequency (Stokes lines). Another possibility with lower probability is that a molecule in an excited state absorbs a photon to a virtual state and then decays to the ground state emitting a photon with higher energies (anti-stokes lines). Elastic scattering is represented by the Rayleigh lines. Adapted from reference \cite{book2}. }
\label{fig:Picture06}
\end{figure}

\begin{figure}[htbp]
\includegraphics[width=\textwidth]{Picture07.jpg}
\makeatletter
\let\@currsize\normalsize
\caption{How a typical Raman spectrum looks like. Data is plotted as Intensity vs. Raman shift in $cm^{-1}$. Adapted from reference \cite{book2}. }
\label{fig:Picture07} 
\end{figure}


\subsection{Instrumentation}
The setup of Raman spectroscopy instruments is shown in fig. \ref{fig:Picture05}. The major components of a Raman setup are a laser source, a dispersive component and a detector. The laser is a source of polarized and coherent EM waves. The dispersive component is used to separate and resolve different frequencies spatially on a detector. A multi-channel detector such as a CCD (Charge Coupled Device) was used in our experiments. Raman setups are more complex with the involvement of many optical pieces to direct, filter and change the beam size. We used commercial spectrometer Horiba Jobin Yvon T64000 that could measure low frequencies as low as 5 \cm (0.8 meV) . Ar$^+$Kr$^+$ laser was used where it has 10 excitation lines from 467 nm to 647 nm. To cool down samples we used Janis cold finger micro cryostat that could go down to 4 K.

In Raman spectroscopy, there are few ways to setup the sample;  we used the pseudo-Brewster angle geometry where the incident EM waves are incident on the sample at Brewster's angle. Depending on the polarization of the incident, scattered radiation and symmetry of the excitation, certain modes could be singled out to investigate. Character tables are used to determine the polarization of incident and scattered radiation to detect the mode of interest. 
\begin{figure}[!htbp]
\includegraphics[width=\textwidth]{Picture05.jpg}
\makeatletter
\let\@currsize\normalsize
\caption{ The major components of any Raman setup. There is the laser source, a component to disperse the scattered light and a detector. Adapted from ~\cite{book2}.}
\label{fig:Picture05}
\end{figure}

\begin{figure}[!htbp]
\includegraphics[width=\textwidth]{Polarization.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{Shows back-scattering setup that shows the polarized incident and scattered light. Adapted from ~\cite{book2}. }
\label{fig:Polarization}
\end{figure}


\clearpage
\subsection{Vibrational Raman}
\begin{comment}
%As mentioned earlier, the scattered light consists of two types: one, called Rayleigh scattering, is strong and has the same frequency as the incident beam (V_{Q}), and the other, called Raman scattering, is very weak (~ 10"^ of the incident beam) and has frequencies vo — Vm and VQ + Vm lines ( called the Stokes and anti-Stokes line), where Vm is a vibrational frequency of a molecule.  Thus, in Raman spectroscopy, we measure the vibrational frequency (v^) as a shift from the incident beam frequency (vo).*



%"The theoretical background of the Raman effect is already extensively described in literature. The Raman effect can be considered as the inelastic scattering of electromagnetic radiation. During this interaction, energy is transferred between thephotons and the molecular vibrations. Therefore, the scattered photons have a different energy to the incoming photons."
\end{comment}
According to classical theory, vibrational Raman scattering can be explained as the scattered radiation due to the change in the polarizability of the molecule as it vibrates under the influence of the oscillating electric field.  When a molecule is positioned in an electrical field , an electrical dipole moment is induced. The relation between this induced dipole moment and the electrical field can be expressed as follows, \cite{book2}:
\begin{equation}
{\bf p}=\alpha.{\bf E}+(1/2).\beta .{\bf E}^{2}+(1/6).\gamma.{\bf E}^{3}
\end{equation}
where $\alpha$, $\beta$ and $\gamma$ are the polarizability, hyper-polarizability and 2nd hyper-polarizability tensors. since $\alpha>>\beta,\gamma$, the electric dipole could be simplified to:
\begin{equation}
{\bf p}=\alpha.{\bf E}
\end{equation}
Because the electric filed is caused by an oscillating EM wave, the electric field vector can be written as
\begin{equation}
{\bf E}={\bf E}_{o}.\cos(2\pi \nu_{o}.t)
\end{equation}
where $t$ is time and $\nu_{o}$ is the frequency of the EM radiation. The polarizability tensor $\alpha$ depends on the shape and dimensions of the chemical bonds as they change during a vibration. Therefore, the polarizability tensor $\alpha$ is dependent on the normal coordinate $Q$ of the molecule. As a result, the polarizability could be expanded as Taylor series in coordinates $Q$ 
\begin{equation}
\alpha=\alpha_{o}+\sum_{k}\Bigg(\frac{\partial \alpha}{\partial Q_{k}}\Bigg)_{o}+\frac{1}{2}\sum_{k,l}\Bigg(\frac{\partial^{2} \alpha}{\partial Q_{k}\partial Q_{l}}\Bigg)_{o}.Q_{k}.Q_{l}+...
\end{equation}
$Q_{l}$ and $Q_{k}$ are normal coordinates that change in the $k^{th}$ and $l^{th}$ normal vibrational modes with frequencies $\nu_{k}$ and $\nu_{l}$. To a first approximation where all vibrational modes are independent of each other, cross terms could be ignored and only the first two terms are retained. Therefore, the equation could be reduced to:
\begin{equation}
\alpha_{\nu}=\alpha_{o}+\alpha^{'}_{\nu}.Q_{\nu}
\end{equation}
where $\alpha^{'}_{\nu}=\frac{\partial \alpha}{\partial Q_{\nu}}$ is the derivative of the polarizability tensor to the normal coordinate $Q_{\nu}$. If we assumed that to a first approximation the normal coordinates are those of a harmonic oscillator, the normal coordinate as a function of time can be written as:
\begin{equation}
Q_{\nu}=Q_{\nu o}.\cos(2\pi. \nu_{\nu}.t+\phi_{\nu})
\end{equation}
where $Q_{\nu o}$ is the amplitude of vibration and $\phi_{\nu}$ is the phase angle. By substitution, we get:
\begin{equation}
{\bf p}=\alpha_{o}.{\bf E}_{o}.\cos(2\pi \nu_{o}.t)+\alpha^{'}_{o}.{\bf E}_{o}.Q_{\nu o}.\cos(2\pi \nu_{o}.t).\cos(2\pi\nu_{\nu}t+\phi_{\nu})
\end{equation}
Applying the trigonometric formula $\cos(A).\cos(B)=1/2[\cos(A+B)+\cos(A-B)]$ we get:
\begin{equation}
\begin{split}
{\bf p}=&\alpha_{o}{\bf E}_{o}\cos(2\pi \nu_{o}t)\\&+1/2 \alpha^{'}_{o}{\bf E}_{o}Q_{\nu o}\cos[2\pi (\nu_{o}+\nu_{\nu})t+\phi_{\nu}]\\&+1/2 \alpha^{'}_{o}{\bf E}_{o}Q_{\nu o}.\cos[2\pi (\nu_{o}-\nu_{\nu})t-\phi_{\nu}]
\end{split}
\end{equation}
Therefore, the induced dipole moment as a function of the vibrational frequencies of the incident radiations $\nu_{o}$ and of the molecule $\nu_{\nu}$ is:
\begin{equation}
{\bf p}={\bf p}(\nu_{o})+{\bf p}(\nu_{o}+\nu_{\nu})+{\bf p}(\nu_{o}-\nu_{\nu})
\end{equation}
where the first term is the elastic Rayleigh scattering, second term and third terms are Raman anti-Stokes and stokes scattering. From equation 3.9, it is obvious that Raman effect only takes place when the change in polarizability with respect to coordinates is not zero; i.e.:
\begin{equation}
 \alpha^{'}_{\nu}=\Bigg(\frac{\partial \alpha}{\partial Q_{\nu}}\Bigg)_{o}\neq 0
\end{equation}
%Raman spectra are plotted in terms of the Raman shift ,$cm^{-1}$,  and is related to the difference in frequency between scattered light and the incident electromagnetic radiation as follows:
%\begin{equation}
%\omega=\frac{\nu_{m}}{c}-\frac{\nu_{o}}{c}
%\end{equation}
%where  $\nu_{m}$ is the scattered light and $\nu_{o}$ is the incident EM wave, and $c$ is the speed of light. 
In reality, with more complex molecules, the polarizability term is a tensor that relates the dipole moment vector and the applied electric field. Certain tensor elements are probed when the incident and scattered are in certain polarization. Therefore, the symmetry of the molecule and vibrational mode determine the polarizability tensor and the choice of incident and scattered polarization of radiation.  In our experiments, more emphasis was placed on the temperature dependence of shape and width of Raman bands rather than their symmetry analysis. Below, I will explain what feature we were interested in and how we used Raman to probe it.

%Imagine in a molecule that have many vibrational modes and every mode is represented by a harmonic oscillator such all vibrational modes are at ground state. a fundamental transition is a transition from a state where all the vibrational modes of molecule is in ground state to another state where one mode is excited to higher energy level. the ground state is:


\subsubsection*{Raman Spectroscopy of BEDT-TTF molecules}

Raman spectroscopy was used to probe the charge degree of freedom for \HgBr ~and \HgCl. In other words, we are interested in the charge dynamics on the BEDT-TTF molecule within the cation layer. As mentioned in Chapter 2, charge order arises in $1/4$ filled systems where charge is not distributed equally among lattice sites leading to broken symmetry. In the metallic state both \HgCl, and \HgBr ~have uniform charge across molecules where holes are exchanged between molecules and dimers. As \HgCl~ goes through MIT, charge disproportionation takes place where one molecule is charge rich and the other is charge poor within the dimer, see Fig \ref{fig:ChargeRaman2a} a). To investigate such change in the charge configuration, we probe certain vibrational modes of BEDT-TTF molecule. 

BEDT-TTF is a flat molecule of 26 atoms with $D_{2h}$ symmetry. It has 72 vibrational modes, few of which are of interest to us. The two modes we investigated were $\nu_{2}$ and $\nu_{3}$, see Fig. \ref{fig:ChargeRaman2a} b). $\nu_{2}$ was of particular importance to us because it is charge sensitive; this is a result of the lengthening of the central C=C bond of the molecule when more charge occupy the highest occupied molecular orbital (HOMO). The on-molecule charge is probed by measuring the frequency of the central C=C molecular bond  vibration which  changes by $\sim$ -140~\cm\ when the charge state changes from (BEDT-TTF)$^0$ to (BEDT-TTF)$^{+1}$~\cite{Dressel04,Yamamoto2005}. Vibrational mode $\nu_{3}$ however, doesn't couple to the HOMO  but we follow its temperature dependence at 1470~\cm\ to emphasize that changes detected in $\nu_{2}$ are due to charge effects only and not due to structural effects that can change the behavior of $\nu_{3}$ with temperature.  %if It stays a narrow single line through the measured temperature range . 


\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{ChargeRaman2a.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{~a) Charge order in $\kappa$ phase where within a dimer there is a charge rich and a charge poor molecule, adopted from Ref.~\cite{Drichko2014}.~b) Vibrational modes $\nu_{2}$ and $\nu_{3}$ and their effect on the stretching of the central C=C bonds, from Ref.~\cite{Yakushi2012_Crystals}}
\label{fig:ChargeRaman2a}
\end{figure}

Fig. \ref{fig:ChargeRaman2b} shows the effect of charge order on the shape of the vibrational band $\nu_{2}$. In the metallic phase $\nu_{2}$  is at $\sim$ 1490 \cm and as the material is cooled down through the transition temperature, the BEDT-TTF molecules differentiate into charge rich and charge poor molecules. This charge ordering is manifested by the splitting of the $\nu_{2}$ band into two bands at 1506 \cm and 1475 \cm. There is a linear relation, determined experimentally in Ref.\cite{Yamamoto2005}, between the charge on the molecule and the location of the band, see fig. \ref{fig:ChargeRaman2b}. Therefore, the charge rich molecule in \HgCl~ at band location 1475~\cm has charge +0.6e and charge poor molecule at band location 1506~\cm has charge +0.4e. %charge on each molecule within the dimer could be inferred from the location of the new bands using a linear relation, fig.\ref{fig: Picture09}. 

\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{ChargeRaman2b.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a)Charge order in $\kappa$ phase where the $\nu_{2}$ is split into two bands $\nu_{2A}$ and $\nu_{2B}$. b)A plot of charge vs. Raman shift that shows a linear relation between energy of vibration and charge on an ET molecule. (from Ref.~\cite{Yamamoto2005}).}
\label{fig:ChargeRaman2b}
\end{figure}

We used the two-state-jump model  \cite{Yakushi2012_Crystals} to fit our Raman spectra to get an estimate of the jump rate between molecules within a dimer. The calculations of the shape of the $\nu_2$ band are done using Eq.~\ref{formula}. According to the model, the shape of the $\nu_2$ depends on $\omega_{EX}$, the frequency of the jumps between the two states BEDT-TTF$^{+0.4}$ and BEDT-TTF$^{+0.6}$. Fig, \ref{fig:TwoSiteJump} shows that $\omega_{EX}$ is $\sim 0$ in charge ordered state \HgCl~ but as the frequency increases, the bands get closer and eventually at very high frequency they merge forming one broad peak.The shift of the bands from the averaged band parameter $\Delta$ = 16~\cm\ is obtained using vibrational spectra in the ordered phase of \HgCl.  $\Gamma$ is the natural width of the vibrational bands experimentally demonstrated in our measurements  by the width of $\nu_3$. 

\begin{equation}
I(\omega)\propto \\
\\
Re [
\begin{pmatrix}a_R, & a_P \end{pmatrix}
\begin{pmatrix} i(\omega-(\omega_{1/2}-\Delta))+\omega_{EX}/2+\Gamma/2 & -\omega_{EX}/2 \\
-\omega_{EX}/2  &    i(\omega-(\omega_{1/2}+\Delta))+\omega_{EX}/2+\Gamma/2 \end{pmatrix}^{-1}
\begin{pmatrix}
 a_R \\ a_P
\end{pmatrix} ]
\label{formula}
\end{equation}

\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{TwoSiteJump.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{Two-states-jump model copied from Ref.\cite{Yakushi2012_Crystals}}
\label{fig:TwoSiteJump}
\end{figure}


% Some organic conductors show charge ordering, fig.\ref{fig: Picture08}, where the  The molecule has $D_{2h}$ symmetry, see fig.


%High resolution Raman measurements have been taken for our material at different temperatures to investigate the change in the $\nu_2$ band. Fig.\ref{fig: fig05} shows the temperature dependence of our spectrum and it is concluded that no charge ordering is taking place. Charge fluctuations could be manifested by broadening of the $\nu_2$ band. However, our data shows no charge fluctuations. 

  

%As mentioned before our material could be mapped into triangular 2D spin 1/2 lattice, see fig.\ref{fig: Picture04}. Thus, two-magnon excitation in Raman spectroscopy is used to estimate the exchange interaction and probe the degree of frustration in $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Br. Two-magnon excitation is the double flipping of two spins; in Raman scattering one photon is absorbed and two electrons are excited to their spin excited states and a photon is scattered. The frequency of the excitation is proportional to the exchange interaction energy. In an anti-ferromagnetic square 2D lattice, $A_{1g}$ is forbidden while $B_{1g}$ band is observed as broad peak with the center at frequency shift of $J/3$. Theoretical work and simulations predict that as the frustration increases, the polarization dependence becomes less pronounced and the Raman shift moves towards lower frequencies. Fig.\ref{fig: Picture10} shows the polarization dependence of Raman spectrum for the spin liquid compound $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$ \cite{Nakamura2014}. Temperature dependence is shown below; as the temperature increase, thermal excitations dominate and two-magnon excitations are suppressed. For parallel polarization, the broad peak is suppressed while at cross polarization the center of the broad band is at around $200 K$. We performed low resolution Raman measurements and as in fig.\ref{fig: fig08} it is obvious that no bands are observable at the frequencies probed at different polarizations. Fig.\ref{fig: fig07} shows the temperature dependence that shows the absence of two-magnon excitations at all temperatures. The narrow peaks are molecular vibrations. More Raman measurements need to be taken to at lower frequencies to check for the presence of spin singlets. 


\begin{comment}
\begin{figure}[htbp]
\includegraphics[width=\textwidth]{Picture08.jpg}
\makeatletter
\let\@currsize\normalsize
\caption{Charge order in $\kappa$ phase where within a dimer there is a charge rich and a charge poor molecule from Ref.~\cite{Drichko2014}.}
\label{fig:Picture08}
\end{figure}
\begin{figure}[htbp]
\includegraphics[width=\textwidth]{Picture09.jpg}
\makeatletter
\let\@currsize\normalsize
\caption{ A plot of charge vs. Raman shift that shows a linear relation between energy of vibration and charge on an ET molecule. (from Ref.~\cite{Yamamoto2005}).}
\label{fig: Picture09}
\end{figure}
\begin{figure}[htbp]
\includegraphics[width=\textwidth]{fig08.jpg}
\makeatletter
\let\@currsize\normalsize
\caption{ The polarization dependence of Raman spectrum of our compound $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Br. No bands were observable in any of the polarizations shown}
\label{fig: fig08}
\end{figure}

\end{comment}


%\cite{Yamamoto2005} for linear relation
%\cite{Yakushi2012_Crystals}
%\cite{Drichko2014}
\clearpage
\subsection{Magnetic Raman Scattering}

In last section, I introduced how inelastic Raman scattering probes molecular vibration. In this section, I will discuss how Raman scattering probes magnetic behavior of materials. In an Anti-ferromagnet, the ground state is an ordered state where every unpaired electron spin in the outermost atomic/molecular energy level is oriented opposite to the spin of electrons on nearest neighbor sites. Raman can probe low energy excitations and those excitation come in the form of spin waves, see fig. \ref{fig:MagRaman1}. Those waves are low energy propagating disturbances in the long range order of spin. These collective excitations are quantized to form quasi-particles called magnon. Magnons carry a certain amount of energy $\omega$ and momentum ${\bf q}$, and have spin-1. Raman scattering can not detect magnons directly; inelastic scattering  probes electric dipole transitions by changing the polarizability of the molecule. Therefore, magnons are detected indirectly through two mechanisms:\\
1) spin-orbit coupling (first order magnetic scattering; main mechanism of one-magnon scattering).\\
2) Exchange interaction (2nd order magnetic scattering; main mechanism of two-magnon scattering).

\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{MagRaman1.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{Spin waves in an anti-ferromagnet copied from Ref.\cite{Loudon}}
\label{fig:MagRaman1}
\end{figure}

The first mechanism is not relevant to my work because organic conductors have very weak spin-orbit coupling and in addition to that, 1st order magnetic scattering has lower intensity than two-magnon scattering. In the two-magnon interaction, two spins are exchanged leaving the system at an excited state. Because the momentum introduced by Raman radiation is very small $K_{I}-K_{S}\approx 0$ and total momentum is conserved, the two-magnon process is simply the propagation of two identical magnons in opposite directions, assuming one magnetic site per primitive cell.  Fig. \ref{fig:MagRaman2} shows the mechanism and how the spin exchange takes place in opposite directions producing two magnon excitations with the same energy but opposite momentum: 


\begin{equation}
\omega_{I}-\omega_{S}=2\omega_{\bf q}
\end{equation}

\begin{equation}
{\bf K}_{I}-{\bf K}_{S}=q_{1}+q_{2}\approx 0
\end{equation}

\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{MagRaman2.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{Propagation of two-magnon excitation in an anti-ferromagnet, inspired by Ref.\cite{Loudon}}
\label{fig:MagRaman2}
\end{figure}


Fig. \ref{fig:MagRaman3} shows the mechanism of spin exchange in detail. The spin exchange takes place between electrons on nearest neighbors ions. In the initial state two electrons are in the ground state $\ket{\mu_{+}({\bf r}_{1})\nu_{-}({\bf r}_{2})}$ where $\ket{\mu}$ and $\ket{\nu}$ are one-electron orbitals on nearest neighbor ions $\mu$ and $\nu$ and $+$, $-$ subscripts indicate spin components of $+1/2$ and $-1/2$. Spin up electron on $\mu$ state is excited by the electric field in the exciting EM wave to higher energy state $\ket{p}$ through electric dipole transition without changing its spin/total angular momentum. Then the spin up electron hops to orbital $\ket{p}^{'}$ on the second ion, due to electron-electron coulomb repulsion, the spin down electron hops to higher energy level in the first ion orbital $\mu$. The spin up electron on state $p^{'}$ then transitions to lower energy orbital and loses energy to produce scattered Raman radiation. As a result of this process, the electrons flip their positions on different ions without changing their spins with final state $\ket{\mu_{-}({\bf r}_{2})\nu_{+}({\bf r}_{1})}$. Such process can be expressed through the transfer matrix element:

\begin{equation}
\braket{\nu_{+}({\bf r}_{1})|e\hat{\textbf{E}}_{S}.{\bf r}_{1} |p^{'}_{+}({\bf r}_{1})}\Braket{p^{'}_{+}({\bf r}_{1})\mu_{-}({\bf r}_{2}) |\frac{e^{2}}{4\pi\epsilon_{0}}\left|{\bf r}_{1}-{\bf r}_{2}\right||\nu_{-}({\bf r}_{2})p_{+}({\bf r}_{1})}\braket{p_{+}({\bf r}_{1})|e\hat{\textbf{E}}_{I}.{\bf r}_{1} |\mu_{+}({\bf r}_{1})}
\end{equation}


where the first and last terms represent the electric-dipole transition of spin up electron from states $\ket{p^{'}}$ to $\ket{\nu_{+}}$ and from $\ket{\mu_{+}}$ to $\ket{p_{+}}$. The middle term represents the exchange due coulomb repulsion between electrons. 

\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{MagRaman3.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{Mechanism of two-magnon excitation due to exchange interaction, From Ref.\cite{ValentineThesis}}
\label{fig:MagRaman3}
\end{figure}

Two-magnon scattering is generally shown in Raman spectrum as a broad peak. The intensity of the band doesn't reflect the two-magnon density of states but rather is a weighted average. This is because of the electric dipole transition term in the transfer matrix, see eq. 3.15. This implies that some two-magnon states will not contribute to the Raman intensity because the corresponding dipole transition is not probed by the polarization of incident and scattered light. The shape and location of two-magnon bands are determined by the magnon interaction. For example, in a 1D chain of spins with anti-ferromagnetic interactions within Ising model, the spin flip of one electron would cost an energy of $2JzS$ where $z$ is the number of nearest neighbor, $S$ is the spin and $J$ is the exchange interaction energy between spins. Therefore, the cost of spin flip of two non-interacting spins is $4JzS=4J$. However, in case of two-magnon scattering, the process could be viewed as the flip of two interacting spins and the energy would be $2J(2zS-1)=2J$, a reduction in the energy of the two-magnon excitation compared to twice the energy of single magnon. The location of the band corresponds to the excitation energy of two-magnon process.

In 2D magnetic lattices, the interactions get more complicated and generally complex models are tested where more terms are included in the Hamiltonian that includes, next nearest neighbors and ring exchange. Such models are studied to predict the shape, location and polarization dependence of two-magnon band in Raman spectrum. Also, few theoretical calculations \cite{Vernay2007,Nakamura2014} were done to study the effect of frustration on the polarization dependence/symmetry and energy of two-magnon bands. In a model of 2D lattice with nearest neighbor interaction $J$ and next nearest interaction $J'$, theoretical calculations were performed, \cite{Vernay2007,Nakamura2014} where the ratio $J'/J$ is increased form 0 (square lattice) to 1 (isotropic triangular lattice). It was shown theoretically and experimentally \cite{Drichko2015,Nakamura2014} that in 2D anti-ferromagnetic square lattice, two-magnon band shows strong polarization dependence where the density of states is much more pronounced in cross polarization, $B_{1g}$ mode, than in parallel polarization, $A_{1g}$ mode, see fig. \ref{fig:MagRaman4}~b) and \ref{fig:MagRaman5}. Such strong polarization dependence decreases as $J'$ increases from 0 to $J$, see Fig.\ref{fig:MagRaman4}~a) and \ref{fig:MagRaman5}. So in frustrated systems, two-magnon excitations appear in Raman spectrum in the $A_{1g}$ mode too. Also, the two-magnon peak position changes from $3J$ to $3J/2$ by changing $J'=0$ to $J'=J$. In particular, frustration-driven spectral downshift was observed experimentally \cite{Vernay2007,Nakamura2014}.  The general trend observed is that spectral softening progressively develops as the system becomes more frustrated. Including other interaction terms in the Hamiltonian such as the ring exchange may predict the broadening of the spectrum due to the increase in low-energy excitations \cite{Vernay2007,Nakamura2014}. 

\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{MagRaman4.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a) Calculation of two-magnon band in an isotropic triangular lattice in parallel (bb) and perpendicular (bc) polarization. b) Calculation of two-magnon band in a square lattice of an anti ferromagnetic material. From Ref. \cite{Nakamura2014}}
\label{fig:MagRaman4}
\end{figure}

\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{MagRaman5.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a) Calculation of two-magnon band in parallel polarization and the effect of frustration $J'/J$ on the location of the band. b) Calculation of two-magnon band in perpendicular polarization and the effect of frustration $J'/J$ on the location of the band. From Ref \cite{Vernay2007}}
\label{fig:MagRaman5}
\end{figure}

\clearpage

%\cite{Vernay2007}
%\cite{Nakamura2014}
%\cite{Drichko2015}
%\cite{Loudon}

\subsection{Experimental Procedure}

Single crystals of $\kappa$-(BEDT-TTF)$_2$Hg(SCN)$_2$Cl (\HgCl) and $\kappa$-(BEDT-TTF)$_2$Hg(SCN)$_2$Br ~(\HgBr) were prepared by electrochemical oxidation of the BEDT-TTF solution in 1,1,2-trichloro-\-ethane (TCE) at a temperature of 40$^{\circ}$ C and a constant current of 0.5~$\mu$A. A solution of Hg(SCN)$_2$, [Me$_4$N]SCN·KCl, and dibenzo-18-crown-6 in 1:0.7:1 molar ratio in ethanol/TCE was used as supporting electrolyte for the \HgCl\ preparation. For the \HgBr\ preparation, a supporting electrolyte Hg(SCN)$_2$/[Me$_4$N]SCN·1.5KBr/ dibenzo-18-crown-6 in 1:0.4:1 molar ratio was used. The composition of the crystal was verified by electron probe microanalysis and X-ray diffraction.

Raman scattering was measured in pseudo-Brewster angle geometry using T64000 triple monochromator spectrometer equipped with the liquid N$_2$ cooled CCD detector.  For the measurements in the 5-400~\cm\ range T64000 in triple monochromator configuration was used, for  the measurements in the range 80-2000~\cm\ single monochromator configuration with the edge filter option was used. Spectral resolution was 2~\cm. Lines of Ar$^+$-Kr$^+$ Coherent laser at 514.nm and 647 nm where used for excitation. Laser power was kept at 2~mW for the laser probe size of approximately 50 by 100~$\mu$m. This ensured that laser heating of the sample was kept below 2~K, as was proved by observing the temperature of ordering transition in \HgCl. Measurements at temperatures down to 10 K were performed using Janis ST500 cold finger cryostat. Samples were cooled from 300~K with cooling rates  between 0.2 and 0.5~K/min. Samples were glued on the cold finger of the cryostat using GE varnish. The experiments were performed on at least 6 samples to ensure reproducibility of the results. A few wave numbers spread in the width of $\nu_2$ band at low temperatures for \HgBr\ was associated with a weak strain originating from GE varnish, all the other parameters of the spectra were reproducible within the error bar of the measurements.
The crystals were oriented using polarization-dependent  Raman scattering measurements.
For the measurements, electrical vector of   excitation $e_L$ and scattered $e_S$ light were polarized along $b$ and $c$ axes. Our notations of polarizations refer to the structure and symmetry of the BEDT-TTF layer, to make an easy comparison to the calculations which refer to D$_{4h}$~\cite{Devereaux07} without loosing the information about the symmetry of the real crystal. Thus A$_{1g}$ symmetry corresponds to the  were measurement in $(b,b)$ and $(c,c)$ geometries, and B$_{1g}$ corresponds to $(b,c)$ and $(c,b)$ geometries ($xy$). %$(x'y')$ polarization is measured with e$_L$ at 45$^{\circ}$ to the $b$ axis.
All spectra were corrected by the Bose-Einstein thermal factor. Intensity of the collective mode was calculated as $I(T) = \int_0^{200~cm^{-1}} \chi''(\omega) d \omega$



%Heat capacity was measured using Quantum Design PPMS system equipped with the DR option. All data were collected on warming, after cooling with the cooling rate of 0.5~K/min.

\section{Heat Capacity}

\subsection{Introduction}


Heat capacity (symbol C) is a measurable physical quantity and is defined as the ratio of the heat transferred to or removed from an object to the resulting temperature change\cite{Halliday2013}. 
\begin{equation}
C_V=(\partial Q/\partial T)_V
\end{equation}
\begin{equation}
C_P=(\partial Q/\partial T)_P
\end{equation}
where the subscripts denotes at constant pressure or constant volume. The unit of heat capacity is joule per kelvin ($J/K$) and that of specific heat used throughout the document is $mJ/K.mole$

Heat capacity measurements of organic conductors were taken to study phase transitions and understand the different contributions to density of states and excitations of the material. The behavior of the heat capacity as a function of temperature has generally three contributions; lattice contributions due to phonons, electronic contribution and magnetic contributions. A typical plot of heat capacity is shown in figure \ref{fig:HCExample} that shows the different contributions to the total heat capacity.

Debye model and Eisenstein models were developed to estimate the phonons contribution to the specific heat in solids. Debye's model treats the vibrations of the atoms in the lattice as phonons in a box while   Einstein's model treats the atoms in a crystal as many 3D non-interacting quantum harmonic oscillators with the same frequency. The Debye model correctly predicts the $T^{3}$ dependence of the heat capacity at low temperature. Both models recover the Dulong-Petit law at high temperatures. In metals, the main low-temperature contribution to the heat capacity comes from the electrons and is calculated by different methods such as Sommerfeld's free electron model. The electron contribution to the heat capacity is proportional to $T$, which dominates the Debye contribution ($\sim T^{3}$) at low temperatures.

\begin{figure}[h]
\includegraphics[width=\textwidth]{TotalHeatCapacity.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{From Reference \cite{Coey2009}. Plot of heat capacity of Nickel. It shows different contributions to total heat capacity.}
\label{fig:HCExample} 
\end{figure}

What we measure experimentally is heat capacity at constant volume $C_{P}$. We used Quantum device PPMS where the relaxation method is the basis of operation to measure the heat capacity. The set up of the puck is shown in figure \ref{fig:PPMSHCPlat}. The PPMS relaxation calorimeter consists of a 3 mm x3 mm alumina platform attached to a heat sink puck using eight thin wires. A thin film RuO heater and a temperature sensor (Cernox sensorTM  from Lakeshore Cryotronics) are attached to the back of the platform. The platform has a well controlled thermal link to a constant temperature heat sink, at temperature $T_{o}$ through the eight thin wires. A radiation shield fits over the top of the puck \cite{HCmanual1}.

%Relaxation calorimetry: ,Adiabatic calorimetry: Equilibrium technique, therefore more accurate . Ac calorimetry 
\begin{figure}[h]
\includegraphics[width=\textwidth]{puck.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{From Reference \cite{HCmanual1}. Plot of the setup of the puck/sample holder of the commercial PPMS. The heater increases temperate of the sample and thermometer measures the temperature. The connecting wires act as the only thermal contact with the puck frame (thermal bath). }
\label{fig:PPMSHCPlat} 
\end{figure}

The basic operation is as follows:
A heater current is applied and the thermometer resistance is recorded at $4 ms$ intervals. The heater voltage is recorded to compute the heater power. A measurement cycle to measure one data point of heat capacity consists of a heating period followed by a cooling period. The platform is heated to temperature $T=T_{0}+\delta T$ and then allowed to relax back to $T_{0}$. The value $\delta T$ of heating can be varied and is one of the parameters of the measurements. Typically,  "short pulses" technique is used (all figures below is of measurements done at short pulses), with  $\delta T=0.5-3\%$ of $T_{0}$. For identification of 1st order phase transitions "long pulse" technique is used, with $\delta T>50\%$ of $T_{0}$. The obtained data on the temperature of sample as a function of time is fitted by two models to extract information about heat capacity. The fit is made by the software during the measurement.

Two models were used to fit the raw data and calculate the heat capacity. The simple model is used if there is a very good contact between the sample and platform and thus, have the same temperature at any point in time. In the simple model, the temperature T of the platform as a function of time t is:
\begin{equation}
   C_{total} dT/dt=-K_w (T-T_{b})+P(t) 
\end{equation}
When the power input is discontinued, the temperature decay of the platform T(t) can be described by: 
\begin{equation}
 T(t)=T_{0}+\delta T e^{-t/\tau}
\end{equation} 

where $C_{total}$ is the total heat capacity of the sample and sample platform; $K_w$ is the thermal conductance of the supporting wires; $T_b$ is the temperature of the thermal bath (puck frame); and $P(t)$ is the power supplied by the heater ($P_o$ during heating and zero during cooling). The solution of this equation is given by exponential functions with a characteristic time constant $\tau$ equal to $ C_{total}/K$. $\delta T$ should be small so that the thermal conductivity between platform and heat sink ($K_{1}$) and $C_{total}$ can be considered T independent. 

If thermal contact between sample and platform $K_2$ is poor such that the condition $K_2>>K_1$ is not  satisfied, then the two-tau model is used. Temperature decay can be expressed as two exponential terms:

\begin{equation}
C_{platform} =(dT_p)/dt=P(t)-K_w (T_p (t)-T_b )+K_g (T_s (t)-T_p (t))
\end{equation} 
\begin{equation}
C_{sample} =(dT_{s})/dt=-K_g (T_s (t)-T_p (t))
\end{equation} 

\begin{equation}
T(t)=T_0+A e^{-t/\tau_1 }+B e^{-t/\tau_{2} }
\end{equation} 

where $\tau_1=C_{pl}/K_1$ , $ \tau_2=C_{sample}/K_2$; $ C_{pl}$ is the heat capacity of the platform and addenda, $ K_1$ is the thermal conductivity between the platform and heat sink, $K_2$ is the thermal conductivity between the sample and the platform, and $C_{pl}$  is the heat capacity of the sample. 
However, this model is valid only when the sample coupling, defined as $100\times(K_2/(K_1+K_2))$ is close to $100 $. The program uses least-square fitting algorithm to determine which model fits the data better. 

In all models, what is measured is the total heat capacity of the sample plus platform. However, we are interested in the heat capacity of the sample only. Thus, before measuring heat capacity of the sample, we measure heat capacity of the platform plus grease (Apizone N), called addenda heat capacity, and then add the sample and do a second measurement with the sample mounted on the platform. The PPMS program automatically subtracts the addenda heat capacity form the total heat capacity to find that of the sample. The amount of grease should be enough to achieve good thermal contact between the sample and the platform without constituting too much of the signal. Also, we preferred to have a flat morphology of the grease to lower the effect of change in heat capacity. This could have large effect especially that our samples are very small. All heat capacity measurements were measured in $mJ/mole.K$. %The molar weight in addition to the number of atoms per molecule were calculated as below: 

\begin{comment}
Apiezon N is used as grease to hold samples to the platform for the following reason:
Inert
Low vapor pressure
Good thermal transport properties: however, it has:
Heat capacity anomaly at 215 K
Heat capacity peak at 290 K
Low temperature heat capacity depends on the grease morphology
\end{comment}

\section{Technical Issues}
\label{sec:TechIssues}

There are many technical issues that can render the data inaccurate measurements of heat capacity. These were the problems faced during our measurements of heat capacity. 
%The technical issues could be classified into two categories:
%1) Technical issues due instrument limitations.
%2) Technical issues due to the sample
%1) pure reproducibility: random error bars 
%2) small mass: large error bar, multiplying factor
\begin{itemize}
\item Reproducibility
\item Sample coupling
\item Effect of mass sample mass on measurement error 
\item Selection of measurement parameters
\item Artifacts due to applied magnetic field 
%\item Bad fits/bad fit parameters
\end{itemize}

\subsection{Reproducibility}

Our first aim was to reproduce published heat capacity results of $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$ and calculate lattice and spinon excitation contribution to confirm that we could get high quality heat capacity results using the relaxation method on a commercial setup and with small sample size (mass is 0.38 mg). For each of the newly studied materials we repeated measurements on a number of crystals to estimate the error in measurements due to electronics, fits and to investigate and exclude sample dependence. Figure  \ref{fig:Reproducibility of Data} shows two measurements of the same crystal ($\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$) done on the same setup immediately one after the other. Measurements were done after sample was cooled at 0.5K/min at zero magnetic field. Both measurements show exactly the same result with the kink at 6 K found in literature \cite{Yamashita2008}. Also, the measurement could be used to have an estimate of errors due instrumentation instability at zero field and random errors of heat capacity. The inset shows that there could be a negligible shift of the kink that could be due to either data sampling or due to the fact that on cooling the temperature needs more time to stabilize or reach the set point.  Figure  \ref{fig:Reproducibility of Data} b) is a measure of the $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl. The blue and red curves were taken on warming the same sample at the same conditions. The measurements were taken consecutively. They show no difference and good reproducibility. The inset on the right is a close up on one data point and rough estimate of the random error is around $10$ mJ/mole.K with ideal sample coupling and parameters. 


\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{Reproducibility}
\makeatletter
\let\@currsize\normalsize
\caption{a) Plot of $C_p$ vs $T$ of $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$. The measurements was taken of the same sample at the McQueen lab at zero field and cooled at 0.5 K/min. Two curves (blue and green) show a kink at 6 K that is detected in Ref.~\cite{Yamashita2008}. The blue curve is taken on warming, the green curse is data taken on cooling. The inset is a closeup at the kink. b) Plot of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl. The blue and red curves are two measurements done at the same instrument with same sample under zero field. The inset on the right is a closeup on data points to have an estimate of random errors (around 10 mJ/mole.K). The inset on the left is a plot of $C_{p}/T^{3} $ vs $T$ of the same measurements. It  shows that even at very specific features, the data is very reproducible. }
\label{fig:Reproducibility of Data} 
\end{figure} 

We also checked reproducibility by comparing measurements done with the same sample but different addendas under the same conditions (same field and cooling rate). This introduced a larger error that could be attributed to features that are sample dependent, or sample coupling or measurement parameters, or setup dependent if different instruments were used. Fig. \ref{fig:Comparing different measurements} a) Is a plot, $C_{p}/T$ vs $T$, of two measurements done at IQM PPMS with the same of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl with mass 1.11 mg. The measurements were taken with different addenda (i.e sample remounted). The result are reproducible with larger errors at phase transitions. The large error at phase transitions could be due to instability of the system or less favorable parameters or bad sample coupling; all are discussed below.

Fig. \ref{fig:Comparing different measurements} b) is a comparison between two measurement of two samples of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Br; two samples of masses 0.8 mg (blue curve) and 0.6 mg (black curve) were measured on the same setup (IQM PPMS). Both samples were zero field cooled at 0.5K/min. The figure is a plot of $C_{p}/T$ vs $T$ at 0 Tesla; no factor was multiplied to either measurements to superimpose on each other. There is a pronounced difference between both measurements in the temperature range 15K to 35K. Looking closely at the parameters of the measurements, it looks like the first measurement (blue curve) has worse values for the "sample coupling" than that of the second measurement which could be cause for such difference. Also, The first measurement has a pronounced instability of values manifested in the zigzag shape of data points where the first measurements gives a vastly different values from subsequent repetitions of one data point, see inset.

Figure \ref{fig:Comparing different measurements} c) is a plot two measurements of different samples of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl; The first measurement, blue curve, was taken for a sample of mass 0.56 mg and was cooled in zero field at 0.5 K/min at the McQueen PPMS.  The second measurement, green curve, was taken of a sample of mass 1.11 mg at the IQM PPMS and was cooled at 1K/min from RT to 85 K and from 65K to 2K, and at 0.5K from 85K to 65K. The plot is of $C_{p}/T$ vs $T$. There is good reproducibility until around 25K where the quality of the first measurement (blue) starts to deteriorate. The inset is a plot of $C_{p}/T^{3}$ vs $T$ because this features shows changes and differences that needs to be studied. The red curve is just the green curve multiplied by a factor of 1.05 to superimpose it on the data taken at McQueen PPMS. A comparison between the blue curve (McQueen PPMS data) and the red curve (IQM PPMS data) shows that there is a slight difference at temperatures less than 8K. However, such difference can't be attributed to true behavior because a comparison between cooling rates in chapter 6 shows no difference in that region. %This difference could be due to-------.   
 In conclusion, comparing different measurements (after accounting for mass error by multiplying by a factor) of the same sample whether taken at same or different setups, or with same or different parameters will in general give good reproducibility of general features but wouldn't be useful for detecting very small features.

\begin{comment}
"The values of the parameter that give the smallest fit deviation determine the heat capacity. The sensitivity of the fit deviation (chi square) to small variations in the fitting parameters is used to estimate the standard errors for the heat capacity."  

"Since the technique used for measuring heat capacity, as described below in Section 1.4.3, is dynamic in nature, the geometry and thermal diffusivity of the sample must be such that the thermal diffusion
time in the sample is small compared to the time constant of the measurement. In cases where the amount of time it takes for the sample to reach internal thermal equilibrium is not small compared to the measurement time, the resulting heat capacity measurement will be too small. Although this problem is indicated in the software as a poor thermal contact between the sample and the sample holder, it is important to use samples that have relatively fast thermalization times to get the correct heat capacity numbers. In cases where the thermal diffusion time in the sample is large, it is necessary to use samples that have a relatively flat geometry, so as to reduce the thermal path through the sample."

Looking closely at the parameters of the measurements, it looks like the first measurement (blue curve) has worse values for the "sample coupling" than that of the second measurement which could be cause for such difference. That was manifested in that the measured time constant at a couple of temperatures was smaller than that of the addenda; this generally happens when the time it takes for a sample to reach thermal equilibrium is too long compared to time of measurement.  Also, The first measurement has a pronounced instability of values manifested in the zigzag shape of measurements where the first measurements gives a vastly different values from subsequent repetitions of the data point, see inset.  
\end{comment}

\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{Comparingdifferentmeasurements.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a) Plot, $C_{p}/T$ vs $T$, of two measurements done at IQM PPMS with the same of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl sample with mass 1.11 mg. The second measurement (red) was re-scaled. The inset is a close up of the phase transition at 30K. b) A plot of $C_{p}/T$ vs $T$ of two different $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Br samples. The inset is the raw data of the two measurements before removing the noisy data. c) Plot, $C_{p}/T$ vs $T$, of two measurements of two different samples of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl. One of the measurements was taken at McQueen PPMS (date June 2016) and the other at the IQM PPMS (May 2017), at zero field. First sample was cooled at 0.5 K/min, the data was re-scaled (x 0.95). Second sample was cooled at 0.5K/min in the temperature range 85K to 65K and at 1K/min at other temperatures. The inset is a plot of $C_{p}/T^{3}$ vs $T$ of the feature at around 8K. The comparison shows slight changes at temperatures lower than 8K.}
\label{fig:Comparing different measurements} 
\end{figure} 

\subsection{Sample Coupling}

Sample coupling was one of the most important parameters to watch while taking the measurement and the most challenging source of problems. The poor coupling will affect the the fitting of data points and subsequently the heat capacity value. Poor thermal coupling could be due to poor mounting of sample or a problem with the sample reaching thermal equilibrium in short time compared to measurement time. Fitting difficulties can also arise when the heat capacity of the sample is not large compared to the platform and this could explain the problems taking place at higher temperature where the sample heat capacity percentage is less than $10\%$. Generally, $100\%$ is considered perfect coupling and anything less than $90\%$ is considered poor sample coupling. It was difficult to achieve good sample coupling because we are restricted with the amount of Apieson N grease. We didn't want the addenda to contribute to the bulk of the signal (even though that was always the case due to the very small size). Also, the samples are fragile and so we couldn't press it hard on grease. In addition to that, we didn't want to change the morphology of the grease to a flatter shape that could greatly affect the addenda and as a result introduce features that could be mistaken for true behavior. Generally, it was noticed that multiple flat pieces of a sample is much better than bulky ones because they have better coupling and faster to reach thermal equilibrium. 

During the measurement, the software calculates sample coupling as one of the parameters. If the coupling values are bouncing around in synchronization with whether it fits it with 1 tau or 2 taus models, then the sample has borderline poor sample coupling. Sample coupling can be tested at room temperature or temperatures higher than the range of interest before the measurement starts. We tested and compared coupling at 260 K and 10 K. Even though the coupling was 100 at the temperatures tested, it was not enough to verify good coupling. A good strategy would be measuring few temperatures to have a better idea on the noise to signal ratio. Fig. \ref{fig:Checking for thermal coupling2} a) is a plot of $C_{p}/T$ vs temperature; it shows a comparison between good and poor quality of data. The blue curve shows the measurement of a sample of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl with mass 0.56 mg. The green curve shows the heat capacity of a sample of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl with mass 1.11 mg. The quality of the blue curve is much worse (larger noise) at temperatures above 20K. One of the reasons that could account to such bad quality is poor coupling (in addition to smaller mass and less data points). Fig. \ref{fig:Checking for thermal coupling2} b) and c) are plots of the sample coupling , time constants tau 1 and tau 2 of the first measurement shown in green curve and second measurement in blue curve. Even though the sample coupling of the first measurement (green curve in fig a) goes to lower values at high temperatures (to around $80\%$) than that of the second measurement (blue curve in figure a) that doesn't go lower than $90 \%$, the quality of the measurement is much better. 

\begin{comment}
ofthere were many measurements done at around 260K with different measurement parameters (measurement time in $\tau$). Not much difference was observed due to different parameters.  Measuring only at one temperatures was not very useful because even when noise looked like compare noise size to signal size.  Fig. \ref{fig:Checking for thermal coupling2} b) shows testing sample coupling at around 10K; this was achieved by cooling and warming the samples few times between 10 and 11 K (sample coupling around 100); it is obvious that the data points do not trace back to superimpose on each other. This could be used to estimate random error (too large I think). Figure \ref{fig: Hg-Br-SampCoupling-0i1Kpermin} shows the coupling constant and corresponding fitting parameters ($\tau_{1} and \tau_{2}$).
\end{comment}




%"Thermal coupling, in percentage, between sample and platform as determined from two-tau model. 100$\%$ indicates perfect thermal contact. 0% indicates no thermal contact. Less than 90% would be considered poor."
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{Checkingforthermalcoupling2.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{Plot of $C_{p}/T$ vs $T$ of two different measurements of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl. One of the measurements was taken at McQueen PPMS (date June 2016), blue curve, and the other at the IQM PPMS (May 2017), green curve, at zero field. First sample was cooled at 0.5 K/min. Second sample was cooled at 0.5K/min in the temperature range 85K to 65K and at 1K/min at other temperatures. b) A plot of fit parameters ,$\tau_{1}$ and $\tau_{2}$, and the sample coupling of the data taken at IQM PPMS, green curve in plot a). c)  A plot of fit parameters ,$\tau_{1}$ and $\tau_{2}$, and the sample coupling of the data taken at McQueen PPMS, blue curve in plot a)}
\label{fig:Checking for thermal coupling2} 
\end{figure} 


\subsection{Effect of sample mass on measurements error}

We mostly worked with very small samples because organic conductors crystals generally have masses less than 1 mg. There are two effects associated with small sample mass:
\begin{itemize}
\item Heat capacity of the sample is a small percentage of the total heat capacity measured.
\item Error bar on the measurement of the mass itself is relatively large.
\end{itemize}

Because the heat capacity measured by the instrument is the total heat capacity of the sample plus the sample holder (called addenda), small sample mass means that the relative heat capacity value with respect to the total heat capacity is very small.  Fig. \ref{fig:Multiplying by factor} shows the  sample heat capacity percentage (sample heat capacity/Total heat capacity) of different measurements. As shown the value is not constant; the general behavior with organic samples was that the percentage was highest in the temperature range T=10K to T=30K. Generally, the sample heat capacity rarely reaches $70\%$ with much smaller contribution at higher temperature. At higher temperatures the sample heat capacity is less than 10 percent of the total heat capacity with the remaining 90 percent due to addenda. Random errors in addenda will have significant effect on the quality of the sample heat capacity if the fractional heat capacity is small relative to the total heat capacity. That means that random errors are going to dominate and thus,  we cannot see any weak features. This could explain the higher noise at higher temperatures. The system will be more tolerant of poor coupling when the sample is a larger fraction of the total heat capacity. %However, We have a limit on the value of the addenda; we can't decrease the amount of grease we add on the sample holder to avoid poor sample coupling. 

The heat capacity error calculated by the program is a combination of the random error due to fitting accuracy and error due to inaccuracy in mass measurement. The latter overwhelms the total error and could be exaggerating the error. Fig \ref{fig:Multiplying by factor} b) is a plot of $C_{p}$ vs $T$ of raw data of two different measurements of $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$ cooled at 0.5K/min (red curve) and 0.1 K/min (blue curve). The green curve is a representation of the same measurement shown by red curve but multiplied by by a factor 1.17. That proves that that the large difference between the red and blue curve is due to error in mass measurements that could be compensated for by multiplying the data by a constant factor. Generally the measurement with the smallest mass (0.2 mg) required to be multiplied by the largest factor (x 1.17) showing that an error in mass could be as large as $20\%$. The inset shows the same figure but with error bars; the two curves agree within error bars. As mentioned earlier the error bars are partly due to error in mass measurements which could reach $20 \%$; therefore, by multiplying a measurements by a constant factor we managed to compare different measurements. 

\begin{comment}
The small size of our samples with respect to the addenda ($\sim 1700-1500 = \sim 200 \mu J/K$ from sample, i.e. 1/8th that of the addenda) means the percent-level random errors in the total (which is $5-10\%$ random errors on the sample value) are going to dominate and means we cannot see any weak features.
\end{comment}

\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{Multiplyingbyfactor.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a) Plot of percentage of sample heat capacity relative to total heat capacity: Sample heat capacity/Total heat capacity x100 of $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$ and $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Br. The sample masses were 0.3 mg and 0.92 mg. The samples were cooled at 0.1 K/min at zero field. b) Comparison between two measurements of $\kappa$-(ET)$_{2}$ Cu$_{2}$ (CN)$_{3}$; red curve is a measurement of a sample of mass 0.38 mg measured in 2015 in McQueen PPMS and cooled at 0.5 K/min. The blue curve is a measurement of a sample of mass 0.2 mg measured in 2016 in IQM PPMS and cooled at 0.1 K/min. The green curve is the 2015 measurement but re-scaled by a factor of 1.17. The inset shows the same measurements but with error bars to show the significant contribution of mass error to error bars. The original data (not re-scaled) agree within the error bars. }
\label{fig:Multiplying by factor} 
\end{figure} 



\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{stabilityofmeasurements2.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a) Plot of $C_{p}$ vs $T$ of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Br cooled at 0.5K/min in zero field. All measurements were done on the same sample, same addenda and setup at IQM PPMS. The measurements were done consequently of different temperature ranges.  The data shows instability due to unknown reasons ( I think mainly change in temp rise parameter). The inset is another measurement of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ cooled at 0.5K/min in zero field.  All measurements were done on the same sample, same addenda and setup at IQM. The three measurement were done consequentially at different temperature ranges. The major difference in the quality of the measurement could be due parameters used. b) The plot of $C_{p}$ vs $T$ of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Br is a comparison between two different measurements done on different samples. It is obvious the lack of instability and irreproducibility of some of the features (jumps) that could be attributed to either poor sample coupling or poor choice of parameters. (However, the jump in the blue curve at around 65 K was detected in another measurement of the same material at 0.1K/min and 0.5 K/min but at a much smaller scale, the jump at 120 K was not seen in any other measurement). c) Plot of $C_{p}/T$ vs $T$ of the phase transition of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl at 30 K. The plot of raw data to show huge difference between the heat capacity value of the first point and subsequent measurement of the same data point. The inset shows the same plot after the first point was removed. }
\label{fig:stability of measurements2} 
\end{figure} 

\subsection{Selection of measurement parameters}
There are few parameters that could be changed to improve quality of the data, some are:
\begin{itemize}
\item Number of data points; that determines the resolution. if fine features or phase transitions are expected to seen at a certain temperature, it is imperative to take many data points. 
\item The temperature rise; this could be set in terms of temperature percentage (for example $5\%$ of the current temperature) or in terms of absolute temperature (for example 5 K). Generally, it is better to follow recommended values. It could be that higher values are better for features like phase transitions, see Fig. \ref{fig:Possible Effect of Parameters and data sampling}.
\item Time of measurements; i.e the time it takes for the heating and cooling cycles in terms of time constant. Also, it is better to follow recommended values. No need for any values longer than 3 tau. 
\end{itemize}
It seems that at phase transitions the measurement parameters are important for the quality of data. Figure \ref{fig:Possible Effect of Parameters and data sampling}  shows two sets of measurements of the phase transition of the same sample of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl (mass 0.93 mg) at 30 K. The measurements were done on the same setup (IQM PPMS) and same addenda (no remounting). The 2nd cooling cycle was measured by heating the sample to $2\%$ of the temperature while the first set was measured by heating to $1\%$ of the temperature. Also, more data points were taken which I think had more effect to get high resolution data. If the temperature rise values were chosen very high ( more than $5\%$ or 2K), the data fit by the program would be inaccurate and values given for capacity heat would be inaccurate. We used very high temperature rise (over $5 K$) to do long pulse measurement. The commercial software is not appropriate to fit such data to produce heat capacity data. So there are two uses for such data:
\begin{itemize}
\item First, to check for first order phase transition; we use the raw data which is a plot of temperatures against time to look at the pattern of the heating cycle and cooling cycle in the temperature range of interest. If first order transition takes place in the temperature range, signatures of latent heat should be seen in the heating and cooling cycles. Fig. \ref{fig:LongPulse} shows the difference between two hypothetical long pulse measurements, the figure on the left is what is expected if no first order phase transition is taking place. The figure on the right shows that first order phase transition takes place at the time when the heater is on but the temperature is not rising with time. Same is happening in the cooling cycle. This is because heat is absorbed as latent heat in first order phase transition rather than in raising the temperature of the sample. 
\item Second, to take measurements similar to short pulse measurements but with faster and better results, the raw data have to be processed by a new program introduced in ref. \cite{HCmanual1}
\end{itemize} 
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{PossibleEffectofParametersanddatasampling.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{ a) Plot of $C_{p}$ vs $T$ of the phase transition of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl at 30 K. It compares two different cooling cycles; both measured the same sample with same addenda (was not remounted) on same setup (IQM PPMS) cooled at 10 K/min in zero field. The only difference is the measurements parameters. b) Plot of $C_{p}/T$ vs $T$ of the phase transition of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl at 30 K. }
\label{fig:Possible Effect of Parameters and data sampling} 
\end{figure} 

\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{k-ET-Hg-Cl_Hyst_CoolingRate_noisy.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{Plot of $C_p vs~ T$ of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Cl. Same as figure above but without subtracting the first point}
\label{fig:k-ET-Hg-Cl_Hyst_CoolingRate_noisy} 
\end{figure} 

\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{LongPulse.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{a) A plot of temperature vs time stamp of the raw data that is later fitted to produce heat capacity values. This figure shows the hypothetical shape of the plot in case of first order phase transition. b) Plot  of temperature vs time stamp of $\kappa$-(ET)$_{2}$ Hg(SCN)$_{2}$ Br in the temperature range 70 K-to 120 K to test for first order phase transition.  }
\label{fig:LongPulse} 
\end{figure} 

\subsection{Artifacts due to applied magnetic field }

Figure \ref{fig:Possible Effect of Addenda} shows the addenda under different fields. The plot is of $C_{p}/T^{3}$ vs $T$ because that is the feature that showed the most change with field while measuring samples. We wanted to check if it is a real feature or an artifact. It is expected to have a magnetic field effect on the addenda but the effect should not be as large. Also, it was interesting that the maximum effect is in the same temperature range (3K to 10 K) that we observe a change in the heat capacity of the sample. This makes us suspect that the effect could be due to the addenda even though the addenda effect should have been subtracted in theory. However, the change in the morphology of the addenda changes its value, and even though extreme care was taken to not change grease morphology when we mounted the sample, there could be slight change that isn't entirely subtracted. The effect could be further magnified by the fact that heat capacity of samples were consistently much smaller than that of addenda .

%Also, it is assumed that due to electronic issues (specify), cooling at more than 2K/min might produce some effects that might affect subsequent heat capacity measurements. (I have No figure) 

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{PossibleEffectofAddenda.pdf}
\makeatletter
\let\@currsize\normalsize
\caption{Plot of $C_p/T^{3} vs~ T$ of addenda with and without field. The effect of the field is pronounced.}
\label{fig:Possible Effect of Addenda} 
\end{figure} 

In conclusion, measuring heat capacity of organic conductors is tricky because they generally have very small masses. As a result of the presented tests and other recommendation from literature \cite{Kennedy2007,HCmanual1}, I summarize the following recommendations for heat capacity measurements of organic insulator samples using Quantum Design PPMS:

\begin{itemize}
\item Sample should be measured  at room temperature to make sure it is well mounted and produces good results at high temperature before cooling down.
\item Addenda measurements should be done at zero field before turning on the field.
\item Most problems are due to either the sample is too small or is poorly coupled to the platform. Because our samples are generally very small on the limit of what could be measured in this setup, many measurements suffered from low quality data.
\item Too much grease may produce a poor measurement of the addenda resulting in an inaccurate sample heat capacity measurement. The grease used depends on the sample’s size and geometry; the grease should be spread out in a thin layer over the area that would be in contact with the sample. Also, samples should be mounted with care to insure that the amount of grease doesn't change. 
\end{itemize}
	


















