Tantalum Arsenide-Based One-Dimensional Photonic Structures

Abstract

Weyl semimetals can be described as the three-dimensional analogue of graphene, showing linear dispersion around nodes (Weyl points). Tantalum arsenide is among the most studied Weyl semimetals. It has been demonstrated that TaAs has a very high value of the real part of the complex refractive index in the infrared region. In this work we show one-dimensional photonic crystals alternating TaAs with SiO₂ or TiO₂ and a microcavity where a layer of TaAs is embedded between two SiO₂-TiO₂ multilayers.

1. Introduction

Tantalum arsenide is a deeply studied Weyl semimetal [1]. Angle resolved photoemission spectroscopy measurements together with ab initio calculations demonstrated that fermions in TaAs can be described as massless chiral particles with spin ½ [2,3,4,5,6]. Recently, Wu et al. [7] have measured a very intense nonlinear optical response of TaAs together with other monopnictide Weyl semimetals, while Weber et al. [8] have studied the ultrafast dynamics of TaAs. Buckeridge et al. [9] have calculated several properties of TaAs, including the dielectric function of the material. Owing to the tetragonal symmetry, an anisotropy of the optical properties arises, with two different dielectric functions when the oscillating electric field is parallel to the a and b axis or parallel to the c axis. In the c direction, at photon energies below 1 eV, the imaginary part of the refractive index decreases significantly, while the real part of refractive index has values above 6. Thus, TaAs can be exploited as a high refractive index material in the infrared region.

For this reason, we want to demonstrate in this work the implementation of tantalum arsenide as a layer in a one-dimensional (1-D) multilayer photonic crystal. One-dimensional photonic crystals are the simplest case of photonic crystals that are materials in which the alternation of high and low refractive indices has a length scale of the light wavelength. In these materials, light is not allowed to propagate for certain photon energies, in correspondence with the so-called photonic band gap [10,11,12,13]. One-dimensional multilayer photonic crystals can be fabricated with many fabrication techniques such as sputtering, spin coating, pulsed laser deposition, chemical etching, and molecular beam epitaxy [14,15,16,17,18]. These structures are useful for several applications, for example lasing [15,19], emission control [20,21,22], and sensing [23,24,25]. Herein, we show one-dimensional photonic crystals alternating TaAs with SiO₂ or TiO₂ and a microcavity where a layer of TaAs is embedded between two SiO₂-TiO₂ multilayers. We show that with very thin layers of TaAs, 16 nm in the photonic crystals and 8 nm in the microcavity, we can achieve very efficient photonic band gaps and cavity defects, due to the high real part of the refractive index of TaAs in the infrared region.

2. Methods

To calculate the optical properties of the different photonic structures studied herein, we have employed the transfer matrix method [26,27]. In our study, the light impinges at normal incidence on a system composed of a glass substrate, the photonic structure, and air.

Briefly, the electric (magnetic) field in air, E₀ (H₀), is related to the electric (magnetic) field in glass, E_m (H_m), through this expression

$$\left[\begin{array}{c}{E}_0\\ H_0\end{array}\right]={\displaystyle \prod }_{j=1}^x{M}_j\left[\begin{array}{c}{E}_m\\ H_m\end{array}\right]=\left[\begin{array}{cc}{m}_{11}& m_{12}\\ m_{21}& m_{22}\end{array}\right]\left[\begin{array}{c}{E}_m\\ H_m\end{array}\right]$$

(1)

where x is the number of layers. M_j is the characteristic matrix of the layer j

$$M_j=\left[\begin{array}{cc}cos\left({\varphi }_j\right)& -\frac{i}{p_j}sin\left({\varphi }_j\right)\\ -i{p}_{j}sin\left({\varphi }_j\right)& cos\left({\varphi }_j\right)\end{array}\right]$$

(2)

${\varphi }_j$ is the phase variation of the light wave passing through the layer, and for normal incidence ${\varphi }_j=\left(2\pi /\lambda \right)n_j{d}_j$, with n_j being the refractive index of the layer, d_j is the layer thickness, and $p_j=\sqrt{{\epsilon }_j/{\mu }_j}$ for a transverse electric wave (TE, taking into account that for normal incidence TE and TM show the same behaviour in 1-D photonic crystals).

The transmission coefficient t and the transmission T are

$$t=\frac{2p_s}{\left(m_{11}+m_{12}{p}_0\right)p_s+\left(m_{21}+m_{22}{p}_0\right)}$$

(3)

$$T=\frac{p_0}{p_s}{\left|t\right|}^2$$

(4)

We have studied the optical response of a TaAs/SiO₂ photonic crystal, of a TaAs/TiO₂ photonic crystal, and of a (TiO₂-SiO₂)₇-[SiO₂-TaAs-SiO₂]-(SiO₂-TiO₂)₇ microcavity in the range 0.2–1.1 eV with a step of 2.5 meV.

3. Results and Discussion

In this work, we studied the optical properties of 1-D photonic crystals made with TaAs alternated with SiO₂ or TiO₂ and of a microcavity in which TaAs was embedded between two SiO₂-TiO₂ Bragg mirrors (seven bilayers), as sketched in Figure 1.

In Figure 2, we show the light transmission spectrum of a photonic crystal made of seven bilayers of TaAs and SiO₂. The thickness of each TaAs layer was 16 nm, while the thickness of each SiO₂ layer was 1200 nm. n and k (real part and imaginary part of the refractive index) of TaAs, in the c direction of the crystal, have been taken from Reference [9]. The Sellmeier equation for the dispersion of the refractive index of silica is [28]:

$$n_{Si{O}_2}^2\left(\lambda \right)-1=\frac{0.6961663{\lambda }^2}{{\lambda }^2-{0.0684043}^2}+\frac{0.4079426{\lambda }^2}{{\lambda }^2-{0.1162414}^2}+\frac{0.8974794{\lambda }^2}{{\lambda }^2-{9.896161}^2}$$

(5)

The Sellmeier equation for the dispersion of the refractive index of titania, reliable for thin films in the range 0.2–1.1 eV, fits the data by Siefke et al. [29,30] and is

$$n_{Ti{O}_2}^2\left(\lambda \right)-1=\frac{4.181{\lambda }^2}{{\lambda }^2-{0.2133}^2}+\frac{5.068{\lambda }^2}{{\lambda }^2-{14.33}^2}$$

(6)

For thicknesses of TaAs above 16 nm, the light absorption of the material became very strong because of the high imaginary part of the refractive index.

If we used TiO₂ instead of SiO₂ to be alternated with TaAs, we observed a photonic band gap in the same spectral position (i.e., 0.35 eV) with a thickness of the TiO₂ layers of 765 nm, resulting in a thinner photonic crystal. In Figure 3, the light transmission spectrum of a photonic crystal made of TaAs and TiO₂ is depicted.

In the spectra in Figure 2 and Figure 3, the transmission valley around 0.65 eV was the second order of the photonic band gap of the photonic crystals. For photon energies above 0.8 eV, the imaginary part of the TaAs complex refractive index increased significantly, such that the transmission valley at 0.9–1.0 eV was mostly due to TaAs absorption [9].

The photonic band gap was less intense with a TaAs-TiO₂ unit cell (minimum at about 0.1) with respect to the one with a TaAs-SiO₂ unit cell (minimum at about 0.03) because of the smaller refractive index contrast in the TaAs-TiO₂ case.

In Figure 4, we show the light transmission spectrum of a (TiO₂-SiO₂)₇-[SiO₂-TaAs-SiO₂]-(SiO₂-TiO₂)₇ microcavity. The TiO₂ layers of the [TiO₂-SiO₂]₇ Bragg mirrors were 377.78 nm thick, while the SiO₂ layers were 590.28 nm thick. In this way, the TiO₂ layers and SiO₂ layers followed the condition n_SiO2d_SiO2 ≅ n_TiO2d_TiO2. The two SiO₂ layers that sandwiched the TaAs layer in the cavity defect were 560 nm thick, while the thickness of the TaAs layer was 8 nm. The microcavity shows in the region 0.31–0.42 eV a photonic band gap with an intense defect peak at 0.376 eV. Since n_SiO2d_SiO2 ≅ n_TiO2d_TiO2, the second order of the microcavity photonic band gap was not observed.

4. Conclusions

In this work, we have exploited the high refractive index of tantalum arsenide in the infrared region (in the region 0.2–1.1 eV), employing the material as a layer in the engineering of a 1-D multilayer photonic crystal and of a 1-D microcavity. We showed one-dimensional photonic crystals alternating TaAs with SiO₂ or TiO₂ and a microcavity where a layer of TaAs was embedded between two SiO₂-TiO₂ multilayers. It is worth noting that with thin TaAs layers, 8 to 16 nm thick, it is possible to build photonic crystals and microcavities in the infrared region with good efficiency. This can be promising for the fabrication of thin filters for this spectral region.