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
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
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
Let us consider the simplest case of a cubic unit cell, for which
At the first sight, it seems that the recipe for the large value of susceptibility is straightforward: if the material has
If we set
Fortunately, one can find representatives of this class of the materials, the example being bulk ReSe2 [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 ReSe2.
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.
The unit cell, side and top views, of the
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–
Bandgap values calculated within LDA and GGA.
Type/Material | LDA/GGA | LDA [11]/GGA [11] | GGA + GW [16] |
---|---|---|---|
Indirect ReSe2 | 1.0043/1.0463 | 0.87/0.99 | – |
Direct (Z) ReSe2 | 1.0714/1.0909 | 0.97/1.00 | 1.38 |
Direct (
|
1.3262/1.3421 | – | – |
Indirect ReSe2 | 1.1424/1.1790 | – | – |
Direct (Z) ReSe2 | 1.1908/1.2147 | – | 1.60 |
Direct (
|
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 ReSe2 and CBM point for ReS2, and the grid 7 × 7 × 7 was used to define the CBM point for ReSe2 and VBM point for ReS2. 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 ReSe2 and ReS2 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.
The band structure of the bulk ReSe2 and ReS2 calculated by the GGA level of DFT for the path going through
In order to get the the optical response, we employ the TDDFT approach, calculating the diagonal terms of the unit cell polarizability
Real and imaginary parts of ε of the bulk ReSe2 (left) and ReS2 (right) for different light polarizations calculated within TDDFT (see details of calculations in Supplementary). It can be seen that the peaks of the imaginary part of the permittivity, and hence the absorption edges, are 1.32 eV for ReSe2 and 1.7 eV for ReS2. TDDFT, time-dependent density functional theory.
It can be seen that for both ReSe2 and ReS2, 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 ReSe2 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 ReS2 is smaller than that of ReSe2, but is still quite large. It should be noted that the imaginary part of the refractive index in this wavelength range is negligibly small.
The calculated dependence of the the real and imaginary parts of refractive indexes of ReSe2 (left) and ReS2 (right) on the wavelength for different cases of light polarizations. The known experimental data for the modern state-of-the-art high-index materials such as c-Si [24], GaP [25], and Ge [26] are shown for comparison.
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.
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 ReSe2 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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