^{1} Centre de Nanosciences et Nanotechnologies (C2N), Université-Paris-Sud, CNRS UMR 9001, Université Paris-Saclay, Orsay 91405, France
^{2} Current address: Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
^{3} Institut des Sciences Analytiques et de Physico-Chimie pour l’Environnement et les Matériaux, CNRS, Université de Pau et des Pays de l’Adour, 64053 Pau Cedex, France
^{4} Laboratoire Charles Fabry, Institut d’Optique Graduate School, CNRS, Université Paris-Saclay, 91127 Palaiseau Cedex, France
^{5} Current address: LP2N, Institut d’Optique Graduate School, CNRS, Univ. Bordeaux, 33400 Talence, France
^{6} Department of Physics, Bridgewater State University, Bridgewater, Massachusetts 02325, USA
^{7} e-mail: laurent.vivien@c2n.upsaclay.fr
Abstract
Nonlinear all-optical technology is an ultimate route for next-generation ultrafast signal processing of optical communication systems. New nonlinear functionalities need to be implemented in photonics, and complex oxides are considered as promising candidates due to their wide panel of attributes. In this context, yttria-stabilized zirconia (YSZ) stands out, thanks to its ability to be epitaxially grown on silicon, adapting the lattice for the crystalline oxide family of materials. We report, for the first time to the best of our knowledge, a detailed theoretical and experimental study about the third-order nonlinear susceptibility in crystalline YSZ. Via self-phase modulation-induced broadening and considering the in-plane orientation of YSZ, we experimentally obtained an effective Kerr coefficient of ${\widehat{n}}_{2}^{\mathrm{YSZ}}=4.0\pm 2\times {10}^{?19}\text{}{\mathrm{m}}^{2}\xb7{\mathrm{W}}^{?1}$ in an 8% (mole fraction) YSZ waveguide. In agreement with the theoretically predicted ${\widehat{n}}_{2}^{\mathrm{YSZ}}=1.3\times {10}^{?19}\text{}{\mathrm{m}}^{2}\xb7\text{}{\mathrm{W}}^{?1}$, the third-order nonlinear coefficient of YSZ is comparable with the one of silicon nitride, which is already being used in nonlinear optics. These promising results are a new step toward the implementation of functional oxides for nonlinear optical applications.
1. INTRODUCTION
Integrated nonlinear photonics has reached the point of meeting demands of on-chip optical processing and transmission, owing to the CMOS compatibility. Silicon in particular presents the ability of strong optical confinement. Nevertheless, some properties of silicon are not adequate for integrated photonics. Notably, it presents a large two-photon absorption (TPA) in the near-infrared, while it is accompanied by a large third-order nonlinear susceptibility. This condition limits the use of silicon for devices exploiting Raman lasing [1], Brillouin amplification [2], or parametric amplification [3]. In this context, the quest for other materials, as compatible as possible with silicon foundries, started several years ago.
In these new plethora of options, promising results have been obtained in silicon nitride [4], amorphous silicon [5], or arsenic-free chalcogenides [6], thus exploiting third-order nonlinearities. More broadly, strongly correlated materials like crystalline oxides have emerged as a promising family for their large variety of properties from ferroelectricity, optical nonlinearities to ferromagnetism, or phase transition. The combination of their tunable properties leads to the development of innovative devices with manifold functionalities in many fields such as solar cells [7], sensors [8], electronics [9], or optics [10], to name a few. However, optical nonlinearities of crystalline oxides are still unknown for the most part.
Among the crystalline oxides, yttria-stabilized zirconia (YSZ) is known for its role as a buffer layer for the integration of ${\mathrm{LiNbO}}_{3}$, ${\mathrm{PbTiO}}_{3}$, ${\mathrm{Pb}(\mathrm{Zr},\mathrm{Ti})\mathrm{O}}_{3}$, and ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7}$ thin films on silicon [11–14" target="_self" style="display: inline;">–14] and as a solid electrolyte due to its high ionic conductivity [15]. In addition, YSZ exhibits extraordinary thermal, mechanical, and chemical stability [1618" target="_self" style="display: inline;">–18] as well as high refractive index about 2.15 in the telecom wavelength range and a wide transparency window from UV to mid-IR [16,1921" target="_self" style="display: inline;">–21].
Thanks to the development of growth technique, YSZ has recently demonstrated its potential as a new material for low-loss optical waveguides and as a flexible substrate for infrared nano-optics [22,23]. However, nonlinear optical properties of YSZ have never been investigated.
We present in this work a theoretical study of the third-order nonlinearities of YSZ and its experimental characterization. First-principle calculations were performed to understand the concentration and doping distribution dependence of the third-order nonlinearities in YSZ. Then, we grew high-quality YSZ thin films and fabricated a low-loss waveguide with an optimized material composition. Finally, thanks to a sensitive technique described in detail in Ref. [24], we have characterized the optical nonlinearities of YSZ. The experimental results presented in this paper agree with the calculations and provide a strong first step toward the promising integration of crystalline oxide in a silicon photonics platform.
2. FIRST-PRINCIPLE CALCULATIONS
Assuming that the optical nonlinearities of YSZ should stem from the intrinsic nonlinear optical (NLO) properties of the doping substrate, namely, the cubic phase of ${\mathrm{ZrO}}_{2}$ (${\mathrm{c}\text{-}\mathrm{ZrO}}_{2}$), we proceeded as follows. First, we focused on the static nonlinear optical properties of pure ${\mathrm{c}\text{-}\mathrm{ZrO}}_{2}$ aiming at a reliable estimation of the order of magnitude of its third-order NLO responses in idealized conditions to be used as a guide for our qualitative conclusions. Because little is known about the true local atomistic structure of YSZ, to circumvent the challenging task [25,26] of determining the most stable local crystal structure of each system considered, the second step comprised computations addressing the importance of the vacancy/dopant distribution on the third-order susceptibilities. For this task, we chose two doping concentrations of 3.2% and 33% (mole fraction, hereinafter the same unless specified otherwise) in ${\mathrm{Y}}_{2}{\mathrm{O}}_{3}$ representing the dilute limit and a relatively high doping concentration, respectively. Finally, the third and last step of the current theoretical investigation involved simulations on YSZ of increasing concentrations in ${\mathrm{Y}}_{2}{\mathrm{O}}_{3}$.
In this work, the relative dielectric $\u03f5$ matrix and nonlinear ${\chi}^{(n)}$ susceptibility tensors were obtained from the electronic part [27] of the polarizability (${\alpha}^{e}$) and the first (${\beta}^{e}$) and second (${\gamma}^{e}$) order hyperpolarizabilities of the unit cell as $${\u03f5}_{ij}={\delta}_{ij}+4\pi {\alpha}_{ij}^{e}/V,$$$${\chi}_{ijk}^{(2)}=2\pi {\beta}_{ijj}^{e}/V,$$$${\chi}_{ijkl}^{(3)}=2\pi {\gamma}_{ijkl}^{e}/(3V),$$where $V$ is the volume of the unit cell and ${\delta}_{ij}$ represents the Kronecker delta. Note that an ideal cubic structure features one independent ${\u03f5}_{ii}$ and two ${\chi}_{iiii}^{(3)}$ and ${\chi}_{iijj}^{(3)}$ components, respectively, while the total second-order nonlinear optical response ${\chi}^{(2)}$ vanishes.
Bearing in mind that the computation of third-order susceptibilities of molecules, polymers, and solids greatly depends on the quantum chemical method applied [2830" target="_self" style="display: inline;">–30], we computed the third-order optical nonlinearities of pure and doped bulk systems using three DFT functionals within the generalized gradient approximation (GGA). These are the pure Perdew, Burke, and Ernzerhof (PBE) [31] exchange-correlation functional; its hybrid counterpart PBE0 [32]; and the Becke, three-parameter, Lee–Yang–Parr exchange-correlation functional (B3LYP) [33]. The first two functionals, namely, PBE and PBE0, have been used in previous studies [25,34], involving the structural and electronic properties of YSZ, while the popular B3LYP was chosen in order to check the robustness of the results. All periodic calculations have been performed at the static limit with a developer’s version of CRYSTAL17 software [35], which allows analytic computations of nonlinear optical (NLO) properties of infinite periodic systems. We also performed benchmark computations focusing on the molecular second dipole hyperpolarizabilities of ${{(\mathrm{ZrO}}_{2})}_{\mathit{n}}$ nanoclusters at CCSD(T) (coupled cluster including single, double, and triple excitations) and MP2 (second-order Möller–Plesset perturbation theory) levels using the GAUSSIAN09 [36] suit of programs.
Third-order nonlinearities of${\mathbf{c}\text{-}\mathbf{ZrO}}_{2}$. The results of the respective computations are summarized in Table 1, where we present lattice constants, bandgaps, the electronic contribution to the dielectric constant, and the two independent ${\chi}^{(3)}$ tensorial components of ${\mathrm{c}\text{-}\mathrm{ZrO}}_{2}$. For the sake of brevity, a more detailed discussion of these outcomes is given in Appendix A. Here, we will only mention that the observed method behavior, characterized by notable oscillations in the computed bulk nonlinearities, is in line with that of previous studies [2830" target="_self" style="display: inline;">–30] performed on finite atomic semiconductor nanoclusters. Interestingly, the current computations reveal that the known pathology of pure DFT functionals effect also concerns the computed dielectric constants for which experimental results are available (see values listed in Table 1). Nonetheless, and in spite of the obvious deviations, the obtained results allow us to make the following deductions. First, for ${\mathrm{c}\text{-}\mathrm{ZrO}}_{2}$, characterized by a rather wide bandgap, all three DFT methods yielded third-order susceptibilities of the same order of magnitude varying between 7.6 and $1.53\times {10}^{-21}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{m}}^{2}\xb7\text{\hspace{0.17em}}{\mathrm{V}}^{-2}$ for the tensorial component ${\chi}_{iiii}^{(3)}$ and between 3.8 and $1.25\times {10}^{-21}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{m}}^{2}\text{\hspace{0.17em}}\xb7{\mathrm{V}}^{-2}$ for ${\chi}_{iijj}^{(3)}$. Second, considering that ${\chi}^{(3)}$ can be written as an infinite sum of terms [42] inversely proportional to the cubic power of transition energies, the obtained outcomes at PBE0, B3LYP, and PBE levels indicate that, as we approach the experimental bandgap value from above/below, one should expect larger/smaller theoretical values for both ${\chi}_{iiii}^{(3)}$ and ${\chi}_{iijj}^{(3)}$. Third, taking into account that pure DFT functionals, as PBE, systematically overshoot the (non)linear optical properties of molecular systems [43], we expect that the upper limit of the theoretically determined components ${\chi}_{iiii}^{(3)}/{\chi}_{iijj}^{(3)}$ of ${\mathrm{c}\text{-}\mathrm{ZrO}}_{2}$ should be considerably lower than 7.6 and $3.8\times {10}^{-21}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{m}}^{2}\text{\hspace{0.17em}}\xb7{\mathrm{V}}^{-2}$, respectively. The latter conclusions are supported by a brief DFT functional performance assessment, on the computation of the molecular second dipole hyperpolarizabilities of three ${{(\mathrm{ZrO}}_{2})}_{\mathit{n}}$ nanoclusters [44], namely, ${\mathrm{Zr}}_{4}{\mathrm{O}}_{8}$, ${\mathrm{Zr}}_{8}{\mathrm{O}}_{16}$, and ${\mathrm{Zr}}_{15}{\mathrm{O}}_{30}$ with respect to CCSD(T) and MP2 ab-initio post Hartree–Fock methods. The respective computations, presented in Appendix A, also exposed the pivotal importance of electron correlation effects on the third-order optical nonlinearities of ${{(\mathrm{ZrO}}_{2})}_{\mathit{n}}$ species.
Table 1. Cell Parameter $a$ (Å), Bandgap ${E}_{g}$ (eV), Electronic Contribution of the Dielectric Constant $\u03f5={\u03f5}_{ii}$, and Third-Order Susceptibility Components ${\chi}_{iiii,iijj}^{(3)}\text{\hspace{0.17em}}({10}^{-21}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{m}}^{2}\xb7\text{\hspace{0.17em}}{\mathrm{V}}^{-2})$ of ${\mathrm{c}\text{-}\mathrm{ZrO}}_{2}$ Computed with the PBE, PBE0, and B3LYP Functionals
Doping and vacancy/dopant distribution effects on the optical nonlinearities of YSZ. To conduct the respective structure property investigation, we considered doped bulk YSZ systems of two concentrations in ${\mathrm{Y}}_{2}{\mathrm{O}}_{3}$, namely, 33% and 3.2%. For the highest dopant concentration, two different compensated unit cells were built upon the face-centered cubic (fcc) $\mathrm{Fm}\overline{3}\mathrm{m}$ cell of zirconium dioxide ($\text{volume}=131.9\text{\hspace{0.17em}\hspace{0.17em}}{\AA}^{3}$, ${E}_{g}=5.37\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{eV}$) by removing one O atom from its tetrahedral site, thus creating an oxygen vacancy. Then, two out of the four nearest-neighbor (NN) Zr atoms of the corresponding vacancy were substituted by an equal number of Y atoms, delivering two nonequivalent NN crystal structures of ${\mathrm{Y}}_{2}{\mathrm{O}}_{3}$, ${2\mathrm{ZrO}}_{2}$. The equilibrium nuclear geometry of each unit cell and their optical nonlinearities were optimized with the PBE0 functional. In the case of 3.2% in ${\mathrm{Y}}_{2}{\mathrm{O}}_{3}$, we considered the 10 most stable nonequivalent configurations reported by Parkes et al. [25], the crystal structures of which were built upon a doped $2\times 2\times 2$ super cell of ${\mathrm{c}\text{-}\mathrm{ZrO}}_{2}$, optimized with the PBE functional.
The computed electronic and optical properties of the two crystal configurations of YSZ 33% in ${\mathrm{Y}}_{2}{\mathrm{O}}_{3}$ are given in Table 2. Both crystal structures are characterized by equal bandgap values and reduced third-order nonlinearities to those obtained for pure ${\mathrm{c}\text{-}\mathrm{ZrO}}_{2}$ (see Table 1). Judging from the relative strengths of each component along the $x\equiv y$ and $z$ axis, we see that, although the creation of an O vacancy and the replacement of two Zr atoms with Y reduces the cubic symmetry of the host, the third-order NLO responses remain quasi-isotropic in character because ${\chi}_{xxxx,xxyy}^{(3)}\approx {\chi}_{zzzz,xxzz}^{(3)}$. Hence, even for a relatively high yttria concentration, YSZ inherits most of the isotropic nonlinear optical nature of ${\mathrm{c}\text{-}\mathrm{ZrO}}_{2}$. A similar property trend is obtained for all configurations of ${\mathrm{Y}}_{2}{\mathrm{O}}_{3}$ 3.2% considered here.
Table 2. Unit Cell Volume V (${\AA}^{3}$), Bandgap ${E}_{g}$ (eV), Electronic Contribution to the Dielectric ${\u03f5}_{xx}={\u03f5}_{yy}$ and ${\u03f5}_{zz}$ Components, and Kerr (IDRI) Effect Third-Order Susceptibility ${\chi}_{xxxx}^{(3)}={\chi}_{yyyy}^{(3)}$, ${\chi}_{zzzz}^{(3)}$, and ${\chi}_{xxyy}^{(3)}$, ${\chi}_{xxzz}^{(3)}={\chi}_{yyzz}^{(3)}$ Components (${10}^{-21}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{m}}^{2}\text{\hspace{0.17em}}\xb7{\mathrm{V}}^{-2}$) of Two Nonequivalent Local Crystal Structures of YSZ 33% in ${\mathrm{Y}}_{2}{\mathrm{O}}_{3}$ (see Fig. 2)^{a}
As shown in Fig. 1, the influence of the vacancy/dopant distribution on the properties of interest is practically negligible. Furthermore, for both concentrations, we also computed the corresponding second-order susceptibilities. In all cases, the computed second-harmonic generation ${\chi}^{(2)}(-2\omega ;\omega ,\omega )$ for $\omega =0.8\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{eV}$ (corresponding to the wavelength $\lambda =1550\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$) has been proven considerably small. Indeed, the largest absolute xyz component value of ${\chi}^{(2)}$ for the most stable YSZ 33% structure is lower than $0.014\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{pm}\xb7{\mathrm{V}}^{-1}$.
Fig. 1. Variation of the third-order susceptibility tensorial components (${\chi}_{iiii,iijj}$) of YSZ 3.2% in ${\mathrm{Y}}_{2}{\mathrm{O}}_{3}$ as a function of the relative position between Y dopants (yellow spheres) and O vacancies (green spheres). The unit cells [25] of each configuration considered, representing symmetry in equivalent vacancy/dopant distributions, are schematically given at the right. Solid lines represent the corresponding susceptibility components of ${\mathrm{c}\text{-}\mathrm{ZrO}}_{2}$. All values have been computed at the PBE0 level of theory.
In comparison with the ${\mathrm{c}\text{-}\mathrm{ZrO}}_{2}$, the weak bandgap opening of about 0.2 eV observed in YSZ 33% in ${\mathrm{Y}}_{2}{\mathrm{O}}_{3}$ should not justify the observed lowering in the third-order susceptibilities. Hence, bearing in mind that the third-order nonlinearities of charge transfer insulators [45] should depend on the amount of electron and hole states lying near the gap, the observed effect could be primarily attributed to the doping-induced reduction of the density of low-lying hole states in the conduction band. This is well illustrated by the projected density of states (PDOS) of YSZ 33% [see Fig. 2(b)], where the DOS projected on Zr atoms (3p in character), located at the bottom of the conduction band, drastically decreases after the replacement of two Zr atoms with an equal number of Y. On the other hand, the creation of an O vacancy does not introduce important changes in the valence band (O-2p-in-character) of ${\mathrm{c}\text{-}\mathrm{ZrO}}_{2}$. These are two enlightening trends, suggesting that the third-order nonlinear optical responses of doped ${\mathrm{c}\text{-}\mathrm{ZrO}}_{2}$ could be further improved with doping agents that are able to deliver a significant increase of low-lying hole states.