Theoretical modeling of quantum confinement effects on thermoelectric properties in nanostructured materials provides critical insights into enhancing the dimensionless figure of merit ZT. Quantum confinement modifies the electronic density of states and phonon dispersion, directly influencing the Seebeck coefficient, electrical conductivity, and thermal conductivity. Advanced computational frameworks, particularly Boltzmann transport theory combined with confined density of states, enable precise predictions of these effects. Case studies on Bi2Te3 superlattices and Si nanowires demonstrate how low-dimensional structures achieve superior ZT compared to their bulk counterparts.

Boltzmann transport theory serves as the foundation for modeling charge and heat transport in nanostructures. Under quantum confinement, the density of states undergoes significant modification. In one-dimensional nanowires and two-dimensional quantum wells, the electron energy states become quantized, leading to sharp peaks in the density of states. This quantization enhances the Seebeck coefficient by increasing the asymmetry of electron distribution near the Fermi level. The electrical conductivity, however, may decrease due to reduced carrier mobility from boundary scattering. The interplay between these factors determines the overall power factor (S²σ), where S is the Seebeck coefficient and σ is the electrical conductivity.

For confined systems, the Boltzmann equation is solved under the relaxation time approximation, incorporating size-dependent scattering mechanisms. The relaxation time τ accounts for electron-phonon interactions, impurity scattering, and surface roughness effects. In nanowires, boundary scattering becomes dominant as the diameter approaches the electron mean free path. The modified density of states alters the electron group velocity and scattering rates, requiring self-consistent solutions to the transport equations. Computational implementations often employ iterative methods to converge on the correct carrier distribution function under an applied electric field or temperature gradient.

Phonon transport is equally critical in determining ZT, as thermal conductivity κ consists of electronic (κₑ) and lattice (κₗ) contributions. Quantum confinement suppresses κₗ by introducing additional phonon-boundary scattering and modifying the phonon dispersion relations. In superlattices and nanowires, the reduction in κₗ outweighs any potential decrease in power factor, leading to net ZT enhancement. First-principles calculations combined with phonon Boltzmann transport equations quantify these effects by computing the full phonon spectrum and scattering rates in confined geometries.

Bi2Te3 superlattices serve as a prominent case study for quantum confinement effects. Theoretical models predict that reducing the superlattice period below 5 nm leads to strong electron state quantization, increasing the Seebeck coefficient by up to 40% compared to bulk Bi2Te3. The power factor optimization occurs when the superlattice period matches the electron wavelength, maximizing the density of states near the Fermi level. Simultaneously, phonon scattering at the interfaces reduces κₗ by over 50%, resulting in ZT values exceeding 2.0 at room temperature. These predictions align with experimental observations, validating the theoretical framework.

Si nanowires provide another illustrative example, particularly for their potential in thermoelectric applications despite silicon's poor bulk thermoelectric performance. Quantum confinement in sub-10 nm diameter nanowires creates discrete electronic sub-bands, enhancing the Seebeck coefficient through energy filtering effects. Theoretical studies show that surface roughness and diameter variations further suppress κₗ by scattering mid-to-long-wavelength phonons. For p-type Si nanowires with diameters around 20 nm, ZT values approaching 1.0 have been predicted, a significant improvement over bulk Si's ZT of 0.01. The models highlight the importance of diameter control and surface passivation in optimizing thermoelectric performance.

The impact of quantum confinement on ZT depends on material-specific parameters, including effective mass, deformation potential, and phonon dispersion. Heavy-hole materials like Bi2Te3 benefit more from density of states engineering due to their large effective mass, while light-electron materials require careful optimization of carrier concentration to avoid detrimental mobility reductions. Theoretical models must account for these nuances when predicting ZT enhancements in different material systems.

Advanced computational techniques, including full-band Monte Carlo simulations and ab initio Boltzmann transport, provide higher accuracy by incorporating realistic band structures and scattering mechanisms. These methods reveal that anisotropic confinement, such as in rectangular nanowires or core-shell structures, can further enhance ZT by selectively suppressing phonon modes while preserving electronic transport along certain crystallographic directions. The computational cost of these methods remains high, but they offer unparalleled insights into designing next-generation thermoelectric nanomaterials.

In summary, theoretical models demonstrate that quantum confinement systematically improves ZT in nanostructured materials through simultaneous enhancement of the power factor and reduction of thermal conductivity. Bi2Te3 superlattices and Si nanowires exemplify these principles, with predictions corroborated by experimental data. Future developments in computational methods will enable more precise design rules for optimizing thermoelectric performance across a broader range of materials and nanostructured geometries. The integration of machine learning for rapid screening of confinement effects may further accelerate the discovery of high-ZT nanomaterials.