Theoretical modeling of anisotropic quantum confinement in semiconductor nanoplatelets provides critical insights into their unique electronic and optical properties, distinguishing them from isotropic quantum dots. Semiconductor nanoplatelets, such as CdSe, exhibit strong quantum confinement along one spatial dimension while maintaining bulk-like behavior in the other two. This anisotropy leads to thickness-dependent electronic structures and optical transitions, which are accurately described using k·p perturbation theory and pseudopotential methods. These approaches contrast with the isotropic confinement observed in spherical quantum dots, where the electronic states are uniformly quantized in all three dimensions.

The k·p method is widely employed to model the electronic structure of semiconductor nanoplatelets due to its efficiency in capturing the effects of quantum confinement. In this framework, the Hamiltonian is expanded around high-symmetry points in the Brillouin zone, typically the Γ-point, to describe the band structure near the bandgap. For CdSe nanoplatelets, the confinement along the thickness direction (typically a few monolayers) results in discrete energy levels, while the in-plane motion remains quasi-continuous. The k·p Hamiltonian incorporates the coupling between conduction and valence bands, including the heavy-hole, light-hole, and split-off bands, which are essential for predicting optical transitions. The thickness dependence of the bandgap is a direct consequence of quantum confinement, with thinner nanoplatelets exhibiting larger bandgaps due to increased quantization energy. For instance, CdSe nanoplatelets with thicknesses of 1.5 nm to 6 nm show bandgaps ranging from 2.5 eV to 2.1 eV, respectively.

Pseudopotential methods offer an alternative approach, particularly for systems where atomistic details are critical. These methods replace the core electrons with an effective potential, reducing computational cost while retaining accuracy in describing the valence electrons. In the context of nanoplatelets, pseudopotential calculations reveal the influence of surface passivation and lattice strain on the electronic structure. For example, the presence of organic ligands or inorganic shells can modify the confinement potential, leading to shifts in energy levels and oscillator strengths for optical transitions. The pseudopotential approach also captures the anisotropy in the effective masses of charge carriers, which is less pronounced in isotropic quantum dots. This anisotropy affects the density of states and the recombination dynamics, resulting in distinct photoluminescence spectra compared to quantum dots.

Optical transitions in nanoplatelets are governed by selection rules derived from their anisotropic confinement. The transition dipole moments are strongly oriented along the confinement direction, leading to polarized emission. This contrasts with isotropic quantum dots, where the emission is unpolarized due to the spherical symmetry. The thickness-dependent exciton binding energy is another key feature, with thinner nanoplatelets exhibiting larger binding energies due to reduced dielectric screening. For CdSe nanoplatelets, exciton binding energies can exceed 200 meV, significantly higher than those in bulk CdSe (approximately 15 meV) or quantum dots (around 30-50 meV). This large binding energy enhances the stability of excitons at room temperature, making nanoplatelets attractive for optoelectronic applications.

The comparison with isotropic quantum dots highlights the unique advantages of nanoplatelets. In quantum dots, the electronic states are described by spherical harmonics, leading to degenerate energy levels labeled by angular momentum quantum numbers. The optical transitions are broadened by size distribution and surface effects, whereas nanoplatelets exhibit narrower emission lines due to their uniform thickness. The absorption spectra of nanoplatelets feature sharp peaks corresponding to transitions between quantized levels, while quantum dots show smoother spectra with less defined features. Additionally, the Auger recombination rates in nanoplatelets are suppressed compared to quantum dots, as the anisotropic confinement reduces the overlap of electron and hole wavefunctions.

Theoretical models also address the impact of dielectric confinement in nanoplatelets, where the dielectric mismatch between the semiconductor and surrounding medium modifies the Coulomb interaction. This effect is less significant in quantum dots due to their isotropic environment. The dielectric confinement enhances the exciton binding energy and alters the radiative recombination rates, further distinguishing nanoplatelets from quantum dots. Computational studies have shown that the dielectric constant of the surrounding medium can tune the optical properties of nanoplatelets, providing a handle for engineering their performance in devices.

In summary, theoretical modeling of anisotropic quantum confinement in semiconductor nanoplatelets using k·p and pseudopotential methods reveals their distinct electronic and optical properties. The thickness-dependent band structure, large exciton binding energies, and polarized emission set nanoplatelets apart from isotropic quantum dots. These insights guide the design of nanoplatelet-based devices, leveraging their unique characteristics for applications in light-emitting diodes, lasers, and photodetectors. The contrast with quantum dots underscores the importance of anisotropy in tailoring nanomaterial properties for specific technological needs.