The two dimensional electron gas subjected to a perpendicular magnetic field is a model system that supports a variety of electronic phases. Perhaps the most well-known are the fractional quantum Hall states, but in recent years there has been an upsurge of interest in the charge ordered phases commonly referred to as electron solids. These solids are a consequence of electron-electron interactions in a magnetic field. While some solid phases form in the lowest Landau level, the charged ordered phases are most abundant in the higher Landau levels. Examples of such phases include the Wigner solids, electronic bubble phases and stripe or nematic phases. Open questions surround the exact role of disorder, confinement potential, temperature and the Landau level index in determining the stability and competition of these phases with other ground states. <div>The interface of GaAs/AlGaAs remains the cleanest host for the two-dimensional electron gas due to the extremely high quality of materials available and the advancement in molecular beam epitaxy growth techniques. As a result, exceptionally high electron mobilities in this system have been instrumental in the discovery of numerous electron solids. </div><div>In this Thesis, I discuss the discovery and properties of several electron solids that develop in such state-of-the-art two dimensional electron gases. These electron solids often develop at ultra low temperatures, in the milliKelvin temperature range. After an introduction to the physics of the quantum Hall effect in two dimensions, in chapter 3, I discuss electron solids developing in the N=1 Landau level. While these solids have been known for some time, details of the competition of these phases xiii with the nearby fractional quantum Hall states remains elusive. A number of reports observe new fractional quantum Hall states at filling factors where electron solids are found in other experiments. We undertook a systematic study to answer some of these unsettled questions. We see evidence for incipient fractional quantum Hall states at 2+2/7 and 2+5/7 at intermediate temperatures which are overtaken by the electronic bubble phases at lower temperatures. Several missing fractional states including those at filling factors 2+3/5, 2+3/7, 2+4/9 highlight the relative stability of the electronic solids called the bubble phases in the vicinity in our sample. </div><div>In chapter 4, I discuss a newly seen electron crystal which manifests itself in transport measurements as a reentrant integer quantum Hall state. Reentrant integer behavior is common in high Landau levels, but so far it was not observed in the lowest Landau level in narrow quantum well samples. In contrast to high Landau levels, where such reentrant integer behavior was associated with electronic bubbles, we believe that the same signature in the N=0 Landau level is due to an electronic Wigner crystal. The filling factors at which we observe such reentrance reveal that it is a crystal of holes, rather than electrons. The discovery of this reentrant integer state paints a complex picture of the interplay of the Wigner crystal and fractional quantum Hall states. </div><div>Finally, in chapter 5, I discuss the observation of a novel phenomenon, that of reentrant fractional quantum Hall effect. In the lowest Landau level, we observe a fractional quantum Hall state, but as the field is increased, we see a deviation and then a return to quantization in the Hall resistance. Such a behavior indicates a novel electron solid. In contrast to the collective localization of electrons evidenced by the reentrant integer quantum Hall effect, such reentrance to a fractional Hall resistance clearly points to the involvement of composite fermion quasiparticles. This property thus distinguishes the ground state we observed as a solid formed of composite fermions. Such a solid phase is evidence for exotic electron-electron correlations at play which are clearly different from those in the traditional Wigner solid of electrons.<br></div>

Graphene, a single atomic layer of graphite, has many exciting electronic and mechanical properties. On a fundamental level, the quasi-relativistic behaviour of the charge carriers in graphene arises from the honeycomb-like atomic structure. Deforming the lattice changes the lengths of the carbon-carbon bonds, breaking the hopping symmetry between carbon sites. Mathematically, elastic strain in a graphene membrane can be described by additional terms in the low-energy effective Hamiltonian, analogous to the vector potential of an external magnetic field. Hence, certain non-uniform strain geometries produce so-called `pseudo-magnetic fields', leading to a predicted zero-field quantum Hall effect. These fictitious magnetic fields are distinct from an external magnetic field in that they are only observed by charge carriers within the membrane, and have opposing polarity for electrons in the K and K' valleys, preserving time-reversal symmetry of the lattice as a whole. Deforming graphene in the non-uniform manner required to produce a homogeneous pseudo-magnetic field has proven to be a huge technological challenge, however, restricting experimental evidence to scanning tunnelling spectroscopy measurements on, for example, highly deformed nanobubbles formed by the thermal expansion of an epitaxially grown sheet on a platinum substrate. These results stimulated a large amount of interest in strain-engineering electron transport in graphene, partly due to the extreme magnitude of the observed pseudo-magnetic field, a direct consequence of the strain components strongly varying over the space of a few nanometres, but the formation of nanobubbles is a highly stochastic process which cannot be reliably reproduced. Subsequent research found a way to fabricate nanobubbles with a high degree of consistency, but the measurements were still limited to local-probe techniques due to the nanoscale size of the devices. As such, a method to reliably induce a homogeneous pseudo-magnetic field within a micron-sized membrane would be an attractive proposition, and is the basis for the work presented within this thesis. The non-uniform strain required precludes a simple bending or elongation of the substrate, hence a more local method is required. A novel nanostructure consisting of suspended gold beams surrounding a graphene membrane will deform upon cooling to cryogenic temperatures, and crucially, the actuation mechanism can be designed to produce any configuration of strain, including uniaxial strain, triaxial strain and a fan-shaped deformation, the latter two of which are predicted to create homogeneous pseudo-magnetic fields within a membrane. Strain patterns which are predicted to produce experimentally significant pseudo-magnetic fields (~1 T) may be generated with complex actuation beams that are physically achievable. Furthermore, the actuation mechanisms may be utilised as electrical contacts to the membrane, allowing its conductivity to be measured in the context of a two- or multi-terminal measurement, in conjunction with an external magnetic field. The design of the devices was developed using finite-element analysis, and the behaviour verified by low-temperature imaging of prototypes. While, after careful annealing, some conventional two-terminal suspended devices exhibited quantum Hall features at very low fields, the fabricated strain-inducing devices did not display pseudo-Landau quantisation, nor Landau quantisation, due to the difficulties of using current annealing to clean devices post-fabrication. The presented work, however, could pave the way towards observing signatures of pseudo-magnetic fields in a range of experimental measurements, as well as creating alternative strain geometries.

The system LiHo_<em>x</em>Y_1-<em>x</em>F₄ is a nearly perfect example of a dilute, dipolar-coupled Ising magnet and, as such, it is an ideal testing ground for many theories in statistical mechanics. At low holmium concentration (<em>x</em> = 0. 045) an unusual spin liquid or "anti-glass" state was discovered in previous work [1]. This state does not exhibit a spin glass freezing transition as is expected for a long-range interaction. Instead, it shows dynamics which are consistent with a collection of low-frequency oscillators [2]. It was also seen to have sharp features in its specific heat [3]. <br /><br /> We present heat capacity measurements on three samples at and around the concentration of the spin liquid state in zero magnetic field and in a temperature range from around 50 mK to 1 K. In contrast to previous measurements, we find no sharp features in the specific heat. The specific heat is a broad feature which is qualitatively consistent with that of a spin glass. The residual entropy as a function of <em>x</em>, obtained through a numerical integral of the data, however, is consistent with numerical simulations which predict a disappearance of spin glass ordering below a critical concentration of dipoles [4]. <br /><br /> Also presented here, is ac susceptibility data on an <em>x</em> = 0. 45 sample which exhibits a paramagnetic to ferromagnetic transition and is found to be consistent with previous work.

The system LiHo_<em>x</em>Y_1-<em>x</em>F₄ is a nearly perfect example of a dilute, dipolar-coupled Ising magnet and, as such, it is an ideal testing ground for many theories in statistical mechanics. At low holmium concentration (<em>x</em> = 0. 045) an unusual spin liquid or "anti-glass" state was discovered in previous work [1]. This state does not exhibit a spin glass freezing transition as is expected for a long-range interaction. Instead, it shows dynamics which are consistent with a collection of low-frequency oscillators [2]. It was also seen to have sharp features in its specific heat [3]. <br /><br /> We present heat capacity measurements on three samples at and around the concentration of the spin liquid state in zero magnetic field and in a temperature range from around 50 mK to 1 K. In contrast to previous measurements, we find no sharp features in the specific heat. The specific heat is a broad feature which is qualitatively consistent with that of a spin glass. The residual entropy as a function of <em>x</em>, obtained through a numerical integral of the data, however, is consistent with numerical simulations which predict a disappearance of spin glass ordering below a critical concentration of dipoles [4]. <br /><br /> Also presented here, is ac susceptibility data on an <em>x</em> = 0. 45 sample which exhibits a paramagnetic to ferromagnetic transition and is found to be consistent with previous work.

This thesis will present research that studies the role of disorder, geometric frustration and the long range dipolar interaction on the collective behaviour of several insulating, rare earth magnets. Experiments were performed at low temperatures to measure the specific heat and magnetic susceptibility of several materials. Susceptibility was measured with a SQUID magnetometer that has been designed and constructed primarily for the study of slow dynamics in glassy systems. Specifically, this thesis will discuss three distinct topics. The first is the series of materials LiHo(x)Y(1-x)F(4), which are manifestations of the dilute, dipolar coupled Ising model. The low-x portion of the phase diagram has become a rather contentious issue in recent years with both theoretical and experimental groups disagreeing on the existence of a spin glass freezing transition and one experimental group arguing for the existence of an exotic "antiglass'' or spin liquid state resulting from quantum entanglement at x=0.045. We present specific heat and dynamical susceptibility measurements on four stoichiometries in this series: x = 0.018, 0.045, 0.080 and 0.012. No evidence of an unusual antiglass state is observed. Instead, our results show evidence, at all dilution levels studied, of a spin glass freezing transition. Interpretation of experimental data is found to be complicated by the anomalously slow dynamics in these materials. The relaxation time scales are found to increase as the concentration of Ho(3+) ions is reduced, an effect which can be attributed to single-ion physics and the importance of the nuclear hyperfine coupling in this system. A second set of materials studied here is a series of several Gd garnet materials, the most famous of which is Gd(3)Ga(5)O(12) (GGG), a material previously argued to be a disorder-free spin glass. Our specific heat experiments reproduce previous experiments on GGG and show that the homologous Gd garnets Gd(3)Te(2)Li(3)O(12) and Ga(3)Al(5)O(12) do not share the same glassy physics but exhibit sharp ordering features. By experimenting with the introduction of random site dilution, it is concluded that a 1-2% off-stoichiometry inherent in GGG is likely a special kind of disorder that is particularly effective in inducing random frustration and the formation of a spin glass. Finally, specific heat measurements on the pyrochlore antiferromagnet Gd(2)Sn(2)O(7) (GSO) are presented. While GSO has generally been found to be a well behaved and well understood model magnet, with long range order developing at around 1 K, like many other geometrically frustrated magnets, it has been discovered to possess persistent spin dynamics down to very low temperatures as measured by μSR and Mössbauer spectroscopy. Measurement of the low temperature limit of the specific heat when compared with linear spin-wave theory, however, presents a consistent picture of gapped magnon excitations that freeze out at low temperatures and make the existence of the proposed dynamic ground state unlikely.

This thesis presents research of two novel magnetic materials, LiHoF4 and Tb2Ti2O7. Experiments were performed at low temperatures and in an applied magnetic field to study thermal expansion and magnetostriction using a capacitive dilatometer designed during this project. This thesis presents 3 distinct topics. This manuscript begins with a thermodynamic description of thermal expansion and magnetostriction. The design of a capacitive dilatometer suitable for use at ultra-low temperatures and in high magnetic fields is presented. The thermal expansion of oxygen free high conductivity copper is used as a test of the absolute accuracy of the dilatometer. The first material studied using this dilatometer was LiHoF4. Pure LiHoF4 is a dipolar coupled Ising ferromagnet and in an applied transverse magnetic field is a good representation of the transverse field Ising model. An ongoing discrepancy between theoretical and experimental work motivates further study of this textbook material. Presented here are thermal expansion and magnetostriction measurements of LiHoF4 in an applied transverse field. We find good agreement with existing experimental work. This suggests that there is some aspect of LiHoF4 or the effect of quantum mechanical fluctuations at finite temperatures which is not well understood. The second material studied is the spin liquid Tb2Ti2O7. Despite theoretical predictions that Tb2Ti2O7 will order at finite temperature, a large body of experimental evidence demonstrates that spins within Tb2Ti2O7 remain dynamic to the lowest temperatures studied. In addition Tb2Ti2O7 also exhibits anomalous thermal expansion below 20K, giant magnetostriction, and orders in an applied magnetic field. Thermal expansion and magnetostriction measurements of Tb2Ti2O7 are presented in applied longitudinal and transverse fields. Zero-field thermal expansion measurements do not repeat the previously observed anomalous thermal expansion. A large feature is observed in thermal expansion at 100mK, in rough agreement with existing experimental work. Longitudinal and transverse magnetic fields were applied to Tb2Ti2O7. Longitudinal magnetostriction measurements show qualitatively di erent behavior than previous observations. These measurements were taken along di erent crystal axes so direct comparison cannot be made. Thermal expansion measurements in an applied transverse field show evolution with the strength of the applied field. This evolution may relate to an ordering transition, however difficulties in repeatability in a transverse field require that these results be repeated in an improved setup.

The fractional quantum Hall (FQH) effect harbors a wealth of unique phenomena, many of which remain mysterious. Of particular interest is the predicted existence of quasi-particles with unusual topological properties, especially in light of recent proposals to observe these properties using electronic interferometers. An introduction to quantum Hall physics and electronic interferometry is given in Chapter 1 of this thesis. The remaining chapters, summarized below, describe a set of experiments in which FQH systems are studied using electronic Fabry-Perot interferometry and related techniques. Since prior studies of electronic Fabry-Perot interferometers revealed unexpected behavior even in the integer quantum Hall (IQH) regime, we began our measurements there. Our initial experiment, presented in Chapter 2, disentangles signatures of Coulomb interaction effects from those of Aharonov-Bohm (AB) interference and provides the first measurement of pure AB interference in these devices. In our next experiment, presented in Chapter 3, we measure AB interference oscillations as a function of an applied dc bias, use their period to study the velocity of the interfering electrons, and study how the oscillations decay as a function of bias and magnetic field. Moving to the FQH regime, applying a similar-sized bias to a quantum point contact leads to long-lasting changes in the strengths and positions of FQH plateaus. The involvement of lattice nuclear spins in this effect, suggested by the long persistence times, is confirmed using NMR-type measurements. Although the exact physical process responsible for the effect remains unclear, its filling-factor dependence provides a striking illustration of composite fermion physics. These measurements are described in Chapter 4. In certain devices, interference oscillations associated with several FQH states are observed. Interpretation of their magnetic-field and gate-voltage periods provides a measurement of quasi-particle charge, and temperature dependence measurements suggest differences between the edge structure of IQH and FQH states. These measurements are described in Chapter 5. Finally, Chapter 6 presents some recent, not-yet-published observations that may shed light on ways to improve the visibility of existing oscillations and potentially observe interference at additional FQH states. This chapter concludes with a discussion of possible next steps toward achieving these goals. / Physics

This thesis describes electronic transport experiments in graphene from the hopping to the ballistic regime. The first experiment studies dual-gated bilayer graphene devices. By applying an electric field with these dual gates, we can open a band gap in bilayer graphene and observe an increase in resistance of over six orders of magnitude as well as a strongly non-linear behavior in the transport characteristics. A temperature-dependence study of resistance at large electric field at the charge neutrality point shows the change in the transport mechanism from a hopping dominated regime at low temperature to a diffusive regime at high temperature. / Physics

Over the past twenty-five years, condensed matter physics has been developing materials with novel electronic characteristics for a wide range of future applications. Two research directions have shown particular promise: topological insulators, and high temperature copper based superconductors (cuprates). Topological insulators are a newly discovered class of materials that can be manipulated for spintronic or quantum computing devices. However there is a poor spectroscopic understanding of the current topological insulators and emerging topological insulator candidates. In cuprate superconductors, the challenge lies in raising the superconducting transition temperature to temperatures accessible in non-laboratory settings. This effort has been hampered by a poor understanding of the superconducting mechanism and its relationship with a mysterious pseudogap phase. In this thesis, I will describe experiments conducted on topological insulators and cuprate superconductors using scanning tunneling microscopy and spectroscopy, which provide nanoscale spectroscopic information in these materials. / Physics

This thesis will present research that studies the role of disorder, geometric frustration and the long range dipolar interaction on the collective behaviour of several insulating, rare earth magnets. Experiments were performed at low temperatures to measure the specific heat and magnetic susceptibility of several materials. Susceptibility was measured with a SQUID magnetometer that has been designed and constructed primarily for the study of slow dynamics in glassy systems. Specifically, this thesis will discuss three distinct topics. The first is the series of materials LiHo(x)Y(1-x)F(4), which are manifestations of the dilute, dipolar coupled Ising model. The low-x portion of the phase diagram has become a rather contentious issue in recent years with both theoretical and experimental groups disagreeing on the existence of a spin glass freezing transition and one experimental group arguing for the existence of an exotic "antiglass'' or spin liquid state resulting from quantum entanglement at x=0.045. We present specific heat and dynamical susceptibility measurements on four stoichiometries in this series: x = 0.018, 0.045, 0.080 and 0.012. No evidence of an unusual antiglass state is observed. Instead, our results show evidence, at all dilution levels studied, of a spin glass freezing transition. Interpretation of experimental data is found to be complicated by the anomalously slow dynamics in these materials. The relaxation time scales are found to increase as the concentration of Ho(3+) ions is reduced, an effect which can be attributed to single-ion physics and the importance of the nuclear hyperfine coupling in this system. A second set of materials studied here is a series of several Gd garnet materials, the most famous of which is Gd(3)Ga(5)O(12) (GGG), a material previously argued to be a disorder-free spin glass. Our specific heat experiments reproduce previous experiments on GGG and show that the homologous Gd garnets Gd(3)Te(2)Li(3)O(12) and Ga(3)Al(5)O(12) do not share the same glassy physics but exhibit sharp ordering features. By experimenting with the introduction of random site dilution, it is concluded that a 1-2% off-stoichiometry inherent in GGG is likely a special kind of disorder that is particularly effective in inducing random frustration and the formation of a spin glass. Finally, specific heat measurements on the pyrochlore antiferromagnet Gd(2)Sn(2)O(7) (GSO) are presented. While GSO has generally been found to be a well behaved and well understood model magnet, with long range order developing at around 1 K, like many other geometrically frustrated magnets, it has been discovered to possess persistent spin dynamics down to very low temperatures as measured by μSR and Mössbauer spectroscopy. Measurement of the low temperature limit of the specific heat when compared with linear spin-wave theory, however, presents a consistent picture of gapped magnon excitations that freeze out at low temperatures and make the existence of the proposed dynamic ground state unlikely.