Chapter 2 Experimental Methods

Magnetometry, measuring the magnetisation of a sample, is ubiquitous in condensed matter physics to study magnetism. In this thesis it is referred to as linear magnetometry to distinguish it from torque magnetometry which is discussed later.

Most experimental methods for measuring magnetisation are based on Faraday’s Law [80]

$\epsilon =-{\displaystyle \frac{d\mathrm{\Phi }}{dt}}=-{\displaystyle \frac{d}{dt}}{\displaystyle {\iint }_{\mathrm{\Sigma }}}𝑩(t)\cdot 𝒅𝑺$

(2.1)

which states that a time-varying magnetic field through a loop will induce an electro-motive force around the loop. If this loop is broken, this can be measured as a voltage across the terminals. The magnetic flux through a coil will have contributions from any “background” magnetic field, and a contribution from the sample magnetisation. Different ways of measuring magnetisation usually involve generating a time-varying sample contribution to the field and measuring the induced $\epsilon$.

There are a number of common methods for measuring magnetisation in samples at low temperatures and in high magnetic fields, including vibrating sample magnetometry, pulsed-field extraction magnetometry, SQUID magnetometry, and gradient-field torque magnetometry. Of these, two were used in this thesis, and these are explained in more depth in the following section.

One method of measuring magnetisation, particularly in DC magnetic fields, is Vibrating Sample Magnetometry (VSM). In order to generate a time-varying magnetic field from the sample, an AC motor moves the sample through the pickup coil. This generates an AC magnetic field at the same frequency, which is picked up with a lock-in amplifier. This can be shown mathematically as the pickup voltage on the coil

	$V$	$=$	$-{\displaystyle \frac{d\mathrm{\Phi }}{dt}}$		(2.2)
		$=$	$-{\displaystyle \frac{d\mathrm{\Phi }}{dz}}{\displaystyle \frac{dz}{dt}}$		(2.3)

If the sample is approximated as a point-dipole and $z=A\cos \omega t$ with oscillation amplitude $A$ and frequency $\omega$

$V$

$=$

$AC\omega m_z\sin \omega t$

(2.4)

with sample total moment written as the integrated magnetisation:

$$m_z={\iiint }_V𝑴\cdot \widehat{𝒛}𝑑V$$

(2.5)

and coupling constant $C$ which will depend on coil geometry and is typically determined by calibration. This is important as over a large temperature range the coil will deform, and so $C$ will be temperature dependent. This pickup voltage is therefore linear in the magnetic moment, and the scaling factors are calibrated out.

The proportionality of this sensitivity to frequency $\omega$ means that higher sensitivity experiments can be performed by increasing the oscillation frequency. The trade-off with increasing $\omega$ is largely increased mechanical stress. For example, in the Quantum Design PPMS VSM used in this thesis, the system is run at $40\mathrm{Hz}$, with an oscillation amplitude of $2\mathrm{mm}$. In this system the sample and holder experience a peak acceleration of $\sim 13g$ where $g$ is the acceleration under gravity on earth’s surface. This provides a substantial mechanical stress on various components of the experiment. As this acceleration scales proportional to the square of the frequency, small increases in sensitivity increase the stress dramatically.

Compared to DC magnets, pulsed fields allow access to much higher magnetic fields. The largest accessible non-destructive pulsed field reaches $100\mathrm{T}$, compared to the largest DC magnetic field of $45\mathrm{T}$. In pulsed magnetic fields, the short pulse time makes VSM unsuitable. A common approach is pulsed-field extraction magnetometry which uses the time-variation of the pulsed field itself to generate the time-varying magnetisation.

The operating principle involves a solenoid containing the magnetic material. The magnetic field inside the coil can be written as the sum of applied (auxiliary) magnetic field $𝑯$ and magnetisation $𝑴$

$$𝑩(t)={\mu }_0\left(𝑯(t)+𝑴(t)\right)$$

(2.6)

the total voltage over the solenoid is summed over each of the coils, which is treated as an integral over the length $l$ of the solenoid:

	$V(t)$	$=-{\displaystyle \frac{d}{dt}}{\displaystyle \int 𝑑l\stackrel{\mathrm{\Phi }}{\overbrace{{\iint }_{\mathrm{\Sigma }}𝑩(t)\cdot 𝒅𝑺}}}$		(2.7)
		$=-{\displaystyle \frac{d}{dt}}{\displaystyle {\iiint }_V}{B}_z(t)𝑑V$		(2.8)
		$=-{\displaystyle \frac{d}{dt}}{\displaystyle {\iiint }_V}{\mu }_{0}H(t)𝑑V-{\displaystyle \frac{d}{dt}}{\displaystyle {\iiint }_V}{\mu }_0{M}_z(t)𝑑V$		(2.9)

where the magnetic field and solenoid axis are taken to be parallel to the $𝒛$ axis. The voltage can then be written as a time varying background

$$V_0(t):=-\frac{d}{dt}{\iiint }_V{\mu }_{0}H(t)𝑑V$$

(2.10)

and a sample-dependent term, assuming that the coil and sample geometry are time-independent,

	$V_s(t)≔$	$-{\displaystyle {\iiint }_V}{\mu }_0{\displaystyle \frac{d{M}_z}{dt}}𝑑V$		(2.11)
	$=$	$-{\displaystyle {\iiint }_V}{\mu }_0{\displaystyle \frac{d{M}_z}{dH}}{\displaystyle \frac{dH}{dt}}𝑑V$		(2.12)

The experimental approach involves measuring an empty coil to determine $V_0(t)$, and a coil with the sample under the same conditions. This can then be subtracted to give $V_s(t)$. If the magnetisation is assumed to be approximately uniform throughout the sample, this gives $\frac{d{M}_z}{dt}$ which is then integrated to give the sample magnetisation. Typically, there is an unknown scaling factor on this determined value, dependent on coil geometry, sample geometry, and filling factor. This can be brought into absolute units by measuring the same material in a much smaller DC field using another technique such as VSM or SQUID magnetometry to scale the units.

As is shown in (2.12), the pickup signal is proportional to $\frac{dM}{dH}$, so the technique is particularly sensitive to sharp changes in the magnetisation such as first-order magnetic phase transitions. The signal is also proportional to $\frac{dH}{dt}$, so the technique is especially suited for fast-pulse magnets - such as the Helmholtz-Zentrum Dresden-Rossendorf $34\mathrm{ms}$, $65\mathrm{T}$ pulsed field magnet, used later in this thesis.

An alternative technique to linear magnetometry is torque magnetometry. Like magnetisation, torque probes a first derivative of the free energy, so it is sensitive to phase transitions. A particular advantage of torque magnetometry is that very small single crystals can be measured, well below the sensitivity of magnetisation measurements [68]. This permits experiments to be performed on materials for which large single crystals cannot be grown. This technique is used throughout the thesis as the primary experimental tool to study magnetism. These measurements are performed with field sweeps, temperature sweeps, and angle sweeps with the sample mounted on a rotator probe.

As I will show in the following section, torque magnetometry probes components of magnetisation orthogonal to the magnetic field. As such the technique is complementary to magnetometry techniques which probe the component parallel to the field, such as VSM or extraction-coil magnetometry.

The two equivalent formulations of magnetic torque commonly quoted in literature are $𝝉=𝒎\times 𝑩$ [66] and $\tau =-{\displaystyle \frac{\partial F}{\partial \alpha }}$ [113] for some free energy $F$ and angle $\alpha$. The equivalence of these two is straightforward to prove and is included here for completeness, including a generalisation for non-uniform magnetic fields.

Consider the system of a crystal as a free energy $F(𝑩(𝒓))$, which is a function of the (non-uniform) magnetic field $𝑩(𝒓)$ at the crystal’s position $𝒓$. If the crystal is free to rotate around some vector $𝒏$, the (scalar) torque is defined to be the negative gradient of free energy with respect to the rotation angle ${\alpha }^{\prime }=-\alpha$ i.e. $\tau =-{\displaystyle \frac{\partial F}{\partial \alpha }}={\displaystyle \frac{\partial F}{\partial {\alpha }^{\prime }}}$, where ${\alpha }^{\prime }$ is the rotation of the magnetic field in the sample frame of reference. Note the change of sign is because the magnetic field in the sample frame rotates in the opposite direction to the sample in the lab frame,

$${\tau }_{𝒏}={\frac{\partial }{\partial {\alpha }^{\prime }}|}_{0}F(R_{𝒏}^{{\alpha }^{\prime }}𝑩(R_{𝒏}^{{\alpha }^{\prime }}𝒓))$$

(2.13)

where $R^{{\alpha }^{\prime }}𝒏$ is a matrix applying a rotation of angle ${\alpha }^{\prime }$ around $𝒏$. This can be written to first order in ${\alpha }^{\prime }$ using the generator matrices of rotations as

	$R_{𝒏}^{{\alpha }^{\prime }}$	$=\exp ({\alpha }^{\prime }\widetilde{R_{𝒏}})$		(2.14)
	${{\displaystyle \frac{\partial }{\partial {\alpha }^{\prime }}}|}_0{R}_{𝒏}^{{\alpha }^{\prime }}𝒖$	$=\widetilde{R_{𝒏}}𝒖=𝒏\times 𝒖$		(2.15)

If this is inserted into (2.13)

	${\tau }_{𝒏}$	$={{\displaystyle \frac{\partial }{\partial {\alpha }^{\prime }}}|}_{0}F(R_{𝒏}^{{\alpha }^{\prime }}𝑩(R_{𝒏}^{{\alpha }^{\prime }}𝒓))$		(2.16)
		$={\nabla }_{𝑩}F\cdot \widetilde{R_{𝒏}}𝑩+{{\nabla }_{𝑩}F\cdot {\displaystyle \frac{\partial }{\partial {\alpha }^{\prime }}}|}_0𝑩(R_{𝒏}^{{\alpha }^{\prime }}𝒓)$		(2.17)
		$={\nabla }_{𝑩}F\cdot (𝒏\times 𝑩)+({\nabla }_{𝑩}F\cdot \nabla )𝑩\cdot (𝒏\times 𝒓)$		(2.18)

using the standard results:

	${\displaystyle \frac{\partial }{\partial \varphi }}f(𝒓)$	$=\nabla f\cdot {\displaystyle \frac{\partial }{\partial \varphi }}𝒓$		(2.19)
	$𝒂\cdot {\displaystyle \frac{\partial }{\partial \varphi }}𝑩(𝒓)$	$=(𝒂\cdot \nabla )𝑩\cdot {\displaystyle \frac{\partial }{\partial \varphi }}𝒓$		(2.20)

Both terms in (2.18) are a vector triple product, and are constant under cyclic permutation, allowing the expression to be rewritten

	${\tau }_{𝒏}$	$=𝒏\cdot \left[𝑩\times {\nabla }_{𝑩}F+𝒓\times ({\nabla }_{𝑩}F\cdot \nabla )𝑩\right]$		(2.21)
		$=𝒏\cdot \left[𝑩\times {\nabla }_{𝑩}F+𝒓\times \nabla ({\nabla }_{𝑩}F\cdot 𝑩)\right]$		(2.22)

Inserting the definition of magnetisation $𝒎=-{\nabla }_{𝑩}F$, this gives the definition of vector torque

	${\tau }_{𝒏}$	$=𝒏\cdot 𝝉$		(2.23)
	$𝝉$	$=𝒎\times 𝑩+\nabla (𝒎\cdot 𝑩)\times 𝒓$		(2.24)

In a uniform field, the second term is zero, such that $𝝉=𝒎\times 𝑩$, and torque magnetometry probes the component of magnetisation perpendicular to the magnetic field. As has been shown here, this is proportional to the derivative of energy with angle, so it is a direct measure of magnetic anisotropy. This makes it a particularly useful technique for measuring anisotropic magnetic samples, where the isotropic magnetisation, which the technique is not sensitive to, may be much larger.

The existence of the second term in this expression $\nabla (𝒎\cdot 𝑩)\times 𝒓$ allows for the operation of the gradient-field torque magnetometer [66]. This technique uses a large magnetic field gradient to measure $\nabla (𝒎\cdot 𝑩)\times 𝒓$, sensitive to $𝒎\cdot 𝑩$. This can allow the linear magnetisation and magnetic torque to be determined for the same sample.

Figure 2.1 shows a schematic view of the experimental setup, the total rotation angle of the sample is split into the equilibrium part $\theta$ and deflection $\delta$. Such that $\alpha =\theta +\delta$. The experiment is performed by mounting a crystal on a cantilever, with an equilibrium angle $\theta$ from the horizontal. A vertical magnetic field $𝑩=B𝒛$ is applied to the apparatus, which can be placed in a cryogenic system for temperature control. The mechanical system can be analysed by writing the energy to leading order as:

$E=K{\delta }^2+F_m(\alpha )$

(2.25)

where $F_m$ is the magnetic energy of the sample. Here, the contribution from the mass of the sample has been neglected.

If the deflection angle $\delta$ is small, the magnetic energy can be expanded around $\delta =0$

	$F_m(\alpha )$	$=F_m(\theta )$	$+$	${{\displaystyle \frac{\partial F_m}{\partial \alpha }}|}_{\theta }\delta$	$+$	${{\displaystyle \frac{1}{2!}}{\displaystyle \frac{{\partial }^2{F}_m}{\partial {\alpha }^2}}|}_{\theta }{\delta }^2$	$+𝒪({\delta }^3)$		(2.26)
		$=F_m(\theta )$	$-$	$\tau (\theta )\delta$	$+$	$\kappa (\theta ){\delta }^2$	$+𝒪({\delta }^3)$		(2.27)

where $\tau$ is the magnetic torque on the sample, and $\kappa$ is the second derivative of energy with angle, which is sometimes called the magnetotropic coefficient [67]. This means the energy of the sample-lever system can be written as a simple harmonic oscillator

	$E$	$\approx \tau (\theta )\delta +K^{\prime }(\theta ){\delta }^2$		(2.28)
		$\approx K^{\prime }{(\delta -{\delta }_0)}^2$		(2.29)

with effective stiffness $K^{\prime }(\theta )=\kappa (\theta )+K$ and equilibrium deflection ${\delta }_0={\displaystyle \frac{\tau }{2K^{\prime }}}$, which can each be measured experimentally. Conventional torque magnetometry, used in this thesis, measures the change in deflection angle ${\delta }_0$. This is linear in magnetic torque when $|\kappa |\ll K$. The two standard methods for measuring this deflection angle are piezoresistive and capacitive cantilever torque magnetometry. An alternative technique is torque-differential magnetometry (TDM) [67, 45], which also measures the change in resonant frequency of the oscillator, sensitive to changes in $\kappa$. An advantage of this technique over conventional torque magnetometry is that for fields along high-symmetry axes, torque is frequently symmetry constrained to zero. The second derivative of free energy does not have the same problem along such axes.

Piezoresistive torque cantilevers [76] (piezocantilevers) were developed for scanning probe microscopy techniques [97] such as atomic force microscopy [77], meaning that they are available commercially. In a piezoresistive lever, (Fig. 2.2a), the "legs" at the base of the cantilever which allow it to bend are coated in a layer of a piezoresistive material, such as silicon, which shows a change in resistance when mechanical strain is applied [47]. This change in resistance is typically measured using a Wheatstone half-bridge, as shown in Figure 2.2b). This circuit measures the difference in resistance between the lever and an identical dummy lever with no sample (lower half of image Fig. 2.2a), allowing the variability of resistance with temperature and/or magnetic field to be minimised. Typically this is measured with a lock-in amplifier.

These piezocantilever measurements are beneficial for their extremely high sensitivity, due primarily to their low stiffness. They are therefore able to measure very small moments from small single crystals. They are however challenging to mount on, especially for crystals which are not plate-like, where the orientation desired to be measured in is the in-plane, out-of-plane orientation. Additionally, they are fragile, and strong torques caused by large anisotropy in high magnetic fields can cause them to break.

Figure 2.2c, d) show an alternative design using a capacitive torque cantilever. This setup works by using the fact that the capacitance of a parallel-plate capacitor depends on the distance between the plates. The sensitivity of the capacitive levers is proportional to the area of the lever and the length of the legs, so are typically larger than piezocantilever measurements. This is beneficial when large crystals need to be mounted, and means specialised mounts can allow crystals to be mounted in orientations which would be challenging on a smaller lever.

In practice, these considerations make capacitive levers a good choice for crystals $\sim 1\mathrm{mm}$ in length, while piezoresistive levers can be made more flexible and thus more sensitive, making them good choices for smaller samples and smaller torques. As is shown later in this thesis, piezocantilever torque magnetometry can be performed on magnetic single crystal samples down to thin plates with diameters , too small for many other techniques. This makes it a particularly effective technique for novel materials, which frequently cannot be synthesised as large crystals. Additionally, torque magnetometry directly probes the magnetic anisotropy of the sample, making it especially useful for studies of anisotropy.

In the simple case of a uniaxial crystal, a paramagnet can be described well in low fields by a susceptibility such that $𝒎=\chi 𝑩$, where the susceptibility tensor has the uniaxial form

(2.30)

When the magnetic field is rotated around the sample such that

$𝑩=B\left(-\sin \theta 𝒙+\cos \theta 𝒛\right)$

(2.31)

producing a magnetisation in the sample

	$𝒎$	$=\chi 𝑩$		(2.32)
		$=B\left(-{\chi }_{⟂}\sin \theta 𝒙+{\chi }_{\parallel }\cos \theta 𝒛\right)$		(2.33)

which produces a torque around the $𝒚$ axis

	$𝝉$	$=𝒎\times 𝑩$		(2.34)
		$={\displaystyle \frac{1}{2}}{B}^2\left({\chi }_{⟂}-{\chi }_{\parallel }\right)\sin 2\theta 𝐲$		(2.35)
	${\tau }_y$	$={\displaystyle \frac{1}{2}}{B}^2\left({\chi }_{⟂}-{\chi }_{\parallel }\right)\sin 2\theta$		(2.36)

The magnetic energy can be written

$$F_m=\frac{1}{4}{B}^2\left({\chi }_{⟂}-{\chi }_{\parallel }\right)\cos 2\theta$$

(2.37)

The angular dependence of both of these values is plotted in Figure 2.3 for the easy-axis ${\chi }_{\parallel }>{\chi }_{⟂}$ and easy-plane cases. Note that these can be distinguished by the sign of the signal.

A particularly informative feature in the magnetic torque are locations of zero torque $\tau (\theta )=0$, and the gradient $\frac{\partial \tau }{\partial \theta }$ through the zeros. These zero-torque locations are equilibria, where energy is a maximum or minimum in angle. These two cases can be distinguished by the sign of the gradient. At a stable equilibrium, ${\displaystyle \frac{{\partial }^2{F}_m}{\partial {\theta }^2}}>0$ and . As such the sign of the gradient of torque through zero shows that the system is in a stable equilibrium. Experimentally determining absolute units for the torque can be very challenging, especially as the samples being measured are often too small to accurately determine the mass, however calibrating the sign of the torque simply requires measuring a sample of a well characterised paramagnet. In this example of a uniaxial paramagnet for instance, the sign of the toque can reveal the order of susceptibilities ${\chi }_{\parallel }$, ${\chi }_{⟂}$.

If the susceptibilities in the material change, for instance as the material orders into an antiferromagnet, this will then be seen as a sharp anomaly in magnetic torque, making the technique useful for identifying the onset of magnetic order.

Heat capacity is a measure of how a material responds to thermal energy. Heat capacity is defined as the amount of heat $\delta Q$ needed to increase the temperature by an infinitesimal temperature $\delta T$ [12]

$$C=\underset{\delta T\to 0}{lim}\frac{\delta Q}{\delta T}$$

(2.38)

The first law of thermodynamics defines the amount of heat in terms of the change in internal energy of the system

$$dU=dQ+dW$$

(2.39)

where the work term includes magnetic contributions

$$dU=dQ-PdV-{\mu }_0𝑴\cdot 𝒅𝑯$$

(2.40)

where $PdV\approx 0$ because the experiments are performed in vacuum.

The most suitable heat capacity to define is the one which is measured experimentally: the heat capacity at constant field

$C_H={\left({\displaystyle \frac{\partial Q}{\partial T}}\right)}_{𝑯}$

(2.41)

The utility of this parameter can be best seen by including the second law of thermodynamics[12], which defines entropy $S$ of the system:

$$dQ=TdS$$

(2.42)

such that heat capacity can be redefined as

$$C_H=T{\left(\frac{\partial S}{\partial T}\right)}_{𝑯}$$

(2.43)

By introducing the Helmhotz free energy $F=U-TS$, $S$ can be written as

$$S=-\frac{dF}{dT}$$

(2.44)

and the heat capacity can be written as

$$C_H=-T{\left(\frac{{\partial }^{2}F}{\partial T^2}\right)}_{𝑯}$$

(2.45)

This is particularly useful as a measure in the region of phase transitions where $F$ is approximated as the Landau free energy $ℱ$, which depends on the order parameters of the phase transition [57]

$$ℱ={ℱ}_0+\mathrm{\Delta }ℱ({\eta }_0,{\eta }_1,\mathrm{\dots })$$

(2.46)

This treatment with many order parameters adds complexity and is not necessary to outline the effect on heat capacity at a phase transition. For the sake of simplicity a single order parameter $\eta$ will be used for the remainder of this section. The entropy then can be written in terms of this Landau free energy

$$S-S_0=\mathrm{\Delta }S=-\frac{dℱ}{dT}=-\frac{dℱ}{d\eta }\frac{d\eta }{dT}$$

(2.47)

where $\mathrm{\Delta }S$ is the contribution to entropy from the order parameter-dependent part of the free energy.

While the free energy is continuous at the phase transition, it is not analytic. The order of a phase transition is defined at the derivative of the Landau free energy where there is a discontinuity. In the case of a temperature-driven phase transition, there is a discontinuity in

$$\frac{d^nℱ}{d{T}^n}$$

(2.48)

such that a 1^storder phase transition will have a discontinuity in $S$ and a latent heat $\lambda =T_c\delta S$ needed to bring the system above the transition, where $\delta S$ is the size of the discontinuity in $S$. A 2^ndorder phase transition will be continuous in $S$ with no latent heat $\lambda =0$ but have a discontinuity in $C$. This allows heat capacity to identify phase transitions, and to differentiate between these two classes of phase transitions, which can provide information about the symmetry of any order parameters involved in the transition [25].

For a simple model of phrase transitions, with Landau free energy

$$\mathrm{\Delta }ℱ(\eta )=\frac{T-T_c}{T_c}{\eta }^2+b{\eta }^4+{\eta }^6$$

(2.49)

will show a first-order phase transition at $T_c$ and $b>0$ will show a second order phase transition at $T_c$. A comparison of $\mathrm{\Delta }ℱ$, $\mathrm{\Delta }S$, and $\delta C$ for these two models is shown in Figure 2.4, showing the difference in form of the heat capacities in both cases.

There are two common methods for measuring heat capacity. One is relaxation-time calorimetry [5], which involves providing a pulse of power to warm the sample weakly coupled to a bath, and measuring the time it takes to cool the sample to the bath temperature. For a given power output, by measuring the peak temperature, the thermal conductivity $K$ between the sample and bath can be determined. The relaxation time constant is then $\tau =\frac{C}{K}$, giving the heat capacity in absolute units. Instead, this thesis uses the AC calorimetry method, which is introduced in more depth in the following section.

The experimental technique used to measure heat capacity in this thesis is AC calorimetry. This involves exciting a sample with a oscillating current, and measuring the size of the ensuing temperature oscillations. The primary advantage of this method is the lower thermal isolation needed for the sample, compared to other methods. Instead, a relatively short thermal relaxation time is an advantage as is allows faster measurement taking. The analysis in this section comes from the original Sullivan and Seidel paper which first discussed this technique [95].

Consider the simple experimental model shown in Figure 2.5, of a sample coupled to a heater, thermometer, and bath by thermal conductances $K_H,K_T,K_B$ respectively. An AC current through the heater generates an oscillating power output $P(t)=P_0\cos \omega t+\frac{1}{2}{P}_0$, with frequency $\omega$.

The equilibrium solution for the measured temperature $T$ on the thermometer is

$$T(t)=T_0+\tilde{T}\cos (\omega t-\alpha )$$

(2.50)

with DC component $T_0=T_b+\frac{P_0}{K_b}$, limiting the base temperature experimentally available, and oscillating component

$$\tilde{T}\approx \frac{P_0}{2\omega C}f(\omega )$$

(2.51)

with total heat capacity $C=C_H+C_T+C_s$ [95]. If characteristic time constants are defined

	${\tau }_T$	$={\displaystyle \frac{C_T}{K_T}}$		(2.52a)
	${\tau }_H$	$={\displaystyle \frac{C_H}{K_H}}$		(2.52b)
	${\tau }_i^2$	$={\tau }_T^2+{\tau }_H^2$		(2.52c)
	${\tau }_b$	$={\displaystyle \frac{C}{K_b}}$		(2.52d)

then the frequency response function is approximately

$$f(\omega )\approx {\left(1+{\omega }^2{\tau }_i^2+\frac{1}{{\omega }^2{\tau }_b^2}\right)}^{-1/2}$$

(2.53)

In the case of ${\tau }_i\ll {\tau }_b$, this frequency response function has a flat plateau in frequency where $f(\omega )\approx 1$. This can be seen in Figure 2.5. Practically, the various time constants are not straightforward to measure for a given experimental setup. One simple approach is to measure the frequency dependence of $\tilde{T}$ and fit it to the functional form of (2.51), giving the optimal frequency ${\omega }_{\text{opt}}$ to perform the experiment at the centre of the plateau region, which is typically assumed to be $f({\omega }_{\text{opt}})\approx 1$, but can also be taken from the fit if ${\tau }_i\ll {\tau }_b$ does not hold. $P_0$ is straightforward to determine experimentally by measuring the voltage on the heater resistor element during the experiment. Given these, in this approximation $C$ can be determined in absolute units without calibration. This total heat capacity includes a sample contribution and an addenda from the apparatus, this can be subtracted by measuring the same apparatus under same conditions without the sample to measure the addenda $C_{\text{ad}}$. Additional experimental considerations such as the effect of finite sample thermal conductivity require a more careful treatment [95], one effect of such considerations is that the optimal frequency response $f({\omega }_{\text{opt}})$ can differ significantly from $1$. This makes the AC technique ineffective in practice of accurately determining $C$ in absolute units.

X-ray crystallography is used throughout this thesis for crystal characterisation and structural determination in novel materials. These crystal structure results are not the primary focus of this thesis, so this section presents an overview of the X-ray diffractometry technique.

In diffraction, i.e. elastic scattering, the incoming photon has energy $E$, and wave vector $𝒌$ and the scattered photon has energy $E^{\prime }=E$ and wave vector ${𝒌}^{\prime }$. The scattering wave vector can then be written $𝑸={𝒌}^{\prime }-𝒌$. The scattering rate from some potential $V(𝒓)$ can be determined from Fermi’s golden rule [91]

(2.54)

with matrix element

		$={\displaystyle \frac{1}{\text{Vol}}}{\displaystyle {\int }_{\text{Vol}}}{e}^{i𝒌\cdot 𝒓}V(𝒓)e^{-i{𝒌}^{\prime }\cdot 𝒓}{d}^3𝒓$		(2.55)
		$={\displaystyle \frac{1}{\text{Vol}}}{\displaystyle {\int }_{\text{Vol}}}{e}^{-i𝑸\cdot 𝒓}V(𝒓)d^3𝒓$		(2.56)

where Vol is the volume of the sample, and is needed to normalise the value. This amplitude can be thought of as the Fourier transform of the scattering potential $V(𝒓)$. In a crystal, the X-rays are scattering off of electrons which are localised to ions on a periodic Bravais lattice. Because of this, the X-rays are scattering off a periodic potential. The scattering potential can be written in terms of the individual ion contributions as

	$V(𝒓)$	$={\displaystyle \sum _{𝒖}}{\displaystyle \sum _j}{V}_j(𝒓-{𝑹}_j-𝒖)$		(2.57)
		$={\displaystyle \sum _{𝒖}}\delta (𝒓-𝒖)\ast {\displaystyle \sum _j}\delta (𝒓-{𝑹}_j)\ast V_j(𝒓)$		(2.58)

where $\ast$ represents a convolution, $𝒖$ indexes all primitive lattice vectors, and $j$ indexes all of the atomic sites in the structural unit cell. Because of this, the convolution theorem allows the matrix element to be rewritten as

(2.59)

The first term in this expression is a (non-divergent) constant if $𝑸$ is a reciprocal lattice vector, and zero otherwise. This is known as the Laue condition, and positions where it is satisfied are known as Bragg peaks. This allows a scattering form factor to be defined as

	$S(𝑸)$	$≔{\displaystyle \sum _j}{e}^{-i𝑸\cdot {𝑹}_j}{\displaystyle {\int }_{\text{Vol}}}{V}_j({𝒓}^{\mathbf{\prime }})e^{-i𝑸\cdot {𝒓}^{\mathbf{\prime }}}𝒅{𝒓}^{\mathbf{\prime }}$		(2.60)
		$={\displaystyle \sum _j}{e}^{-i𝑸\cdot {𝑹}_j}{f}_j(𝑸)$		(2.61)

where the atomic form factor has been inserted:

	$f_j(𝑸)$	$≔{\displaystyle {\int }_{\text{Vol}}}{V}_j({𝒓}^{\mathbf{\prime }})e^{-i𝑸\cdot {𝒓}^{\mathbf{\prime }}}𝒅{𝒓}^{\mathbf{\prime }}$		(2.62)
				(2.63)

as in X-ray scattering, the scattering potential is proportional to the electron density $\rho ({𝒓}^{\mathbf{\prime }})$. The scattering intensity is then proportional to the square of the amplitude:

(2.64)

Experimentally, by finding the $𝑸$ vectors where reflections occur, the lattice constants can be determined. Higher space group symmetries cause systematic absences, that is $𝑸$ vectors where the Laue condition is fulfilled but $S(𝑸)=0$, this allows the Laue class of a crystal to be determined. Finally, by fitting the intensities of the observed peaks to those predicted from theory, a full crystal structure can be refined.

In practice, there are a number of further considerations which much be made to accurately refine these parameters. One important example comes from the fact that in finite temperature, atoms throughout the crystal will vary in position from their equilibrium position. This will produce an incoherent scattering background, and reduce the magnitude of the coherent scattering intensities. The physics of this can be captured in Debye-Waller factors [23, 101] which can be estimated from anisotropic displacement parameters by assuming a Gaussian probability distribution function [100].

All X-ray crystallography experiments reported in this thesis were performed on an Mo-source Oxford Diffraction Supernova diffractometer. In this apparatus, X-rays are produced by a molybdenum X-ray tube, producing a collimated beam of monochromatic X-rays with wavelength $\lambda =0.707\mathrm{\AA }$. X-rays are detected by a CCD area detector.

In this set-up, the incoming beam $𝒌$ is fixed, as is the detector, so a small solid angle of scattered X-rays can be detected. As this is measuring diffraction, $|{𝒌}^{\prime }|=|𝒌|$, so all of the possible scattered $𝑸$ vectors lie on an Ewald sphere [28], and the detection solid angle is simply a region of this sphere. By rotating the crystal on a goniometer, Bragg peaks can be brought onto this region of the Ewald sphere where they can be detected.

By keeping track of the sample orientation throughout the experiment, the scattering wave vectors where reflections are detected can be transformed into the sample frame. A fitting procedure can then determine the highest symmetry lattice which indexes the observed peaks. In order to accurately determine intensities of individual peaks, longer experiments can measure the scattering intensity over a very large volume of the space of accessible $𝑸$ vectors. Intensities can be determined by integrating over a small region around individual Bragg peaks. In this thesis, refining the structure from these intensities is done with the FULLPROF [83] analysis software.

Air and moisture sensitive samples are able to be measured without exposure to air by sealing the sample in inert gas or vacuum in a borosilicate glass capillary. This is done by first attaching the sample with paraffin to the outside of a  $400\mathrm{\mu }\mathrm{m}$ rectangular capillary in inert gas in a glove box. This is then placed in a  $500\mathrm{\mu }\mathrm{m}$ cylindrical capillary which may be evacuated and is then sealed by flame. As borosilicate glass is non-crystalline, it does not contribute coherent scattering to the measurement. The additional, and inevitably anisotropic, absorption and incoherent scattering from the capillary cannot be corrected, and so will reduce the accuracy of determined reflection intensities.