Heavy fermion compound UTe₂, as a recently discovered unconventional superconductor, has received significant attention due to its potential spin-triplet superconducting pairing, high-field re-entrant superconducting phases, and unique quantum critical characteristics. However, experimental results of this system show significant changes and discrepancies, primarily due to difference in sample quality. The key unresolved issues include whether the system exhibits multi-component superconducting order parameters, whether time-reversal symmetry is spontaneously broken, and whether multiple field-induced superconducting phases share a common origin. These unsolved issues hinder an in-depth understanding of the intrinsic superconducting pairing mechanism in the UTe₂ system.

This paper reviews recent advances in single-crystal growth methods for UTe₂, including chemical vapor transport (CVT), Te-flux, molten salt flux (MSF), and molten salt flux liquid transport (MSFLT). We systematically analyze how growth conditions influence superconductivity and crystal quality. Although the CVT method was initially employed in UTe₂ studies, the samples grown by this method exhibit poor quality and significant compositional inhomogeneity, even in individual samples. Consequently, the CVT method has been progressively supplanted by the recently developed MSF method. In contrast, the MSF method and MSFLT method yield high-quality UTe₂ single crystals with T_c achieving a value as high as 2.1 K and residual resistivity ratio (RRR) reaching up to 1000; however, the sample sizes are smaller than those grown by the CVT and Te-flux methods. Notably, MSF-grown samples occasionally contain magnetic impurities such as U₇Te₁₂, so careful screening is required in the sample collection process. The MSFLT combines the advantages of CVT and MSF methods to grow high-quality UTe₂ single crystals while producing larger sample sizes than MSF. Our research findings highlight the importance of optimizing growth parameters such as Te/U ratio, temperature gradient, and cooling rate. For instance, lower growth temperature and precise control of the Te/U ratio can significantly enhance T_c and sample quality. Several controversies have been identified regarding high-quality MSF and MSFLT samples, including clarifying the single-component nature of the superconducting order parameter and confirming the absence of time-reversal symmetry breaking in optimized samples.

This review underscores the pivotal role of advanced single-crystal growth techniques in advancing the study of UTe₂. Future research should focus on utilizing these high-quality UTe₂ samples grown by MSF and MSFLT methods to accurately determine superconducting order parameters, elucidate mechanisms behind high-field re-entrant superconducting phases, and explore topological properties, such as potential Majorana fermions. These efforts will deepen our understanding of unconventional superconductivity, spin fluctuations, and quantum critical phenomena in the UTe₂ system.