# Abstract

We establish a simple quantitative criterium for the search of new dielectric materials with high values of refractive index in the visible range. It is demonstrated, that for light frequencies below the bandgap, the latter is determined by the dimensionless parameter *η* calculated as the ratio of the sum of the widths of conduction and valence bands and the bandgap. Small values of this parameter, which can be achieved in materials with almost flat bands, lead to dramatic increase of the refractive index. We illustrate this rule with a particular example of rhenium dichalcogenides, for which we perform ab initio calculations of the band structure and optical susceptibility and predict the values of the refractive index

## 1 Introduction

All-dielectric photonics [1], [2] is arguably the most rapidly evolving field of modern nano-optics. The basic component in all-dielectric photonics, a dielectric nanoantenna, supports optical Mie resonances [3], with properties that can be flexibly controlled by the geometry of a nanoantenna. In particular, the variation of its shape allows changing the nature of the lowest energy optical resonance from electric dipole to magnetic dipole [4], [5], the functionality inaccessible in plasmonic devices. With the current stage of technology, the fabrication of specifically designed arrays of resonant antennae with finely controlled shape and lattice geometry became a routine task, which paved the way toward unprecedented control over linear [6], [7], [8] and nonlinear [9], [10] light manipulation.

The key factor defining the functionality of resonant all-dielectric nanostructures, besides their shape, is the value of the refractive index of the material forming resonant nanoantennas. Indeed, the quality factor of the fundamental Mie resonance scales is

While virtually purely real refractive indices

In this work, we show the sum of the widths of the conduction

is the key parameter which defines the value of the refractive index for the frequencies slightly below the bandgap. We derive a simplified estimation for the susceptibility and demonstrate that the refractive index can be substantially increased if *η* becomes small, as it happens in materials with flattened valence and conduction bands. This situation takes place in

## 2 Theoretical model

We start from considering a simplified model of interband polarization in bulk material and show that the condition of the weak dependence of the bandgap on the wave vector in the whole Brillouin zone (BZ) (the extreme case being a material with flat bands) results in a high value of optical susceptibility close to the absorption edge.

We start from the simplest expression for the susceptibility tensor given by the Kubo formula

where *f* is the Fermi–Dirac distributions,

where

Let us consider the simplest case of a cubic unit cell, for which *D* is the lattice period. We limit ourselves to the case of single conduction and valence bands separated by the bandgap

where

At the first sight, it seems that the recipe for the large value of susceptibility is straightforward: if the material has *P* are directly related to each other. This can be easily seen if we use

where the last equality follows from two band *χ* on the bandwidths is therefore determined by the competition between the numerator and denominator in Eq. (4).

If we set

where *η*, as it contains the universal decaying function of this parameter. Therefore, although the matrix element of the interband transition decays with decrease in *η*, maximal values of the susceptibility should be still expected for materials with small *η*, i.e., for those having large gap and narrow conduction and valence bands.

Fortunately, one can find representatives of this class of the materials, the example being bulk ReSe_{2} [11]. This layered material belonging to the rhenium dichalcogenide family has been gaining recently increasing attention, mainly owing to its pronounced in-plane anisotropy and suppressed interlayer van der Waals coupling [12]. To check the results of our qualitative analysis, in the next section, we provide the data of ab initio modeling of the linear optical response of bulk ReSe_{2}.

## 3 Results of the ab initio modeling

All ab initio calculations were performed using QUANTUM ESPRESSO package [13]. The analysis of the optical response was performed in three steps.

In the first step, we determined the equilibrium positions of the ions in the lattice by full self-consistent geometry optimization within density functional theory (DFT). The top and side views of the resulting structure are presented in Figure 1. The obtained lattice parameters and their comparison to experimental XRD data are given in Supplementary.

### Figure 1:

In the second step, we employed two different DFT approaches (LDA and GGA) to calculate the three-dimensional (3D) band structure.

In the third step, we calculated the diagonal components of the unit cell polarizability tensor

Most of the previous calculations of the band structure of the considered materials focused only on the dispersion along the special sets of directions, example being the 2D projection into the layers plane [14], [15]. In the other cases, highly symmetric paths were chosen in the 3D BZ connecting 3D special points, without exploring the structure of the whole BZ [16], [17] and thus missing the true band edges, which are crucially important for the determination of the optical response. The possible drawbacks of this approach were analyzed in detail in the recent work [11], wherein the necessity of the calculation of the band energies everywhere within the BZ and determination of the constant energy surfaces was stressed. In the present paper, we used the *k*-path, developed in the study by Gunasekara et al. [11], and calculated the band structure using the following path: –CBM–CBM′–*Z*–*Z*–*Z* point run along the surface of the BZ containing the *Z* point. To find the k-points corresponding to the VBM and CBM, we used the different grids in BZ–6 × 6 × 6 and 7 × 7 × 7. The calculated values of the direct and indirect bandgap are presented in Table 1 and compared with the previously reported data [11], [16].

### Table 1:

Type/Material | LDA/GGA | LDA [11]/GGA [11] | GGA + GW [16] |
---|---|---|---|

Indirect ReSe_{2} | 1.0043/1.0463 | 0.87/0.99 | – |

Direct (Z) ReSe_{2} | 1.0714/1.0909 | 0.97/1.00 | 1.38 |

Direct (_{2} | 1.3262/1.3421 | – | – |

Indirect ReSe_{2} | 1.1424/1.1790 | – | – |

Direct (Z) ReSe_{2} | 1.1908/1.2147 | – | 1.60 |

Direct (_{2} | 1.6809/1.7441 | – | 1.88 |

CBM, conduction band minimum; VBM, valence band maximum.

The grid 6 × 6 × 6 was used to define the VBM point for ReSe

_{2}and CBM point for ReS_{2}, and the grid 7 × 7 × 7 was used to define the CBM point for ReSe_{2}and VBM point for ReS_{2}. This choice was dictated by the reasons of grid parity and symmetry of the BZ. All energies are given in electron volt.

The obtained band structures of ReSe_{2} and ReS_{2} are presented in Figure 2. It can be seen that the energy of the lowest conduction and the highest valence bands depend only weakly on the wavevector, and the bands are close to the flat ones.

### Figure 2:

In order to get the the optical response, we employ the TDDFT approach, calculating the diagonal terms of the unit cell polarizability *V*,

### Figure 3:

It can be seen that for both ReSe_{2} and ReS_{2}, the absorption sharply increases only when the photon frequency approaches the bandgap in

One can see that the real parts of the in-plane components of the permittivity of ReSe_{2} exceed 25 in the broad frequency range, which should correspond to the high refractive index. Since for biaxial crystals, the refractive index depends on the light propagation direction, we limit ourselves to the case when light travels along one of the three principal axes. In this case, the refractive index for each of the two polarizations is just a square root of the corresponding permittivity. The spectra of the refractive index components are shown in Figure 4. One sees that in the broad wavelength range, the refractive index exceeds 5, which is substantially larger than the index of the modern state-of-the-art high-index materials such as c-Si [24], GaP [25], and Ge [26]. The refractive index of ReS_{2} is smaller than that of ReSe_{2}, but is still quite large. It should be noted that the imaginary part of the refractive index in this wavelength range is negligibly small.

### Figure 4:

In our analysis, we have not accounted for the excitonic contribution to the dielectric permittivity, which can be substantial and would lead to the additional absorption losses at the frequencies slightly below the bandgap. Nevertheless, since the range of the large refractive index covers more than 0.5 eV below the bandgap, it should be possible to detune from the exciton absorption lines to probe the lossless large refractive index.

## 4 Conclusions

We established a simple quantitative criterium for the search of high-refractive-index dielectric materials, expressed in terms of the single dimensionless parameter calculated as the ratio of the sum of the widths of conduction and valence bands and the bandgap. With the use of it, we found that ReSe_{2} is low-loss material, possessing a record high refractive index

**Funding source: **Russian Science Foundation

**Award Identifier / Grant number: **20-12-00224

**Funding source: **Ministry of Education and Science of the Russian Federation10.13039/501100003443

**Award Identifier / Grant number: **14.Y26.31.0015

**Funding source: **Icelandic research fund

**Award Identifier / Grant number: **163082-051

**Author contribution:**All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.**Research funding**: The authors acknowledge the support from the mega-grant No. 14.Y26.31.0015 of the http://dx.doi.org/10.13039/501100003443, “Ministry of Education and Science of the Russian Federation.” I.A.S. acknowledges the support from the Icelandic research fund, grant No. 163082-051.**Conflict of interest statement:**The authors declare no conflicts of interest regarding this article.

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## Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph-2020-0416).

**Received:**2020-07-23

**Accepted:**2020-09-29

**Published Online:**2020-10-28

© 2020 Anton A. Shubnic et al., published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.