A modified Moss rule highlights underexplored classes of high refractive index materials

Technical Summary: A Modified Moss Rule for High Refractive Index Materials

Problem Statement
High refractive index ($n$) dielectric materials are critical for advancing photonic technologies, including nanoresonators, metalenses, and nonlinear optical devices. However, the empirical Moss rule establishes a fundamental trade-off between high refractive index and optical transparency, expressed as $E_g n_0^4 \approx 95 \text{ eV}$, where $E_g$ is the fundamental bandgap and $n_0$ is the sub-bandgap refractive index. This relationship implies that high transparency and large refractive indices are typically mutually exclusive. Conventional screening methods based solely on the fundamental bandgap often overlook materials where physical mechanisms—such as symmetry and momentum selection rules or negligible wavefunction overlap—suppress optical absorption well above the electronic gap. Consequently, many potential "super-Mossian" materials (those surpassing the rule's predictions) remain unidentified.

Methodology
The authors propose a modified Moss rule anchored to the optical absorption edge ($E_{edge}$) rather than the fundamental bandgap ($E_g$). $E_{edge}$ is defined as the photon energy at which the absorption coefficient ($\alpha$) reaches $10^4 \text{ cm}^{-1}$. The modified Moss factor is defined as:
$$M_{edge} = \frac{E_{edge} n_0^4}{95 \text{ eV}}$$
Materials with $M_{edge} > 1$ are classified as super-Mossian.

To identify these materials, the authors queried the Materials Project database for compounds satisfying two criteria:

1. A modified Moss factor ($M_{edge}$) greater than 1.2.
2. A PBE-calculated absorption edge ($E_{edge}$) above 1.6 eV.

Thermodynamic stability was enforced by selecting only materials within 0.05 eV/atom of the convex hull. The study focused on ternary and quaternary compounds, excluding binary systems where the method was already validated. For the most promising candidates, specifically the $(Hf,Zr)_2(S,Se)N_2$ family, the authors performed frequency-dependent density functional theory (DFT) calculations using the hybrid HSE06 functional (25% Hartree-Fock exchange) to obtain accurate refractive index and absorption spectra. These calculations utilized the independent particle approximation within the LOPTICS framework in VASP.

Key Contributions and Results

1. Identification of Recurring Chemical and Structural Prototypes:
   Screening revealed that super-Mossian materials are not random but cluster around specific chemical compositions and structural prototypes. Key recurring elements include alkali/alkaline earth metals (notably Be, though its toxicity is noted), Si, and early transition metals in the $d^0$ configuration (Zr, Hf). The analysis highlights that moving beyond single-anion oxide chemistry is crucial, as many high-performing compounds are multi-anion chalcogenides, nitrides, and phosphides.
   Three primary structural prototypes emerged:

   • Chalcopyrites (e.g., $ZnSiP_2$, $AlCuTe_2$).
   • Zintl pnictides (e.g., $Mg(BeAs)_2$).
   • Multi-anion chalconitrides and oxychalcogenides of early transition metals.

2. Discovery of the $(Hf,Zr)_2(S,Se)N_2$ Family:
   The study identifies the chalconitride family $(Hf,Zr)_2(S,Se)N_2$ as a unique and largely unexplored material class. These compounds crystallize in two hexagonal polymorphs ($\alpha$ and $\beta$), differing only in the stacking sequence of $Zr_2N_2$ layers.

   • Optical Performance: Hybrid DFT calculations show that the $\beta$-phase is consistently more transparent than the $\alpha$-phase due to slightly wider bandgaps, weaker transition dipole moments, and stricter symmetry selection rules.
   • Specific Candidates:
     • $\beta$-$Hf_2SeN_2$: An experimentally unreported compound with a calculated $M_{edge}$ of 1.97 and an absorption edge in the UV. It exhibits a sub-bandgap refractive index of 2.74 and a stable birefringence ($\Delta n$) of +0.10 below 2 eV.
     • $\beta$-$Zr_2SeN_2$: Features an ultra-high sub-bandgap refractive index approaching 2.9 ($n_0 \approx 2.84$) with an absorption edge at 2.97 eV.
     • $\beta$-$Hf_2SN_2$: Offers the widest transparency window in the family, with an HSE absorption edge exceeding 3.7 eV.
   • Anisotropy: The materials exhibit significant optical anisotropy. The $c$-axis direction consistently displays a higher absorption onset and a larger Moss factor compared to the $a$- and $b$-axes. Controlling crystal orientation could further enhance performance.

3. Electronic Properties:
   The $(Hf,Zr)_2(S,Se)N_2$ compounds exhibit low effective masses for both holes and electrons ($0.39 < m^* < 0.46$). Combined with previous predictions of ambipolar conductivity and observed degenerate conductivity in $Zr_2SN_2$, this suggests the family's potential as high-refractive-index transparent conductors.

Significance and Claims
The paper establishes that anchoring the Moss rule to the absorption edge rather than the fundamental bandgap is a physically meaningful figure of merit for identifying high-$n$, low-loss materials. This approach successfully recovers known high-performance materials (e.g., $Zr_2SN_2$, $TiO_2$) while uncovering new candidates.

The authors claim that the $(Hf,Zr)_2(S,Se)N_2$ family represents a unique, largely unexplored material system for next-generation photonic applications. These compounds offer a combination of ultra-high refractive indices, low effective masses, and transparency windows extending deep into the UV, surpassing state-of-the-art materials like $TiO_2$ and $SiC$ in specific metrics.

The study emphasizes that while the $\beta$-phase generally outperforms the $\alpha$-phase, the minimal difference in formation enthalpy ($< 18 \text{ meV/atom}$) suggests that kinetic effects during thin-film growth may lead to composite structures. However, since both polymorphs exhibit large Moss factors, such intergrowth is unlikely to critically undermine optical performance. The authors conclude that experimental realization of these compounds as thin films, with a focus on controlling polymorphism and crystal orientation, is the necessary next step to validate their potential.