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Uniaxial stress is a powerful technique to tune the electronic structure of very pure materials. The novel piezoelectric-based techniques developed by our group, which allow application of large and homogeneous uniaxial pressure in a continuously-tunable manner, make uniaxial pressure an independent axis in the parameter space for the study of quantum materials. Many exciting experiments have been performed that combine different measurement methods with this uniaxial stress technique in the past few years. In this thesis, I demonstrate the first electrical transport measurement under uniaxial pressure of a free-standing microstructure single-crystalline sample patterned by focused ion beam (FIB) milling. With the microstructuring technique that I developed, the transport properties transverse to the force direction can be more accurately probed. The ability to resolve the anisotropy introduced by the uniaxial pressure lets us have a better understanding of how the electronic structure of Sr₂RuO₄ changes under uniaxial stress. Moreover, the microstructure technique opens new roads for smaller crystals (∼ 100 µm) to be studied under uniaxial pressure. In addition, higher stresses and better sample homogeneity could be achieved by working with smaller samples. For Sr₂RuO₄, one of the three Fermi-surface sheets can be driven through a Lifshitz transition by applying uniaxial stress along the [100] direction. Superconductivity and resistivity have been observed to be strongly enhanced at the singularity. In addition, a spin-density wave (SDW) has been observed at stresses beyond the Lifshitz transition. Measurement of the Hall effect under uniaxial stress allows us to probe Hund’s metal physics in Sr₂RuO₄. The Hall coefficient of unstressed Sr₂RuO₄ goes through two sign reversals, at 30 K and 120 K. Under the Hund’s metal scenario, this temperature dependence has been proposed to result from orbital differentiation of the inelastic scattering rate, which is a key property expected of Hund’s metals. In the present study, it is shown that at a temperature where electron-electron scattering dominates (≳ 5 K), the Hall coefficient becomes less electron-like while approaching the VHS, which is consistent with increased scattering in the d_xy band. Beyond the transition, the Hall coefficient becomes much more electron-like, which is opposite to expectations from the change in Fermi surface topology, but can be explained by a combination of Hund’s metal physics and strong suppression in the d_xy scattering rate. At very low temperature (0.5 K), the Hall coefficient is essentially unchanged across the Lifshitz transition, despite the change in the Fermi-surface topology. In contrast to the longitudinal resistivity that has a strong peak at the VHS but does not respond to the SDW, the resistance transverse to the force direction shows a strong response to the SDW, but only a small response at the VHS. In addition, I obtain ρ(T) at the Lifshitz transition below Tc by subtracting off the magnetoresistance and find that T² ln(1/T) fits better than T^3/2, which suggests a saddle point rather than an extended saddle point at the VHS.:1. Introduction to Sr2RuO4 1.1. Normal-State Properties Van Hove Singularity and Lifshitz Transition in Sr2RuO4 1.2. Hall Effect in Sr2RuO4 Weak-field Hall Coefficient Experimental Hall Coefficient in Sr2RuO4 and Related Systems 1.3. Hund’s Metal Scenario Dynamical Mean-Field Theory Experimental Evidence for Orbital Differentiation in Sr2RuO4 Hall Coefficient of Sr2RuO4 within Hund’s Metal Scenario 1.4 Uniaxial-Pressure Projects on Sr2RuO4 2. Experimental Setup 2.1. Stress and Strain 2.2. Uniaxial Stress Technique Uniaxial-Stress Cell Sample Carrier 2.3. Imperfections of the Stress Cells 2.4. Sample Preparation Needle Sample Preparation Microstructure Sample Preparation Comparison of the Two Samples 2.5. Measurement Setup 3He Cryostat Transport Measurement Setup 3. Hall Coefficient and Resistivity Measurements 3.1. Basics of Resistivity Measurement Stress Ramps 3.2. Basics of Hall Measurement Setup Field Dependence of Hall Resistivity Temperature Dependence of Hall Coefficient 3.3. Stress Ramps under Constant Magnetic Field 3.4. Stress Dependence of Hall Coefficient and Resistivity 3.5. Resistivity Measurements below Tc 3.6. Field Sweeps within the Magnetic Phase 3.7. Summary 4. Measurements Transverse to the Stress Axis 4.1. Setup for Transport Measurements Transverse to the Uniaxial Stress 4.2. Simulations Based on Finite Element Method 4.3. Resistance Measurements Transverse to Applied Stress 4.4. Summary 5. Data Analysis and Discussion 5.1. A Tight-Binding Model under Uniaxial Pressure 5.2. Analysis of Hall Coefficient across the Lifshitz Transition Hall Coefficient Analysis under the Isotropic-l or Isotropic-τ Approximations Hall Coefficient Analysis under Hund’s Metal Scenario 5.3. Magnetoresistance Subtraction in Temperature Ramps 5.4. Transport Properties at 5 K 5.5. Summary 6. Conclusions and Outlook Appendices A. Si-Gap-Platform Microstructure Project A.1. Si-Gap Platform A.2. Sample Preparation with PFIB-Microstructuring A.3. Microstructure Stress Cells B. Other results B.1. Hall Effect from the Hall Pair 2 B.2. Magnetoresistance in Longitudinal and Transverse Configurations B.3. Toward -1.5 GPa B.4. Comparison of RH(T) in Sr2RuO4 Compressed along [100] Direction and YBa2Cu3O6.67 Compressed along the b-axis Bibliography