Theoretical investigations into miniband formation in quantum dot superlattices have provided critical insights into the electronic and optical properties of these nanostructured systems. Quantum dot superlattices, composed of periodic arrays of semiconductor nanocrystals such as PbS and CdSe, exhibit unique miniband structures due to inter-dot coupling. These minibands arise from the overlap of quantum-confined electronic states in neighboring dots, enabling delocalized charge transport and tunable optoelectronic behavior. Theoretical frameworks, including the Hubbard model and density functional theory (DFT), have been instrumental in predicting the effects of inter-dot coupling on miniband formation, charge carrier mobility, and optical absorption.

The Hubbard model offers a simplified yet powerful approach to describe electron-electron interactions in quantum dot arrays. In this model, the system is treated as a lattice of sites, each representing a quantum dot, with on-site Coulomb repulsion and inter-site hopping terms. For PbS quantum dot superlattices, the Hubbard model predicts that miniband width is highly sensitive to inter-dot spacing and the dielectric environment. Studies show that reducing the inter-dot distance below 1.2 nm enhances hopping integrals, leading to broader minibands and increased charge delocalization. For instance, in PbS arrays with 5 nm diameter dots and 1 nm ligand spacing, the calculated miniband width can reach 120 meV, facilitating efficient electron transport. The Hubbard model also captures the role of Coulomb blockade effects, which become significant at small inter-dot separations, potentially localizing charges and narrowing minibands if the on-site repulsion exceeds the hopping energy.

Density functional theory provides a more detailed, first-principles description of electronic structure in quantum dot superlattices. DFT calculations for CdSe quantum dot arrays reveal that miniband formation depends critically on the crystallographic alignment of dots and the nature of surface ligands. For example, in face-centered cubic (FCC) packed CdSe superlattices with oleic acid ligands, DFT predicts a conduction band miniband dispersion of approximately 80 meV for a 6 nm dot spacing. The valence band miniband is typically narrower due to heavier hole effective masses. Substituting shorter ligands, such as ethanedithiol, increases the electronic coupling between dots, widening the minibands to over 150 meV. DFT also highlights the influence of structural disorder on miniband formation. Random deviations in dot positions or sizes by just 5% can introduce localized states within the miniband, reducing charge mobility.

Inter-dot coupling not only affects electronic transport but also modifies optical properties. Theoretical studies demonstrate that miniband formation in PbS superlattices leads to a red-shift in the absorption edge compared to isolated quantum dots. This shift arises from the reduced effective bandgap due to state delocalization across multiple dots. For PbS arrays with 4 nm dots and 0.8 nm spacing, the absorption onset shifts by 0.15 eV, consistent with experimental observations. In CdSe superlattices, the oscillator strength of optical transitions is enhanced in the miniband regime due to increased wavefunction overlap. DFT calculations predict a two-fold increase in absorption coefficient for coupled CdSe dots compared to isolated ones at energies corresponding to the miniband states.

Charge transport in quantum dot superlattices is governed by miniband conduction at high coupling strengths and hopping transport at weaker couplings. The Hubbard model identifies a transition between these regimes when the miniband width exceeds the thermal energy kT. For PbS arrays at room temperature, this requires miniband widths greater than 26 meV, achievable with inter-dot spacings below 1.5 nm. Beyond this threshold, electron mobility increases sharply, with theoretical predictions reaching 10 cm²/Vs for optimally coupled PbS superlattices. In contrast, CdSe arrays typically exhibit lower mobilities due to stronger hole localization, with calculated values around 2 cm²/Vs even in tightly packed configurations.

Optical properties are further influenced by excitonic effects in coupled quantum dot systems. The Hubbard model extended to include exciton binding predicts that inter-dot coupling reduces the exciton binding energy by up to 30% in PbS superlattices, as the electron-hole pair becomes less confined. DFT calculations support this, showing a decrease from 150 meV in isolated PbS dots to 100 meV in closely spaced arrays. This reduction has important consequences for photovoltaic applications, where exciton dissociation is critical. In CdSe superlattices, the situation is complicated by the presence of dark exciton states, which DFT reveals can become optically active through miniband-induced mixing with bright states.

Theoretical studies also address the impact of external fields on miniband properties. Under an applied electric field, DFT simulations show that PbS superlattices exhibit Stark shifts in the miniband edges, with a linear shift rate of 0.5 meV/(kV/cm) for fields below 50 kV/cm. At higher fields, miniband narrowing occurs due to localization effects, reducing conductivity. Magnetic fields, on the other hand, induce Zeeman splitting of miniband states, with g-factors calculated to be 2.5 for electrons and 0.8 for holes in CdSe arrays.

Temperature effects on miniband stability have been investigated through molecular dynamics simulations coupled with electronic structure calculations. These studies reveal that thermal vibrations can modulate inter-dot distances by ±0.1 nm at 300 K, causing fluctuations in miniband width by ±10 meV for PbS superlattices. While this does not completely destroy miniband coherence, it contributes to temperature-dependent mobility reductions. Phonon scattering rates calculated for CdSe arrays indicate that optical phonons with energies around 25 meV dominate the electron relaxation processes in minibands.

Comparative studies between PbS and CdSe superlattices highlight material-specific differences in miniband formation. PbS dots generally exhibit stronger inter-dot coupling due to higher dielectric constants and smaller effective masses, leading to broader minibands at equivalent spacings. DFT calculations show that PbS minibands are less sensitive to surface chemistry variations compared to CdSe, where ligand type strongly influences coupling. However, CdSe arrays benefit from more precise size control, enabling narrower miniband distributions in monodisperse systems.

Theoretical approaches continue to evolve in modeling these complex systems. Recent advances combine DFT with many-body perturbation theory to more accurately describe excitonic effects in minibands. These methods predict exciton binding energies within 5% of experimental values for both PbS and CdSe superlattices. Machine learning techniques are being employed to rapidly screen large parameter spaces of dot sizes, spacings, and materials to identify optimal configurations for specific applications.

In summary, theoretical studies using Hubbard models and DFT have established a comprehensive framework for understanding miniband formation in quantum dot superlattices. These approaches quantitatively describe how inter-dot coupling influences electronic structure, charge transport, and optical properties in materials such as PbS and CdSe. The insights gained guide the design of quantum dot arrays for applications ranging from photovoltaics to quantum computing, where precise control over miniband engineering is essential. Future work will likely focus on extending these methods to more complex superlattice geometries and hybrid material systems.