Universal relation between energy gap and dielectric constant

Abstract

We establish a universal relation between the energy gap and the static dielectric constant for all insulating states. This relation yields an upper bound on the energy gap, which only depends on the electron density and electronic dielectric constant. We identify two types of energy gaps associated with transverse and longitudinal excitations at long wavelength, which correspond to the optical gap and the plasmon energy respectively. Their upper bounds are set by the dielectric constant and its inverse respectively. The transverse gap bound is calculated for a wide range of materials and compared with the measured optical gap. A remarkable case is cubic boron nitride, in which the direct gap reaches \SI{72}{\percent} of the bound. Our results are derived from the Kramers-Kronig relation and the $f$-sum rule, and therefore rest on general physical principles.

6 pages + references + Supplemental materials (4 pages). 5 figure, 1 table. Significantly expanded in v2

Cited by in corpus (14)

• Quantum weight: A fundamental property of quantum many-body systems
• Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals
• Instantaneous response and quantum geometry of insulators
• The multi-state geometry of shift current and polarization
• Quantum entanglement and quantum geometry measured with inelastic X-ray scattering
• Quantum Metric in Step Response
• Optical bounds on many-electron localization
• Quantum Geometry and the Hidden Scales in Materials
• Quantum geometric bounds for observables: Linear responses, Drude weight, and orbital magnetization
• Universal Wilson Loop Bound of Quantum Geometry
• Accurate Electron-phonon Interactions from Advanced Density Functional Theory
• Enhancing the Hyperpolarizability of Crystals with Quantum Geometry
• Reinforcement Fine-Tuning for Materials Design
• Electronic bounds in magnetic crystals