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Abstract
Dielectric resonators with high light confinement offer low optical losses, which is essential for the development of high-performance optoelectronic devices. Among the different compounds, Zn2GeO4 has emerged as a promising wide band gap transparent conductive oxide with a low-dielectric constant. In order to control and optimize their optoelectronic properties, a good knowledge of their optical constants, such as the refractive index, is required. Here, we exploit the behaviour of Zn2GeO4 microrods as optical cavities to estimate experimentally for the first time, so far as we know, the refractive index dispersion of this compound. In addition, first principle calculations by density functional theory (DFT) of the dielectric function as a function of the light propagation direction have been performed, which have revealed a strong optical anisotropy. The results show that there is also an influence of native defects on the optical propagation properties. These physical aspects are crucial to tailor the optical features of Zn2GeO4 and exploit them in optoelectronic devices with bespoke properties.
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1. Introduction
Light confinement plays a pivotal role in the advancement of optoelectronic devices, enabling precise control and enhanced efficiency of light-matter interactions [1]. Solar cells [2], waveguides [3], optical modulators [4], and nanolaser sources [5] are just a few examples of the many devices that use this crucial feature. Among the several possibilities for achieving light confinement, dielectric and metallic cavities present robust capabilities, while dielectric resonators provide an intriguing alternative, offering high light confinement with minimal optical losses [6]. However, the size of these resonators is limited by the fundamental laws of diffraction [7]. To exploit the potential of semiconductor micro- and nanowires in the aforementioned devices, a prerequisite is a thorough understanding and accurate control over their electrical and optical properties at these reduced scales. In pursuit of this goal, the energy band gap and the refractive index represent two fundamental physical parameters that characterize the optoelectronic behavior of semiconductors, as they determine the photon absorption threshold and the transparency of the material to incident radiation.
In this context, Zn$_2$GeO$_4$ has emerged as a promising wide band gap semiconductor, combining the properties of visible-light transparency (E$_g$ $\sim$ 4.5 eV), high conductivity ($\sim$ 10$^4$ S cm$^{-1}$), and high electron mobility ($\sim$ 80 cm$^2$ V$^{-1}$ s$^{-1}$) [8,9]. Furthermore, Zn$_2$GeO$_4$ ceramics exhibit a low-dielectric constant (relative permittivity) and high-quality factors in the microwave range [10,11], which makes this oxide attractive for the downscaling of integrated circuits [12–14]. Despite these promising attributes, the optical parameters of Zn$_2$GeO$_4$ as a one-dimensional material have remained largely unexplored in the literature. Mostly, ellipsometry studies or transmittance measurements have been applied to determine the optical constants of films or arrays of nanowires, but not of a single microwire [15,16]. However, oxides microwires may behave as optical Fabry-Pèrot (FP) resonators [17]. Therefore, analyzing these FP resonances within a single Zn$_2$GeO$_4$ microrod offers a unique opportunity to estimate the wavelength dependence of the refractive index [18].
In light of these considerations, the present work aims to comprehensively explore the optical properties of Zn$_2$GeO$_4$ microwires. Observed fingerprints occurring in cathodoluminescence (CL) spectra recorded from Zn$_2$GeO$_4$ microrods have been analyzed in the framework of FP optical cavities. The results have allowed us to determine the parameters in both Cauchy and Sellmeier formula for the dispersion relation. On the other hand, we have confronted our experimental results with ab initio Density Functional Theory (DFT) calculations in the 500 to 1000 THz range, where vibrational and optical polarizations are the principal contributions to the dielectric permittivity of Zn$_2$GeO$_4$. The calculations have been performed as a function of the light propagation direction in defect-free crystals as well as in material containing the native points defects. A discussion of both experimental and calculated results is presented.
2. Results and discussion
2.1 Zn$_2$GeO$_4$ microrods as optical microcavities
Zn$_2$GeO$_4$ luminescence studies report an ultraviolet (UV) and/or a visible emission depending on the intrinsic defect landscape of this material [19,20]. In particular, CL spectra show a main UV emission centred at 360 nm, which has been attributed to oxygen vacancies and zinc interstitials [18,21]. Figure 1(a) shows the scanning electron microscopy (SEM) image of a single Zn$_2$GeO$_4$ microrod obtained by a thermal evaporation method (see Figure S1, Supporting Information) on which the CL spectrum displayed in Fig. 1(b) was acquired. The CL spectrum exhibits a series of interference maxima, which are suggestive of the resonant modes of a FP-cavity, overlapping the broad UV emission. The origin of these resonances is the successive reflection of light between the parallel faces of the hexagonal microrod, as has been previously reported [18]. The peak maxima positions can be used to estimate the refractive index dispersion of the Zn$_2$GeO$_4$ microrod over the entire visible range. The spacing between modes $\Delta \lambda$, the refractive index $n$, and the cavity length $L_{FP}$ are related according to the relationship [22]:
where ${dn}/{d\lambda }$ is the wavelength dependence of refractive index. The optical path ($L_{FP}$) can be simply determined by directly measuring the top face of the hexagonal microrod and assuming it possesses a uniform cross-sectional shape (see the inset of Fig. 1(b)). In this case, the measured optical path was 3.205 $\mu$m.
Fig. 1. (a) SEM image of a single Zn$_2$GeO$_4$ microrod. (b) CL spectrum from the Zn$_2$GeO$_4$ microrod shown in (a). (c) Calculated Cauchy (blue solid line) and Sellmeier (red dashed line) dispersion curves.
It is widely accepted that within transparent materials, dispersion remains minimal in regions far from the band gaps (visible range). However, as the wavelength decreases and approaches the optical band gap (typically found in the UV region), a notable increase in dispersion is expected. In such cases, the Cauchy and Sellmeier equations stand as the two commonly employed models for describing refractive index dispersion [23]. Both models are valid in the optically transparent regions (away from band gaps) where the imaginary part of refractive index is very small and there is normal, not anomalous, dispersion. However, unlike the Cauchy equation, the Sellmeier equation can be applied to materials with multiple band gaps and other material absorption bands in the frequency range of interest. Therefore, given that Zn$_2$GeO$_4$ harbors a radiative center within the UV region, specifically between 300-400 nm, which has the potential to influence the dispersion relation, our study employs both approaches to investigate the refractive index of Zn$_2$GeO$_4$.
For the Cauchy approach, we will consider it sufficiently accurate to include only the first two terms and two constants. Accordingly, the Cauchy equation is given by:
On the other hand, for most solid materials, a two-term Sellmeier model, with one term in the UV and another in the infrared, suffices for frequency ranges away from the absorption peaks [23]. Thus, the double-oscillator Sellmeier equation is expressed as follows:
Through an iterative fitting procedure, similar to the approach introduced by Bordo et al. [24], we merge Eq. (1)—initially incorporating the experimental values of $L_{FP}$ and $\Delta \lambda$—with either Eq. (2) or (3). This systematic process enables us to determine the coefficients associated with each dispersion model. The optimal fitting result is determined by selecting the value that minimizes the chi-square statistic, ensuring the best fit.
For the Cauchy model, the optimal fit is reached with values of $A = 1.675$ and $B = 20358$ nm$^2$. In contrast, our fitting procedure for the Sellmeier equation yields the following values: $B_1 = 0.9929$, $C_1 = 30062$ nm$^2$, $B_2 = 0.8449$ and $C_2 = 28900$ nm$^2$. Figure 1(c) shows the estimated Cauchy and Sellmeier dispersions (solid blue and red dotted lines, respectively). Both curves overlap in the region 400-650 nm, while Sellmeier dispersion is slightly higher than the Cauchy’s one for lower wavelengths. This could be due to radiative centres present in the 300-400 nm region, which could influence the Sellmeier curve. In any case, both models propose a refractive index that vary from 1.90-1.93 in the UV range up to 1.72 in the visible region. These results provide the first experimental estimation of the Zn$_2$GeO$_4$ refractive index dispersion in single microwires. The results may be of great interest to exploit the optical properties of Zn$_2$GeO$_4$, such as its waveguiding behaviour when it is used as a one-dimensional material [25].
2.2 First principles calculations
One of the challenges in the light guiding properties in inorganic materials is the eventual anisotropy related to the crystalline structure. Herein, first principles calculations have been performed to determine the optical constants of Zn$_2$GeO$_4$ in the framework of the DFT [26]. While previous DFT-based calculations have explored the optical parameters of Zn$_2$GeO$_4$, they often underestimated the wide band gap of this compound (due to the existing flaw of well-known deficiencies of DFT-GGA functional) and overlooked the consideration of different propagation directions [27]. Here, we employ the B3LYP hybrid functional, which provides very accurate results characterizing oxides [28]. As a result, we obtain a calculated band gap of approximately 4.64 eV, which aligns more closely with experimental values. Furthermore, our investigation encompasses light propagation calculations in the $x$, $y$, and $z$ directions.
The optical parameters have been obtained by using the Coupled Perturbed Hartree-Fock (CPHF) method [29,30] through the frequency-dependent dielectric tensor $\varepsilon (\omega )=\varepsilon ^{'}(\omega )+i\varepsilon ^{''}(\omega )$. The CPHF method provides the microscopic tensor $\alpha$, which is the second derivative of the total energy with respect to the electric field $\mathcal {E}_{\beta }^{0}$ components:
The polarization vector $P$ is the induced dipole moment per volume unit:
where $V$ is the volume of the unit cell. By inserting (7) in (6), and comparing with (5), we obtain the link between the microscopic tensor $\alpha$ and the corresponding macroscopic tensor $\varepsilon$: where $\delta _{\beta \gamma }$ stands for 1 when $\beta = \gamma$ and 0 otherwise.Figure 2 shows the real and imaginary part of the dynamic (wavelength dependent) dielectric function for perfect and defective Zn$_2$GeO$_4$ crystals for different light propagation directions. The point defects considered are oxygen vacancies and zinc interstitials (see Figs. 2(a)-c), which are mainstream defects in Zn$_2$GeO$_4$, and are linked to the main luminescence band [21]. The real part of the dielectric function (Figs. 2(d)-f) corresponds to the polarization response of the material to an external electric field. On the other hand, the imaginary part (Figs. 2(g)-i) describes the ability of the material to absorb energy of the external electric field. In all cases studied here, the dielectric function is anisotropic, as it strongly dependent on the crystal direction. Zn$_2$GeO$_4$ has a rhombohedral unit cell with space group $\overline {R}$3, and its crystalline structure is constituted by corner-sharing GeO$_4$ and ZnO$_4$ tetrahedra bridged by oxygen atoms, aligned parallel to the $c$-axis in a pattern of $\{\textrm {Zn} + \textrm {Zn} + \textrm {Ge} + \textrm {Zn} + \textrm {Zn} + \textrm {Ge} +\ldots \}$ (see Fig. 2(a)). Therefore, there is a distinction between the $z$-axis and $x$- and $y$-axis. Moreover, the calculations reveal a lifting of degeneracy between the $x$- and $y$-axis in the case of oxygen vacancies.
Fig. 2. (a)-(c) Zn$_2$GeO$_4$ unit cell view from $c$-axis and considered native defects. Zn, Ge, and O are represented as blue, magenta and green spheres. Calculated (d)-(f) real and (g)-(i) imaginary part of the dynamic (wavelength dependent) dielectric function for: the perfect Zn$_2$GeO$_4$, Zn$_2$GeO$_4$ with a Zn interstitial atom, and Zn$_2$GeO$_4$ with an O vacancy.
Once the dielectric function is obtained, the refractive index can be simplified to $n^2(\omega )\approx \epsilon (\omega )$. Thus, taking into account the real and the imaginary part of the dielectric permittivity, it results as:
Figure 3 illustrates the wavelength-dependent refractive index for both the perfect Zn$_2$GeO$_4$ crystal and its defective counterpart. A comparative analysis of the values computed through this DFT approach with those experimentally derived from Zn$_2$GeO$_4$ microrods reveals that the DFT-calculated refractive index tends to yield slightly higher values when compared to those estimated using the Cauchy or Sellmeier models (shown in Fig. 1(c)). Nevertheless, it is noteworthy that this overestimation is comparatively lower than the refractive index values reported in previous DFT studies. For instance, in the study develped by Xie and co-workers, the calculated refractive index values for the visible range were approximately 1.5 for defect-free material and 1.3 for cases involving oxygen vacancies [27], both of which fall below the refractive index values obtained experimentally. The enhanced agreement observed between our experimental data and the DFT calculations in this study underscores the significance of employing hybrid functionals when investigating oxides like Zn$_2$GeO$_4$.
Fig. 3. Calculated refractive index as a function of the wavelength for: (a) the perfect Zn$_2$GeO$_4$, (b) Zn$_2$GeO$_4$ with a Zn interstitial atom, and (c) Zn$_2$GeO$_4$ with an O monovacancy.
In addition, optical anisotropy is observed in all the studied cases in concordance with has been obtained from the dielectric function curves. The refractive index exhibits higher values along the $z$-axis as compared to the $x$- and $y$-axes, where the values remain identical. This degeneracy is lifted in the case of the presence of oxygen vacancies, resulting in the most pronounced variation in refractive index among the different orientations. In addition, the decreasing functional dependence on wavelength as it increases, as observed in the Cauchy formula, is not preserved in the calculated dispersion laws. In particular, there are slight maxima in the 400-500 nm range that are not resolved in the Cauchy or Sellmeier formulations.
It is worth noticing that the Zn$_2$GeO$_4$ microrods are aligned with the $c$-axis. The microrods are acting as a FP-cavity, in which light is confined between the parallel faces of the hexagonal structure within the basal plane of the unit cell, essentially perpendicular to the $z$-axis. Consequently, the observed FP resonances detected by CL originate from the propagation of light within the $x-y$ plane. Finally, our DFT calculations of the Zn$_2$GeO$_4$ optical properties reveal a relative low absorption coefficient (imaginary part of the dielectric function), which is in agreement with our consideration that a simplify model of Cauchy and Sellmeier model can be used to calculate the refractive index dispersion in the UV-visible range. Our results show an anisotropy in the refractive index, yielding a higher values in the $z$-axis than in the basal plane ($x$- and $y$-axis) for all the under consideration cases. These anisotropic properties of Zn$_2$GeO$_4$ can be used to control and manipulate the polarization and guiding of light passing through the material.
3. Conclusions
A thorough knowledge of the Zn$_2$GeO$_4$ optical constants is essential to exploit this compound in optoelectronic applications. However, information on these parameters is scarcely reported. In this work, we have exploited the behaviour of a single Zn$_2$GeO$_4$ microrod as optical microcavity to estimate for the very first time, to the best of our knowledege, the refractive index dispersion based on experimental data. The values obtained go from around 1.90 in the UV range to 1.72 in the longest wavelength of the visible region. Knowing the dispersion relation allows to optimise the implementation of Zn$_2$GeO$_4$ microrods in optical devices taking advantage of their waveguide performance. In addition, we confront our experimental results with first-principles calculations of the dielectric function of perfect and defective Zn$_2$GeO$_4$. Our results reveal an anisotropy in the refractive index, yielding a higher values in the $z$-axis than in the basal plane ($x$- and $y$-axis), which is particularly pronounced in the case of oxygen vacancies. These results shed light on the understanding of the Zn$_2$GeO$_4$ optical properties, which could be of great interest for the incorporation of this compound in high-performance optoelectronic devices.
4. Experimental section
- • Synthesis: Zn$_2$GeO$_4$ microrods have been growth by a single-step thermal evaporation method conducted at 800$^{\circ }$C for 8 hours under an Ar flow of 1.5 l/min. The material precursors were a compacted mixture of ZnO:Ge:C powders (2:1:2 wt.%) The morphological and structural characterization were previously performed (see XRD pattern of Figure S1, Supporting Information). The results showed high-quality microrods, with hexagonal cross-section, well ended surfaces and oriented following the $c$-axis of the orthorhombic crystalline structure of the Zn$_2$GeO$_4$ [18].
- • Cathodoluminescence measurements: CL studies were carried out at 20 keV in a Hitachi S2500 scanning electron microscope (SEM) equipped with a PMA-12 charge coupled device camera. Spectra are acquired in a single pass. The measurements were conducted within a wavelength range from 200 nm to 950 nm, with a wavelength accuracy of less than 0.75 nm and a resolution (FWHM) of less than 2 nm. In order to minimize thermal noise, the sensor was actively cooled to a temperature as low as −15 $^{\circ }$C.
- • Procedure for resonance analysis: i) Detection of maxima: The first step involves identifying the positions of the modulation peaks observed in the CL spectrum. To accomplish this, we employ the well-established ’Unipolar Peak Area’ technique, which detects local maxima and minima while considering local variations in the background adjustment. ii) Geometry parameters adjustment: After identifying the maxima, we incorporate experimentally measured cavity geometry parameters into the analysis. We employ the Eq. (1) for adjustment, considering that we lack prior knowledge of the resonance indices. The script iteratively adjusts, allowing the resonance indices to vary until the best possible fit to our observed maxima is achieved. iii) Refractive index determination: Once the resonance indices of the optical cavity have been determined, we proceed to calculate the refractive index using either the Cauchy or Sellmeier approximation within the desired wavelength range. In our case, this range aligns with the experimental spectrum (300 – 700 nm). Importantly, no constraints are imposed on the refractive index values, except that they must exceed unity. The script continues adjusting until it identifies the absolute minimum, signifying the point at which the adjustment of resonance indices yields the lowest error (chi-square).
- • Simulation details: DFT calculations have been performed using the CRYSTAL program [26], in which the crystalline orbitals are expanded as a linear combination of atom-centered Gaussian orbitals, the basis set. The zinc, oxygen, and germanium ions are described using all-electron basis sets contracted as s(8) sp(64111) d(41), s(8) sp(611) d(1), and s(9) sp(76) d(511) sp(31), respectively. Electronic exchange and correlation were approximated by using the B3LYP hybrid functional (static calculations) and the PBE functional (wavelenght dependent calculations). Hybrid functionals are not yet implemented in the CRYSTAL program for dynamical calculations and therefore the PBE functional has been used to perform them. The band gap obtained with the PBE functional (2.5 eV) is smaller than the one provided by B3LYP (4.6 eV), but the electronic and optical properties are not affected by this change. Integration over the reciprocal space was carried out using Monkhorst-Pack (MP) meshes of 6 $\times$ 6 $\times$ 6. The self-consistent field (SCF) algorithm was set to converge at the point at which the change in energy was less than 10$^{-7}$ Hartree per unit cell. The internal coordinates have been determined by minimization of the total energy within an iterative procedure based on the total energy gradient calculated with respect to the nuclear coordinates. Convergence was determined from the root-mean-square (rms) and the absolute value of the largest component of the forces. The thresholds for the maximum and the rms forces (the maximum and the rms atomic displacements) have been set to 0.00045 and 0.00030 (0.00180 and 0.0012) in atomic units. Geometry optimization was halted when all four conditions were satisfied simultaneously.
Funding
Ministerio de Ciencia e Innovación (PID2021-122562NB-I00).
Disclosures
The authors declare no conflicts of interests.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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