Phase transitions in low-dimensional semiconductor systems such as quantum dots, nanowires, and two-dimensional materials exhibit unique behaviors distinct from their bulk counterparts. These phenomena arise due to quantum confinement, surface effects, and dimensional constraints, leading to altered thermodynamic stabilities and transition pathways. Understanding these effects is critical for designing next-generation electronic, optoelectronic, and quantum devices.

In quantum dots, the finite size and discrete electronic states impose significant modifications on phase transition thermodynamics. For instance, the melting temperature of semiconductor nanocrystals like CdSe or Si exhibits a strong diameter dependence. Experimental studies show that CdSe quantum dots below 5 nm in diameter may experience melting point depressions exceeding 200 K compared to bulk CdSe. This arises from the increased surface energy contribution and the reduced coordination of surface atoms. Similarly, structural phase transitions, such as the wurtzite-to-rock-salt transition in CdSe, occur at higher pressures in quantum dots due to the additional energy required to overcome surface strain. The critical pressure for this transition can increase by several gigapascals for dots smaller than 4 nm.

Nanowires present anisotropic confinement effects, where the phase stability depends on both diameter and crystallographic orientation. In Si nanowires, the diamond-cubic to beta-tin phase transition under pressure is delayed compared to bulk silicon, with the exact pressure threshold varying with wire diameter. For sub-10 nm wires, the transition pressure may increase by 2-3 GPa. Additionally, surface reconstructions in nanowires can stabilize metastable phases not typically observed in bulk materials. Ge nanowires, for example, exhibit a size-dependent stabilization of the hexagonal phase, which is unstable in bulk germanium at ambient conditions. The high surface-to-volume ratio also leads to pronounced premelting effects, where the nanowire surface becomes disordered below the core melting point.

Two-dimensional materials demonstrate layer-dependent phase transitions, with monolayer and few-layer systems displaying markedly different behavior from bulk crystals. Molybdenum disulfide (MoS2) serves as a prominent example, where the semiconductor-to-metal transition and the trigonal prismatic to octahedral structural transition are strongly influenced by layer count. Monolayer MoS2 exhibits a higher phase transition temperature compared to bulk due to suppressed interlayer interactions and enhanced in-plane bonding. Strain engineering further modulates these transitions; applying biaxial tensile strain above 10% can induce a direct-to-indirect bandgap transition in monolayer MoS2. Similarly, tungsten diselenide (WSe2) shows a layer-dependent excitonic phase transition, where the binding energy of excitons increases significantly in monolayers due to reduced dielectric screening.

Confinement effects also alter the kinetics of phase transitions in low-dimensional systems. In quantum dots, nucleation barriers increase due to the limited volume available for critical nucleus formation. This results in pronounced supercooling or superheating during first-order transitions. For example, the solidification of germanium quantum dots from the melt requires undercooling of several tens of degrees Kelvin more than bulk germanium. Nanowires exhibit similar kinetic constraints, with phase front propagation along the wire axis differing from three-dimensional growth due to geometric confinement.

The stability of polymorphic phases in these systems is highly sensitive to surface chemistry and environment. In CdSe quantum dots, surface ligands can stabilize either the wurtzite or zinc blende phase, with thiol-terminated ligands favoring the zinc blende structure. For MoS2 monolayers, environmental doping can trigger reversible phase transitions between the semiconducting 1H and metallic 1T phases. This tunability enables applications in non-volatile memory and reconfigurable electronics.

Thermodynamic modeling of these systems requires modifications to classical nucleation theory and phase field models to account for finite-size effects. The Gibbs free energy landscape becomes distorted in confined geometries, with additional terms for surface energy and quantum confinement contributions. For quantum dots, the melting temperature follows a linear relationship with the inverse diameter in the size range where surface effects dominate. In nanowires, the transition temperature depends on both diameter and aspect ratio due to the competing effects of surface and strain energies.

Experimental characterization of phase transitions in low-dimensional systems presents unique challenges. Conventional techniques like differential scanning calorimetry must be adapted for small sample volumes, while in situ electron microscopy provides atomic-scale insights but may introduce beam-induced artifacts. Raman spectroscopy proves particularly useful for tracking pressure-induced transitions in quantum dots and 2D materials through shifts in vibrational modes. For example, the E2g mode softening in MoS2 under pressure serves as a sensitive indicator of the layer-dependent phase transition.

Applications leveraging these phenomena include phase-change memory devices using confined GeSbTe alloys, where the reduced dimensions enable faster switching and lower power consumption. In photovoltaics, the tunable phase stability of perovskite quantum dots allows optimization of their optoelectronic properties. Strain-engineered 2D materials find use in flexible electronics, where controlled phase transitions enable novel device functionalities.

The study of phase transitions in quantum-confined systems continues to reveal new physics at the intersection of thermodynamics, quantum mechanics, and materials science. As synthesis techniques advance to produce ever more uniform nanostructures, and characterization methods improve in spatial and temporal resolution, our understanding of these phenomena will deepen, enabling precise control over material properties for technological applications. Future directions include exploring nonequilibrium phase transitions in these systems and developing predictive models for multicomponent low-dimensional materials. The interplay between electronic, structural, and thermal properties in confined geometries remains a rich area for fundamental investigation and applied research.