Unconventional Superconductivity in Uranium Ditelluride Zheyu Wu Department of Physics University of Cambridge This thesis is submitted for the degree of Doctor of Philosophy Jesus College September 2025 I would like to dedicate this thesis to my loving mom. Declaration This thesis is the result of my own work and includes nothing which is the outcome of work done in collaboration except as declared in the preface and specified in the text. It is not substantially the same as any work that has already been submitted, or is being concurrently submitted, for any degree, diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the preface and specified in the text. It does not exceed the prescribed word limit for the relevant Degree Committee. The research projects described in this thesis were undertaken with the supervision of F. M. Grosche and A. G. Eaton. Several of the research projects presented in this thesis were the work of collaboration. All measurements on UTe2 single crystals in this thesis were the work formed on samples grown by A. Cabala of the Valiŝka group, Charles University, Czech Republic. The pulsed field PDO measurements with rotator probe were probed in collaboration with Z. Zhu of Wuhan High Magnetic Field Center, Huazhong University of Technology. The pulsed field PDO measurements up to 80 T were performed in collaboration with Y. Skourski of Hochfeld-Magnetlabor Dresden. The resistivity measurements in resistive magnets were performed in collaboration with D. Graf of National High Magnetic Field Lab. The modelling of metamagnetic fluctuation mediating field-reinforced superconduc- tivity and quantum critical line were performed in collaboration with Dmitry Chichinadze from National High Magnetic Field Lab and Daniel Shaffer of University of Wisconsin- Madison. The current-voltage measurement in magnetic field measurements were performed in collaboration with G. Li and R. Zhou from Synergetic Extreme Condition User Facility. Sample preparation and characterisation was performed by myself, H. Chen and M. Long, with some additional assistance from A. G. Eaton. Spot welding of samples was performed by myself, H. Chen and T. I. Weinberger. The resistivity data presented in Chap. 3 in applied magnetic field of strength ≤ 14 T were taken by myself. The 3He magnetization measurements were performed by J. Chen and C. Liu. The AC magnetic susceptibility measurement under pressure was performed in collaboration with P. Alireza and J. Chen. The heat capacity under pressure measurements were performed by T. I. Weinberger. The contactless resistivity and magnetic oscillation experiment in Fig. 3.3 and Fig. 3.12 were performed by A. Hickey and A. G. Eaton at SCM 4 and Cell 6 in NHMFL, Tallahassee. The vi contactless resistivity experiment measuring metamagnetic transition was performed by A. G. Eaton and T. I. Weinberger in 70 T pulsed magnet in Dresden. The contacted resistivity experiment from Fig. 4.10 to Fig. 4.14 that reveals the spillover of SC3 and non-ohmic behaviour of SC3 onset were performed by A. G. Eaton and M. Long at Cell 6 in NHMFL, Tallahassee. All other data presented in this thesis at magnetic field strengths > 14 T were collected at high magnetic field facilities, as part of a group effect. Analysis on the high-field data presented in this thesis was performed primarily by myself, with some performed jointly with A. G. Eaton. These contributions are also acknowledged in the text where applicable. The results presented in Chap. 3 from Sec. 3.1 to Sec. 3.3 about careful measurement of ultraclean UTe2 single crystal are published in Z. Wu et al. Proc. Natl. Acad. Sci. 121 (37) e2403067121 (2024). The results on measuring AC magnetic susceptibility of UTe2 under pressure, which is shown in Sec. 3.4, is published in Z. Wu et al. Phys. Rev. Lett. 134 (23), 236501 (2025). The results concerning the discovering the quantum critical line of metamagnetic transition surface and spillover and non-ohmic behaviour of SC3 phase in UTe2, as shown in Sec. 4.1, and Sec. 4.2, is published in Z. Wu et al. Phys. Rev. X 15 (2), 021019 (2025). The result about enhanced Tc of field-induced SC3 phase, as in Sec. 4.3, is published in Z. Wu et al. Proc. Natl. Acad. Sci. 122 (2), e2422156122 (2025). Zheyu Wu September 2025 Acknowledgements The work presented in this thesis has been carried out in the Quantum Matter group, Cavendish Laboratory, from October 2021 to September 2025. I would like to express my sincere thanks to my PhD supervisor Prof. Malte Grosche and my advisor Dr. Alex Eaton. Without their guidance and help nothing in this thesis would ever have happened. I thank Prof. Gil Lonzarich for his endless wisdom and priceless guidance. I would like to thank my close collaborators and friends, Mengmeng Long, Hanyi Chen, and Theo Weinberger for their help and care in this journey. I would also like to thank Dmitry Chichinadze, Daniel Shaffer, and Gil for their excellent theory and insight. Special thanks goes to Patricia Lebre Alireza, Jiasheng Chen, and Cheng Liu for their great help in experiments and life choices. My sincere gratitude goes to Alex Hickey for his wise advice and Indy Liu for his great help in PDO techniques. I would like to thank Jinxu Pu for his participation and help. I would also like to thank all other QM members, Oliver, Christian, Max, Ran, Stephen, Riley, Mads, Leszek. Many thanks to the friends on the other side, Then, Gilles, Matt, Nick and Jess. Thanks for going through all these with me. I am especially grateful to Dave Graf in Tallahassee, Yurii Skourski in Dresden, Gang Li and Rui Zhou in Beijing, and Gangjian Jin, Huakun Zuo and Zengwei Zhu in Wuhan. I have carried out 45 weeks of magnet time during my PhD in total. All the support and kindness in these magnet times made a difference to my research and life. I am indebted to Cambridge Trust and Jesus College for their generous financial support. I would like to thank the friends who visited me during these trips for magnet time, they are Yurou Chen, Ling Fu, Yu Ling, Wei Xie, Shuo Zou, Xiaodong Guo, Jing Zhang, Yuhao Ye, Fenglin Mao, Zijian Wang, Zihao Wang, Yiyang Ye, Zhengzhi Wu, Kailin Wang, Ding Gu, Haodong Hu, Yaozhang Zhou and Yang Liu and many other friends. Special thanks to my loving girlfriend, Rou Deng, for her support. Finally, I would like to sincerely thank my mother, for supporting me all the way. Abstract Superconductivity remains one of the most important topics in condensed matter physics, arising from the pairing of two fermions, in either a singlet or triplet spin configurations. Spin-triplet superconductivity has attracted a lot of interest due to its intriguing physics and potential application in recent years. While superfluid helium-3 has been rigorously established to possess triplet character, no triplet superconducting analog has yet been conclusively identified. This dissertation investigates the heavy fermion compound uranium ditelluride (UTe2), one of the most promising candidates for spin-triplet superconductivity. UTe2 exhibits extremely high upper critical fields of up to 70 Tesla, minimal change in the local spin susceptibility upon the crossing superconducting transition temperature, and a remarkable phase diagram comprising multiple distinct superconducting phases intertwined with complex magnetic fluctuations, metamagnetic transitions, and ordered states. Nevertheless, conflicting results between numerous experimental studies have impeded progress in understanding the nature of the superconductivity in UTe2. Initial studies on low- quality UTe2 samples suggested time-reversal symmetry breaking and a multi-component superconducting order parameter, based on polar Kerr effect and specific heat measurements. Subsequent work using cleaner samples failed to reproduce these results, attributing the discrepancies to inhomogeneities in early-generation crystals. One part of this dissertation focuses on providing a detailed and systematic experimental study of the superconducting and normal state properties of UTe2 in the ultraclean limit to firmly elucidate – in the absence of impurities or inhomogeneities – the intrinsic properties of this material. Remarkably, the field-reinforced superconducting phase shows acute sensitivity to the impurity level in the sample and sheds light on its origin from Cooper pairs mediated by metamagnetic fluctuations. The pressure-dependent magnetic susceptibility study of UTe2 single crystals shows a superconducting transition within another superconducting state, highlighting the difference between these superconducting states. Another remarkable result in this dissertation is the discovery of a quantum critical line in the Ha-Hb-Hc Cartesian field space. UTe2 undergoes a metamagnetic transition in which a large field along the b-axis is applied. Application of transverse field components x along the a-axis and c-axis was found to suppress the critical endpoint of the metamagnetic transition towards zero temperature. Two-axis rotational studies discover that the critical endpoint is suppressed into a series of quantum critical endpoints. These quantum critical endpoints connect into a quantum critical line in the three-dimensional field space and bound the metamagnetic transition surface. The quantum fluctuations accompanying this quantum critical line likely mediate the pairing of the field-induced superconducting phase, resulting in its exotic toroidal shape and enhanced critical temperature. In combination, this dissertation establishes a comprehensive framework for the discus- sion of the intrinsic properties of UTe2. The careful mapping of the phase diagram reveals a new type of enhanced-dimensional quantum critical phase boundary and provides new insights into the mechanisms underlying spin-triplet superconductivity. Table of contents 1 Introduction 1 1.1 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Quantum Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 U-based superconductor materials . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Uranium Ditelluride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.2 Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.3 Normal State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4.4 Ambient superconducting phase . . . . . . . . . . . . . . . . . . . 19 1.4.5 Superconducting phase diagram in field . . . . . . . . . . . . . . . 19 1.4.6 Phase diagram of UTe2 under pressure . . . . . . . . . . . . . . . . 26 1.5 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2 Methods 33 2.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1.1 Growth methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1.2 Sample Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.2 Moissanite anvil cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3 DC field measurement techniques . . . . . . . . . . . . . . . . . . . . . . 37 2.3.1 Spot welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3.2 Electrical transport . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3.3 Current-voltage measurement . . . . . . . . . . . . . . . . . . . . 39 2.3.4 AC magnetic susceptibility . . . . . . . . . . . . . . . . . . . . . . 40 2.4 Pulsed field measurement techniques . . . . . . . . . . . . . . . . . . . . . 42 2.4.1 Transport measurement . . . . . . . . . . . . . . . . . . . . . . . . 44 2.4.2 Radio frequency measurement . . . . . . . . . . . . . . . . . . . . 45 2.5 Uncertainty Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 xii Table of contents 3 Intermediate Magnetic Field Superconducting Phase Diagrams of UTe2 51 3.1 Enhanced Superconductivity in MSF-grown UTe2 . . . . . . . . . . . . . . 51 3.1.1 Sample Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.1.2 Upper Critical Field (Hc2) . . . . . . . . . . . . . . . . . . . . . . 52 3.1.3 Lower Critical Field (Hc1) . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Intrinsic Metamagnetic Transition in UTe2 . . . . . . . . . . . . . . . . . . 60 3.3 Enhanced Angular extent of SC2 . . . . . . . . . . . . . . . . . . . . . . . 66 3.4 Phase diagram under pressure with ACMS . . . . . . . . . . . . . . . . . . 68 3.5 Understanding SC2 in UTe2 . . . . . . . . . . . . . . . . . . . . . . . . . 76 4 High Magnetic Field Phase Landscape of UTe2 79 4.1 Discovery of a Quantum Critical Line in UTe2 . . . . . . . . . . . . . . . . 79 4.2 Spillover of SC3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.3 Angular dependence of Tc in SC3 of UTe2 . . . . . . . . . . . . . . . . . . 93 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5 Conclusions 107 References 109 Chapter 1 Introduction 1.1 Superconductivity Superconductivity has long been a central topic of condensed matter physics research. Since its discovery in 1911 in mercury by Heike Kamerlingh Onnes [1, 2], five Nobel Prizes have been awarded for research related to superconductivity [1, 3–6]. In recent years, it has also attracted a new burst of interest due to the discoveries of (i) extremely high temperature superconductivity in pressurized hydrides [7–9], (ii) a dome-like temperature-gate phase diagram in moiré systems [10–12], and (iii) materials of new unconventional superconducting pairing symmetry, including UTe2 and CeRh2As2 [13, 14]. Superconductivity emerges when the material cools to below the critical temperature Tc and is suppressed by a magnetic field more than the upper critical field Hc2. The phe- nomenology of the superconductivity effect could be defined with three aspects: (i) zero electrical resistivity, (ii) perfect diamagnetism, and (iii) zero electronic thermal transport. These key observations depict three aspects of superconductivity and apply constraints on the definition. The central phenomenon of the superconductor is dissipationless electrical transport. However, in theory quantum mechanical electrons would propagate in an ideal periodic lattice without impedance. What uniquely sets superconductivity apart from the “ideal conductor” is that superconducting materials are perfectly screened by the supercurrent and form a perfectly diamagnetic state, namely the Meissner effect [15]. The theoretical framework of superconductivity was built by Bardeen, Cooper, and Schrieffer in 1957 [16]. In addition to the experimental discovery of zero resistivity and the Meissner effect, another important experimental result that sheds light on the understanding of superconductivity is the discovery of the isotope effect [17, 18]. By substituting the superconducting elemental material with their isotope, the critical temperature scales with 2 Introduction the inverse square root of the isotope mass, which gives Tc √ Mc = Constant Inspired by the isotope effect and other experimental discoveries, Bardeen and cowork- ers proposed that electron-phonon interactions could give rise to an effectively attractive interaction [3]. A paired electron state, known as Cooper pairs, becomes the energy ground state with the attractive interaction. The resulting Cooper pair behaves as a bosonic-like quasiparticle, and the ensemble condenses into a macroscopic quantum state characterised by a well-defined quantum phase (φ in ψ = Aeiφ ). This condensate exhibits quantum phase rigidity, meaning that any change in the phase of a single particle wave function must involve a collective change in the phase of the Cooper-pair condensate. As a consequence of the phase rigidity, electron scattering within the superconducting state can only occur elastically. Inelastic scattering, which would require altering the quantum phase of an individual electron, is forbidden because it would necessitate a simultaneous phase change of a large number of electrons. Neither electron–electron nor electron–phonon scattering processes possess the energy required to perturb the condensate. This collective protection provides a natural explanation for the zero-resistivity property of superconductors. Since the Cooper pairs consist of two fermions, the wave function must be antisymmetric due to the Pauli exclusion principle. BCS theory requires that the two electrons have opposite spin polarisation [19]. The spin-singlet configuration is antisymmetric; thus, the orbital configuration is symmetrical. The order parameter describes the superconducting gap, which could be labeled as s-, p-, d-, etc. For a spin-singlet Cooper pair, only even parity gap symmetry, including s-wave and d-wave symmetry, is possible. Superconducting materials have many practical applications. Fig. 1.1 shows several examples of applications. Strong magnetic fields have many practical applications, including medical magnetic resonance imaging and nuclear fusion. The power of a 20 T resistive water-cooled magnet is about 8 MW. The energy cost of running such a magnet for an hour is equivalent to the energy cost of 3 UK families for a year, impeding any real applications due to the huge energy cost. However, by winding the superconductors into a coil, a superconductor could maintain a magnetic field without any energy cost. Therefore, superconducting magnets are necessary for any practical application that requires a magnetic field in order to save energy. Another major application of the superconductor is quantum computing. As a macro- scopic quantum phenomenon, superconductors have also been applied to form Josephson junctions and are used to perform quantum computation [20]. However, because of the short 1.1 Superconductivity 3 coherence time, quantum computation based on a conventional superconductor is still a long way to go from a practical application. One possible way to improve quantum computation is by building a quantum computer with a topologically protected superconductor. One possible material realization is a multi- component chiral p-wave superconductor. Since p-wave is an odd-parity gap symmetry, the spin configuration is required to be spin-triplet. Therefore, the search for spin-triplet superconductor candidates is a pressing topic [21]. Medical imaging Tokamak fusion Power cable Quantum computing Fig. 1.1 Examples of major applications of superconductor materials. Superconductor materials could be applied to make superconducting magnets which can be used in medical imaging and nuclear fusion [22, 23]. Development of kilometer-long superconducting thin film belts allow lossless DC current transportation. The quantum nature of superconductors enables quantum computation [20]. 4 Introduction 1.2 Quantum Phase Transition A “phase transition” usually refers to a substance transforming between different states of matter [24]. For example, water boils into vapour or freezes into ice. Typically, the phase transition is driven by the change in temperature of the substance. For example, nickel would go through a phase transition from ferromagnetic (FM) to paramagnetic (PM) when its temperature goes above the Curie point [25]. The phase transition can be classified according to the behaviour of the thermodynamic free energy [26]. The order of the phase transition is determined by the lowest derivative of the free energy with respect to the thermodynamic variable that is discontinuous in the phase transition [24]. A first-order phase transition indicates that the first derivative of the free energy is discontinuous. The metamagnetic (MM) transition is a first-order phase transition. Here, the MM transition is defined as a sudden change in the magnetization due to the change of the external magnetic field. Since magnetization is the first derivative of the free energy with respect to the magnetic field and is discontinuous, the metamagnetic transition is a first-order phase transition. A finite-energy change or latent heat is related to a first-order phase transition. As for the second-order phase transition, a famous example is the superfluid phase transition of 4He at the lambda point T = 2.2 K [27]. The phase transition occurs without latent heat. Physical variables that are related to the second derivatives of free energy, including heat capacity and magnetic susceptibility, are discontinuous at this phase transition. With knowledge of the state of the material versus the temperature and other control parameters such as pressure and chemical doping, a phase diagram could be constructed. The phase transition temperature could be tuned by non-thermal parameters. For example, the boiling point of water decreases as the pressure decreases. This explains why it is hard to cook rice on the top of the mountain. The phase transitions, defined by pairs of temperature and control parameters that distinguish two phases, form a phase boundary. In Fig. 1.2, a schematic representation of the temperature - control parameter phase diagram is shown. As the value of the control parameter changes, the phase transition temperature decreases. In certain cases, the phase boundary would end at a certain point in the phase diagram. This point is referred to as the “critical endpoint” (CEP) for a first-order phase boundary and is known as the “critical point” for a second-order phase boundary [28]. A well-known example of critical point is in the (T, p) phase diagram of water. The liquid-vapor phase boundary terminates at temperature T = 647 K (374°C) and pressure p ≈ 0.2 kbar. In the vicinity of the critical point, the physical properties of the matter would deviate dramatically from normal conditions [29]. Under normal conditions, water is incompressible and is a 1.2 Quantum Phase Transition 5 good solvent for electrolytes. When it is near the critical state, these properties go into the exact opposite: water becomes compressible and is a bad solvent for electrolytes. Ordered Not Ordered Quantum Critical Point Control Parameters (Doping, Pressure, …) Te m pe ra tu re (K ) Fig. 1.2 Left panel: A scheme of quantum phase transition in the temperature - tuning parameters. Dashed line marks the phase boundary of the thermal phase transition. The star marks the quantum critical point (QCP) at the quantum phase transition. A superconducting dome occurs in the vicinity to the QCP. Right panel: An example of superconducting dome happening near the quantum critical point in ferromagnetic UGe2. The dashed line connects the Curie temperature TC data point at different applied hydrostatic pressure. Superconducting dome is represented by the solid line bounded by the superconducting transition temperatures T SC c under each applied pressure. Figure adapted from ref. [30]. reproduced with permission of Springer Nature. When the phase boundary is suppressed and terminates at zero kelvin, the termination point in the phase diagram is known as a quantum critical point (QCP) [31] for the second- order phase transition or a quantum critical endpoint (QCEP) [32] for the first-order phase transition. It could be viewed as a phase transition driven by the non-thermal parameter instead of the temperature. Physical properties in the vicinity of the QCP would also change drastically. As shown in Fig. 1.2, the superconductivity dome emerges next to the QCP due to suppression of the ferromagnetic transition Curie temperature TC to zero kelvin by the application of non-thermal hydrostatic pressure in the heavy fermion ferromagnetic superconductor UGe2 [30]. Observation of the superconductivity dome close to the QCP or QCEP is ubiquitous among many different quantum material systems, including copper- based high-Tc superconductor [33–35], iron-based high-Tc superconductor [36, 37] and heavy fermion systems such as UGe2, URhGe, CeSb2, CePd2Si2 and CeRh2Si2 [30, 38–40]. The superconducting phase near the QCP could be mediated by a pairing mechanism beyond the 6 Introduction BCS theory [41] and have non-trivial topological properties that would be intriguing to study, as discussed in Sec. 1.1. C D B B (T) A Hole Doping p Fig. 1.3 Signatures of quantum phase transitions and quantum critical points. (A) Magnetic quantum oscillation frequency of CeRhIn5 as a function of applied pressure. The quantum phase transition is tuned by pressure and happens at critical pressure pc. Reproduced with permission from ref. [42]. (B) Hall number versus hole doping p of Ca-doped YBa2Cu3O7+x superconductor. Quantum phase transition is at critical doping pc = 0.192. Figure adapted from ref. [43] and reproduced with permission of Springer Nature. (C) A Tc - hole doping p superconducting phase diagram of YBa2Cu3O7+x under different magnetic fields indicated by the colour code. Yellow stars and white diamonds marks the inverse effective mass 1/m∗ as a function of hole doping. Two quantum critical points are noted by the blue triangles at zero temperatures. Figure adapted from ref. [33] and reproduced with permission of AAAS. (D) Colour contour plot for resistivity power exponent ε (from ρ = AT ε +ρ0) of YbRh2Si2 in temperature - magnetic field for B//c. Quantum critical point at 0.7 T is indicated by the linear-in-T resistivity (ε = 1). Figure adapted from ref. [44] and reproduced with permission of Springer Nature. 1.3 U-based superconductor materials 7 Fig. 1.3 shows three types of signatures accompanying the quantum phase transition and the quantum critical point. Firstly, a Fermi surface reconstruction occurs when the material is tuned through the QCP or QCEP. As shown in Fig. 1.3(A) and (B), the quantum oscillation (QO) frequency and Hall number change abruptly at the QCP for CeRhIn5 and cuprate YBCO at their QCP induced by pressure and chemical doping. The quantum oscillation corresponds to the extremal area of the Fermi surface perpendicular to the magnetic field [45] and the Hall number corresponds to the carrier density that reflects the Fermi surface volume [46]. Both of these observations indicate that a change in the Fermi surface geometry occurs at the quantum phase transition. There are many examples of Fermi surface reconstruction at a QCP, including electron-doped cuprate NCCO [47] and heavy fermion superconductor URhGe [48], about which further discussion will be made later. Secondly, the effective mass will diverge near the QCP. As shown in Fig. 1.3C, the yellow stars that represent 1/m∗ approach zero as the chemical doping in the YBCO approaches the second QCP (p ≈ 0.19) in the phase diagram. A similar divergence in effective mass is also observed for doping p near the first QCP (p ≈ 0.9). This divergence in effective mass is also measured by heat capacity in YBCO [49] and iron-based superconductors [37]. However, there is a controversial report indicating that the effective mass divergence near QCP might be interrupted by “other first-order transitions and or novel emergent phases”, as illustrated in the case of NiS2 [50]. Finally, linear in temperature resistivity known as strange metal behaviour is usually observed near the QCP [51]. As shown in Fig. 1.3D, the colour map plot shows that the temperature exponent for resistivity is close to 1 in the QCP with H//c at 0.7 T. The linear- in-T resistivity persists down to the lowest temperature in this measurement. However, no consensus has been reached for the understanding of strange metal phenomena up to now [51]. 1.3 U-based superconductor materials In this section, three similar U-based superconductor materials, including UGe2, URhGe and UCoGe, are introduced. In many aspects, these materials share similar properties with UTe2, which will be the focus of this thesis. Understanding their temperature - field phase diagram and field-reinforced superconducting phases would be helpful for the study of UTe2. To start with, Fig. 1.4 shows the crystal structures of these U-based systems. UGe2 is the first material discovered to have a superconducting phase coexisting with a ferromagnetic state [30]. Before its discovery, ferromagnetism and its corresponding magnetic field were believed to impede the formation of superconducting Cooper pairs. 8 Introduction UGe2 crystallizes in an orthorhombic structure with the space group Cmmm (No.65), as shown in Fig. 1.4A. Its U atoms form zigzag chains along the a-axis and its FM moments are polarised along the magnetization easy a-axis. At ambient pressure, it has a Curie temperature TC = 52 K. The inset of Fig. 1.4B shows the hysteretic behaviour when a 0.1 T field is applied along the a-axis at 4.5 K [30, 52], indicating its FM ground state at ambient pressure. As described in Sec. 1.2, a superconducting dome appears next to the QCP when the TC of the FM phase is suppressed to zero-kelvin by applying pressure [30]. 0GPa A C D F B E Fig. 1.4 Crystal structure of (A) UGe2, (B) URhGe and UCoGe. Arrows indicate the direction of spin polarisation. (C) Magnetic property of UGe2, temperature dependent magnetization measurement shows a ferromagnetic transition at about 55 K. Inset: moment versus field showing a ferromagnetic state. (D) Projection of URhGe / UCoGe crystal structure from b-axis. (E) Sommerfeld coefficient and Neel / Curie temperature as a function of the nearest distance between two U atoms. (F) Magnetization measurement showing the ferromagnetic nature of URhGe. Because the application of pressure is not a simple technique, efforts were made to substi- tute Ge with other transition metals to seek ambient pressure coexistence of ferromagnetism and superconductivity. The resulting materials are URhGe and UCoGe. They both have a 1.3 U-based superconductor materials 9 TiNiSi-type orthorhombic structure with space group Pnma, as shown in Fig. 1.4A. The U atoms also form zigzag chains along the a-axis dU−U for both URhGe and UCoGe. However, the FM moments are polarised along the c-axis for both of them. Thus, the c-axis now becomes the easy axis and the a-axis is the hard axis for URhGe and UCoGe. The nearest distance between two U atoms is 3.497 Å and 3.481 Å for URhGe and UCoGe, respectively. Magnetic ordering temperature and Sommerfeld coefficient as a function of dU−U show that URhGe and UCoGe are at the border of the FM phase, as shown in Fig. 1.4C. A B C Fig. 1.5 Field - temperature phase diagram showing field-induced and field-reinforced superconductivity in U-based materials. (A) Hc2 versus temperature for H//a in UGe2 [53] when p = 1.35 GPa pressure is applied. Here, the Hx denotes the magnetic field that induces a ferromagnetic phase. Hc2 versus T for H//b in (B) URhGe [38] and (C) UCoGe [54]. Reproduced with permission from ref. [55]. Now, the focus turns to the properties of superconductivity in these U-based ferromagnets. Fig. 1.5 shows the upper critical field Hc2 versus temperature for (A) UGe2, (B) URhGe, and (C) UCoGe. For all three materials, magnetic field-reinforced superconductivity emerges. In UGe2, the abrupt increase in Hc2 has been explained as the field switches between sub-FM phases, resulting in two superconducting phases with different Tc [53]. The superconducting phase diagram of URhGe and UCoGe is more interesting. The FM moments and Ising spins are polarised along the easy c-axis in URhGe and UCoGe, and the transverse field along the b-axis drives the field-induced superconducting phase in URhGe and the field-reinforced superconducting phase in UCoGe. This could be understood 10 Introduction as the transverse field along b would lead to the collapse of the spin-polarised Ising moment along c and thus drive a FM-PM crossover. The fluctuations in the vicinity of the FM-PM quantum phase transition then help to mediate the superconducting pairings or make the superconducting phase energetically favorable. URhGe A B C QCP Fig. 1.6 Field-induced superconductivity enclosing the QCP in URhGe. (A) Magnetization and resistivity versus magnetic field applied near the b-axis of URhGe, revealing a metamag- netic transition at 12 T and an accompanying field-induced superconducting phase at low temperature T = 40 mK with resistivity. Figure adapted from ref. [38] and reproduced with permission of AAAS. (B) Illustration of the ferromagnetic wings and QCP in the (T , Hb, Hc) space in URhGe, where Hb and Hc are magnetic field components along b and c axis. Arrow marks the field orientation determined by the angle θ between b- and c-axes. Field-induced superconducting phase are indicated by the red region. (C) Quantum oscillations with field rotated from b towards c by 10° at T = 20 mK. Resistivity kink and vanishing magnetic oscillation indicates the location of QCP. Figure adapted from ref. [56, 48] and reproduced with permission of Springer Nature. 1.4 Uranium Ditelluride 11 Fig. 1.6 presents more measurements on the field-induced superconducting phase in URhGe. In Fig. 1.6A, torque cantilever and neutron scattering found an abrupt change in the magnetization when HM = 12 T of magnetic field was applied along the b-axis of URhGe. Resistivity also picks up a peak feature in the ρ(H) curve at HM for high temperature T = 500 mK. At lower temperatures T < 350 mK, field-induced superconductivity emerges above and below HM. Thus, a first-order metamagnetic phase transition, as described in Sec. 1.2, and accompanying field-induced superconductivity were observed in URhGe [56]. When the field is tilted away from b toward c, i.e. when a H//c component is applied, even when a small rotation angle θ = 2.5° would soften the transition and change it into a continuous transition. As shown in Fig. 1.7B, the field along c would suppress the metamagnetic transition and lead to FM wings in the temperature - H//b - H//c space and result in a QCEP. The QCEP is located at Hc ≈ 1.1 T and Hb ≈ 13.5 T [57]. The existence of QCEP was corroborated by the observation of an abrupt change in the Shubnikov-de Haas oscillation as the field sweeps across the HM, as illustrated in Fig. 1.6C [48]. The T -B superconducting phase diagram of URhGe of field along b is shown in Fig. 1.7. An important feature of the phase diagram is that the field-induced superconducting phase persists up to a temperature (0.4 K) that is higher than the Tc = 0.25 K of the superconductivity in the ambient field. This illustrates that the field-induced superconducting phase in URhGe is mediated by the magnetic fluctuation accompanying the QCEP, and thus its Tc is enhanced [56, 38]. 1.4 Uranium Ditelluride UTe2 is a paramagnetic metallic material that was first discovered in 2006 [58]. Ever since spin-triplet superconductivity was discovered in this material in 2019 [13, 59], it has attracted a surge of interest - over 600 research articles have been published on UTe2 to date. The rich superconducting phase diagram and potential spin-triplet superconducting phase is the central topic of relevant research. In this section, important background information and previous results on which this dissertation is based on, are introduced. Firstly, the crystal structure and electronic structure will be introduced. Then, important physical properties of the normal state will be introduced. Finally, diverse superconducting phase diagrams under different magnetic field and pressure will be discussed. 12 Introduction Fig. 1.7 Superconducting Phase diagram of URhGe. Colour map of the resistance value versus temperature and magnetic field along b-axis. Two black region marks the two superconducting phases. Inset shows the raw R(H) curve that constructs the colormap. Figure adapted from ref. [38] and reproduced with permission of AAAS. 1.4.1 Basic properties UTe2 crystallises in a body-centered orthorhombic structure with the symmetry group Immm (No.71, D2h). The lattice constants at ambient temperature and pressure are a = 4.159 Å, b = 4.159 Å, and c = 13.945 Å [60]. There are four formula units per unit cell, as shown in Fig. 1.8. The only natural cleavage plane is (011), which corresponds to the surface perpendicular to the direction tilted by 23.7° from b to c-axes [61, 62]. It should be noted that the crystal structure of UTe2 is body-centered, with its inversion center located in the center of the crystal unit cell rather than at the uranium site. This indicates that the local inversion symmetry is broken. This situation is similar to another recently discovered heavy fermion superconductor CeRh2As2 [14]. The superconducting single crystal UTe2 was first grown by the chemical vapour transport (CVT) method [13, 59, 63]. The CVT-grown crystal was large in volume, but the quality was not good. Firstly, the residual resistivity and mean free path were low, interfering with the superconductivity critical temperature and impeding the observation of magnetic quantum oscillation. Moreover, CVT-grown single crystals in an early stage have different sub-phases 1.4 Uranium Ditelluride 13 that have different Tc, resulting in debates on many questions relevant to UTe2 including its superconducting order parameter [64, 65] and phase diagram [66, 63]. Higher quality samples were then grown by the molten salt flux method [67]. The new generation of samples settled these debates and revealed the intrinsic properties of this material. The details of the MSF-grown samples will be discussed in Chapter 3. a b c a b c a b cA CB Fig. 1.8 (A) Orthorhombic crystal structure of UTe2. Crystal axes are noted by the coloured axes. U atoms have one equivalent location while Te have two non-equivalent location in a unit cell. (B) Zoom-in on the nearest neighbors of U atoms dimer. The arrows indicate how the inter-atomic distances change as the temperatures is lowered [60]. (C) A photo of a UTe2 single crystal with axes marked by the the arrows. Figure adapted from ref. [68] and reproduced under a Creative Commons license. 1.4.2 Electronic Structure In this section, theoretical and experimental studies of the electronic structure of UTe2 are discussed, with particular emphasis on understanding its Fermi surface structure. Transport experiments found that UTe2 is a metallic material and should have a Fermi surface. However, a naive density functional theory calculation (DFT) based on the local density approximation (LDA) predicts that UTe2 is an insulator with a small gap ∆ = 13 meV. When the generalized gradient approximation with a Coulomb on-site U interaction (GGA+U) calculation is applied, a metal-insulator transition could be realized by tuning the on-site U [69]. This indicates the strongly correlated nature of the electronic structure of UTe2. When U = 2.0 eV, DFT generates a quasi-2D Fermi surface that qualitatively agrees with the angle-resolved photoemission spectroscopy (ARPES) result, with a dichotomy 14 Introduction on the existence of a three-dimensional Fermi surface pocket [70]. However, ARPES is a surface-sensitive technique, thus it is challenging for ARPES to probe the bulk property of the material. There have been many debates on the Fermi surface structure between the ARPES results and bulk-sensitive techniques, for example, spin-triplet superconductor candi- date Sr2RuO4 [71, 72], high-Tc superconductor YBCO [73, 74] and altermagnet candidate RuO2 [75, 76]. Moreover, the interpretation of the ARPES measurement of UTe2 is even more challenging due to the heavy core spectrum of the U atom. The presence or absence of a three-dimensional Fermi surface pocket in UTe2 is crucial, as it determines the topological property of the superconducting gap symmetry. Thus, bulk-sensitive quantum oscillation studies to understand the Fermi surface of UTe2 were indeed necessary. High quality UTe2 single crystals that showed quantum oscillations were grown by the MSF method in 2022 [67]. Both initial reports that measured QOs in UTe2 qualitatively agreed on the quasi-2D Fermi surface structure, similar to the one generated by the U = 2.0 eV DFT+U calculation [77, 78]. As shown in Fig. 1.9, the QO frequency - rotation angle analysis reveals that the Fermi surface of UTe2 is composed of two cylindrical sheets oriented along the c-axis. The Fermi surface is constructed by tuning the parameters in first-principle DFT calculations to match with the experimental QO frequency vs. angle in c− a and c−b rotations [78]. The Fast Fourier transformation (FFT) spectrum of oscillation shows a degenerate 3.5 kT frequency with H//c and splits into two similar frequencies as the field tilts away from the c-axis, as shown in Fig. 1.9. This observation indicates that the two cylindrical Fermi sheets share the extremal orbit area when H//c. The temperature- dependent Lifshitz-Kosevich study shows that the effective mass for H//c is 40 me. The Sommerfeld coefficient predicted by the Fermi surface volume and the effective mass matches well with the experimental results, indicating that the two cylindrical pockets account for the correct volume of the density of states and the existence of an additional Fermi surface pocket is unlikely. There was a controversial report on resolving a low-frequency f ≈ 200 T oscillation in any measured field orientations [79]. Broyles et. al. interpreted this observation as the signature of a three-dimensional Fermi surface pocket. Weinberger et. al. measured a similar 220 T frequency oscillation for H//a [80]. However, as the field tilted away from the a-axis by 20°, this low-frequency oscillation disappeared. This argues strongly against the existence of the three-dimensional Fermi surface pocket. This has been corroborated by two further magnetoresistance studies measuring focused-ion-beam-milled UTe2 structures [81, 82] In conclusion, careful magnetic quantum oscillation study has shown a quasi-2D cylin- drical Fermi surface with heavy effective mass, and no indication of any 3D Fermi surface pocket has been confirmed. 1.4 Uranium Ditelluride 15 A B C D 77 Fig. 1.9 Observation of quantum oscillation in high quality UTe2 single crystal. (A) Exam- ples of background subtracted de-Haas van Alphen (dHvA) oscillation spectrum at several selected angles in the c−a rotation plane. (B) Corresponding FFT spectrum for the quantum oscillation spectrum shown in (A). Experiments determined and density functional theory (DFT) calculated quantum oscillation frequency as a function of rotation angle for (C) c−a rotation plane and (D) c−b rotation plane is shown. The data from ref. [77] are put on top of the simulated frequency as well. Figure adapted from ref. [78] and reproduced under a Creative Commons license. 16 Introduction Fig. 1.10 Schematic figure of the Fermi surface structure of UTe2 constructed by the QO frequency versus angle study. Yellow and blue colour indicates the electron and hole nature of the Fermi surface cylinder respectively. Figure adapted from ref. [78] and reproduced under a Creative Commons license. Fig. 1.11 Observation of low frequency magnetic oscillation for H//a. There is a contra- dictory report attributing this observation as a hidden 3D Fermi surface pocket [79]. The observation of losing this low frequency oscillation with a 20° rotation away from a-axis shows decisive evidence of absence of a 3D Fermi surface pocket relevant to this magnetic oscillation. Figure adapted from ref. [80] and reproduced under a Creative Commons license. 1.4 Uranium Ditelluride 17 1.4.3 Normal State In this section, important information about the normal state of UTe2 is introduced. Fig. 1.12 shows the evidence of UTe2 being a paramagnetic material. Measurements were performed with a CVT-grown sample [13]. For moment vs. field, no hysteresis was observed for the field along any of the three crystal axes. At low temperature T < 10 K, the easy magnetization axis is the a-axis and the hard magnetization axis is the b-axis. For the moment versus temperature curve at constant magnetic field H = 1000 Oe with H//a, the susceptibility agrees with the Curie-Weiss law for temperature above T = 50 K. At low temperature, the M vs. T curve deviates from the 1/T behaviour and diverges faster, indicating existence of magnetic impurity in the early growth batches of this material. This divergence was later eliminated by MSF-grown ultraclean UTe2 samples [83]. Fig. 1.12 Magnetic susceptibility with field along a-, b, c-axes as a function of temperature and field. At low temperature, the M vs. T curve shows that the easy magnetization axis is a and hard magnetization axis is b. Inset: M vs. H curve showing paramagnetic ground state of UTe2. Figure adapted from ref. [13] and reproduced with permission of AAAS. 18 Introduction For H//b, M(T ) has a maximum around T ∗= 35 K. Following Maxwell relations, we have the following. ( ∂M ∂T ) H = ( ∂S ∂H ) T (1.1) For T < T ∗, H//b, the slope of the M(T ) curve is positive. From the Maxwell relation, this indicates that as the magnetic field increases, the entropy increases. This indicates the magnetic fluctuation enhanced by applying the Hb component. Similar T ∗ maximum behaviour was found by nuclear magnetic resonance (NMR) measurement of the spin relaxation time scale as well [84]. Fig. 1.13 Resistivity anisotropy of UTe2 normal state. Left panel: Photo of a focused-ion- beam milled UTe2 sample device and crystal structure of UTe2 for corresponding axis of the device. Right panel: Resistivity as a function of temperature on log− log scale from room temperature 300 K to low temperature for current along a-, b- and c-axes. Figure adapted from ref. [81] and reproduced under a Creative Commons license. Regarding the resistivity of the normal state of UTe2, the resistivity of UTe2 is anisotropic for different current orientations. The anisotropy originates from the quasi-2D nature of the Fermi surface, while the intermediate value of the anisotropy (maximally ρc/ρb = 5) comes from the warping in the cylinder, which could contribute to the electrical transport channel for current j//c like a three-dimensional Fermi pocket [78]. At low temperature for j//a and j//b, as temperature decreases, resistivity reaches a shallow maximum around 60 K and starts to rapidly decrease below 50 K. For j//c, there is a sharp peak in the ρc(T ) curve at T ≈ 12 K. These features have been attributed to the formation of Kondo coherence [68]. 1.4 Uranium Ditelluride 19 1.4.4 Ambient superconducting phase For new generation UTe2 single crystals grown by the molten salt flux method [67, 83], the bulk superconducting transition at ambient pressure and zero field is at Tc = 2 K and is evident by the measurements of resistivity ρ , magnetization M and heat capacity γ as shown in Fig. 1.14. The zero resistivity, diamagnetism, and heat capacity jump all indicate sharp transition of the SC1 superconducting phase. As discussed in Sec. 1.1, spin-triplet superconductivity is of great physical interest and potential technical applications. Therefore, the most intriguing property of the superconduc- tivity in UTe2 is the small decrease in Knight shift ∆Kb when a low magnetic field µ0H = 1 T is applied [85]. Knight shift is a shift in the nuclear magnetic resonance frequency that measures the spin susceptibility in the material [86, 87]. For a conventional spin-singlet superconductor, the drop in Knight shift would be 100% as T approaches to zero. For param- agnetic normal state above Tc, spin would be polarised along the external field orientation. As the temperature cools down, spin-singlet Cooper pairs would form and screen all the spin polarisation and result in a zero Knight shift state. Here, the change in Knight shift for the UTe2 SC1 phase is only 6%. The small decrease could be accounted for because the spin-triplet Cooper pairs have a preferred polarisation orientation that is different from the magnetic field and the ∆Kb measures the rotation of the triplet electron pairs. 1.4.5 Superconducting phase diagram in field One of the most remarkable features of the UTe2 phase diagram at ambient pressure is the presence of three distinct superconducting phases for the magnetic field aligned along certain orientations [88, 89]. For H applied along the b-axis, at low temperatures (T < 0.5 K), zero resistance is observed up to 34.5 T [90]. Remarkably, at higher temperatures (T ≈ 1 K) and for the field applied along a slight tilt away from the b-axis, multiple different measurements including resistivity and heat capacity of CVT samples have shown that there are two distinct superconducting phases present over this field interval [42, 88, 91], with the higher-field phase (SC2) having been referred to as a “field-reinforced” superconducting state [68]. Fig. 1.15 presents the resistivity measurements of the SC2 phase at incremental temperatures and compares two different field orientations. For H//b, ρ(H) at base temperature T = 0.2 K remains zero up to 34 T. At higher temperature T > 0.4 K, as the field increases, the normal state with finite resistivity above Hc2 of SC1 appears. Then ρ(H) decreases and enters the field-reinforced SC2 zero resistivity state. The crossover between the SC1- SC2 at low temperature is hard to directly measure by resistivity as it is screened by zero 20 Introduction 0 100 200 300 T (K) 0 200 400 (µ c m ) 0 2 4 T (K) 0 4 8 12 (µ c m ) Tc = 2.01 K UTe2, j // a 0 1 2 T (K) 0 0.2 0.4 C (µ J m ol -1 K -2 ) 0 1 2 3 4 T (K) -0.3 -0.2 -0.1 0 M ( 10 -3 em u) Ambient pressure µ µ µ Kb (% ) T (K) 10 Oe H // b A B C D Fig. 1.14 Superconductivity transition in high quality MSF-grown UTe2 at ambient field and pressure. (A) Resistivity measurement of normal state and superconducting transition. Red line is a Fermi liquid fitting ρ = AT 2 +ρ0 from 2.2 K to 4 K. (B) Magnetization M as a function of temperature with field along b-axis. Sample was cooled down to 0.5 K in zero field. Then µ0H = 10 Oe field is turned on at base temperature. M(T ) was measured during the warm up. (C) Heat capacity at zero field measures the bulk superconducting transition from the jump in γ(T ). (D) Knight shift ∆Kb as a function of temperature with different magnetic field applied. Arrow indicates the superconducting transition of corresponding magnetic field. Panel (D) reproduced with permission from ref. [85]. Copyright (2023) by the American Physical Society. resistivity. Meanwhile, heat capacity has found a bulk phase transition between these two superconducting phases [91]. 1.4 Uranium Ditelluride 21 As shown in Fig. 1.15, when the magnetic field is rotated away from b-axis by 4° towards a-axis, the temperature stability of the SC2 phase weakens. By comparing the two data sets between H//b and 4° towards a at 0.5 K, the ρ(H)|H//b remains zeros above 22 T up to 35 T, while there is only a negative magnetoresistance feature but no zero resistivity for ρ(H)|4deg. By taking the (H,T ) location of the phase transition, the resulting Hc2−T phase diagram at different orientations is as shown in Fig. 1.15. In the phase diagram, Hc2 −T curve shows an S-shape for different rotation angles in b−a rotation plane. The temperature at which the sample enters the SC2 state with µ0H field could be denoted as T SC2 c (H). The T SC2 c (H) is higher when the applied magnetic field is higher. For each orientation, there is a maximum for the T SC2 c (H) and this maximum decreases as the field tilts towards the a-axis. This observation illustrates that the transverse components of the field Hc and Ha are destructive to the formation of the SC2 phase. NMR Knight shift measurements above 15 T in high magnetic field show that there is no change of spin susceptibility ∆Kb when the temperature is cooled through the Tc of the SC2 phase, indicating a triplet state and with no change in spin orientation upon entering the SC2 phase, as shown Fig. 1.14D. This is direct evidence of the spin-triplet pairing structure for the SC2 phase in UTe2. Fig. 1.15 Left panel: Resistivity of UTe2 versus field applied near b-axis at different tem- peratures. Field re-entrant superconducting phase could be seen in higher temperature ρ(H) curve. Right panel: Temperature - field phase diagram of UTe2 with field applied along different angle rotated away from b-axis. Reproduced with permission from [42]. 22 Introduction The pairing of SC2 is likely due to the strong magnetic fluctuation in H//b. Evidence for strong spin fluctuations has been reported from measurements of NMR and magnetization as shown in Fig. 1.16. In the left panel, as the magnetic field increases, the A-coefficient and 1/T1, 1/T2 increase and show the signature of divergence. Here, the A-coefficient is the prefactor in the Fermi liquid fitting ρ = AT 2 +ρ0, T1 is the spin-lattice relaxation time scale, and T2 is the spin-spin relaxation time scale. The NMR measurement from ref. [92] reports that the relaxation timescale was too short such that their measurement could not measure it properly, indicating the divergence of 1/T1, 1/T2. The A-coefficient is closely related to the effective mass of the electrons, and the NMR relaxation time scale T1 and T2 measures the spin fluctuation. These measurements illustrate that the spin fluctuation is getting more pronounce as the magnetic field along the b-axis increases. This observation is corroborated by the observation of an increase in the entropy as the field increases. The entropy is inferred by measuring the magnetization M as a function of temperature under different constant magnetic fields. Following the Maxwell relation as in Eq. 1.1, (∂M ∂T )H = ( ∂S ∂H )T . The entropy could be calculated by integrating the ∂M/∂T over the magnetic field H. Fig. 1.16 Left panel: the field dependent A-coefficient (ρ = AT 2 + ρ0, from ref. [42]) compare with the nuclear spin-lattice relaxation rate (1/T1) and nuclear spin-spin relaxation rate (1/T2). Inset: Schematic superconducting phase diagram for H//b and H < 34 T. Right panel: Colour map of the magnetic entropy increment ∆S of UTe2 in temperature - field phase diagram with H//b. Solid line marks the maximum of ∆S(T ) as a function of temperature. CEP denotes critical endpoint of metamagnetic transition. Figures reproduced with permission from ref. [92, 93]. Copyright (2023, 2024) by the American Physical Society. As the field along the b-axis increases to above 34 T, a metamagnetic transition occurs. Fig. 1.17 presents the observation of the metamagnetic transition, measured using the pulsed field extraction magnetization technique and resistivity at different temperatures. For M vs H at low temperature, M grows linearly to 0.4 µB (Bohr magneton) as the field sweeps 1.4 Uranium Ditelluride 23 up to 34 T. Then a jump in magnetization of 0.5 µB size occurs when UTe2 enters the polarised paramagnetic state. The magnetization continues to grow with the field after the metamagnetic transition, indicating that the polarised moment is not saturated. Magnetization did not show signs of saturation even up to 45 T. As the temperature increases and remains isothermal during the field sweep, the MM transition increases in width and decreases in height, indicating that the temperature is approaching the critical endpoint. A B Fig. 1.17 Experimental signature of metamagnetic transitions with HM = 34 T with field along hard b-axis. (A) Magnetization as a function of field along b at incremental temperatures. Reproduced with permission from [94]. (B) Resistivity of UTe2 versus magnetic field with H//b at different temperatures. Arrows indicate the location of metamagnetic transitions at corresponding temperature. Reproduced with permission from [42]. In terms of resistivity measurements, there is a pronounced jump in resistivity at HM = 34 T for H//b. As the temperature increases, the first-order MM transition softens and becomes a continuous transition, showing a maximum resistivity when the temperature is above the critical endpoint TCEP = 9 K. The ρmax moves to a lower magnetic field as the temperature increases. In this dissertation, the three distinct superconducting phases are denoted as: • SC1: The superconducting phase at zero field and persisting up to intermediate magnetic fields (µ0H < 20 T) • SC2: The field-reinforced superconducting phase, which occurs when the magnetic field is applied near the b-axis. • SC3: The field-induced superconducting phase, which appears at high magnetic fields (µ0H > 40 T) when the magnetic field is tilted away from the b-axis towards the c-axis. When the magnetic field is rotated away from the b-axis, a diverse superconducting field- angle phase diagram emerges [88]. Fig. 1.18 presents the superconducting phase diagrams 24 Introduction Fig. 1.18 Top: Field - angle superconducting phase diagram of UTe2 in b− c and b− a rotation planes. Three distinct superconducting phases are observed. Polarised paramagnetic state above the MM transition is marked by the red region in the phase diagram. Bottom: contacted and contactless resistivity measurement of UTe2 at incremental rotation angles θ in b− c plane. Both type of measurements resolved field-induced SC3 phase when field is tilted away from b-axis. Figure adapted from ref. [88] and reproduced with permission of Springer Nature. mapped by contacted and contactless resistivity measurements with field tilt in two orthogonal rotation planes: from b to c and from b to a initially reported by ref. [88]. For µ0H < 15 T, the ambient superconducting phase SC1 shows an anisotropic upper critical field as a function of the rotation angle. For rotation angles within 10° in both b− c and b−a rotations, the field-reinforced SC2 phase is present. As discussed earlier, field rotation away from the b-axis would suppress the critical temperature of the SC2 phase. This superconducting phase vanishes for the rotation angle θb−c > 10° [95] and θb−a > 5° [88]. 1.4 Uranium Ditelluride 25 For the MM transition, the lowest value of HM occurs with H//b. HM increases when the field is rotated away [88]. For rotation in the b− c plane, HM cos(θ)≈ 34 T for the highest rotation angle, indicating that the amount of Hb remains unchanged in the b− c rotation plane, and the transverse field component Hc does not affect HM for the b-axis. For the b−a rotation, HM grows faster than 1/cos(φ), where φ denotes the rotation angle from b towards a [96]. When the magnetic field is rotated by an intermediate amount of rotation angle (20° < θ < 40°) in the b− c rotation plane at a high magnetic field of µ0H > 40 T, a distinctive field-induced superconducting phase, denoted SC3, emerges and was observed to persist up to at least 60 T for the optimal field orientation of (θ = 35°) [97]. The upper critical field shows a dome shape as a function of θ [88, 97]. The SC3 phase shows great stability against impurities. Ref. [98] presented the result of a poor quality sample, showing an extremely low residual resistivity ratio (RRR) of 3. Although the SC1 phase of the sample was fully suppressed, the field-induced SC3 phase emerges for µ0H > 45 T and persists up to 52 T and has a critical temperature of approximately 1 K when the field is along θb−c = 35°. Fig. 1.19 Left panel: Illustration of UTe2 (H-θbc-θa) superconducting phase diagram. Three distinct superconducting phases are shown in blue. Right panel: High field H −θa phase diagram at different θbc shows the slides of the superconducting halo. Figure adapted from ref. [96] and reproduced with permission of AAAS. When the magnetic field rotates away from the b− c plane toward the a-axis with an offset angle, the SC3 phase extends in the θa rotation for ≈ 15°. The phase diagram shows a 26 Introduction halo-like shape in the (H-θbc-θa) phase diagram as shown in Fig. 1.19. The entrance field, when zero resistivity is reached, for the SC3 phase as a function of θb−c and θa can be modeled as: Hm = Hb m cos(θbc) +α2 sin2 θa +α4 sin4 θa (1.2) 1.4.6 Phase diagram of UTe2 under pressure In previous sections, the basic properties of superconductivity and the high magnetic field phase diagram of UTe2 at ambient pressure were presented. In this section, the pressure dependence of these properties of UTe2 is discussed. Fig. 1.20 Temperature - pressure phase diagram of UTe2 measured by heat capacity (solid symbols) and resistivity (open symbols). Two distinct superconducting phases emerge in the T -p phase diagram, denoted as SC1 and SC2. As pressure increases, critical temperature of SC1 decreases while critical temperature of SC2 increases. Both superconducting phases disappears pressure goes above the critical pressure pc ≈ 1.5 GPa. Signatures of antiferro- magnetic phases (MO, magnetic ordered phase) were observed when p > pc. Left panel: Figure adapted from ref. [99] and reproduced under a Creative Commons license. Right panel: Figure adapted from ref. [66] and reproduced with permission of AAAS. To start with, the zero-field superconducting phase diagram mapped by heat capacity under pressure is presented in Fig. 1.20. The major feature is that as the pressure increases, the Tc of the ambient pressure zero field SC1 phase has been suppressed. When pressure p > p∗ = 0.3 GPa is applied, a pressure induced superconducting SC2 phase appears. In Sec. 3.5, a discussion of the pressure-induced and field-induced SC2 phase will be made. Its Tc shows a dome-like feature as a function of applied pressure and has the highest Tc = 3 K 1.4 Uranium Ditelluride 27 with p = 1.2 GPa [99, 66]. Since the measurements were conducted with low quality CVT- grown UTe2 samples showing multiple heat capacity jumps even when no pressure was applied [64], the pressure-induced superconducting phase was interpreted as originating from one of the ambient superconducting phases [66]. Later studies showed that the two heat capacity jumps were artifacts, and high-quality samples exhibit a single component and a single superconducting transition. The pressure-dependent superconducting phase diagram with high-quality MSF-grown UTe2 displays a similar phase diagram and will be discussed in the results of this dissertation. Therefore, the pressure-induced superconducting phase should be interpreted as a separate phase different from the ambient pressure SC1 phase. This imposes another problem: at the bifurcation point p = 0.3 GPa, there are three second-order phase transitions joining at one point, which is not allowed by thermodynamic arguments [100]. One possibility is another hidden phase boundary that separates the superconducting phases for p < 0.3 GPa and p < 0.3 GPa at low temperatures. Both superconducting phases terminate at critical pressure pc = 1.5 GPa. The magnetic ordered phase (MO) emerges for pressures slightly higher than pc, as evidenced by heat capacity measurements. The MO phase is an antiferromagnetic phase, as inferred from the broad specific heat transition and the inflection in resistivity - temperature curve [68, 66]. Linear-in-T resistivity was observed in the vicinity of the critical pressure [66]. However, a recent study about the pressure dependence of quantum interference oscillations showed no signature of Fermi surface reconstruction in the high field paramagnetic state across the critical pressure [101]. As shown in Fig. 1.21, the NMR experiment of the pressure-induced SC2 shows no change in Knight shift after cooling through the Tc. For lower temperatures, a 6% decrease in ∆Kb was observed. The drop in ∆Kb agrees quantitatively with the superconducting transition of the ambient pressure SC1 phase. This indicates that the two superconducting phases under pressure have the same ∆Kb when compared to the SC1 phase with zero field of ambient pressure and the SC2 phase induced by the field. The focus now turns to the (p, H, T ) superconducting phase diagram of UTe2. Fig. 1.22 shows the pressure evolution of the temperature-field superconducting phase diagram with the field applied along the b-axis mapped by heat capacity [103]. When pressure increased, Tc and Hc2 of the SC1 phase were suppressed. Moreover, as pressure increased, the lowest field in which the SC2 superconducting transition emerged moved lower on the field axis and reached zero above p ≈ 0.2 GPa. The evolution of the SC2 phase as a function of pressure illustrates that the pressure induced superconducting phase occurring at higher Tc above p∗ ≈ 0.2 GPa is actually connected with the field-induced SC2 phase at ambient pressure in a magnetic field. Therefore, the NMR Knight shift and the superconducting phase diagram 28 Introduction Fig. 1.21 NMR Knight shift study of different superconducting phases of UTe2. (A) Field- temperature superconducting phase diagram at ambient pressure and when p = 1.2 GPa is applied. Blue, orange, and red arrows shows the three temperature sweeps in panel (B), (C) and (D). (B) Knight shift ∆K as a function of temperature when small amount of magnetic field is applied. This measures the change in ∆K between SC1 phase and the normal state. A 6% change of ∆K is observed approaching 0 K. (C) ∆K(T ) curve of field-induced SC2 transition is shown when 24 T of field along b-axis is applied. No change in ∆K(T ) is observed. (D) Temperature sweep of ∆K under 1.2 GPa of pressure. When temperature is cooled through the higher Tc, no change in ∆K(T ) is observed. When cooling through the lower Tc, a 6% change of ∆K is observed. Figure adapted from ref. [102] and reproduced under a Creative Commons license. under pressure indicate that the SC2 phase in the field has a similar origin and shares many superconducting properties. As discussed previously, there appear to be three second-order superconducting transitions in the phase diagram meeting at a point, which are not allowed by thermodynamics [100]. Ref. [103] gives another possible answer to this problem. The data presented in Fig. 1.22 near p∗ indicate that the heat capacity jump for the SC2 phase is close to vanishing. Thus, the phase boundary of SC2 in the p−T phase diagram could be a third-order phase boundary, which is allowed to meet two other second-order phase transitions. 1.5 Outline of this thesis 29 Fig. 1.22 Pressure dependent field - temperature phase diagram measured by heat capacity. As the pressure increases, the re-entrant magnetic field of SC2 phase decreases as pressure increases and reaches zero when p = 0.2 GPa is applied. Figure reproduced with permission from ref. [103]. Copyright (2025) by the American Physical Society. Finally, the pressure dependence of the high magnetic field phase diagram is discussed. In Fig. 1.23, the p−H phase diagram for the field along b− c θ = 35° is shown in the top panel. As the magnetic field is rotated away from the b-axis, the field-induced SC2 phase is no longer observed. The metamagnetic transition field and the field-induced SC3 entering field HM are suppressed by the increasing applied pressure p. For p > 1.1 GPa, the polarised paramagnetic state and the SC3 phase are disconnected, and the SC3 phase occurs at higher magnetic fields above the metamagnetic transition. The field-induced SC3 phase survives above the zero field critical pressure pc = 1.5 GPa. The phase diagram is constructed using contactless and contactless resistivity measurements of ρ(H) at different temperatures and applied pressures. 1.5 Outline of this thesis This dissertation is structured as follows. Chapter 2 introduces the details of the methods that was applied during my PhD study. Methodologies of sample selections and measurement techniques for high magnetic field are elaborated. In chapter 3, experimental results of MSF-grown ultraclean UTe2 in low magnetic field (< 35 T) and under pressure are discussed. Chapter 4 reveals a discovery of an enhanced dimensional quantum critical line that terminates 30 Introduction Fig. 1.23 Pressure dependent field-temperature phase diagram of CVT-grown UTe2 up to 45 T. Top: Schematic field-pressure phase diagram with H rotated by θ = 35° from b-axis towards c-axis. FP means “field-polarised” phase, and HFP is the metamagnetic transition field. SCPM1 and SCPM2 correspond to SC1 and SC2 phases (see Fig. 1.22). M stands for magnetic ordered phase. Bottom: Evolution of field-temperature phase diagram as pressure increases for magnetic field rotated. The colormap was measured by contactless and contacted resistivity and shown by the colour contour. Metamagnetic transition field HM decreases as pressure increases. Figures adapted from ref. [104] and reproduced under a Creative Commons license. the metamagnetic transition surface and encloses the toroidal SC3 phase in three-dimensional field space. Moreover, temperature and angle dependent study shows that the Tc of the field-induced SC3 phase is higher than the Tc of zero field SC1 phase, indicating the nature 1.5 Outline of this thesis 31 of magnetic fluctuation mediated mechanism of SC3. Chapter 5 gives a summary of this dissertation. Chapter 2 Methods 2.1 Sample preparation This section describes the methods to grow, select and prepare high quality UTe2 samples. After a brief introduction of the new growth method, important methods for characterising the quality of UTe2 single crystals are introduced. 2.1.1 Growth methods With the recent development of an innovative growth method using a molten-salt flux (MSF) [67], the sample quality of UTe2 has been largely enhanced. In this dissertation, all the UTe2 single crystals were grown using the molten salt flux technique by Dr. Michal Valiŝka’s group from Charles University, Czech Republic. An equimolar mixture of powdered NaCl and KCl salts was used as a flux, which had been dried at 200°C for 24 hours. Non- radioactive depleted uranium metal, having an initial purity of 99.9% was further refined by the solid-state electrochemical method under ultra-high vacuum (10−11 mbar) to prevent oxide formation. Following the purification process, a piece of uranium with a typical mass m ≈ 0.35 g was etched using nitric acid to remove the surface oxide. Then it was placed in a carbon crucible of inner diameter 13 mm together with pieces of tellurium (99.9999%) at a molar ratio of 1:1.71. Subsequently, the equimolar mixture of NaCl and KCl was added. The molar ratio of uranium to NaCl, KCl mixture was 1:60. This process was conducted in a glovebox filled with an argon protective gas atmosphere to avoid oxide formation. The carbon crucible was plugged by quartz wool, placed in a quartz tube, and heated to 200°C under dynamic high vacuum for 12 hours. Later, the quartz tube was sealed and placed in the furnace. It was initially heated to 450°C in 24 hours. Then it was heated to 950 °C at 0.35°C/min rate and stayed for another 24 hours. Finally, it cooled down to room temperature 34 Methods after removing the bar-shaped single crystals of UTe2 from the ampoules, samples were rinsed with acetone, and stored under protective argon atmosphere [67, 78]. Samples were then characterised by the methods described in the following section. 2.1.2 Sample Selection In this section, the methodology of sample selection will be described. The purpose of the sample selection is threefold: (i) verifying the superconductivity in the samples; (ii) probing the crystallographic orientation; and (iii) selecting the highest quality samples for further measurements, including magnetic quantum oscillations. The main methods of characterising the sample quality are measuring magnetic moment and heat capacity versus temperature. In order to quickly screen the samples of a growth batch typically containing 40-50 samples of adequate sizes (≈ 2 mm× 0.5 mm× 0.3 mm), I usually carry out the magnetic moment measurement with a Quantum Design Magnetic Property Measurement System (MPMS) with a built-in superconducting-quantum-interference-device (SQUID), which could pick up the change in magnetic field flux to highest sensitivity. Due to its short bore size, it could cool to 1.8 K with a rate up to 35 K/min, thus the samples could be screened in a short period of time. The working mechanism is shown in Fig. 2.1. The single crystal sample is secured on a low magnetic background quartz tube and moves across the SQUID pick-up coil by a driving motor. The magnetic moment is inferred by fitting the captured flux - location curve. As shown in Fig. 2.1 right panel, the normal state susceptibility χ(T ) of UTe2 below 50 K shows a positive dM/dT only when the field is applied along the b-axis, as discussed in Sec. 1.4.3. This feature is utilized to verify the crystal orientation. At lower temperature, the sample is first cooled below the superconducting Tc in zero field and then a small field is applied to tune the sample to the diamagnetic state. The sample Tc is determined by the first data point where the field warming curve depart from the zero field cooling curve. After quickly evaluating the critical temperature and orientation of the samples, the heat capacity of the sample was measured to obtain the bulk superconductivity property of the UTe2 samples. Heat capacity is measured using a Quantum Design Physical Property Measurement System (PPMS) heat capacity module. The sample is secured on the heat capacity puck with N-grease to obtain a good thermal link between the thermometer and the sample at low temperature. addendum measurement were performed before the sample measurement to subtract the background heat capacity contribution from the N-grease. In Fig. 2.2, example data of several different samples are displayed. The Tc is determined by the mid-point of the superconducting transition. Higher Tc indicates higher sample quality since impurities / dislocations scatter electrons that would otherwise form superconducting Cooper 2.1 Sample preparation 35 Magnet SQUID Sample z z M 0 100 200 300 T (K) 0 100 200 300 400 500 M om en t ( em u/ m ol ) H // a H // c H // b 0H = 1 T 1.8 2 2.2 T (K) -80 -40 0 M om en t ( em u/ m ol ) 0H = 10 Oe Tc = 2.05 K Fig. 2.1 Left: a scheme of operation methodology of superconducting-quantum-interference- device (SQUID). Sample moves through the SQUID coil and the change of magnetic flux could be captured. Left hand side curve is a data curve indicating the moment versus location z. Right: example data of a 300 K to 3 K moment versus temperature curve with 1 T field applied. Inset: example of diamagnetic transition of UTe2 sample with 10 Oe field applied. pairs. The width of the superconducting transition is taken to evaluate the inhomogeneity of the sample. In addition to the critical temperature Tc measured by magnetization and heat capacity, another important parameter that could indicate the quality of the sample is the “residual resistivity ratio” (RRR). The RRR is defined by ρ(300 K)/ρ(0 K). For a normal metal, as the temperature cools, scattering of conduction electrons, which induce the impedance, would reduce. When the temperature reaches zero kelvin, the only origin of the residual resistivity (ρ0 = ρ(0 K)) is the scattering between conduction electrons and the defects in the single crystal. Therefore, the lower the residual resistivity, the better the sample quality [105]. However, determining ρ0 value in absolute units requires the determination of sample dimensions, which gives a large error bar to the measured value of ρ0 between different samples. When the temperature is at room temperature T ≈ 300 K, the origin of the resistivity is dominated by scattering between electrons and phonons, which does not depend on the quality of the sample. As the sample dimension does not have much temperature dependence, the RRR is thus dimensionless and reflects the sample quality. Therefore, in practice, the RRR is calculated by R(300 K)/R(0 K), where R is the resistance of the sample. And R(0 K) is extrapolated by fitting the measured R(T ) curve with quadratic form R = R0 +AT 2 from 2.5 K to 5 K. When RRR of a sample measured by applying the current J along a-axis is larger than 100, the sample would be able to probe the quantum oscillation. This criterion is set to select the high-quality samples. 36 Methods 1.8 2 2.2 2.4 T (K) 0.1 0.2 0.3 0.4 C ( J m ol -1 K -2 ) UTe2 Thermometer Heater µ Fig. 2.2 Example data of heat capacity as a function of temperature comparing multiple UTe2 samples showing clear superconducting transitions. Inset: schematic DC heat capacity measurement techniques. 2.2 Moissanite anvil cell Part of the my PhD research involved applying hydrostatic pressure to the sample and measuring the sample with the AC magnetic susceptibility measurement. In this section, the tool for applying pressure to the sample is introduced. The pressure is obtained by a moissanite anvil cell. It is a setup consisting of two concentric opposing anvils that have a small space between. By filling the space between the anvils with pressure transmitting medium (Daphne oil in this study) and squeezing the two anvils, pressure could be obtained. As shown in Fig. 2.3(A), the diameter of the region, which is within the outer coil, is less than 1 mm. This shows that the cross sectional area between the anvils is extremely small, resulting in a large pressure with a small amount of force. As the sample is in liquid pressure transmitting medium in the pressure range p < 1.5 GPa, which is the range at work for this thesis, the sample is applied to a hydrostatic pressure. A 2.3 DC field measurement techniques 37 A B 1 mm Fig. 2.3 (A) Photo of moissanite anvil cell with AC magnetic susceptibility measurement setup. The black line indicates the scale of 1 mm length. (B) Scheme of a working moissanite / diamond anvil cell with the gasket, pressure medium, pressure marker, and the opposing anvils. Figure reproduced with permission from ref. [106]. Copyright (2018) by the American Physical Society. ruby is placed inside the pressurized region to serve as a pressure marker. The frequency of the fluorescence of ruby being measured at room temperature each time before the pressure cell cools down. As the frequency would shift due to the hydrostatic pressure, the frequency after pressing would serve as an indicator of the sample pressure [106, 107]. 2.3 DC field measurement techniques Several experiments in this dissertation were based on electrical transport. In this section, I will describe the methods for making electrical contacts and the basic aspects of measuring electrical transport behaviour. 2.3.1 Spot welding Due to the complicated surface chemistry of UTe2, the typical way of making electrical contact with silver paste is not feasible with UTe2. Therefore, the way of making electrical contacts on UTe2 is via polishing the surface to remove any oxide layer and then carry out spot welding. The bottom side of the sample requires fine polish to obtain good contact between samples and the copper plate as shown in Fig. 2.4. A photo of the sample that has electrical contact is shown by spot welding. During the spot welding process, the needle would press the gold wire on the sample and press the sample onto the conductive copper plate to form a closed circuit. A multimeter is usually connected in series in the circuit to observe the resistance between the needle / gold wire and 38 Methods the sample. After the resistance in the circuit of the multimeter is below 5 Ω, I would apply a pulse of voltage of around 5 - 7 volts. Such a pulse would melt the gold into the sample and thus form good electrical contacts. After the first pulse, I would usually move the needle by several microns with the micro manipulator and apply another pulse to form a stronger contact. After the spot welding contact, Devcon epoxy was applied to mechanically secure the contact. Copper Board Sample Sample Gold wire NeedleA B Gold wireCopper BoardCopper plate Copper plate Fig. 2.4 (A) Scheme of spot welding method of making electrical contact with low contact resistance. (B) Example of a sample that has spot weld contacts in a four-probe fashion on the spot weld station. 2.3.2 Electrical transport The continuous electrical transport resistivity measurement in low field / DC field is done by using lock-in amplifier. The lock-in amplifier is an instrument that extracts tiny signals of a certain frequency from a noisy background. If the system is excited with a DC current and the resistance is inferred by measuring the resulting voltage, then the whole noise spectrum is picked up. When using a lock-in amplifier, the sample is usually excited with a sinusoidal waveform, and the response voltage of the same frequency is measured. The lock-in amplifier extracts the weak response signal from the noisy background by phase-sensitive detection. The instrument multiplies the measured voltage by a reference signal which is of the same frequency as the excitation. This multiplication shifts the component of the input signal that matches the reference frequency to zero frequency, while all other frequency components (noise) are shifted to higher frequencies. The resulted product signal is then passed through a low-pass filter and integrated over a period known as time constant. During this integration, the DC component, which corresponds to the signal at the reference frequency and phase, is retained, whereas the contributions from noise average to zero, in accordance with the 2.3 DC field measurement techniques 39 Fourier principle. Consequently, the output of the lock-in amplifier represents the amplitude of the signal that is coherent with the excitation. Typically, for measurement in steady field magnets, the frequency of measurement is selected to be a prime number in a range between 10 Hz to 1 kHz. Depending on the country, the frequency of the power line cycle should be avoided. 2.3.3 Current-voltage measurement Current-voltage (IV) measurement is a useful tool for extracting information about supercon- ductivity, such as superconducting critical currents. However, performing such measurements could be challenging especially in high magnetic field and low temperature due to several reasons. Firstly, according to the Biot-Savart law [108], F = µ0HIL, where F is the exerted form, H is the magnetic field intensity, I is the applied current, and L is the length of the wire, applying 30 mA to 1 mm long gold wire at 30 T would exert a force equivalent to weight of 10 g, e.g., a AAA battery or a tea bag. This force could break the 25 µm thick golden wire or the contact with the spot weld on the sample. Secondly, for a typical contact resistance of 1Ω, 30 mA of current would give a heating power of 1 mW, which could surpass the cooling power of the cryostat, especially at dilution fridge temperature, resulting in warming and, thus, less trustworthy data. To address these problems, measurements were performed using the pulse delta sweep mode of the Keithley 6221 - 2182A system. In Pulse mode, the excitation current is applied only during a small fraction of the power line cycle, thereby minimizing overall Joule heating. The delta mode eliminates thermal electromotive force contributions at low current levels by performing voltage measurements before and during the current pulse. For additional information, see the Keithley 6221 current source manual [109]. In all of the current-voltage measurements, a pulse delta sweep with 5 ms of pulse width was applied. Each data point takes 1 second to measure and communicate with the computer, thus, the duty cycle of the pulse is 0.5%. The multimeter is controlled by a python code based on pymeasure package [110]. The impedance of the wires in the dilution fridge are usually 200 Ω, thus the compliance voltage of the Keithley 6221 is changed correspondingly to successfully source the current pulse into the sample. For the variable tuning study, we define the shape of the excitation pulse using the pymeasure code with the “arbitrary waveform generator” feature of the Keithley 6221 and check the waveform by sourcing the waveform to a 1 kΩ test resistor and trigger a snapshot of the voltage with a digital oscilloscope. 40 Methods Current Measure Measure 1st Cycle 2nd Cycle 3rd Cycle A B Fig. 2.5 Scheme of pulse delta mode measurement. The horizontal axis is the time and the vertical axis is the applied current. Arrows indicate where the measurement of the multimeter is taken place. 2.3.4 AC magnetic susceptibility Magnetic susceptibility is a useful probe of superconductivity and magnetic structure. The spin of the electrons would align in an external magnetic field and give rise to magnetization M. Magnetic susceptibility χ is defined as the response of the net magnetic moment M to the applied field H: χ = ( ∂M ∂H )∣∣∣∣ T . (2.1) In the previous Sec. 2.1.2, I described the measurement of the magnetic moment with a quantum design MPMS system. The coil sets that are applied to measure AC magnetic susceptibility will be introduced. The measurement was carried out with lock-in. The magnetic field inside the material is the sum of the dc magnetic field and the AC magnetic field: H = Hdc +Hac cosωt (2.2) 2.3 DC field measurement techniques 41 The susceptibility is defined by χ = ∂M/∂H, where M is the magnetization of the material. Since the magnetic field is small, the induced magnetization could be written as M = χH. Therefore, the AC magnetic susceptibility is the quadrature term of the AC signal, χ ∝ Hac sinωt (2.3) Thus, the out-of-phase component in the lock-in measurement would be proportional to the magnetic susceptibility. The typical setup for measuring AC magnetic susceptibility is shown in Fig. 2.6. It usually consists of three coils, with a large background coil as the “drive” coil that gives the oscillatory AC field and two smaller coils inside that serve as the “pick-up“ coil and the “compensation” coil. The compensation coil has the same cross sectional area as the pick up coil and is in the opposite polarity to the pick up coil. It aims to balance the induced voltage from the pick-up dH/dt which would interfere with the signal. In order to obtain good compensation, we usually turn on oscillatory excitation in the drive coil, and add or decrease the number of turns in the compensation coils to obtain as small pick-up voltage as possible at room temperature. Such a setup could be combined with pressure cell and high magnetic field. Depending on the applied environment, the specification would vary. In Fig. 2.6C and D, I show the ACMS coil setup for the ambient pressure high magnetic field rotation measurement and high pressure measurement correspondingly. For the ambient pressure measurement, the drive coil usually has a diameter of several millimeters and up to 2000 turns. The pick up coil and compensation coil are both inside the drive coil and have a similar number of turns. The sample would try to fill the pick-up coil in order to increase the filling factor of the coil. For setup under pressure, due to the limited space inside the anvils, the pick-up coil is usually 300 µm in diameter and would be 8-10 turns. The drive coil would be outside the anvil, which is 2.5 mm in diameter and would have 130 turns. This would give a magnetic field of 0.3 mT/A at the center of the drive coil [107]. The sample would usually be cut out with a diamond scribe to obtain a sufficiently small specimen, whose diameter would be around the same of the pick-up coil. Since the measurement in the pressure cell does not have a varying field, the compensation coil was ignored to save space. The space inside the coil would help to hold the pressure medium, such as glycerol or Daphene oil, which would help to obtain a hydrostatic pressure. We usually drive the signal with a frequency of 0.8 kHz to 5 kHz. The signal usually depends positively on the driving frequency. The excitation amplitude is usually between 2 mA to 8 mA. In Fig. 2.6B, an example data curve of AC-magnetic susceptibility χ as a function of temperature is shown to measure the superconductivity. The kink around 7 K is the superconducting transition of Sn in the solder joints that connect the coil and the signal 42 Methods wires. The two drops in susceptibility are the superconducting transitions of UTe2 under pressure, which will be discussed in Sec. 3.4. DC Magnet Drive Pick-up & Compensation 1 mm Pick-upDrive DC magnetic field Drive Compensation Pick-up Sample Sample A B C D Fig. 2.6 Schematic illustration of experimental setup of AC magnetic susceptibility (ACMS) measurement. (A) Schematic of the ACMS measurement, where the big coil is the drive coil and two small coils inside are the pick-up coil and the compensation coils correspondingly. (B) Example data of ACMS measuring superconducting transition. (C) Photo of ACMS setup for ambient pressure, high magnetic field rotation. (D) Photo of ACMS setup for ACMS under pressure. 2.4 Pulsed field measurement techniques In order to obtain a higher magnetic field to probe exotic physics, such as the field-induced superconducting phase SC3 above 40 T in UTe2, we have to utilize international user facilities. Usually, the magnet field strength is limited by the heat produced by the current in the giant resistive coil. Usually for a continuous operating magnet, the heat is taken away via the 2.4 Pulsed field measurement techniques 43 water cooling system. The state-of-the-art steady field magnet is a hybrid magnet up to 45 T combining a 11 T superconducting magnet and a 34 T water cool resistive magnet at the National High Magnetic Field Lab (NHMFL), Tallahassee, USA. The amount of heat is proportional to the quadratic energy of the magnetic field. Thus, ramping to an even higher field with a limited amount of cooling power would result in melting the magnet itself. Therefore, pulse magnet ramps to higher field in a transient fashion, and sit at zero field state for much longer time to cool down. The amount of peak field depends on the maximum voltage of the capacitor bank. Instead of connecting into a power supply that would ramp a resistive magnet or a superconducting magnet, when running the pulsed magnet, energy is first charged into the large capacitor bank and then discharge into the pulsed magnet within several hundreds of milliseconds. Here, the magnetic field µ0H as a function of time t profile is shown in Fig. 2.7 at their highest peak field available. The magnet in Wuhan High Magnetic Field Center (WHMFC) could be ramped to 60 T, with a full-width half maximum (FWHM) equal to 30 ms, and the High Field Lab Dresden (HLD) has a maximum field of 70 T and 50 ms of FWHM. The different pulse widths are due to the design of the coil and the number of capacitor banks used to generate the pulse. On the one hand, a longer pulse would give a smaller dH/dt and thus less heating from the eddy current. Furthermore, a long pulse would give more time to collect data, which could result in a lower frequency required and more data points collected, which would be crucial to optimize the data. However, a longer pulse width and a higher magnetic field lead to a larger energy density. Thus, the 70 T wide pulsed magnet has a much longer cooling time compared to the 60 T, requires 3 hours to cool down before performing the following pulse, while the cool down time for the 60 T magnet at Wuhan is 1 hour. Due to the very short period of time for a pulse, a quick measurement acquisition rate is usually needed. For contacted electrical transport measurements, tens of kHz frequency is usually applied and a mega sample per second sampling rate is usually used with a high-speed oscilloscope. In order to obtain the magnetic field as a function of time, a pick-up coil is used to measure the induced voltage from the magnetic flux during the magnetic field pulse. Fig. 2.8 presents the way to obtain the magnetic field during the pulse. As shown in the photo, the pick-up coil is a many turn coil that is secured in a slot at the bottom of the probe. Usually, the pick-up coil should be within 10 mm of the sample, which is usually the homogeneous region of the magnet. In Fig. 2.8B, the induced voltage from the pick-up coil is measured with a high-pass filter and then measures the voltage difference by a two-point method. This measured voltage is proportional to the dH/dt of the Faraday’s Law [111]. We then integrate 44 Methods 0 50 100 150 Time (ms) 0 20 40 60 80 0H ( T ) WHMFC 60 T HLD 70 T HLD 80 T Fig. 2.7 Magnetic field as a function of time in the three main pulsed magnets of this study. The 60 T and 70 T magnets are single coil magnet while the 80 T magnet is a dual coil setup. the pick-up voltage and get the raw integrated voltage as a function of time as shown in (B). The resulted curve is usually tilted due to the DC voltage offset in the circuit. By fitting the linear background before the start of the pulse, we can subtract background and divide the voltage by the field factor and get the magnetic field - time curve. The field factor is defined by averaging multiple low-energy (usually 2 kV) pulses and corrected by the designed value of the pulses. 2.4.1 Transport measurement Resistivity is a powerful probe of exotic quantum behaviour such as superconductivity and magnetic transitions. In previous sections, I have introduced methods for measuring the resistivity via AC lock-in amplifier or dc delta mode in a steady magnetic field environment. In order to extend the capability to measure the resistivity up to higher magnetic fields, in pulsed magnet, due to the limit of the very short period of time, large dH/dt background and electrically and mechanically noisy (vibrating) environment, we have to use high frequency numerical lock-in measurement techniques. Usually a frequency between 7 kHz and 70 kHz is applied to accumulate enough data points to resolve the ρ(H) behaviour due to the short period of the pulse. While a higher excitation frequency could help get more data points, and 2.4 Pulsed field measurement techniques 45 0 100 200 Time (ms) 0 20 40 60 B (T ) 0 100 200 Time (ms) 0 0.1 0.2 0.3 In te gr at ed P ic k- up (V ) 0 100 200 Time (ms) 0 2 4 6 pi ck -u p (m V) A B C Pick-up Coil Background 0H (T ) Fig. 2.8 Procedure of obtaining the correct field - time profile. (A) Raw voltage versus time profile measured by the pick-up coil due to the flux change in the coil. Inset: photo of a pick-up coil in a pulsed field rotator probe. (B) The raw integrated voltage versus time. (C) The magnetic field versus time curve after subtracting the background and dividing by the field factor specific to this system. thus help smoothing the data curve, it could induce a large out-of-phase quadrature voltage and introduce artifacts into the signal. Therefore, in general, a frequency of the order of 10 kHz is selected. During measurement, the voltage induced by the excitation is recorded by a data acquisi- tion card (DAQ) with a rate of up to 3 MS/s. The excitation amplitude is also recorded by measuring the voltage of a reference resistor with known resistance that is put in series to the sample. The frequency, phase, and amplitude of the excitation could then be inferred. Fig. 2.9 presents an example of pulsed field resistivity measurements of UTe2. In panel (A), raw excitation and the voltage signal near the metamagnetic transition are presented over a 5 ms window. The red dashed line marks the midpoint of the metamagnetic transition. It is clear that the voltage and current are well in phase and the abrupt increase in resistivity due to the metamagnetic transition could be seen in the contour of the voltage-time signal. The voltage signal convolved with the excitation signal then produces the resistance curve. 2.4.2 Radio frequency measurement Proximity detector oscillator (PDO) is a radio frequency experiment technique that measures the frequency shift in an LC circuit to infer the change in resistivity and magnetic property of the material under investigation [112]. During the measurement, the sample is mounted on a hand-wound coil as shown in the inset of Fig. 2.10B. The sample is coupled to the coil 46 Methods -1 0 1 I ( m A) -20 0 20 V (m V) 24 25 26 27 28 29 Time (ms) 40 45 50 0H (T ) 10 20 30 40 50 60 0H (T) 0.05 0.1 0.15 R (m ) UTe2 0 50 100 150 200 Time (ms) 0 20 40 60 0H (T ) A B C Fig. 2.9 Example of pulsed magnetic field contacted electrical transport measurement carried out in 65 T magnet in HLD Dresden, Germany. (A) raw signal as a function of time for a short period of time (5 ms). These signals give data points near the red arrow indicated in the resistance vs. field and field vs. time signal. The red dashed line at 26.5 ms marks the center of the metamagnetic transition in UTe2. From top panel to the bottom panel are AC-current, AC-voltage and magnetic field as a function of time. (B) Up sweep resistance signal as a function of magnetic field after digital lock-in. The red arrow marks the location as shown in panel (A) at the metamagnetic transition. (C) Magnetic field profile in time domain with red arrow indicated position for the data point in panel (A). and the coil is connected into a LC circuit with a coaxial cable. The purpose of the proximity detector is to balance the dissipation of resistance in the LC circuit. The ringing frequency of the PDO circuit is typically between 25 MHz and 35 MHz. The physical quantities measured by the PDO technique are not straightforward to interpret. In order to better understand what is measured by the PDO technique, a tank coil setup is easier for theoretical consideration. The planar coil can be viewed as a combination of many single turn tank coils. Consider a long cylinder with a radius of the coil denoted R, there is a cylindrical sample inside the coil with a radius r. When the radio frequency signal 2.4 Pulsed field measurement techniques 47 0 20 40 60 B (T ) 0 40 80 120 Time (ms) -0.4 -0.2 0 0.2 0.4 V PD O (V ) 27.79 27.794 27.798 Time (ms) -0.4 -0.2 0 0.2 0.4 V PD O (V ) 0 1 2 3 4 Frequency (MHz) 0 0.5 1 FF T (a rb .) 50.78 50.8 50.82 B (T ) 40 50 60 B (T) 1.9 2 f PD O (M Hz )A B C D 0H (T) 0H (T )0H (T ) Fig. 2.10 Example of pulsed magnetic field proximity detector oscillator (PDO) measurement carried out in a 60 T magnet at WHMFC Wuhan, China. (A) Data of magnetic field and raw PDO signal in the vicinity to the metamagnetic transition. (B) Corresponding fast Fourier transformation (FFT) of the data within a window in (A). (C) Raw magnetic field profile as a function of time, and (D) raw PDO signal throughout the period of pulse. The dashed line marks the location of the window of data in panel (A) and (B). The resulted curve of PDO frequency as a function of magnetic field is as shown in the inset of (C), where red triangle marks the location of the metamagnetic transition as indicated by the dashed line in the raw signal panel (D). is driven in the coil, it will emit an EM wave on the sample with a skin depth δ . δ = √ 2ρ ωµrµ0 (2.4) where ρ and µr = 1+χs are the resistivity and relative susceptibility of the samples, respec- tively. ω is the angular frequency of the driving signal, which is in the range of 10 MHz to 100 MHz. Assume that λ is the depth of sample that was coupled to the PDO coil. λ is of the same order as skin depth δ . For the radio frequency driven by PDO, the skin depth is about 1µm for a good metal. Thus, the skin depth is small compared to the sample radius. 48 Methods When an external magnetic field µ0H0 is applied, the magnetic flux of the empty coil is φ0 = µ0πH0R [113]. When the sample is loaded into the coil, the resulting magnetic flux φ can be written as φ = µ0πH0[(R2 − r2)+2µrrλ ] (2.5) The susceptibility and skin depth are both vulnerable to change. Therefore, the full differentiation of the magnetic flux is as follows. ∆φ = 2πrµ0H0(µr∆λ +λ∆µr) = πµrλ rµ0H0 ( 2∆λ λ + 2∆µr µr ) = φ0η λ r ( µr ∆ρ ρ +∆χs ) (2.6) where η = r2/R2 is the filling factor of the sample. Since λ ≈ δ , so that 2 lnλ = lnρ− ln µs+const. Taking the derivative on both sides of the equation, we find that 2∆λ/λ = ∆ρ/ρ − ∆µr/µr. The quantity directly measured by PDO is the change in the ringing frequency. For the LC circuit, ω = 1/LC. Therefore, ∆ f f =−∆L 2L =− ∆φ 2φ0 =−η λ d ( 2∆λ λ + 2∆µr µr ) =−η λ d ( µr ∆ρ ρ +∆χs ) (2.7) here, d = 2r is the diameter of the sample. Above all, the change in PDO frequency is ∆ f f ≈−η δ d ( µr ∆ρ ρ +∆χs ) (2.8) For a non-magnetic metal, ∆ f/ f = −η δ d ∆ρ ρ . This explains why the PDO technique is often referred to as “contactless resistivity” [13, 98]. The sensitivity of the PDO technique is better than the contacted resistivity since the voltage amplitude is much larger and due to the high ringing frequency of PDO. Typically, the contribution to the inductance can be 1% of the whole LC circuit without too much optimization. A superconducting transition would fully reduce the inductance contribution, and give a ∆ f ≈ 30 MHz × 0.01 = 300 kHz size of signal. The noise of a PDO circuit is less than 30 Hz when the setup is correct. Thus, it would give a signal-to-noise ratio of 10,000. For a typical contacted resistivity measurement, the noise is usually ≈ 20 nV, and the size of the superconducting transition is about 1 µV. This estimate gives a signal-to-noise ratio of 50. However, the drawback of the contactless resistivity technique is that it is challenging to extract a quantitative value for resistivity. For example, no zero resistivity is observed in contactless resistivity sweeping field, a constant slope line is observed instead, due to the pick-up of magnetic flux. Fig. 2.10 presents the scheme of pulsed field PDO measurement. Due to the short pulse period, to obtain enough data points, the usual limiting factor is the acquisition of the DAQ. 2.5 Uncertainty Analysis 49 In order to suppress the noise of the signal and mitigate the requirement of the sampling rate, the raw signal of f ≈ 30 MHz is typically mixed down to fPDO ≈ 2 MHz. For the measurement in Wuhan, the system was limited by the National Instrument PXIe-5922 card sampling rate of 30 MS/s. The PDO signal was amplified twice and then mixed down to 10.8 MHz with a function generator and a mixer. The signal was then passed through a band-pass filter ranging from 9.8 MHz to 11.8 MHz. The filter signal was mixed to 2 MHz and then filtered by a low-pass filter. For experiments in Dresden, the Teledyne Lecroy HDO8180A oscilloscope was applied for PDO measurements. Ultrahigh 500MS/s sample rate was applied and thus, only one mixing down to 10.8 MHz was performed. 2.5 Uncertainty Analysis This section explains the protocols of extracting the uncertainty of different types of phase transitions in later chapters. For the superconducting transitions measured by the resistivity versus temperature ρ(T ) at critical temperature Tc as shown in Fig. 2.11(A), the uncertainty of the critical temperature is estimated by fitting the ρ(T ) data points to the sigmoid function: ρ = ρn 1+ exp ( −T−Tmid ∆T ) (2.9) where, ρn is the normal state resistivity, Tmid is the fitted mid-point of the transition. The ∆T which corresponds to the width of the transition is taken as the estimate of the uncertainty for the value of Tc. The uncertainty of the lower critical field is extracted in a similar way. For the magnetic field induced superconducting transition and metamagnetic transition as shown in Fig. 2.11(B), ρ(H) data curve before, at, and after the transition are fitted with a linear function correspondingly. The range of the transition is determined by where the linear functions meet. The width of the transition, ∆H is taken as the error bar of the transition field. For the pulsed field PDO measurements, the field-induced metamagnetic transition and re-entrant superconductivity (as in Fig. 2.11(C)), the range of the transition is determined by location of the maximum and minimum of the fPDO. The error bar of the MM / SC entrant transition is the width of the transition range. To extract the upper critical field from pulsed-field PDO measured field-induced super- conducting phase, the derivative of the frequency vs. magnetic field ∂ f/∂H is calculated. A Gaussian function is fitted to derivative data curve near the transition as indicated by the local maximum. The σ of the resulted Gaussian fitting is taken as the error bar of the upper critical 50 Methods field. Similar procedure is also applied to analyze the uncertainty of the superconducting transition measured by ac magnetic susceptibility under pressure. 1.9 2 2.1 2.2 2.3 T (K) 0 1 2 3 (µ c m ) 20 30 40 0H (T) 0 20 40 60 (µ c m ) T = 0.4 K b-a = 9° 30 40 50 60 70 0H (T) 0 20 40 60 80 f PD O (k Hz ) T = 0.6 K b-c = 33° 40 50 60 0H (T) 0 0.1 0.2 0.3 0.4 0.5 f/ H (a rb .) µ µ A B C D Fig. 2.11 Uncertainty analysis for different type of data. (A) Resistivity as a function of temperature ρ(T ) measures the superconducting transition. The dashed line is the sigmoid function fitting to the transition. The function is written next to the data curve. (B) Resistivity versus magnetic field ρ(H) measures the superconducting transition and the metamagnetic transition. Dashed lines are the linear fitting of different part of the data curve. The width of the transition is marked. (C) PDO frequency is plotted as a function of magnetic field which shows the field-induced superconductivity and its upper critical field. The arrow marks the field re-entrant superconducting transition. (D) Derivative of PDO frequency over magnetic field ∂ f/∂H is shown as a function of magnetic field. The red dashed line Chapter 3 Intermediate Magnetic Field Superconducting Phase Diagrams of UTe2 This chapter presents the experimental results obtained from measurements on high-quality UTe2 single crystals grown by the molten salt flux (MSF) method. Comparison between these results and data from earlier studies on chemical vapour transport (CVT)-grown single crystals is made to highlight the effects of enhanced sample quality. The discussions focus on ambient-pressure superconductivity and field-reinforced superconductivity, that is, SC1 and SC2. This chapter contains four main aspects: (i) sample quality metrics, (ii) enhanced SC1 superconducting properties, (iii) field-induced phenomena including metamagnetic transitions and multiple superconducting phases up to 35 T, and (iv) superconducting phase diagram under pressure. 3.1 Enhanced Superconductivity in MSF-grown UTe2 3.1.1 Sample Quality In Sec. 1.1 and Sec. 1.4, the basic concepts of superconductivity and the prior studies mapping out the superconducting phases in UTe2 grown by chemical vapour transport are introduced. Here, results from the new generation of UTe2 crystals grown by the molten salt flux technique are the focus. In Fig. 3.1, an overlay comparison of resistivity on three MSF-grown samples and the original report of superconductivity in CVT-grown UTe2 [13] is presented. The MSF-grown samples exhibit an enhanced critical temperature (Tc), an increased residual resistivity ratio (RRR, as defined in Sec. 2.1.2), and a reduced residual resistivity (ρ0) compared to CVT-grown samples. From the measured resistance - temperature curve, the resistivity was obtained by assuming the ρ(300 K) = 300µΩ cm from ref. [114] to avoid the 52 Intermediate Magnetic Field Superconducting Phase Diagrams of UTe2 error from obtaining the geometry of the sample. The residual resistivity ρ0 was obtained by fitting the ρ(T ) curve between 4 K and the onset of superconducting transition with ρ = AT 2 +ρ0. In table 3.1, ρ0, RRR and Tc of MSF-grown and CVT-grown UTe2 samples are listed. In order to make a consistent comparison, all the resistivity vs. temperature curves were collected from the corresponding data repository and through digitization of the figure. Then the values of Tc, ρ0, and RRR were re-extracted in a consistent manner. A clear positive dependence between Tc and RRR could be observed. Since room-temperature resistivity is sample independent, RRR value is determined by the residual resistivity ρ0, which reflects the sample quality. Thus, a high RRR is indicative of high sample purity. Higher-purity crystals contain less disorder, which scatters the quasiparticles. The exceptional purity of the MSF-grown UTe2 samples examined in this study is further highlighted by the observation of magnetic quantum oscillations, both de Haas-van Alphen (dHvA) and quantum interference oscillation effects, at high magnetic fields and low temperatures. All measurements reported in this chapter were performed on crystals grown by the same method as described in ref. [78, 80, 67], which demonstrated high-frequency quantum oscillations, indicative of a long mean free path and, consequently, excellent crystalline quality. 3.1.2 Upper Critical Field (Hc2) As enhanced sample quality has been demonstrated, the superconducting properties of SC1 under magnetic fields are examined. As shown in Fig. 3.2, resistivity versus temperature at various different magnetic fields applied along the a, b, and c-axes were measured up to the magnetic field of 14 T and down to the temperature of 350 mK. It is worth noting that when the temperature is around 1 K, the PPMS 3He module has lower cooling power in the circulation mode. Therefore, smaller amplitude (<0.1 mA) of excitation was applied to maintain the temperature stability, which resulted in a smaller signal and a higher noise level. The critical temperature at a specific field is determined by where the ρ(T ) curve first reaches zero. The results of ρ(H) and Tc(H) with field along the three different a-, b- and c-axes are shown in Fig. 3.2. All of these data were taken from the same sample in separate rounds of cool down. The field orientation is varied by mounting the sample differently with respect to the magnetic field. The upper critical field (Hc2) values at higher magnetic field and lower temperature were measured with the contactless conductivity method as described in Sec. 2.4.2 in NHMFL, USA. In Fig. 3.3, skin depth as a function of magnetic field measured by PDO techniques is presented. It should be noted that the curve of 0.1 K is measured in SCM4 superconducting 3.1 Enhanced Superconductivity in MSF-grown UTe2 53 0 1 2 3 4 T (K) 0 4 8 12 16 20 24 (µ c m ) 0 = (0.48 0.02) µ cm RRR = 904 20 Tc = (2.10 0.01) K RRR = 406 10 Tc = (2.08 0.02) K RRR = 105 7 Tc = (2.02 0.02) K CVT RRR = 40 5 Ran et al. [13] Tc = (1.4 0.01) K µ µ Fig. 3.1 Comparison of superconducting transition measured by electrical resistivity ρ versus temperature (T ), between three samples grown by molten-salt-flux (MSF) method and data from the original report of superconducting UTe2 grown by chemical-vapour-transport (CVT) method [13]. Critical temperature (Tc) is determined by the highest temperature data point that reaches zero resistivity within resolution. The residual resistivity ρ0 is calculated by fitting the normal state data above the superconducting transition up to T = 4 K using a Fermi liquid resistivity model, expressed as ρ = ρ0 +AT 2. The residual resistivity ratio (RRR) is defined as ρ(T = 300 K)/ρ0. The uncertainty of the critical temperature is extracted in the way described in accordance with Fig. 2.11. The uncertainty of the residual resistivity is extracted by analyzing the covariance of the fitting. The error bar of the RRR is propagated from the error bar of the residual resistivity. magnet at NHMFL with a dilution fridge option, while all other curves were measured in Cell 6 water-cooled resistive magnet at NHMFL with a He-3 cryostat. Although the noise was largely suppressed in the superconducting magnet, the superconducting transitions are prominent in both sets of measurements, as marked by a drastic change in the sample skin depth from the superconducting state into the resistive normal state. 54 Intermediate Magnetic Field Superconducting Phase Diagrams of UTe2 Table 3.1 Data of Tc, ρ0, and RRR comparing different UTe2 samples grown with MSF and CVT methods from my work (as shown in Fig. 3.1) and other reports. FIB is the acronym for “focused ion beam” milled samples. In Sakai et al. [67], the sample showing RRR = 1000 was too small to measure the sample dimensions, and thus, ρ0 was not reported. The error is calculated by the method described in Fig. 3.1. Growth method Tc (K) ρ0 (µΩ cm) RRR Reference 2.10 ± 0.01 0.48 ± 0.02 904 ± 25 MSF 2.08 ± 0.02 1.1 ± 0.02 406 ± 10 This study 2.02 ± 0.02 4.7 ± 0.04 105 ± 7 MSF 2.06 ± 0.01 1.7 ± 0.02 220 ± 7 Aoki et al. (2022) [77] MSF 2.10 - 1000 Sakai et al. 2.04 ± 0.02 2.4 ± 0.02 170 ± 5 (2022) [67] 2.00 ± 0.01 7 88 CVT 1.95 ± 0.02 9 70 Rosa et al. (2022) [63] 1.85 ± 0.01 12 55 CVT 1.44 ± 0.01 16 40 Ran et al. (2019) [13] CVT 1.55 - 1.60 19 35 Aoki et al. (2019) [59] CVT 1.55 - 1.60 16 35 - 40 Helm et al. CVT FIB 1.55 - 1.60 27 25 - 30 (2022) [97] For contacted resistivity measurement, we define the upper critical field as the field value where the resistivity is larger than 0.1 µΩ cm at constant temperature. In the same way, Tc(H) is extracted from the first data point that goes below 0.1 µΩ cm for the ρ(T ) cooling curve. The extracted transitions agree in the H −T phase diagram as shown in Fig. 3.5. This is also corroborated by ref. [115]. 3.1 Enhanced Superconductivity in MSF-grown UTe2 55 0 1 2 T (K) 0 2 4 (µ c m ) A H //a 0 5 10 0H (T) 0 0.5 1 1.5 2 T (K) 0 5 10 0H (T ) H //a MSF CVT D 0 1 2 T (K) 0 2 4 (µ c m ) C H //c 0 5 10 150H (T) 0 0.5 1 1.5 2 T (K) 0 5 10 15 0H (T ) H //c MSF CVT F 0 1 2 T (K) 0 2 4 (µ c m ) B H //b 0 5 10 150H (T) 0 0.5 1 1.5 2 T (K) 0 5 10 15 0H (T ) H //b MSF CVT E µ µ µ Fig. 3.2 Resistivity curves as a function of temperature for the RRR = 406 sample from Table 3.1 at intermediate magnetic fields with H applied along the (A) a-axis, (B) b-axis, and (C) c-axis. The strength of the applied field is indicated by the colour bar. The corresponding profile of Tc(H) is provided in panels (D-F). This figure also presents a comparison between MSF-grown and CVT-grown UTe2, utilizing CVT data from ref. [13]. The error is calculated by the method described in Fig. 3.1. To illustrate the criteria of extracting Hc2 consistently in the contactless resistivity mea- surement, contacted and contactless resistivity measurements were performed simultaneously on the same sample. In Figure 3.4, the contacted / contactless resistivity and the derivative of skin depth versus magnetic field are plotted on the same scale. A Gaussian function is fitted to the transition of the derivative curve, with dashed lines indicating the location of the Gaussian maximum, 0.5 σ and 1 σ . The location of 0.5 σ (the middle dashed line) agrees with the value of Hc2 extracted by the resistivity data curve from the criterion described 56 Intermediate Magnetic Field Superconducting Phase Diagrams of UTe2 0 10 20 30 0 H (T) S ki n de pt h (a rb .) 0.1K 0.4K 0.7K 1.1K 1.5K 6.5K 16KH // c f H Fig. 3.3 Proximity diode oscillator (PDO) measurement of skin depth δ as a function of magnetic field H. Here, the magnetic field is applied along c axis at incremental temperatures. the derivative of δ (H) curve at 0.1 K, i.e. ∂ fPDO ∂H , is added to explain the criteria of extracting the upper critical field Hc2 with PDO data. earlier in these simultaneous measurements and also in the phase diagram. Thus, the criteria were empirically determined as the center of the Gaussian fitting of the measured transition plus 0.5 σ . The extracted Tc(H) temperature-field phase diagram is as shown in Fig. 3.5. Here, a comparison between two generations of UTe2 samples with different levels of impurity is shown. The open symbols are extracted from [115] for the field along the b-axis. A clear enhancement of the upper critical field / Tc(H) along both the a and the c orientations is shown. The kink feature of the Hc2(Tc) curve in the temperature-field phase diagram for the field along b also shifts to higher temperature in higher purity specimens. The enhancement of Hc2 along both the a- and c-axes qualitatively scales with the quadratic increase in Tc. The extrapolated Hc2 to T = 0 K for all three orientations are listed in table 3.2. These data agree with other independent reports on MSF-grown UTe2 [116, 115]. The observation of an increase in Hc2 as the sample purity is improved is not surprising. 3.1 Enhanced Superconductivity in MSF-grown UTe2 57 0 1 2 3 R ( m ) A 0 H (T) 2 3 4 5 6 f (M H z) B 0 10 20 30 0 H (T) f / H ( ar b. ) C Fig. 3.4 Simultaneous measurement of (A) contacted and (B) contactless resistivity were performed on the same sample during a field sweep. The derivative of the skin depth data versus magnetic field is presented in (C). This illustrates the method for extracting Hc2(T ) from contactless resistivity, ensuring consistency with the contacted measurements. The Hc2 mark how the magnetic field breaks the superconducting pairs and the strength of the Cooper pairs is reflected by the value of Tc. Similar observations in ruthenates [117], cuprates [118], and heavy fermion superconductors [119, 120] have also been reported. 3.1.3 Lower Critical Field (Hc1) In addition to the upper critical field that has been enhanced as a result of the increase in purity, the lower critical field (Hc1) is also enhanced. Magnetization measurements (see Sec. 2.1.2) to determine the lower critical field (Hc1) were obtained using the helium-3 option of a QD MPMS, with the data presented in Figure 3.6. The sample was mounted inside a Kapton tube with GE varnish, with the field aligned along the a-axis. For each isothermal 58 Intermediate Magnetic Field Superconducting Phase Diagrams of UTe2 0 0.5 1 1.5 2 T (K) 0 5 10 15 20 0H ( T ) H // a H // c H // b This study Sakai et al. [115] Ran et al. [13] Aoki et al. [59] CVT MSF UTe 2 Fig. 3.5 Comparison of magnetic field - temperature superconducting phase diagram for MSF-grown (bold symbols) and CVT-grown (pale symbols) UTe2 at low magnetic field for three field along a-, b- and c-axes [13, 115, 59]. The upper critical field and Tc(H) are extracted in a consistent way within our dataset and from references with digitized data. Lines are guide-to-eye. The field orientation of each Hc2(T ) curve is noted by the text next to the curve and indicated by the colour. Table 3.2 Comparison of anisotropic Hc2 values between CVT-grown samples and enhanced quality MSF-grown UTe2. The Hc2 values of SC1 extrapolated to zero kelvin of field along a-, b- and c-axis are listed, together with their corresponding Tc. For MSF-grown sample, the H//a and H//c values are from resistivity measurements and the H//b value is from PDO measurement. CVT MSF Tc (K) Hc2 (T) Tc (K) Hc2 (T) H//a 1.5±0.05 7 ± 0.1 [13, 59] 10.1±0.2 [13, 59] H//c 10 ± 0.1 2.08 ± 0.02 16 ± 0.6 H//b 1.91.5±0.02 20 ± 1 [91] 22 ± 1 field sweep, the sample was first heated above its Tc, after which both the instrument magnet and the Superconducting Quantum Interference Device (SQUID) magnetic coil were turned 3.1 Enhanced Superconductivity in MSF-grown UTe2 59 0 50 100 0 H (Oe) -2 -1 0 M A 0.5 1.0 1.5 2.0 T (K) 0 20 40 0 H (Oe) M ( ar b. ) 0.4K 0.5K 0.6K 0.7K 0.8K 0.9K 1.0K 1.1K 1.2K 1.3K 1.5K 1.7K 1.8K B 0 1 2 T (K) 0 5 10 15 20 25 0H ( O e) MSF CVT H // a C Fig. 3.6 Measurement of lower critical field as a function of temperature for field along easy a-axis via field-dependent magnetization measurement. This set of data was obtained with the RRR = 105 sample from Table 3.2. (A) Magnetic moment M versus applied magnetic field H at various temperatures for the magnetic field H along the a- axis. (B) Background-subtracted moment versus temperature data curves at different temperatures. The subtracted background corresponds to the superconducting diamagnetic effect regime in the superconducting UTe2 at the corresponding temperatures. The temperature values of each curve are listed next to the curve. The lower critical field Hc1 value were determined by the first data point that deviates from the linear ground line of superconducting diamagnetic state. (C) Hc1 values at different temperatures for the field along a-axis. Here, the orange colour data curve represents the extracted Hc1 values for MSF sample, while the black data points correspond to the Hc1(T ) curve for the CVT sample for H//a [121]. The error bar is extracted in the way as described in Sec 2.5. off and restarted at temperature above Tc (T > Tc = 2 K). The purpose of this procedure is to clear out the trapped flux line inside the instrument and secure a zero magnetic field environment. The sample was then cooled down to a certain temperature and stabilized. DC magnetic moment measurements at incremental stable magnetic field values were then performed to obtain isothermal M(H) curves for each temperature. When a sample is in the superconducting phase, it will be in a diamagnetic state with constant negative susceptibility [122]. Therefore, in terms of moment versus field, a straight line with a constant negative slope M(H) (χ = ∂M/∂H) is thus expected within the dia- magnetic state, indicating a constant susceptibility diamagnetic state corresponding to the perfect screening due to the super current. The flux penetration field Hp may therefore be identified as the lowest field value where the M vs. H curve deviates from linearity (with 60 Intermediate Magnetic Field Superconducting Phase Diagrams of UTe2 a correction for the demagnetization effect, as detailed in, e.g. refs. [123, 124]). A linear function is fit to the data below 5 Oe at each temperature, which is then subtracted from each curve. The background-subtracted data for each temperature are shown in Fig. 3.6B. The flux penetration field Hp is then extracted by finding the first point that deviates from the flat line at each temperature. Following the discussion in ref. [125], Hc1 may be related to Hp through the expression: Hc1 = Hp tanh √ 0.36t/w , (3.1) where t is the sample thickness and w is the sample width. For this measurement, with H ∥ a, t = 3.46 mm (along the a direction) and w = 0.51 mm. Hc1 is enhanced for this new generation of higher quality samples (Fig. 3.6C, similar to the elevated values of Hc2 shown in Fig. 3.5. The Hc1 value of ≈ 20 Oe for H ∥ a agrees well with a recent report of a similar study on MSF-grown UTe2 [126]. With the upper critical field and the lower critical field value known, important parameters of the ambient superconducting phase SC1 including, i.e., coherence length ξ and penetration depth λ could be calculated with these equations: Hc1 = Φ0 4πλ 2 ln ( λ ξ ) Hc2 = Φ0 2πξ 2 (3.2) From the values of the upper critical field Hc2 and the values of the lower critical field Hc1 extracted from the data shown in Fig. 3.6 and the report [126], the calculated penetration depth λ and the coherence length ξ are shown in table 3.3. Table 3.3 A table of penetration depth λ and coherence length ξ of the UTe2 SC1 phase for field along a-, b- and c- axis. The error was calculated by propagating the errors from Hc1, Hc2 in Eq. 3.2. Penetration Depth (nm) λ Coherence Length (nm) ξ a 636 ± 40 6 ± 0.1 b 480 ± 35 4 ± 0.2 c 351 ± 26 3.5 ± 0.2 3.2 Intrinsic Metamagnetic Transition in UTe2 In previous sections, significantly enhanced quality of MSF-grown UTe2 samples is demon- strated by increasing RRR. Both Tc and Hc2 of the SC1 superconducting phase at zero field 3.2 Intrinsic Metamagnetic Transition in UTe2 61 have been enhanced. In this section, the measurement of the metamagnetic (MM) transition (as defined in Sec. 1.2) of MSF-grown UTe2 with contactless and contacted conductivity is presented. The data indicate that the field at which the metamagnetic transition occurs (HM) has remained unchanged within our measurement accuracy between the new and old generation of UTe2 samples. Figure 3.7 shows the skin depth of UTe2 measured in pulsed magnetic fields up to 70 T, for field applied along b-axis at various different temperatures. The MM transition to the polarised paramagnetic state is clearly observed by a sharp step in the skin depth at µ0HM ≈ 35 T for all temperatures, similar to CVT samples [68]. It could be seen from both the raw data and the extracted phase diagram that the metamagnetic transition does not have much temperature dispersion below T = 4.2 K and that the metamagnetic transition field HM behaviour agrees with the previous results from the CVT-grown samples at this temperature range. The agreement of HM between old and new generation of UTe2 suggests that the metamagnetic transition in UTe2 is independent of sample quality. Figure 3.8 tracks the MM transition as the magnetic field is rotated from b-axis towards the c-axis, with prior PDO measurements on a CVT specimen reported in ref. [88] at the corresponding angles. At θ = 0° and θ = 20°, for both MSF-grown and CVT-grown samples, the MM transition, as notified by the abrupt increase in the contactless resistivity, happens at the same magnetic field HM. At θ = 33°, a notable decrease in skin depth is evident for both types of UTe2 samples. This comparison indicates that the MM transition field HM is not affected by the sample quality, unlike superconducting properties of SC1. The sharp drop in frequency, caused by the abrupt increase in resistivity and magneti- zation, which is characteristic of entering the polarised paramagnetic phase [89] – occurs at the same value of H for CVT and MSF samples within the experimental resolution. At θ = 33°, both types of samples exhibit a jump in skin depth at the same magnetic field strength. However, in this case, the jump is in the opposite direction due to the presence of field-induced superconductivity SC3. 62 Intermediate Magnetic Field Superconducting Phase Diagrams of UTe2 10 30 50 70 0H (T) Sk in d ep th (a rb .) 0.6 K 0.9 K 1.0 K 1.2 K 1.6 K 1.8 K 1.9 K 2.3 K 2.5 K 4.2 K 0.1 K H // b 0 1 2 3 T (K) 0 10 20 30 40 0H (T ) 0 1 2 3 T (K) 0 10 20 30 40 0H (T ) 0 1 2 3 T (K) 0 10 20 30 40 0H (T ) 0 1 2 3 T (K) 0 10 20 30 40 0H (T ) MSF PDO MSF MSF (115) CVT (13, 90) CVT Cp (91) MSFCVT Polarised paramagnet Normal paramagnet A B SC2 SC1 Fig. 3.7 Mapping the magnetic field - temperature superconductivity and metamagnetic transition phase diagram for the field along the hard b-axis. (A) PDO measurements for H//b at various temperatures. The T = 0.1 K curve was replicated from Fig. 3.12. The drastic changes in skin depth at H ≈ 35 T represents metamagnetic transitions, as verified by extraction magnetization experiments for H//b [94]. The arrows indicate features in the PDO signal, attributed to the SC1-SC2 crossover. (B) Magnetic field - temperature phase diagram for UTe2 with the field applied along the b-axis. The coloured data points represent phase boundaries determined by multiple contacted / contactless conductivity measurements on MSF-grown samples. The gray data points are adapted from the corresponding references [13, 90, 91, 115]. The lines and colour fill are guides to the eye. The errors of superconducting transition are adapted from Fig. 3.5, the errors of metamagnetic transitions are the width of the transition in terms of magnetic field as described in Sec. 2.5. 3.2 Intrinsic Metamagnetic Transition in UTe2 63 10 30 50 70 0 H (T) S ki n de pt h (a rb .) MSF, T = 0.6 K CVT [88], T = 0.45 K = 0° = 20° = 33° b - c Fig. 3.8 Intrinsic metamagnetic transition and field-induced SC3 of CVT- and MSF-grown UTe2. Skin depth measurement up to 70 T at different field angles in the b-c rotation plane, where θ = 0° indicates the magnetic field is along b-axis. The location of the MM transition and field-induced SC3 remain the same between MSF samples (solid lines) and CVT samples (dashed lines, adapted from ref. [88]). The metamagnetic transition is indicated by the abrupt increase of the skin depth at θ = 0°and θ = 20°while the field-induced superconductivity SC3 is indicated by the decrease of the skin depth at θ = 33°. 64 Intermediate Magnetic Field Superconducting Phase Diagrams of UTe2 0 10 20 30 40 0 H (T) 0 100 200 300 ( µ c m ) A 1.8 K 5.1 K 12 K 20 K 34 K RRR = 406 b - c, = 29° 0 10 20 30 40 T (K) 0 20 40 H m ( T ) B MSF = 29° CVT = 0° Knafo et al. [42] Fig. 3.9 (A) DC field resistivity data tracking the evolution of the metamagnetic transition at HM (indicated by markers). (B) Comparison of the temperature progression of HM from panel (A) with that reported for a CVT sample in ref. [42]. The red symbols indicate the first measured temperature point of each study at which the resistivity maximum or first-order jump is no longer observed. The HM above critical end point is identified by the location of maximum resistivity. As the temperature increases, the MM transition becomes a second-order-like crossover behaviour. This occurs as the temperature rises above the critical end point (CEP). The location of the metamagnetic crossover HM above the CEP, which determines the Widom line, is determined by the maximum resistivity, marked by the arrow in Fig. 3.9. Here, we compare the temperature dependence of HM between MSF-grown and CVT-grown UTe2 samples and find that they are consistent with each other. The metamagnetic transition could be understood with a free-energy expansion model. Fig. 3.11 shows a free-energy expansion with respect to magnetization M. The free energy expansion at zero magnetic field could be expressed as: F = αM2 +βM4 + γM6 (3.3) 3.2 Intrinsic Metamagnetic Transition in UTe2 65 0 10 20 30 40 0 H (T) 0 10 20 30 T ( K ) H || b Ref. [94] This Study Fig. 3.10 Comparison of metamagnetic transitions in MSF-grown and CVT-grown UTe2 at different temperatures for magnetic field applied along hard b-axis. Data points for CVT samples are as digitized from [94] When the parameters follows that α > 0, β < 0 and γ > 0, the free energy F(M) has two local minima, one at M = 0 and the other at finite magnetization M. When an external magnetic field is not applied, the global minimum is in the M = 0 state. When an external magnetic field is applied, a −M ·H term is added to the free energy expansion. When a small amount of magnetic field is applied, the global energy minimum is in the vicinity of M = 0. This explains the paramagnetic property of UTe2 when the magnetic field H is less than 34 T. When the external magnetic field is large enough, the local minimum at the finite magneti- zation state becomes a new ground state. In the case of UTe2 when the external magnetic field is along H//b and the field strength is larger than HM = 34 T, the energetically favorable state jumps from the M = 0 state into the finite polarized magnetized state. Since magnetization is a localized property and the metamagnetic transition stems from the free-energy landscape, it is not surprising that the MM transition does not show much sample quality dependence. 66 Intermediate Magnetic Field Superconducting Phase Diagrams of UTe2 Magnetization M F re e E ne rg y F F = M2+ M4+ M6-M H > 0, < 0, > 0 Paramagnetic H = 0 PPM H > 34 T Fig. 3.11 Free energy expansion model explaining the origin of metamagnetic transition. A schematic free energy expansion that explains the origin metamagnetic transition. In zero field H = 0 with certain selection of expansion parameters, there is a local energy minimum at finite magnetization M while the global minimum is at M = 0. When magnetic field is increased, the finite moment state becomes energetically favorable and the material’s magnetic moment becomes polarised. 3.3 Enhanced Angular extent of SC2 In this section, results of the field-induced superconducting phase with magnetic field H//b, i.e. the SC2 phase, are presented and discussed. An interesting feature in our PDO measurements (see Sec. 2.4.2) is the presence of an anomalous kink, marked with arrows in Fig. 3.7A (and in the inset of Fig. 3.12A, which appears to separate SC1 from either SC2 or the normal state, depending on the temperature. The extracted kink data points are plotted as purple circles in Fig. 3.7, along with resistivity and specific heat data from previous studies [13, 90, 91, 115]. By Eq. 2.8 the change in frequency of the PDO circuit is sensitive to both the electrical resistivity and the magnetic susceptibility of the sample. Thus, this observation is consistent with recent reports [115, 127], where a kink in the χ(H) curve has been attributed to marking transition between SC1 and SC2, which is visible in our skin depth measurements even though the resistivity remains zero as the material transitions from SC1 to SC2. As shown in Fig. 3.7, the field - temperature dependence of the kink feature is qualitatively consistent with the phase boundary set by the measurement of the heat capacity [91]. This shows the ability to map the SC1-SC2 transition 3.3 Enhanced Angular extent of SC2 67 using the PDO method, which is not possible for the contacted resistivity measurement. The data point of the SC1-SC2 phase transition with H//b for the MSF sample is marked by a black oval symbol in the magnetic field - angle phase diagram in Fig. 3.14. Figure 3.13 shows the resistivity of MSF-grown UTe2 measured in a DC magnet over the field range 0 T ≤ µ0H ≤ 41.5 T at T = 0.4 K for various magnetic field tilt angles away from b-axis. The data curves in the b− c plane were taken from the RRR = 406 sample in Table 3.1 while those in the b−a plane are from the RRR = 105 sample. At T = 0.4 K, for small tilt angles within 5° from the b-axis in both rotation planes, zero resistivity persists until the magnetic field strength exceeds 34 T, whereupon the resistivity increases rapidly at the MM transition as SC2 terminates and the polarised paramagnetic state is entered. In the b− c rotation plane, this remains the case for angles up to 19° away from b; however, by 25° non-zero resistivity is observed for µ0H as low as 20 T (Fig. 3.13A. Above 20 T, the resistivity at this angle then remains small but nonzero up to 38 T. At this point, the SC3 phase is accessed and zero resistivity is observed up to the highest applied field strength. These results were then utilized to construct the superconducting phase diagram in Fig. 3.14. In contrast to the SC2 range for CVT-grown samples as shown in Fig. 3.13(C)(D), our measurements on MSF-grown UTe2 yield zero resistivity over the entire field interval 0 T ≤ µ0H ≤ 34.5 T for successive tilt angles up to and including 19° away from b towards c. Notably, our measurements in the b− c plane were performed in a 3He system, at a temperature an order of magnitude higher than those reported by Knebel et al. from dilution fridge measurements [95]. This indicates a remarkable expansion of the SC2 angular range in the b−c plane resulting from the enhancement of purity in the new generation of MSF-grown UTe2 crystals. A similar trend is observed in the b−a rotation plane. Previous measurements in a CVT sample reported by Ran et al. [88] found a strong sensitivity to the field orientation (that is, the H//a component) of SC2 within a very small angular range of only 0.3°, with markedly different ρ(H) observed for 4.7° compared to 5.0° (Fig. 3.13d). For comparison, at 5°, zero resistance persists to µ0H > 34 T, while at 9° and 10° the field value for the resistive transition of SC2 to normal state is notably sensitive to small changes in angle, indicating that the SC2 boundary for the MSF samples lies close to this angle. Interestingly, it appears that the angular extent of SC2 in both rotation planes appears to be approximately doubled for MSF compared to CVT samples – for angles in the b− c plane from approximately 12°to between 19° - 25°, and for b−a from 5° to around 10°. This is better illustrated in Fig. 3.14. 68 Intermediate Magnetic Field Superconducting Phase Diagrams of UTe2 A B Fig. 3.12 Skin depth measurements of pristine UTe2. (A) PDO measurement (see Sec. 2.4.2) of the skin depth of UTe2 for magnetic field applied along the b-axis (blue) and c towards b 15°(orange) at T = 100 mK. A dip feature in the H//b curve at 22 T is marked by the triangle is shown. The zoom-in of this feature in the skin depth as a function of magnetic field curve between 21 T and 24 T is plotted in the inset, identifying the superconducting transition between SC1 and SC2 phase. (B) Oscillatory component of the PDO signal at 20 mK for magnetic field rotated from c to b by θ = 15°, showing prominent quantum oscillations of frequencies ≈ 3.5 kT, consistent with prior studies [77, 78]. All data in this figure were collected on the same sample. 3.4 Phase diagram under pressure with ACMS After going through the results of mapping the UTe2 superconducting phase diagram at ambient pressure and in high magnetic field, we turn to the results for UTe2 under pressure. As introduced in Sec. 1.4.6, previous experiments of CVT-grown UTe2 under pressure measured the heat capacity [66] and the Knight shift [102]. At p > 0.3 GPa, heat capacity C(T ) revealed two distinct phase transitions. As pressure increases, one critical temperature increases and the other decreases [66]. When the pressure goes above critical pressure pc = 1.5 GPa, both superconducting phases vanish. NMR Knight shift measurements found that the pressure-induced SC2 phase exhibits a vanishing spin susceptibility suppression as temperature cools through Tc [102]. By comparing the pressure - temperature superconducting phase diagram with the H//b magnetic field - temperature phase diagram below the MM transition, as shown in Fig. 3.15, it could be seen that the superconducting critical temperature shows a similar dependence to the tuning parameters for both Hb and p. As the field along b-axis and pressure increase, the observed Tc shows a “S” shape behaviour. The superconducting transitions are truncated at the critical pressure pc =1.5 GPa, and the metamagnetic transition HM =34 T. The heat 3.4 Phase diagram under pressure with ACMS 69 0 10 20 30 40 0H (T) 0 10 20 30 40 (µ c m ) A b - c 34 36 38 0H (T) 0 10 20 30 (µ c m ) 25° 19°17° 13° 9° 1° 0 10 20 30 40 0H (T) 0 20 40 60 (µ c m ) b - a 5°9°10°20°40°90° B 34 35 36 37 0H (T) 0 25 50 (µ c m ) 5° 9° 10° 10 20 30 40 0H (T) 0 10 20 30 40 (µ c m ) C 25° 19° 17° 13° 12° 16° 24° MSF 400 mK CVT (15) 30 mK b - c 10 20 30 40 0H (T) 0 2 4 (µ c m ) 25° 13° 12° 10 20 30 40 0H (T) 0 20 40 60 (µ c m ) D MSF 0.40 K 5°9° 10° CVT (14) 0.35 K 4.7°5.0° 7.3° b - a 15 25 35 0H (T) 0 1 2 3 (µ c m ) 5°9° 10° 4.7° 5° Fig. 3.13 Angular dependent ρ(H) curves in (A) b to c and (B) b to a rotation planes, focusing on the evolution of the SC2 superconductivity phase. The angles are indicated by the numbers next to the data curves and are colour-coded. The inset figures zoom in to present the metamagnetic transition and field-induced superconducting phase SC3 in greater detail. (C) and (D) compare the angular extent of SC2 measured by resistivity method between our data from MSF sample and previous studies [95, 88]. For the data of MSF samples, b - c rotation planes were performed on the sample with RRR = 406 and for b - a rotation planes were performed on the sample with RRR = 105 from Table. 3.1. capacity measurements also revealed a second superconducting transition under a certain applied pressure [66] or H//b [91]. These observations raise two questions about the phase diagram: (1) What is the origin of the SC2 phase in the pressure - temperature phase diagram. (2) What is the nature of the mixed superconductivity regime. To address these intriguing questions related to UTe2 under pressure, an AC magnetic susceptibility experiment was performed with new generation UTe2 under pressure. The temperature - pressure superconducting phase diagram was carefully mapped with ultraclean MSF-grown UTe2 single crystal from the same batch that exhibited magnetic quantum oscillations [78]. The measured sample was cut from a parent sample characterised by MPMS, as shown in Fig. 3.16A. The platelet inside the cell has a diameter of 200 µm and a thickness of 50 µm. The sample was mounted in a moissanite anvil cell (MAC) on a 70 Intermediate Magnetic Field Superconducting Phase Diagrams of UTe2 MSF Fig. 3.14 High magnetic field phase diagram for pristine quality MSF-grown UTe2. The phase boundary between SC1 and the normal state locates at higher magnetic field strengths for MSF samples compared to previous studies on CVT specimens (blue region). Additionally, the angular extent of SC2 is significantly enhanced for MSF samples (pink region). The polarised paramagnetic state (orange region) exhibits the same angular profile for both types of samples. Lines and shading serve as a guide to the eye. In this figure, all the MSF related data points (diamond) measured by resistivity were measured at 0.4 K. The PDO data point for SC3 were measured at 0.7 K and for SC1 were measured at 20 mK. CVT data points from refs. [97, 95, 88]. The uncertainty of each transitions were analyzed with the methods described in Sec. 2.5. micro-coil set, consisting of a balanced pick-up coil of 10 turns and a driving coil of 130 turns, as described in Fig. 2.6. The drive coil would give approximately 0.3 mT of magnetic field for each Ampere applied. The MAC was then pressurized using a hydraulic ram and the pressure was secured with screws. The pressure medium employed in this study was Daphne oil 7474, which remains liquid up to the highest pressure in this experiment (2 GPa) at room temperature. The assigned pressure value was determined at room temperature by measuring the fluorescence frequency shift of a ruby inside the pressure cell. After determining the pressure value, the MAC was mounted on an ICEoxford 3He probe and cooled to the base temperature of T = 400 mK. The details and method of AC magnetic susceptibility measurement are described in Sec. 2.3.4. For this study, the driving excitation was set at I = 8 mA and 1.1 kHz. 3.4 Phase diagram under pressure with ACMS 71 Fig. 3.15 Schematic superconducting phase diagram of pressure p, magnetic field H and temperature T of UTe2. For both increasing magnetic field along b-axis and pressure, critical temperature Tc decreases and then increases before being truncated by a magnetic transition. The dashed line indicates transition within superconducting state. The mixed regimes are marked by a question mark. Here, the PPM is the abbreviation of the polarised paramagnetic state, and the AFM means the antiferromagnetic state. The ACMS data χ(T ) at incremental temperatures are presented in Fig. 3.16B and the derivative ∂ χ/∂T is shown in (C). A single superconducting transition is observed at ambient pressure and at the lowest applied pressure in the dataset, indicated by a blue arrow in the derivative curve. As pressure increases, a second superconducting transition emerges when p > 0.25 GPa. In the χ(T ) curve, the higher critical temperature increases, while the overall size of the signal decreases. The Tc1 of the lower temperature transition decreases as the pressure increases. The pressure dependence of the critical temperatures agrees qualitatively with the previous results of CVT-grown UTe2 under pressure [66, 99]. Here, the criteria for determining the critical temperature of our ACMS signal χ(T ) are discussed. A direct comparison of three different measurement techniques of the supercon- ducting transition at ambient pressure is presented in Fig. 3.17. All these measurements were performed on the same UTe2 specimen with the heat capacity and magnetization characteri- zation performed on the original sample, and the resistivity measured on that sample after electrical contacts were made, as described in Sec. 2.3.1. The midpoint of the measured heat 72 Intermediate Magnetic Field Superconducting Phase Diagrams of UTe2 0 2 4 T (K) -0.3 -0.2 -0.1 0 M ( 10 -3 em u) A Ambient pressure 0 2 4 T (K) -1 -0.5 0 (a rb .) 0.25 GPa 0.62 GPa 0.79 GPa 0.99 GPa 1.26 GPa 1.50 GPa B 1 2 3 4 T (K) / T (a rb .) C Tc1 Tc2 1 Tc1 Tc2 1 Fig. 3.16 (A) DC magnetization, M, measuring superconducting transition of a UTe2 single crystal at ambient pressure. The sample was mounted on a quartz sample holder, cooled to below critical temperature with 10 Oe external field applied. (B) AC magnetic susceptibility, χ , as a function of temperature T at incremental applied hydrostatic pressure. The UTe2 specimen was pressurized by a moissanite anvil cell (MAC). Arrows marks the two super- conducting transition related anomalies at Tc1 (blue) and Tc2 (magenta) in the χ(T ) curve. (C) Derivative with respect to temperature of the AC magnetic susceptibility ∂ χ/∂T of the data in panel (B). At p = 0.25 GPa, only one derivative peak could be identified with one arrow marking the SC1 transition. At all higher pressures, an additional derivative peak is identified as labeled by magenta arrow of SC2. capacity transition matches well with the onset of the dip in magnetic susceptibility χ(T ) and where the resistivity reaches zero. The critical temperature values as a function of applied pressure were then extracted by taking the onset of the dip in the AC magnetic susceptibility data curve in the later discussion. In Fig. 3.18, an aligned comparison between ACMS and resistivity and heat capacity under pressure is shown. The resistivity and heat capacity measurements were performed on the same sample in a piston cylinder cell (PCC) by Theodore Weinberger. The data is included here to compare with the superconducting transitions measured by ACMS, including the χ(T ) curve and its derivative with respect to temperature ∂ χ/∂T . The pressure value of PCC was determined by the Tc of a lead sample inside the teflon tube under the same pressure as the UTe2 sample at low temperature. The pressure value for the MAC is determined by the ruby fluorescence frequency shift at room temperature. Therefore, the assigned pressure value of the MAC is a higher estimate than the pressure value at low temperature due to thermal contraction. Thus, the pressures of these two datasets are close in pressure value, which can 3.4 Phase diagram under pressure with ACMS 73 0.1 0.2 0.3 0.4 C p /T ( J/ K 2 m ol ) UTe 2 Ambient Pressure -30 -20 -10 0 ( em u/ O e) 1.8 2 2.2 T (K) 0 20 40 ( µ c m ) Fig. 3.17 Comparison of measuring superconducting transition between resistivity ρ , mag- netic susceptibility χ and heat capacity Cp. All these measurements were all taken with the same sample. This shows consistency of criteria for determining the Tc between different measurement techniqeus. be seen in the comparison between the onset of the decline of AC susceptibility χ(T ) and the Tc determined by zero resistivity. The χ(T ) data curve is quite intriguing, as it captures the second drop in magnetic susceptibility at lower temperature, while resistivity remains zero. Both peaks in the ∂ χ/∂T curve coincide with the two superconducting transitions measured by the heat capacity. This indicates that the transition at lower temperature captured by the ACMS is the superconducting transition of SC1. The observation of the lower-temperature superconducting transition is unconventional. The drop in the ACMS signal is due to the diamagnetic screening of the sample, as our experiment was carried out at zero magnetic field and the applied AC magnetic field due to the ACMS measurement was lower than the lower critical field Hc1 inferred from Fig. 3.6. The sample is in a perfect diamagnetic state and all AC magnetic responses should be screened by the supercurrent of the superconducting SC2 phase. Such an unusual magnetic signature of the superconducting transition inside another superconducting state could be accounted 74 Intermediate Magnetic Field Superconducting Phase Diagrams of UTe2 ( ar b. ) A PCC 0.74 GPa 0 1 2 3 4 T (K) ( ar b. ) C MAC 0.79 GPa 0 1 2 3 4 T (K) / T ( ar b. ) D MAC 0.79 GPa C p / T ( ar b. ) B PCC 0.74 GPa Fig. 3.18 Comparison of measuring superconducting transitions between different mea- surement methods. The resistivity ρ and heat capacity Cp/T were measured in a piston cylinder cell (PCC) at 0.74 GPa, while AC magnetic susceptibility was measured in a MAC at 0.79 GPa. The derivative with respect to temperature, ∂ χ/∂T , is also presented for comparison with heat capacity. From the comparison, we can see the two anomalies in the χ(T ) data happen at consistent position measured by heat capacity, indicating χ(T ) captures a bulk thermodynamic phase transition. for by the fact that the diameter of the sample being measured is small enough that the change in penetration depth of the superconducting state is sensitive to the ACMS probe. The penetration depth of the UTe2 sample is of the order 1 µm as discussed in Table. 3.3. The sample radius R ≈ 100 µm is small enough to be comparable to the penetration depth. As a type-II superconductor, magnetic fields penetrate the perimeter of the UTe2 specimen into the surface layer of thickness ≈ λ . The screened volume thus would be Vscr ≈ π(R−λ )2t. Since the single-crystal platelet is placed inside the pick-up coil, a large demagnetizing effect is induced. The geometric demagnetizing factor could be written as N−1 ≈ 1+0.8t/(R−λ ). 3.4 Phase diagram under pressure with ACMS 75 The ACMS signal U could be written as: U = π(R−λ )2t × ( −1 (1−N) ) = π(R−λ )2t × 0.8(R−λ ) t ∝ π(R−λ )3 (3.4) Thus, the relative change in the ACMS signal is proportional to 3λ/R. When the sample is small enough that its radius R of the sample is comparable to the change in penetration depth, the ACMS could detect the change in penetration depth. Similar observation of the double transitions resolved in χ(T ) has been reported in UPt3 [128], which coincides with the bulk thermodynamic phase transition between the su- perconducting A and B phases. Similar results were also reported for U0.97Th0.03Be13 [129]. In light of these reports on similar multiphase superconducting systems,the observation of double kink features in the χ(T ) curve is interpreted as a change in London penetration depth when the UTe2 went through the transition from SC2 into SC1 at Tc1. Fig. 3.19 presents the resulting superconducting phase diagram constructed from the ∂ χ(T )/∂T data. Tc2 reaches its maximum of 3.2 K with 1.2 GPa pressure applied to UTe2 and Tc1 continues to decrease as the applied pressure increases. An important observation is that both superconducting transitions remain resolvable up to 1.42 GPa, with Tc1 = 2.92 K and Tc1 = 0.85 K. However, no signature of any transitions is observed at p = 1.5 GPa, marking the determination of the critical pressure pc. This indicates that for both superconducting states SC1 and SC2 under pressure, their Tc − p phase boundary does not converge at the critical pressure for high-quality UTe2 samples. Our measurements of ACMS and observation of the lower-temperature superconducting transition at Tc1 within a superconducting state suggest that further study of quantitative penetration depth measurement to determine λ (T ) would be important to understand the nature of the superconducting states SC1 and SC2. 76 Intermediate Magnetic Field Superconducting Phase Diagrams of UTe2 p c Fig. 3.19 The temperature - pressure superconducting phase diagram of UTe2 is mapped based on our pressure-dependent AC magnetic susceptibility measurements. The SC1 (blue triangles) and SC2 (magenta diamonds) phases are identified by our AC magnetic susceptibility χ(T ) measurements. The orange oval indicates the antiferromagnetic (AFM) state, which is reproduced from ref. [66] and corroborated by our AC magnetic susceptibility measurements, s no signature is seen at the corresponding transition temperature as shown in Fig. 3.16B. The black triangle marks the first pressure point where superconducting transitions vanish. The critical pressure pc is indicated by the gray dashed line. The white dashed line between SC1 and SC1+SC2 regime at p = 0.29 GPa is determined by the first signature of the SC2 phase from ref. [130]. The error was extracted by fitting the ∂ χ/∂T at transition with Gaussian function as shown in Fig. 3.16C, and the error bar equals to σ of the gaussian fitting as per Fig. 2.11. 3.5 Understanding SC2 in UTe2 In this section, the implications of our experiments on SC2 and how it helps to understand the origin of this phase are discussed. Important results relevant to SC2 from the measurements 3.5 Understanding SC2 in UTe2 77 performed on new-generation ultra-clean MSF-grown samples from this study and other reports could be summarized as follows. • The field-induced superconducting phase SC2 arises when an intermediate amount of magnetic field is applied in close vicinity to the b-axis. For H//b, the SC2 phase emerges when the field is larger than 21 T, much above the Pauli paramagnetic limit. It is then truncated by the metamagnetic transition as shown in Fig. 3.7B. When the magnetic field rotates away from the b-axis, its maximum Tc decreases and vanishes when the field rotates too much away from the b-axis. In other words, the SC2 phase hangs beneath the metamagnetic transition and centers at H//b. • At ambient pressure, the angular range of the SC2 phase shows an acute sensitivity to the impurity level of the sample; the SC2 phase in MSF samples survives to an angular range twice that in CVT samples. Meanwhile, the metamagnetic transition HM does not show a dependence on sample quality. However, the metamagnetic transition field HM could be suppressed by applying pressure. The pressure of p = 0.4 GPa suppressed HM from 34 T to 30 T, and p = 0.8 GPa suppressed HM to 20 T for CVT- and MSF-grown samples. [103, 131]. • There is a thermodynamic phase transition between the SC1 and SC2 phases [66, 91] for both ambient pressure in field measurement and under pressure measurement. Such a phase transition induces a change in the penetration depth of the sample and is probed by magnetic susceptibility. These results indicate that SC1 and SC2 have a different character. • Nuclear magnetic resonance (NMR) Knight shift measurements reveal that spin sus- ceptibility reduces by 6% for the SC1 phase, while there is almost no change when it enters SC2 for both field-induced case. A similar zero reduction in spin susceptibility upon phase transition was also found in SC2 with zero magnetic field applied under pressure [85, 102]. • Further heat capacity under pressure and in magnetic field measurement reveals that the phase diagram of the SC2 phase in the magnetic field Hb connects to the SC2 phase at a zero field under pressure [103]. These observations imply that SC2 in the magnetic field and under pressure is the same superconducting phase. In order to have a unified theoretical framework that could explain all these observations, the assumption that the electron pairs for the SC2 phase come from the fluctuation accompa- nying the MM transition may be taken. The fluctuation comes from the fact that the spins 78 Intermediate Magnetic Field Superconducting Phase Diagrams of UTe2 are polarised along the b-axis when it crosses the MM transition and enters the polarised paramagnet state. By applying a similar approach as described in Fig. 3.11, the Ginzburg-Landau free energy expansion could be applied to explain the metamagnetic transition in UTe2 with field along b-axis. The spins are fully polarised after the MM transition. However, the spin are free before the MM transition and thus, this gives a large spin fluctuations. Such spin fluctuation could give rise to an effectively attractive interaction [132, 41]. This effectively attractive interaction could give rise to the electron pairing and result in the SC2 phase. From Fig. 3.11, the fluctuation reaches its maximum when Hb is slightly below HM. At this point, the free energy of the polarized magnetized metastable state with a finite moment is nearly zero, making it easiest for the system to tunnel into the polarized state. This mechanism could account for the appearance of SC2 below the MM transition, and applying the magnetic field along the a and c axes is detrimental to this phase. The disorder level in UTe2 single crystal is modeled by varying the relaxation time in the model. By solving the linearized superconducting gap function of different relaxation time, the theoretical model could predict the angular range of SC2 in the field - angle phase diagram. As for the SC2 phase under pressure, the MM transition is suppressed to lower field. Thus, the metamagnetic fluctuation is suppressed to the lower field, as it is always below the MM transition. This could explain the emergence of SC2 under pressure that the MM transition is suppressed to lower and the magnetic fluctuation is large enough even when no external magnetic field is applied. Chapter 4 High Magnetic Field Phase Landscape of UTe2 This chapter focuses on the behaviour of MSF-grown UTe2 in the high-field regime (above 35 T). Firstly, the data that mapped the metamagnetic transition surface and its critical endpoints (CEPs) in multiple rotation planes of three-dimensional field space are presented. The data illustrate that the CEP temperature is suppressed by increasing the transverse components of the field, that is, by the Ha and Hc components. This suppression leads to the discovery of a series of quantum critical endpoints (QCEPs), as introduced in Sec. 1.2. These QCEPs connect to form a continuous quantum critical line (QCL) that marks the termination of the MM transition surface. Then additional data concerning the high magnetic field re-entrant superconductivity SC3 phase are presented. On the one hand, a dome-like dependence of the critical temperature on field orientations in the b-c rotation plane was found. On the other hand, resistivity measurements in the b−ac ([010]-[101]) rotation plane revealed that SC3 extends beneath the metamagnetic transitions. In combination, these measurements provide evidence that the enigmatic SC3 state of UTe2 is mediated by quantum critical magnetic fluctuations. 4.1 Discovery of a Quantum Critical Line in UTe2 As discussed in Sec. 1.2 and Sec. 3.2, a metamagnetic (MM) transition occurs in UTe2 when a large magnetic field is applied along the b-axis. As the temperature increases, the first-order MM transition approaches its CEP at 9 K [42] and gradually smears into a continuous second-order-like phase transition at higher temperatures, as shown in Fig. 3.9A. In general, a CEP may be suppressed to lower temperature by applying non-thermal parameters [133]. 80 High Magnetic Field Phase Landscape of UTe2 For UTe2, the non-thermal parameter that suppressed the CEP of MM transitions is the transverse field components, i.e., Ha and Hc. When the CEP is suppressed to zero kelvin, it becomes a QCEP. Fig. 4.1 shows a schematic model of tuning toward QCEP with non-thermal parameters. H T T H Tuning, e.g. transverse field Critical endpoint Metamagnetic transition Polarized paramagnet Quantum critical endpoint Widom line Fig. 4.1 A pathway to the QCEP. Left: Schematic of temperature T - magnetic field H visualization of the critical endpoint (CEP) of a first-order metamagnetic transition. The solid orange line marks the MM phase boundary and the solid orange oval marks the CEP. The dashed line marks the second-order-like crossover behaviour regime at higher temperatures. Right: Suppression of CEP towards lower T by applying a non-thermal tuning parameters, i.e. transverse magnetic field for UTe2. Green star marks the QCEP when CEP is tuned to absolute zero. The dashed line indicates the Widom line accompanying the QCEP. Here, experimental evidence of suppressing the CEP of the MM transition in UTe2 is presented. As shown in Fig. 4.2, a direct comparison of the temperature dependence of the MM transitions between H//b and H rotated toward ac ([101]) by θb−ac = 42° orientation is made. These measurements were performed with the sample mounted on an in-situ rotator probe. The sample and the PDO coil remained fixed during this experiment. Therefore, the frequency change could be qualitatively compared to indicate the difference of skin depth (magnetization and resistivity) between normal paramagnetic and polarized paramagnetic (PPM) states across the metamagnetic transition. For the dataset of H//b, the first-order MM transition remains clear and the derivative peak persists up to 9 K. As the temperature increases above the CEP, the transition smears into a maximum turning behaviour in the skin depth versus the field curve. This agrees with the result measured by contacted resistivity, as in Fig. 3.9, and with the literature [134, 94, 135, 93]. However, when the magnetic field is rotated away from the b-axis towards ac at a high angle (θb−ac = 42°), the MM transition is softened at all measured temperatures and the CEP is suppressed clearly to below 4.2 K. When the prominence of the peak in the ∂ f ∂H is less than 20% of the difference of 4.1 Discovery of a Quantum Critical Line in UTe2 81 the derivative before and after the MM transition. The extracted value coincides with e.g. ref. [42]. 20 40 60 0H (T) 0 50 0 50 100 150 f PD O (k Hz ) 1.5 K 4.2 K 9.0 K 12 K H || b ( b-ac = 0°) 1.5 K4.2 K 8.0 K 12 K b-ac = 42° f/ H (a rb .) b-ac = 42° b-ac = 0° a b c H Fig. 4.2 Left panel: Temperature evolution of MM transition measured by PDO techniques comparing two separate field orientation. For field along b, the first-order transition persists up to T = 9 K. When field rotates 42° from b towards ac, the first order transition softens into second-order-like transition down to 1.5 K. The derivative curves are shifted in x direction for clarity. Right panel: The model of sample performing b-ac rotation plane. H//b is as displayed in the model figure, while θb−ac is marked by the rotation axis in the figure. In Fig. 4.3, data that illustrate the process of suppressing the MM transition and the CEP by varying the transverse field components are presented. Fig. 4.3A was obtained by measuring the change in contactless resistivity with the PDO method introduced in Sec. 2.4.2 in a pulsed magnet at liquid 4He base temperature T = 1.7 K. The sample was mounted on a 17° wedge to obtain an initial b-c offset angle, after which the field orientation was incrementally rotated towards the a-axis. A schematic model is shown on the right-hand side of Fig. 4.3. Each angle was determined by measuring the angle between the sample and the edge of the probe from the photo taken after the sample was secured. Since the edge of the probe is parallel to the magnetic field, this would give the offset angle between the sample and the magnetic field. The value of rotation angle is noted next to the corresponding data curve. When the magnetic field orientation is tilted by θb−c = 6° from b to c, the MM transition appears to be a sharp increase in skin depth at 37 T. As the field rotates away from the b-c plane (θa increases), the MM transition field (HM) shifts to a higher field, and the sharp first-order MM transition gradually weakens in intensity and broadens. Thus, this isothermal rotation suppresses the MM transition and tuned it into a second-order-like 82 High Magnetic Field Phase Landscape of UTe2 transition. At θa = 20°, there is no signature of MM phase transition or crossover behaviour up to 79 T, indicating that the CEP at this magnetic field direction is well below the experiment temperature T = 1.7 K. 30 40 50 60 70 80 0H (T) 0 50 100 f PD O (k Hz ) 6° 10° 14° 17° 18° 19° a = 20° MM T = 1.7 K b-c = 17° Experiment 30 40 50 60 70 80 0H (T) 0 0.5 1 M ag ne tis at io n (a rb .) 5° 10° 20° 30° 40° b-c = 17° a = Landau TheoryB A b a c H Fig. 4.3 Experimental probe of CEP suppression towards QCEP. (A) Contactless resistivity measurements of UTe2 with PDO techniques at 4He base temperature. Data are vertically offset for clarity. The sample mounted with field tilted 17° from b-axis to the c-axis, and subsequently rotated towards the a-axis at the indicated by the note next to the data curve. As the rotation angle increases, the MM transition evolves from a sharp, first-order jump into a less prominent, second-order like continuous crossover. Dashed lines mark the suppression of the MM transition and terminate at QCEP (green star). (B) Theoretical model of the suppression of MM towards QCEP as shown in Eq. 4.1. (C) A schematic model figure of the sample mounted on a wedge to obtain a b - c angle offset. The whole device was then rotated by remounting the sample rotated with respect to magnetic field to get incremental θa. Fig. 4.4 maps the suppression of the MM transition in several additional rotation planes in the Ha-Hb-Hc space, they are: (A) from b to a, (B) from b to c, (C) from b to ac as illustrated in Fig. 4.2, (D) from θb−c =7° to a and (E) from θb−c =17° to a as illustrated in Fig. 4.3. The θb−c =17° to a dataset is replicated from Fig. 4.3A to get a better comparison. For all of these rotation planes, as the field is rotated away from the b-axis, i.e. when transverse field components Ha and Hc are increasing, the MM transition shifts to a higher magnetic field and loses its sharpness. This softening behaviour could be explained that as the CEP is 4.1 Discovery of a Quantum Critical Line in UTe2 83 suppressed toward the experiment temperature, the MM transition softens and changes into a continuous phase transition as the CEP at the corresponding (Ha, Hb, Hc) approaches. 40 60 80 0H (T) 0 40 80 f PD O (k Hz ) 12° 7° 0° 17° 23° 24° 25° 26° b-a = 28° b - a 40 60 80 0H (T) 0 40 80 f PD O (k Hz ) 15° 24° 36° 40° 44° 47° 48° 0° 14° b-c = 60° b - c 40 60 80 0H (T) 0 40 80 f PD O (k Hz ) 0° 12° 22° 27° 32° 35° 39° 42°b-ac = b - ac 0 1 M M s ize (a rb .) 40 60 80 0H (T) 0 50 100 f PD O (k Hz ) 0° 15° 17° 18° a = 20°T = 1.7 K b-c = 7° 30 40 50 60 70 80 0H (T) 0 50 100 f PD O (k Hz ) 6° 10° 14° 17° 18° 19° a = 20° MM T = 1.7 K b-c = 17° A B C D E F Fig. 4.4 Mapping QCEPs in various different rotation planes in three-dimensional Ha,b,c space. (A) - (E) PDO data for b−a ([010]–[100]), b−ac ([010]–[101]), b− c ([010]–[001]), θb−c = 7° −a and θb−c = 17° −a correspondingly. All data for maximum field less than 60 T were measured at WHMFC, Wuhan at T = 4.2 K and the data for maximum field larger than 60 T were measured in a dual coil magnet up to 80 T at HLD, Dresden at T = 1.7 K. Due to the nature of two pulses on top, it would produce artifact at 39 T when the second pulse coil is turned on, as described in Method chapter. The data curve from dual coil system is rescaled by a factor of 0.57 to match the size of the MM transition of single coil system for easy comparison. For all rotation planes, the MM softens in abruptness as H tilts away from b, when Ha and Hc components becomes larger. (F) A bird-eye view of MM transition landscape with guide-to-eye indicating the MM data projected on the Ha-Hc plane. The green stars marks the location of the QCEPs in the Hc −Ha plane. The green line connecting the QCEPs is the quantum critical line. The orange ovals are the location of metamagnetic transition. The color of the ovals and the color filling qualitatively indicate the amplitude of the metamagnetic transition. The line on the data point indicates the uncertainty of the MM transition by analyzing the width of the transition with respect to magnetic field as per Sec. 2.5. It is easier to see the suppression of the MM transition in derivative with respect to the magnetic field, that is, ∂ f/∂H. In Fig. 4.5, the raw and derivative of the PDO data for three rotation planes b− c, b−a and b−ac (as explained in Fig. 4.2) at 4.2 K are shown. The data were measured by PDO techniques in pulsed magnetic fields up to 57 T. The peak location of the derivative curve matches the center-field location of the MM transition at this field orientation and agrees with data from Fig. 3.14. The prominence and full width at half maximum of the peak quantitatively show the abruptness of each transition. As the field is 84 High Magnetic Field Phase Landscape of UTe2 rotated away from the b-axis and the HM of the MM transition shifts to a higher magnetic field, the height of the derivative peaks diminishes, and the width of the derivative peak broadens. The diminishing derivative peaks better illustrate the suppression of the CEP and capture that the CEP had been suppressed to lower than 4.2 K at θb−ac = 42°. 20 40 60 0H (T) 0 100 200 300 400 f PD O (k Hz ) 0° 15° 24° 32° 36° 37° 40° 41° 44° 47° 48° 49° 51° 30 40 50 0H (T) f/ H (a rb .) b - c T = 4.2 K 20 40 60 0H (T) 0 100 200 300 400 500 f PD O (k Hz ) 33° 22° 21° 19° 17° 13° 12° 11° 7° 3° 0° 30 40 50 0H (T) f/ H (a rb .) b - a T = 4.2 K 20 40 60 0H (T) 0 100 200 300 400 500 f PD O (k Hz ) 0° 12° 22° 27° 35° 39° 42° 46° 30 40 50 0H (T) f/ H (a rb .) b - ac T = 4.2 K A B C Fig. 4.5 PDO data and derivative with respect to the magnetic field ∂ f/∂H that tracks the softening of MM transition in (A) b− c, (B) b− a and (C) b− ac rotation planes. The derivative curves clearly show the location and abruptness of the MM transition. 4.1 Discovery of a Quantum Critical Line in UTe2 85 Despite the fact that both Ha and Hc can suppress the first-order MM transition and the CEP, the effect is drastically different. Fig. 4.6 compares the temperature-dependence of the MM transition and the suppression of CEP between b− c and other rotation planes. The upper row includes three different orientations in the b− c plane. The CEP remains unchanged through the b− c rotations. For θb−c = 47°, while the MM transition occurs at 55 T, which is at the edge of the field range, the CEP is still higher than 9 K. The lower row compares three different field orientations where HM ≈ 55 T with selected temperatures. The approximate value of CEP for θb−c = 47° is 9 K. In comparison, the CEP for the θb−a = 21° is 7.5 K and for the θb−ac = 43° is 1.5 K. These measurements illustrate that the suppression effect on the CEP of the Ha component is larger than the Hc component and the combination of both two transverse components could largely suppress the CEP compared to any one of the transverse field components. Due to the stiffness of the MM transition against the tuning of the pure Hc component, the first-order MM transition persists up to θb−c = 60° and HM = 70 T. Therefore, the QCEP in the b−c rotation plane is inferred by extrapolating the size of the MM transition, as shown in Fig. 4.7. 86 High Magnetic Field Phase Landscape of UTe2 20 40 60 0H (T) 1.5 K 2.8 K 4.2 K 6.4 K 9.0 K 10 K 12 K 15 K 20 K 35 K 50 K b-c = 47° 20 40 0H (T) 4.2 K 6.0 K 9.0 K 12 K 15 K 20 K 35 K 50 K b-c = 32° 20 40 60 0H (T) 0 100 200 300 f PD O (k Hz ) 1.5 K 3.0 K 4.2 K 6.0 K 9.0 K 12 K 15 K 20 K 50 K b-c = 0° 20 40 60 0H (T) 50 150 250 350 f PD O (k Hz ) 1.5 K 2.8 K 4.2 K 6.4 K 8.5 K 10 K 12 K 15 K 20 K 35 K 50 K b-c = 47° 20 40 60 0H (T) 1.5 K 1.9 K 3.0 K 4.2 K 6.0 K 7.5 K 9.0 K 10 K 11 K 12 K 15 K b-a = 21° 20 40 60 0H (T) 1.5 K 4.2 K 6.0 K 8.0 K 11 K 12 K 15 K b-ac = 43° A B C D E F Fig. 4.6 Temperature evolution of MM transition at different magnetic field orientation as an extended dataset for Fig. 4.2. PDO data at incremental temperatures for each of these magnetic field angles away from b-axis. Top row: magnetic field rotates in b-c rotation plane, the CEP value is not suppressed to below 9 K up to the highest accessible MM field (55 T). Bottom row: Temperature evolution at field orientation away from b-c plane, the CEP is suppressed to below 9 K and even lower for b−ac rotation. These data show that the QCEP is lower in field when field is rotated away from b− c plane. Panel (D) is a replication of (C) for the ease of comparison. The previous dataset covered multiple different rotation planes and mapping the supercon- ducting and magnetic phase diagram of UTe2 with respect to θb−c and θa. In order to better understand the three-dimensional property of UTe2, all the phase boundaries and QCEPs are converted into data points in the Cartesian magnetic field axes of Ha-Hb-Hc. The resulting phase diagram with data points is shown in Fig. 4.8. SC1 and SC2 are the ambient superconductivity and the field-induced superconducting phase near the b-axis as described in Sec. 3.3, which are marked by the blue and pink regions. The purple region corresponds to the field-induced SC3 with a large offset angle away from the b-axis. Further measurements on this superconducting phase will be shown in later sections. 4.1 Discovery of a Quantum Critical Line in UTe2 87 40 60 80 0H (T) 0 40 80 f PD O (k Hz ) 0° 60° b - c 35 55 75 0Hm (T) 0 0.5 1 M M s ize (a rb .) MM size Fig. 4.7 Tracking MM transition in b− c plane and extrapolation of the (potential) QCEP in this rotation plane to high magnetic field. The amplitude of the data ∆ fPDO from the dual coil magnet are rescaled by 1.1 to match with the MM size at θb−c = 24°, enabling quantitative comparison. This rescaling may come from the difference in the sample, PDO coil and the background due to different ∂H/∂ t. Since this sample is different from that in Fig. 4.4 thus, the scaling factor is different. The size of the MM transition is defined as the change in frequency after background subtraction, as illustrated here. The MM size versus HM follows a linear fitting, which extrapolates to QCEP (marked by the open green star) at µ0Hc = 69 T, which corresponds to the QCEP at µ0HM = 76.7 T. For the MM landscape, it has a minimum value of µ0HM,b = 34 T at H//b. The value of HM is almost constant when Hc is applied, while it increases drastically when Ha is applied. All the data points about HM with different θb−c and θa are from the various different rotation planes from Fig. 4.5 and Fig. 4.4. These data points span a MM transition surface (orange), which separates the normal paramagnetic state and the polarised paramagnetic state (PPM). Throughout the isothermal rotation dataset as in Fig. 4.4 and the temperature dependent dataset at different field orientations as in Fig. 4.6, the CEP value is at its maximum of 9 K with H//b. No matter what rotation plane the magnetic field rotates in, once sufficient amounts of Ha and Hc components are applied, the CEP would be suppressed to lower temperatures. As shown in Fig. 4.4, all these different rotation planes lead to a QCEP when H is rotated away from b-axes. These QCEPs connect into a continuous quantum critical boundary in the three-dimensional field space, termed as the “quantum critical line” (QCL), as indicated by the green line. This construction sets the dimensionality of the high magnetic field phase diagram of UTe2. 88 High Magnetic Field Phase Landscape of UTe2 In more conventional cases such as UGe2 [30], the tuning parameter hydrostatic pressure (p) is a one-dimensional quantity which leads to one single QCP in the T − p phase diagram. For URhGe as introduced in Sec. 1.3, a similar MM transition is induced by Hb and the MM transition is tuned by Hc. Since its MM transition does not respond to Ha, it is a two-dimensional case for URhGe. Therefore, it has two QCPs in the Ha-Hc phase diagram as shown in Fig. 1.6. For UTe2, transverse field components Ha and Hc both tune the MM transitions, resulting in a three-dimensional MM transition surface. The enhanced dimensionality of UTe2 phase diagram leads to infinitely many QCEPs that connect to form the QCL, which could give rise to exotic physics and the accompanying field-induced SC3 phase. PPM SC1 SC2 SC3 SC3 spillover SC3 onset Fig. 4.8 Construction of the high magnetic field phase diagram. Here the positive magnetic field Cartesian quadrant of the schematic phase diagram is displayed, including data points used to bound the SC1, SC2, SC3 and PPM phases. Each data point corresponds to either a DC field sweep or a pulsed field measurement. Each green star marks a QCEP identified by a isothermal rotation from Fig. 4.4. The green line connecting all the QCEPs is the QCL that terminates the MM transition surface at T = 1.7 K. The open square point marking the top of SC3 is taken from ref. [97] at 0.7 K. Data points of all three superconducting phases (SC1, SC2, and SC3) were taken at 0.4 K. The SC3 spillover and onset regions are identified from Fig. 4.10. The line on the data points indicate the corresponding error bar as per Sec. 2.5. The orientation of the error bar indicates the orientation of field sweep by which the data point of phase boundary is taken. Here, the MM transitions were measured at 1.8 K. The SC3 were measured at 0.7 K. The SC1, SC2 and SC3 spillover were measured at 0.4 K. 4.1 Discovery of a Quantum Critical Line in UTe2 89 Fig. 4.9 Numerical simulation of the MM transition surface and QCL based on a Landau free energy expansion as in Eq. 4.1. The orange surface correspond to the MM transition surface and the green line is the QCL. The MM transition jump at a certain field orientation could be captured by numerical simulation based on Landau free energy expansion, as shown in Fig. 4.3. The free energy expansion model is as shown in Eq. 4.1. F [M](H) = 1 2 χ −1 i M2 i + 1 4 βi jM2 i M2 j + 1 6 δi jkM2 i M2 j M2 k −M ·H (4.1) where i, j,k = a,b,c and βi j and δi jk are the model parameters. Similarly to the toy model illustrating a possible origin of MM as in Eq. 3.3, the free energy is expanded with respect to magnetization and field in a three-dimensional field space. The locations of MM jumps for each H are determined by minimizing the free energy. The QCL is determined by finding the field orientation along which the MM jump vanishes. By minimizing the free energy in the form of Eq. 4.1, the resulted HM as a function of the field orientation is shown in Fig 4.9, which agrees qualitatively with the experimental result in Fig. 4.8. 90 High Magnetic Field Phase Landscape of UTe2 4.2 Spillover of SC3 As discussed previously in Sec. 1.3, in the ferromagnetic superconductor URhGe, a system similar to UTe2, the magnetic field induces a superconducting phase in the vicinity of a QCP at µ0H ≈ 10 T, which is also related to a magnetic transition [56]. Notably, the field-induced superconductivity emerges on both sides of the magnetic transition, with its zero resistivity occurring predominantly in its FM ground state below the magnetic transition with some region of spillover into the paramagnetic state above the magnetic transition as shown in Fig. 1.7. In this section, evidence of a similar phenomenon in UTe2, where field-induced superconductivity spills over into the normal paramagnetic phase below the MM transition, is presented. Fig. 4.10(A) presents the resistivity as a function of magnetic field at selected incremental angles θb−ac away from the b-axis in the b−ac rotation plane. These data were obtained in a single cool-down. The minimal HM among these MM transition is 34 T, indicating that the rotation plane passed the b-axis. The geometry of the rotor setup is shown in Fig. 4.2. In this configuration, a combination of the Ha, Hb and Hc field components was applied to UTe2. As the sample rotated through the MM-SC3 phase boundary at θb−ac = 16.5°, the MM transition marked by the abrupt increase in resistivity was replaced by SC3, indicated by zero resistivity. The zero resistivity all the way up to 41.5 T was due to the coincidence of SC1-SC2-SC3 at this field direction at T = 0.4 K as magnetic field sweeps up. As the angle continued to increase, the SC2 gradually lost its strength and moved to lower field. The normal paramagnetic state emerges between the SC1/SC2 and the SC3 onset regime. The SC3 onset regime is defined by the negative magnetoresistivity regime starting at where the ρ(H) curve reaches its maximum and ends at where the resistivity reaches zero and the sample enters SC3 state. The location of the SC3 onset where the ρ(H) curve reaches its maximum is highlighted by a guide-to-eye in Fig. 4.10. These results are better summarized in the field - angle phase diagram on the right-hand side. This experiment reveals a dome-like behaviour of the SC3 onset regime as a function of the rotation angle and a field orientation where SC1-SC2-SC3 phases co-align in the b−ac rotation plane. 4.2 Spillover of SC3 91 SC3 onset SC3 UTe2 0 10 20 30 40 5010 20 30 40 0H (T ) b acb-ac (°) 10 20 30 40 0H (T) 0 2 4 6 (µ c m ) = 20° = 35° A B 59.0 MM 0.0 16.5 34.5 17.5 PPM SC2 SC1 µ DC c b a ac Fig. 4.10 Mapping the field angle phase diagram of SC3 spillover outside the polarized paramagnetic phase. (A) Contacted resistivity ρ(H) at selected rotation angles in the b−ac plane at 3He base temperature T = 0.4 K. The angle of each curve is colour coded and indicated in the colour bar. Angle value for important orientations are also marked by numbers. (B) The magnetic field - field angle phase diagram for UTe2 in b− ac rotation plane. The superconducting phase boundaries and the SC3 onset are identified by panel (A). The MM transitions marked by red ovals are extracted from data at 6 K in Fig. 4.11B. The spillover region determined by zero resistivity happens at lower field than the metamagnetic transition at the same angle at 6 K is clearly seen. A clear inverse dome like shape of the SC3 onset region, marked by the onset of the ρ(H) decreasing, could be seen. The angular dependence of both SC1 and SC2 phases follows the trend of the b− c rotation plane. The dashed line is a guide-to-eye indicating the angle of the selected field sweeps in panel (D). The error bars are extracted with the methods as in Sec. 2.5 (C) Schematic model explains the b−ac rotation plane. The sample (black) is mounted on a 16.5° wedge, while the magnetic field rotates from along b-axis to along ac direction.(D) Selected field sweeps at θ = 20° and θ = 35°. In order to illustrate the spillover of SC3 below the MM transition, a comparison of the ρ(H) dataset in the b− ac rotation plane between T = 0.4 K and 6 K is presented in 92 High Magnetic Field Phase Landscape of UTe2 Fig. 4.11. The 6 K dataset lies above the critical temperature of SC3 and therefore could help highlight the MM transitions HM. The most important example of SC3 spillover occurs at θb−ac = 31°, marked in blue. The low-temperature dataset shows that the zero-resistivity starts at µ0H = 40 T while at 6 K, the MM transition occurs above 41.5 T. For other angles, the MM transition field HM is identified by the maximum resistivity location, which corresponds to the location of the crossover between the normal paramagnetic state and the polarised paramagnetic state. It should be noted that HM at a higher temperature above the CEP is lower than HM at a lower temperature as shown in Fig. 3.9; therefore, this criterion underestimates HM, making the evidence for spillover even stronger. The HM values at different rotation angles θb−ac is plotted in the angle phase diagram with the red oval, and the superconducting phase boundary of leaving / entering zero resistivity is plotted by triangles of the corresponding colours, SC1 - blue, SC2 - pink, and SC3 - purple. It is clear that the phase boundary for entering the SC3 zero-resistivity state lies at significantly lower magnetic fields compared to the MM transition HM at this angle. The data at θb−ac = 31° for both T = 0.4 K and T = 6 K are marked with blue to show that zero resistivity is observed at low temperature while the MM transition at high temperature is above the maximum field of the experiment. The extracted HM - θb−ac curve is plotted in Fig. 4.10B and it is clear that for 20° < θb−ac < 35°, the SC3 entrance phase boundary is lower in magnetic field compared with the MM transition field HM at the corresponding angle. This marks the observation of the SC3 spillover regime in the angular dependent study. It is worth noticing while the resistivity remains zero due to the superconductivity SC3 at low temperature, the metamagnetic transition persists as observed by extraction magnetization [135]. To further illustrate the spillover of SC3, a careful temperature-dependent study of SC3 - MM at fixed angles is presented. Fig. 4.12 shows two temperature-dependent datasets for θb−ac = 0° and θb−ac = 35°, measuring ρ(H) and presenting the derivative with respect to the magnetic field ∂ρ/∂H. The HM of each temperature is determined by the point where the ∂ρ/∂H curve changes sign, marked by the arrows of the corresponding colour. Most importantly, for θb−ac = 35°, zero-resistivity happens at 40 T for T < 0.7 K, while the HM remains to be higher than 41.5 T up to T = 22 K. These results are summarized in the temperature-magnetic-field phase diagram shown in Fig. 4.14B. For θb−ac = 35°, the SC3 and its onset regime extend into the field range where the derivative curve indicates the HM is higher in field, as marked by the orange cross. Moreover, a further temperature-dependent dataset directly observed the zero resistivity of SC3 happens at lower field comparing with the MM transition HM, as shown in Fig. 4.13. These further demonstrate a clear spillover of the SC3 outside the PPM regime. 4.3 Angular dependence of Tc in SC3 of UTe2 93 10 20 30 40 70H (T) 0 4 8 12 16 ; (µ + c m ) T = 0.4 K 3b-ac 31° -30° 0° 30° 60° 10 20 30 40 70H (T) 0 50 100 150 200 ; (µ + c m ) T = 6 K 27.5° 31° 0° 20° 40° 3b-ac µ µ Fig. 4.11 (A) full dataset of ρ(H) at successive rotation angles at T = 0.4 K. The blue line marks the first angle where clear spillover SC3 is observed. (B) rotation dataset at T =6 K. The location of MM transition is determined by the maximum of resistivity. For the θb−ac = 31° (marked by blue), the MM transition happens at higher magnetic field than 41.5 T while the field-induced zero-resistivity happens below 40 T as shown in (A). 4.3 Angular dependence of Tc in SC3 of UTe2 For any superconductor, the critical temperature is an important property. For the field- induced superconducting phase SC3, Tc of a given magnetic field orientation could be defined as the highest temperature at which superconductivity still persists above the MM transition. A careful mapping of Tc as a function of the field angle is essential to understand the physics of SC3. Contacted and contactless electrical resistivity measurements in steady and pulsed mag- nets were conducted at different temperatures and field orientations in the b− c rotation plane. The Tc and upper critical field at each field orientations were measured at multiple different field angle from θb−c = 21° to θb−c = 39°. An isothermal rotation at 1.5 K was also performed to map the Hc2 as a function of rotation angle θb−c. The resulted field - angle superconducting phase diagram focusing on SC3 is presented in Fig. 4.15(E). For each isothermal rotation study at 0.7 K, 1.5 K and 2.2 K respectively, Hc2 vs. θb−c shows a dome-like dependence with its maximum at θb−c ≈ 35°. 94 High Magnetic Field Phase Landscape of UTe2 0 10 20 30 40 0H (T) 0 100 200 300 (µ c m ) 5 K 15 K 22 K 28 K H || b 0 20 40 0 3 6 0.4 K 0.7 K 1.2 K 1.5 K 0 10 20 30 40 0H (T) 0 100 200 300 (µ c m ) 5 K 15 K 22 K 28 K b-ac = 35° 0 20 40 0 3 6 0.4 K 0.7 K 1.2 K 1.5 K µ µ 0 20 40 0H (T) -20 -10 0 10 20 / H (a rb .) 5 K 15 K 22 K 28 K 0 10 20 0 20 40 0H (T) -2 0 2 4 6 8 / H (a rb .) 5 K 15 K 22 K 28 K || A B C D Fig. 4.12 Temperature dependent study of relation between field-induced SC3 and meta- magnetic transition. (A) Resistivity ρ with respect to magnetic field H for H//b and its corresponding ∂ρ/∂H in (B). The inset shows the resistivity measurement of SC2 at low temperatures. (C) ρ(H) at θb−ac = 35°, and (D) its derivative versus field ∂ρ/∂H. For temperature above CEP, the MM field HM is determined by the maximum of ρ(H), which is the zero value of the derivative, marked by the arrow. The spillover of SC3 could be seen in (C) by the observation of zero resistivity at low temperature above 40.5 T while the resistivity is still growing at 41.5 T for 5 K, indicating that the HM is higher than the onset field of SC3 for this orientation. 4.3 Angular dependence of Tc in SC3 of UTe2 95 10 20 30 40 0H (T) 0 10 20 (µ c m ) 0.4 K 0.8 K 1.1 K 1.3 K 1.6 K 8 K 10 K = 25° ÷ 8 ÷ 8 µ Fig. 4.13 A direct observation of SC3 spillover. ρ(H) curve at incremental temperature with magnetic field applied along θ = 25° in b− c rotation plane. The two temperature curves at high temperatures, 8 K and 10 K, are rescaled by a factor of 1/8 to better compare the field-induced superconducting phase SC3 and MM transition. To settle the location where Tc of SC3 reaches its maximum, Fig. 4.16 shows a detailed comparison of the sensitivity of SC3 to temperatures between two close angles θb−c = 33° and θb−c = 35°. Since the angles of these two measurements were fixed with a careful machined wedge, the error bar of the angle is less than 0.5°. It is apparent that the “upper critical field” Hc2 measured at the lowest temperature and the extrapolated Tc are both larger for θb−c = 35°. A detailed set of measurement of ∆ f (H) with finer steps in temperature values is made for θb−c = 35°. In Fig. 4.17A, contactless resistivity measurements with PDO are shown at this orientation θb−c = 35° at a series of temperatures through the transition temperature of SC3. In a single sweep, the transition into the SC3 phase is identified by a sharp decrease in the derivative of the signal. This dip is still clearly visible at T = 2.3 K, but is gone at T = 2.5 K. For T ≥ 2.5 K, the sample enters the PPM through a MM at the same HM ≊ 45 T. The “upper critical field” Hc2 is identified by the Gaussian-like peak in the ∂ f/∂H derivative curve. As the temperature increases, the value of Hc2 decreases and the Gaussian-like peak above the SC3 transition ceases to exist as the temperature increases above Tc. This gives a consistent way of extracting the Tc for a dataset with ∆ fPDO at multiple different temperatures for a certain magnetic field orientation. 96 High Magnetic Field Phase Landscape of UTe2 20 40 0H (T) b-ac = 35° Tc > 1.8 K 0 1 2 K 0 20 40 0H (T) 0 2 4 6 8 (µ c m ) b-ac = 18° Tc 0.5 K 0 20 40 0H (T) 0.4 1 2 5 10 20 30 b-ac = 0° H || b Normal paramagnet Polarised param agnet 20 40 0H (T) b-ac = 35° Normal paramagnet SC3 SC3 onset T (K ) µ A B SC2 SC1 SC1 Fig. 4.14 (A) Comparison of the “critical temperature” of the field-induced SC3 at the edge (left) and in the center (right) of the field angle phase diagram. The ρ(H) at incremental temperatures at θb−ac = 18° and θb−ac = 35°. The ρ(H) measurement temperature is as indicated by the colour bar. The Tc is marked by the temperature of the data curve at which the zero resistivity is not observed. (B) Evolution of the temperature - magnetic field phase diagram for H//b and θb−ac = 35°. The superconductivity phase boundaries are decided by the edge of zero resistivity from Fig. 4.12. The SC3 onset is the field location where the resistivity reaches the maximum. The first-order MM transition is marked by the solid orange oval and the second-order like crossovers above the CEP are marked by open oval. The cross marks that the MM transition is not observed up to the highest magnetic field but is expected to happen as shown in Fig. 4.12(D). Remarkably, spillover zero resistivity SC3 is measured to below 40 T while the MM at 5 K is above 41.5 T. The error of superconducting and metamagnetic transition is determined by the width of the transition as per Fig. 2.11. With the criterion of determining T SC3 c , the critical temperatures were extracted from various different datasets as shown in Fig. 4.15 from θb−c = 21° to θb−c = 46°. The resulting Tc - θb−c relation is shown in Fig. 4.17. T SC3 c (θb−c) exhibits a dome-like angular dependence, being only ≈ 0.6 K in θb−c = 21°, extending to ≈ 2.4 K in θb−c = 35°, and then reducing to ≈ 1.8 K in θb−c = 39°. From the isothermal rotation mapping of SC3 with pulse-field PDO measurements as shown in Fig. 4.18, the dip in the PDO frequency as a signature of SC3 only survives up to θb−c = 45°. This sets the upper angle limit of SC3 at 0.7 K in Fig. 4.15E. 4.3 Angular dependence of Tc in SC3 of UTe2 97 It should be noted that the temperature values quoted for the pulse field curve ∆ f (H) were decided by the temperature reading of the RuOx thermometer immediately before the pulse. Because pulsed fields induce heating due to eddy currents, the actual sample temperature during the pulse is systematically higher than the recorded thermometer value. Furthermore, the ∆ fPDO data curves above µ0HM = 45 T is transformed into a colourmap by setting the end of the frequency drop superconducting transition at base temperature T = 0.6 K as the zero point of the PDO frequency shift. This colourmap is presented in Fig. 4.17C. The extent of SC3 in the field - temperature phase diagram is restricted by the dashed line connecting the Hc2(T ) data points. This pinpoints the critical temperature for the field along θb−c = 35° to be T SC3 c = 2.4 K. From the Tc - θ dome in Fig. 4.17(E) and the comparison between the H-T superconducting phase diagram with field along θb−c = 35°, one thing is well demonstrated: the critical temperature of field-induced SC3 phase (Tc = 2.5 K) is enhanced comparing with zero-field SC1 phase (at clean limit Tc = 2.1 K, see ref. [83]) in pristine quality UTe2 single crystal. Similar enhancement of Tc for the field- induced superconducting phase has been seen in URhGe (introduced in Sect. 1.3), which was related to the magnetic fluctuation associated with a transverse field-induced QCEP at µ0H = 13.5 T [57]. In UTe2 for field tilted by θb−c, the SC3 phase starts at HM = 45 T and persists up to remarkably 70 T [97]. This similarity indicates that the SC3 phase in UTe2 originates from a similar mechanism of field-induced quantum criticality. 98 High Magnetic Field Phase Landscape of UTe2 A B C D E 2.2 K 1.5 K 0.7 K 0.4 K Fig. 4.15 Angular evolution of critical field of SC3. (A) Contacted resistivity versus magnetic field at incremental temperature at the edge of SC3 for field along θb−c = 21°. T SC3 c ≈ 0.6 K. Above this temperature the drastic jump of resistivity at HM due to MM instead of zero resistivity due to SC3 is observed. The inset shows superconductivity due to SC2 up to 32 T at 0.4 K. (B), (C), temperature dependent sweeping field measurement of contactless resistivity at θb−c = 32° and θb−c = 39°. The critical temperature of SC3 at these two angles are 2.2 K and 1.7 K correspondingly. From the raw data in (A), (B) and (C), a non-monotonic dependence between SC3 Tc and θb−c could be seen. (D) Isothermal rotation of ∆ fPDO through the SC3 dome at different θb−c as indicated by the number next to the corresponding curve. The upper boundary of SC3 could be seen from the slope change of ∆ f (H) curve. The dome-like dependence of upper boundary of SC3 could be seen for different temperatures. The green triangles marks the “upper critical field” of SC3. (E) field - field angle θb−c phase diagram focusing on SC3 dome at high magnetic field. Hc2 measured at different temperatures are indicated by the colour and text. Orange region indicates the polarised paramagnetic regime, pink region marks the field-reinforced SC2 phase and the purple region marks the field-induced SC3 phase. The colour gradient marks the SC3 phase at different temperatures. The uncertainty is analyzed in the way as described in Sec. 2.5. The MM transition was determined at 4.2 K as shown in Fig. 4.5. The SC2 was measured at 0.4 K. 4.3 Angular dependence of Tc in SC3 of UTe2 99 30 40 50 60 70 0 H (T) 0 0.05 0.1 f P D O ( kH z) 0.7 K T = 1.7 K 0.7 K1.8 K b-c = 33° b-c = 35° 40 50 60 0 H (T) -0.5 0 0.5 f/ H ( ar b. ) 0 1 2 T (K) 40 55 70 0H ( T ) b-c = 33° T c 2.3 K 0 1 2 T (K) 40 55 70 0H ( T ) b-c = 35° T c 2.4 K Fig. 4.16 Fine angular mapping of lower- and upper- bound of SC3 in temperature-magnetic field phase diagram near the optimal field orientation. Here a direct comparison of the field-temperature phase diagram of SC3 between two close θb−c near the center of the SC3 dome is made. The angles of these two datasets were obtained by mounting the same samples on an angular wedge. Thus, the error of these angle values is less than 0.5°. Top panel: direct comparison of SC3 at two selected temperatures at θb−c = 33° and 35° is made. The gaussian-like peak in ∂ f/∂H indicates the SC3 upper boundary of corresponding θb−c and temperature. Lower panel: magnetic field - temperature phase diagram of two different field orientations. Error bars are extracted as per Sec. 2.5. 100 High Magnetic Field Phase Landscape of UTe2 A B C D E bc - plane SC1 SC3 Fig. 4.17 Enhancement of the critical temperature for field-induced SC3 phase. (A) PDO data of skin depth as a function magnetic field at different temperatures. This measurement was taken at θb−c = 35°, at which the critical temperature is the highest as indicated by rotation study. The drastic transition of skin depth was due to the field-induced superconductivity SC3 (MM) transition at low (high) temperature, marked by a decrease (increase) of skin depth at the transition HM. (B) The derivatives of skin depth with respect to field H. The sign of derivative shows the nature of transition. The hump after the superconducting transition indicates the “upper critical field”. The critical temperature Tc is determined by the highest temperature value below which the decrease of skin depth at the transition happens. (C) The colour map magnetic field - temperature phase diagram of SC1 - SC3, for θb−c = 35°. The colour above transition are constructed from the data in (A). The purple triangles that mark the superconducting transition HM upper critical point Hc2 are determined by the peak location of the derivative curve. The phase boundary of SC1 was determined by a separate dataset from a resistivity measurement in a 14 T PPMS with a 3He module. Error bar of the upper critical field is determined by the σ of the gaussian fitting of ∂ f/∂B near the upper critical field range as indicated by the triangle in panel (B) as per Sec. 2.5. (D) The critical temperature as a function of the rotation angle θb−c. (E) The critical temperature as a function of the MM field HM. It shows a dome like feature centering at the region in close vicinity to θb−c = 35°. The error bar of the critical temperature indicates the difference between temperatures of the measurements before and after the one at which the SC3 - MM transition happens. 4.3 Angular dependence of Tc in SC3 of UTe2 101 Moreover, SC3 is mapped in multiple other rotation planes at T = 0.7 K. In Fig. 4.18, the result of PDO measurement in a pulsed magnet up to 57 T at the incremental rotation angles in (A) b− c, (B) b−ac and (C) θb−c = 33° towards a is shown. From the derivative curve ∂ f/∂H, it can be seen how the entrance magnetic field HM indicated by the dip in the PDO frequency and the upper critical field Hc2 (the Gaussian-like peak) evolve with the rotation angle. Here, the angles are determined by the angle pick-up coil and therefore have an uncertainty of less than 0.5°. From the b−ac dataset, the dome-like non-monotonic angular dependence of Hc2 is observed as the field is tilted away from the b-axis. From the bc−a rotation, the SC3 regime exists only within a narrow range of tilted angle towards the a-axis. All these extracted SC3 entrance fields and rotation angles are transformed into data points of Ha-Hb-Hc field space and are presented in Fig. 4.18D. These data points further constrain the shape of the SC3 toroidal shape in three-dimensional field space. 0 20 40 60 0H (T) 0 200 400 600 f PD O (k Hz ) 25° 26° 28° 37° 39° 35° b-ac = 41° UTe2, b - ac T = 0.7 K 30 40 50 60 0H (T) f/ H (a rb .) 0 20 40 60 0H (T) 0 200 400 600 f PD O (k Hz ) 2° 5° 7° 9° 11° 12° a = 22° b-c = 33° T = 0.7 K 30 40 50 60 0H (T) f/ H (a rb .) 0 20 40 60 0H (T) 0 200 400 600 800 f PD O (k Hz ) 25° 26° 30° 32° 40° 42° 44° 45° 46° UTe2, b - c T = 0.7 K 30 40 50 60 0H (T) f/ H (a rb .) A B C D Fig. 4.18 Mapping the magnetic field - field angle phase diagram of SC3 in pristine quality UTe2 with skin depth measurements via PDO. In (A) and (C), the onset Hm and upper critical field Hc2 of SC3 are mapped at incremental angles away from b-axis in b− c and b− ac rotation planes correspondingly. As the magnetic field H is tilted away from b-axis, the onset field increases and the Hc2 shows a dome-like dependence of the rotation angle. (D) Bird-eye view of the SC3 toroidal-shape in the first quadrant. The greens stars and green line marks the QCEPs and QCL correspondingly. The orange and purple color region are metamagnetic transition landscape and the SC3 dome. The solid triangles are SC3 phase boundary determined by pulsed-field PDO measurements. The open triangles are extracted from the data in ref. [96]. The error bar is extracted by calculating the width of the transition as per Sec. 2.5. 102 High Magnetic Field Phase Landscape of UTe2 4.4 Discussion PPM SC2 SC1 SC3 Fig. 4.19 High field MSF-grown UTe2 phase diagram for magnetic field orientations in the b− c and b− a rotational planes. The phase boundaries of the SC1, SC2 phases are determined by my measurements reported in previous chapter. 3 as indicated by the open symbols of corresponding colour. The phase boundary of polarized paramagnetic (PPM), i.e. the HM as a function of θb−c and θb−a at two different temperatures are depicted by the solid oval of corresponding colour. The black line is a fitted curve of the MM transition data points and truncates at a quantum critical endpoint (QCEP) as indicated by the green star determined by the softening of the MM transition measured by PDO. The blue symbols shows the temperature evolution of the entrance and “upper critical field” of SC3 centering at θb−c = 35°. The 0.4 K curve is a guide-to-eye, while other three temperatures are quadratic fitting. Error bar is determined by the method described in Fig. 2.11. Spillover of SC3 was observed in b− c rotation plane later with higher quality sample. In order to be consistent, the phase region is not included here. As a summary of this chapter, a complete superconducting phase diagram in three- dimensional field space with respect to Ha - Hb - Hc Cartesian axes, and an updated field angle phase diagram including the three superconducting phases, the MM transition landscape, and the QCL have been constructed, as shown in Fig. 4.19 and Fig. 4.20. These phase diagrams are inspired by and directly constrained by experimental data points from various rotation studies presented in Fig. 4.8. In the 3D phase diagram with Ha - Hb - Hc Cartesian axes in Fig. 4.8 and Fig. 4.20, the magnetic field can be seen as a vector starting from the origin, and 4.4 Discussion 103 the length of this vector reflects the strength of the applied magnetic field. The magnetic field vector rotates in Cartesian space and passes phase boundaries. For the ambient pressure superconductivity in UTe2 (SC1) marked by the blue region, its boundary could be naturally modeled by a paraboloid with different intercept along the Ha, Hb, and Hc axes. The intercepts were determined by the Hc2 value extracted from field-dependent ρ(T ) studies as discussed in Chap. 3. This shape is also implied by the Hc2 data in the b−ac rotation planes in Fig. 4.8. As the magnetic field increases while remaining near the b-axis, UTe2 enters the field- induced superconducting phase SC2, marked by the pink region. Its phase boundary of SC2 is also fitted by a paraboloid, with its center at Hb = 20 T, indicated by the heat capacity data from ref. [91] and the PDO characteristic of Fig. 3.7. The paraboloid is restricted by the angular range of SC2 in b− c, b−a, and b−ac rotation planes and the MM transition surface from experiments. The MM transition landscape is bounded by the quantum critical line (QCL) connected by the QCEPs. The QCEPs are marked by a green star and determined by the location where the isothermal rotation study of the pulsed-field PDO measured MM transition is suppressed. The MM transition landscape is then interpolated based on the QCL and the data points from the experiments of the b− c, b−a, b−ac and bc−a rotation planes. The high magnetic field-induced superconducting SC3 phase is marked by a purple region. From our b− c and b−ac rotation study, the phase boundary of SC3 shows a dome shape in the polar angle ϕ tilted away from the b-axis and centers at ϕ = 35° for two measured azimuthal angles θb−c = 0° and θb−c = 17.5°. As shown in the dataset for rotating towards the a-axis with an offset angle θb−c in Fig. 4.18, the SC3 region is a torus shape surrounding the b-axis. With these constraints, the SC3 is modeled by a torus-shaped region with a dome rotating around the b-axis centering at an offset polar angle and shrinks as the azimuthal angle away from the Hc axis increases. Quantum criticality is behind many exotic phenomena, including unconventional super- conductivity and strange metallicity, as discussed in Sec. 1.2. In conventional cases, the quantum phase transition occurs at one point in the phase diagram at zero kelvin. Strikingly, in UTe2 as discussed in Sec. 4.1, there are infinitely many QCEPs and they connect into a QCL that terminates the MM transition surface. This discovery expands the horizon for understanding the quantum criticality in higher dimensional space. Another enigma of UTe2 is the field-induced SC3 phase. As discussed in Sec. 1.4.5, it requires the magnetic field to be tilted by an angle away from b-axis with no clear crystal symmetry [88] and shows surprising stability against impurity [98]. One proposal for the origin of the SC3 phase is the Jaccarino-Peter effect [136]. It has successfully explained 104 High Magnetic Field Phase Landscape of UTe2 field-induced superconductivity in several other materials [137, 138]. It assumes that UTe2 have a strong internal exchange magnetic field after the MM transition, and the external magnetic field compensates the internal magnetic field and thus reveals the superconducting ground state [97]. Our observation of SC3 spillover below the MM transition strongly argues against this scenario, as the observation of zero resistivity before the MM transition imposes the restriction that SC3 can occur before the magnetic structure of UTe2 changes drastically due to the MM transition. Furthermore, careful mapping of SC3 at different temperatures and tilt angle in the b− c rotation plane shows that the maximum Tc of SC3 with field along θb−c = 35° is enhanced compared to the zero field SC1 phase at the clean limit [83]. This enhancement of Tc for the field-induced superconducting phase is similar to URhGe, whose superconductivity was driven by a field-induced QCEP. This similarity indicates that the SC3 phase is driven by some magnetic fluctuation. Considering that the toroidal shape of the SC3 phase in the Ha- Hb-Hc Cartesian axes is enclosed by the QCL, all three discoveries discussed in this chapter come together to the conclusion that the SC3 phase is driven by the magnetic fluctuation accompanying the novel enhanced dimensional QCL consisting of infinitely many QCEPs. 4.4 Discussion 105 Polarised paramagent QCL SC3 SC2 SC1 QCL QCEPs A B C SC3 PPM Fig. 4.20 The high magnetic field superconducting phase diagram of UTe2 in Ha −Hb −Hc cartesian, featuring the quantum critical line (QCL). (A) The 3D superconducting phase diagram of UTe2. This 3D phase diagram is inspired and constrained by experimental data. For the ground state superconducting state SC1, the blue rotating paraboloid intercept the Hc and Ha axis at the corresponding Hc2 value of MSF-grown UTe2 samples as discussed in Fig. 3.4. The top point of the SC1 paraboloid was determined by the Hc2 for H//b. The shape of SC2 pink paraboloid was constrained by the angular extent of the SC2 in b− c and b−a rotation planes. For the MM transition surface, it’s fitted by the MM transition’s HM data for multiple different rotation planes and was bounded by the QCL. The QCL was determined by a spline interpolation based on the 5 discovered QCEP. For SC3, the polar angle range of SC3 onset for each azimuthal angle is firstly determined through the constraint of data. Then a parabola for each constant azimuthal angle rotation away from H//b is fitted. (B) One quadrant of UTe2 high magnetic field superconducting phase diagram. Triangles marks the data points of entering the SC3 regime from pulse field PDO measurements at T ≊ 0.7 K. The dashed line is a guide-to-eye explaining the b− ac rotation plane in this phase diagram. The solid triangles are from our experiments; the open triangles are adapted from ref. [96]; the open square point in b− c plane is adapted from ref. [97]. These data points constraints our 3D modeling of the SC3 superconducting torus. Error bar is adapted from Fig. 4.18. (C) Top view along Hb of superconducting phase diagram. This shows how the series of QCEPs connects into the QCL and bounds the SC3 torus. Chapter 5 Conclusions This dissertation presented multiple new results in mapping the superconducting phase dia- gram of the spin-triplet superconductor UTe2. Mapping the intrinsic behaviour of ultraclean UTe2 In experiments performed on first-generation of CVT-grown superconducting UTe2, there were controversial results regarding the signatures of time-reversal symmetry breaking and the nature of the superconducting order parameter in UTe2. Progress towards resolving these issues was made through measurements on the ultraclean UTe2 single crystals grown by the molten salt flux (MSF) method [65, 67]. A comprehensive set of measurements on these new-generation samples demonstrates that superconducting properties such as Tc, Hc2, and Hc1 are largely enhanced with improved sample quality, as quantified by the increased residual resistivity ratio (RRR). Understanding field-reinforced superconducting phase SC2 Careful mapping of the field - angle phase diagram of high quality UTe2 reveals that the angular extent of the field-reinforced superconducting phase SC2 is highly sensitive to impurity concentration. In CVT-grown samples, SC2 persists up to tilts of θb−c ≈ 10° or θb−a ≈ 5° as magnetic field is tilted away from the b-axis [88, 95]. In contrast, ultraclean MSF-grown crystals exhibit an extended angular range of SC2 changes to θb−c ≈ 20° and θb−a ≈ 10°, respectively. These findings are consistent with a theoretical model in which SC2 is mediated by magnetic fluctuations near the metamagnetic transition, with the angular enhancement explained by variations in quasiparticle relaxation time. Pressure-dependent susceptibility measurements further reveal that SC1 undergoes a transition into SC2 under 108 Conclusions pressure, supporting the interpretation that the suppression of the metamagnetic transition field drives the emergence of SC2 at zero field. Discovery of a quantum critical line that encloses the field-induced superconducting phase SC3 Field rotational studies through multiple rotation planes show that the critical endpoints of the metamagnetic transition are suppressed by transverse field components, reaching zero temperature at quantum critical endpoints (QCEPs). These QCEPs connect to a one- dimensional quantum critical line in the three-dimensional (Ha-Hb-Hc) space. 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Table of contents 1 Introduction 1.1 Superconductivity 1.2 Quantum Phase Transition 1.3 U-based superconductor materials 1.4 Uranium Ditelluride 1.4.1 Basic properties 1.4.2 Electronic Structure 1.4.3 Normal State 1.4.4 Ambient superconducting phase 1.4.5 Superconducting phase diagram in field 1.4.6 Phase diagram of UTe2 under pressure 1.5 Outline of this thesis 2 Methods 2.1 Sample preparation 2.1.1 Growth methods 2.1.2 Sample Selection 2.2 Moissanite anvil cell 2.3 DC field measurement techniques 2.3.1 Spot welding 2.3.2 Electrical transport 2.3.3 Current-voltage measurement 2.3.4 AC magnetic susceptibility 2.4 Pulsed field measurement techniques 2.4.1 Transport measurement 2.4.2 Radio frequency measurement 2.5 Uncertainty Analysis 3 Intermediate Magnetic Field Superconducting Phase Diagrams of UTe2 3.1 Enhanced Superconductivity in MSF-grown UTe2 3.1.1 Sample Quality 3.1.2 Upper Critical Field (Hc2) 3.1.3 Lower Critical Field (Hc1) 3.2 Intrinsic Metamagnetic Transition in UTe2 3.3 Enhanced Angular extent of SC2 3.4 Phase diagram under pressure with ACMS 3.5 Understanding SC2 in UTe2 4 High Magnetic Field Phase Landscape of UTe2 4.1 Discovery of a Quantum Critical Line in UTe2 4.2 Spillover of SC3 4.3 Angular dependence of Tc in SC3 of UTe2 4.4 Discussion 5 Conclusions References