Breaking the Moss rule

TL;DR

This work addresses the Moss rule, which links band gap and sub-bandgap refractive index, by focusing on super-Mossian dielectrics that achieve high n with wide transparency. It clarifies the physical origin of this behavior as a large joint density of states near the band edge, governed by band-structure features such as joint critical points and band tracking, and it surveys experimental realizations across conventional and emerging materials. The review highlights first-principles discovery approaches (DFT/TDDFT, GW-BSE, and the BSE+ embedding scheme) and summarizes their accuracy and limitations, emphasizing the need to include lattice defects and phonons for realistic predictions. Finally, it connects refractive index to key photonic device performance—nanoresonators, waveguides, and metasurfaces—demonstrating how higher n can enhance confinement, Q factors, and diffraction efficiency, and outlining an actionable, interdisciplinary roadmap for discovering and implementing high-index dielectrics in next-generation photonics.

Abstract

Photonic devices depend critically on the dielectric materials from which they are made, with higher refractive indices and lower absorption losses enabling new functionalities and higher performance. However, these two material properties are intrinsically linked through the empirical Moss rule, which states that the refractive index of a dielectric decreases as its band gap energy increases. Materials that surpass this rule, termed super-Mossian dielectrics, combine large refractive indices with wide optical transparency and are therefore ideal candidates for advanced photonic applications. This Review surveys the expanding landscape of high-index dielectric and semiconductor materials, with a particular focus on those that surpass the Moss rule. We discuss how electronic band structures with a large joint density of states near the band edge give rise to super-Mossian behavior and how first-principles computational screening can accelerate their discovery. Finally, we establish how the refractive index sets the performance limits of nanoresonators, waveguides, and metasurfaces, highlighting super-Mossian dielectrics as a promising route toward the next performance leap in photonic technologies.

Figures (4)

• Figure 1: Super-Mossian dielectric materials. Sub-bandgap refractive index $n_\textrm{s}$ as a function of band gap energy $E_\textrm{g}$ for 388 semiconductors, of which 108 are experimental values (see Supplementary Table S1) and 280 are calculated using density functional theory (DFT) Svendsen2022. The experimental indices are shown as a function of the fundamental band gap energy, while the calculated indices are as a function of the direct band gap energy. The computed materials are categorized according to their anisotropy. For both experimental and computed materials, the component of the refractive index tensor with the highest value is shown. Frequency-dependent complex refractive indices for all computed materials are available in the CRYSP database CRYSP. The dashed lines show Moss relations with Moss factors of $M=3$ and $M=5$. Note the change in scales on the $E_\textrm{g}$ and $n_\textrm{s}$ axes at 3.5 eV and 6, respectively.
• Figure 2: Optical properties of semiconductors. a The optical response of a two-level system under weak excitation is captured by the two-level response function $\chi_{21}$, while the optical response of a semiconductor is the sum of responses from each pair of states. Joint density of states (JDOS) is the total number of state pairs available for optical transitions. b Singularities in JDOS or zeros of k-space gradient of ($E_\textrm{c}-E_\textrm{v}$) shape the optical properties of semiconductors. These singularities occur at the joint extrema of bands (type-1) and parallel stretches of bands (type-2). c Band structure of silicon, highlighting the singularities in JDOS. d Refractive index of silicon relating the prominent features to the singularities in JDOS shown in c.
• Figure 3: Experimental demonstrations of super-Mossian materials in nanophotonics.a Dark-field scattering map of WS$_2$ nanodisks with varying disk radii. The anapole resonance is indicated by the green dashed line. Inset: scanning electron micrograph of the fabricated WS$_2$ nanodisks with a residual resist top layer. b Reflection spectrum of a FeS$_2$ metasurface supporting an electric dipole resonance. Inset: scanning electron micrograph of the fabricated metasurface. c Dark-field scattering (black) and simulated scattering efficiency (blue) of a BP nanoparticle supporting multiple Mie resonances. Inset: scanning electron micrograph of BP nanoparticle. d Extinction map of MoS$_2$ nanodisks with varying disk radii supporting multiple Mie resonances. Inset: scanning electron micrograph of fabricated MoS$_2$ nanodisks. e Dark-field scattering map of HfS$_2$ nanodisks with varying disk diameters supporting multiple Mie resonances. Inset: scanning electron micrograph of fabricated HfS$_2$ nanodisks. f Transmission spectrum of diamond waveguide coupled to a ring resonator showing dips due to excitation of ring resonator modes. Inset: scanning electron micrograph of waveguide-coupled ring resonator. Figures adapted with permission from: a Ref. Verre2019, Springer Nature; b Ref. doiron2022super, Wiley-VCH GmbH; c-e Refs. Svendsen2022Green2020Zambrana-Puyalto2025 under a Creative Commons License https://creativecommons.org/licenses/by/4.0/; f Ref. Hausmann2012, American Chemical Society.
• Figure 4: Refractive index impact on nanophotonic devices.a-c Maximal scattering efficiency, quality factor, and stored energy enhancement for the magnetic and electric dipole resonances supported by a spherical dielectric resonator as a function of refractive index, respectively. d-f Effective mode index, single-mode cut-off, and fraction of power guided in the waveguide core of a dielectric slab waveguide as a function of refractive index, respectively. The slab waveguide has thickness $2h$ and is surrounded by a cladding refractive index of $n_\textrm{cl}=1$. The wavenumber-thickness product is set to $k_0h=1$. The inset in f shows a schematic of the electric and magnetic field profiles of the TE$^0$ and TM$^0$ modes. g Diffraction efficiency (+1 order) of phase-discretized blazed grating as a function of refractive index. The grating diffracts normally incident light into an angle of $70^{\circ}$. h Minimum meta-atom thickness to achieve $2\pi$ phase shift as a function of refractive index. The red circles in g-h show the performance of topology-optimized dielectric meta-gratings from Ref. Yang2017. i Metalens focusing efficiency as a function of refractive index for different numerical apertures (NAs) from Ref. Bayati2019. The dashed lines are guides to the eye.