Energy Regimes, Thresholds and Delimitation in Ellipsoidal Quantum Dots with a Modified Hellmann Potential

This study explores energy regimes, thresholds, and delimitation in ellipsoidal quantum dots with a modified Hellmann potential, combining Coulomb and Yukawa interactions. Using ellipsoidal coordinates, we solve the Schrödinger equation numerically via the Finite Difference Method (FDM). The radial distance is approximated as , valid for moderate eccentricities but introducing errors for highly elongated dots. Three energy regimes are identified: low-energy ( ) with shallow confinement, moderate-energy 7) with intermediate confinement, and high-energy ( ) with strong confinement and localized states. Threshold analysis reveals critical values for eccentricity, Coulomb strength, Yukawa repulsion, and screening parameters, determining bound state stability. Delimitation analysis shows increased energy level spacing and discrete density of states with higher eccentricity, confirming stronger quantum confinement in elongated dots. The material properties of InAs, including its small effective mass and low bandgap, significantly influence energy levels and wavefunctions. These findings have practical implications for designing quantum dot-based devices, such as infrared photodetectors, lasers, and tunable bandgap systems. This work provides a robust framework for understanding and engineering ellipsoidal quantum dots, highlighting the interplay between geometry, potential, and material properties in quantum confinement.