Universal 3:1 Scaling of Quantum-Confined Stark Spectra Revealed by a Three-Dimensional Profile

Abstract

We report that the quantum-confined Stark effect spectrum exhibits a nearly rigid redshift while preserving its characteristic peak spacing patterns when increasing the electric field strength F. Using InGaN as a model system, we uncover two electric-field-independent scaling laws for the spectral peaks in both the sub-bandgap and above-bandgap regions and the coefficient ratio is near 3:1. With a novel three-dimensional (3D) visualization, we reveal that the sub-bandgap peak spacings scale as $\frac{12\ddot{I}€{\hslash }^2}{L^2\sqrt{m_e{m}_h}}$ while the above-bandgap peak spacings scale as $\frac{4\ddot{I}€{\hslash }^2}{L^2\sqrt{m_e{m}_h}}$, explaining the origin of the 3:1 ratio. This scaling behavior, validated in both InGaN and GaAs systems and at electroluminescence working conditions, shows that increasing F only expands the energy range and increases the number of peaks without altering the spacing. Beyond these laws, the 3D profile offers new insights into the Tauc background, Franz-Keldysh oscillations and coherence length, providing a powerful tool for the design and diagnostics of electro-optic devices.