Fig. 8: Bandgap engineering by dielectric environment.

a Survey of recent experimentally observed optical gaps of 2D materials in different dielectric environments. For TMDC, lowest energies correspond to samples encapsulated in h-BN. In general, energies increase as the substrate is changed from h-BN to SiO₂, CYTOP, MgO, Al₂O₃, and LaAlO₃, respectively¹⁶⁷. For N-BP, lower energies correspond to samples on sapphire covered by h-BN¹⁷³, whereas uncovered samples on Si/SiO₂ substrate¹⁸¹ exhibit higher energy. b Difference between optical bandgaps (top) of the suspended monolayer (ε = 1) and those in a medium with an effective dielectric constant \({\varepsilon }_{m}=\frac{{\varepsilon }_{1}+{\varepsilon }_{2}}{2}\) for four TMDC, as calculated by solving SBE for the gapped Dirac Hamiltonian⁷². The parameter r₀ was fitted for each TMDC as to obtain the experimental optical bandgap: 1.91 eV for suspended MoS₂²⁷⁹ using r₀ = 108 Å, 1.66 eV for MoSe₂ deposited on SiO₂²⁸⁰ with r₀ = 90 Å, 1.74 eV for WSe₂ deposited on SiO₂¹⁷⁸ with r₀ = 47 Å, and 2.04 eV for WS₂ deposited on SiO₂²⁸¹ with r₀ = 23 Å. The other parameters were obtained from the GGA calculations of ref. ²⁸². The effect on the QP bandgap, as calculated by the method explained in the Supplemental Material, is shown in the bottom panel.