Computational studies of bandgap engineering in quantum dots have become indispensable for understanding and predicting how quantum confinement effects dictate their electronic properties. The ability to precisely control the bandgap through size, shape, and composition adjustments is central to applications in optoelectronics, photovoltaics, and quantum computing. Theoretical approaches such as the effective mass approximation, tight-binding models, and density functional theory provide frameworks for modeling these effects, offering insights that complement experimental observations without direct overlap with synthesis or optical characterization.
The foundation of quantum confinement lies in the spatial restriction of charge carriers within dimensions comparable to their de Broglie wavelength. As quantum dots shrink below the bulk exciton Bohr radius, their electronic states become discrete, leading to a size-dependent bandgap. The effective mass approximation offers a simplified yet powerful method to estimate these changes. By treating electrons and holes as particles with effective masses moving in a potential well, this model predicts that the bandgap increases inversely with the square of the quantum dot radius. For example, in CdSe quantum dots, calculations using this approach show a bandgap shift from 1.74 eV in bulk to over 2.5 eV for dots smaller than 3 nm, aligning closely with experimental trends.
However, the effective mass approximation has limitations, particularly for very small quantum dots or those with anisotropic shapes. Tight-binding models address these shortcomings by accounting for atomic-scale interactions. This method discretizes the quantum dot into a lattice of atoms, with electronic states calculated based on hopping integrals between neighboring sites. Tight-binding simulations of InAs quantum dots reveal how deviations from spherical symmetry—such as elongation or faceting—introduce additional splitting of energy levels. For instance, a 5 nm elongated InAs dot exhibits a 0.15 eV reduction in bandgap compared to its spherical counterpart, highlighting the role of shape in bandgap engineering.
Density functional theory provides the most rigorous framework, incorporating electron-electron interactions and material-specific details. DFT calculations excel in predicting compositional effects, such as alloying or doping, on band structure. Studies on ZnS-CdSe core-shell quantum dots demonstrate how interfacial strain and charge redistribution modulate the bandgap. A ZnS shell of 1 nm thickness on a 3 nm CdSe core reduces the bandgap by approximately 0.3 eV due to strain-induced orbital hybridization, a finding validated by later experimental measurements. DFT also captures the impact of surface passivation; for example, ligand-capped PbS quantum dots show a 0.2 eV higher bandgap than uncapped ones due to suppressed surface states.
Case studies of II-VI semiconductors like CdTe and HgTe illustrate the interplay between quantum confinement and strong spin-orbit coupling. DFT simulations reveal that HgTe quantum dots undergo a transition from semiconducting to semi-metallic behavior below 4 nm, a consequence of inverted band ordering exacerbated by confinement. This prediction was crucial for understanding their anomalous optical properties. Similarly, computational studies of III-V materials such as GaAs and InP highlight the role of stoichiometry. InP quantum dots with phosphorus-rich surfaces exhibit a 0.4 eV wider bandgap than indium-rich ones, as shown by GW corrections to DFT, which improve quasiparticle energy estimates.
The choice of computational method often depends on the trade-off between accuracy and computational cost. While DFT provides high fidelity, its scalability limits applications to quantum dots with fewer than a few thousand atoms. Tight-binding models strike a balance, enabling simulations of larger systems while retaining atomic detail. For rapid screening of size-dependent trends, the effective mass approximation remains useful despite its simplifications. Recent advances in machine learning potentials promise to bridge these gaps, offering DFT-level accuracy at reduced costs by training on existing datasets.
Multiscale modeling approaches combine these methods to address complex scenarios. For example, a hybrid DFT-tight-binding study of CdSe-ZnS heterostructures elucidated how interfacial defects introduce mid-gap states, explaining experimental observations of non-radiative recombination. Another study on CuInS2 quantum dots used DFT to parameterize a tight-binding model, enabling simulations of size effects across a 2-10 nm range that matched spectroscopic data within 5% error.
Challenges persist in accurately describing excitonic effects, which are critical for optical applications. The Bethe-Salpeter equation, often coupled with DFT, improves exciton binding energy predictions. In PbSe quantum dots, this method revealed that exciton binding energies exceed 0.5 eV for sub-3 nm dots, significantly impacting their absorption spectra. Such insights guide the design of quantum dots for specific applications, such as tunable lasers or single-photon emitters.
Validation against experimental data remains a cornerstone of computational studies. For instance, theoretical predictions of bandgap trends in alloyed CdSe1-xTex quantum dots across the entire composition range (x=0 to 1) showed excellent agreement with absorption measurements, with deviations of less than 0.1 eV. Similarly, computational studies of strain effects in lattice-mismatched core-shell systems like CdSe-CdS provided explanations for observed spectral shifts, enabling rational design of heterostructures with desired optoelectronic properties.
Future directions include integrating machine learning for high-throughput screening of novel quantum dot materials and exploring many-body effects beyond standard DFT approximations. Computational studies will continue to play a pivotal role in unraveling the intricate relationship between quantum confinement and electronic properties, accelerating the development of next-generation nanomaterials. By bridging theory and application, these efforts underscore the transformative potential of bandgap engineering in quantum dots.