The superlattice-on-insulator wafer-， such as the 300-mm silicon wafer-， is presently a theoretical-physics vision that promises high performance in several opto-electronics areas¹. Formed by wafer bonding, this heterogeneously integrated wafer would be diced into chips offering uniquely capable photonic and opto-electronic integrated circuits (PICs and OEICs). Our theoretical analysis indicates that the chips will have strong Pockels effect, large optical nonlinearities, and large electric-field-induced three-wave mixing, EFIM, which is the topic of this paper.

In principle, the wafer-scale short-period superlattice stacks can be engineered to give high performance chips by (1) the choice of the undoped lattice-matched semiconductors A and B that comprise the superlattice, (2) the choice of crystallographic growth direction, such as [111], (3) the choice of the atomic-monolayers numbers N and M used to create the superlattice stack, (4) the choice of transverse electric (TE) or transverse magnetic (TM) polarization of modes traveling in the superlattice waveguides, (5) the direction of the applied d.c. electric field E(dc) in the present EFIM focus, and (6) the choice of the wavelength-of-operation λ in relation to the superlattice band-gap wavelength λg at which band-to-band absorption of photons begins to occur, assuming λ>λg for low-loss superlattice waveguiding. The buried oxide in the superlattice-on-insulator platform assures high index-contrast waveguiding. The SLOI wafer is believed to be compatible with high-volume manufacture in a CMOS foundry.

We have identified GaP/Si, AlP/Si, ZnS/Si, GaAs/Ge, ZnSe/Ge, AlAs/Ge, and AlN/3C-SiC as the key platforms. Speaking generally, all seven of these possess large or very large second-order and third-order nonlinear optical susceptibilities ${\chi }^{\left(2\right)}$ and χ${\chi }^{\left(3\right)}$. However, the inherent ${\chi }^{\left(2\right)}$ by itself does not ensure phase-matching of waves in a three-wave mixing process, and because of that, we are focusing instead in this paper on the inherent ${\chi }^{\left(3\right)}$ of the superlattice, on the ideas that we can convert this ${\chi }^{\left(3\right)}$ into a highly effective ${\chi }^{\left(2\right)}$ by means of properly chosen $E_{\left(dc\right)}$ direction, and that we can obtain at the same time high-efficiency phase matching of three waves using our new periodically reversed applied voltage (PRAV) technique applied to a straight or a racetrack SL strip waveguide. Numerical simulations of this PRAV quasi-phase matching (QPM) are presented here.

Practical applications of this EFIM are projected for both classical and quantum photonics, such as classical second-harmonic generation EFISHG and quantum frequency conversion QFC. The spectral translation of a quantum state of light plays a crucial role in enabling photonic interconnects in quantum networks². The development of quantum chips will be boosted by efficient difference-frequency generation DFG and by efficient optical parametric oscillation OPO, which are frequency down-conversion processes. These integrated-chip applications will also be enabled by sum-frequency generation SFG, such as second-harmonic generation SHG. Our PRAV QPM is projected to provide such DFG, OPO, SFG and SHG.

There is considerable prior art in the photonics literature on E-field-induced and poling-induced χ(2) effects in Si, SiO₂, Si₃N₄, glass fiber, polymer, BaTiO₃ and LiNbO₃^3-20. However, for years, the periodically poled lithium niobate photonics platform, PPLN, has been the acknowledged leader in QPM frequency conversion. In the studies given here, we make comparisons of the SL PRAV to the present PPLN art, and we conclude that superlattice performance for the “representative” ZnS/Si stack will be comparable to, or competitive with, the PPLN performance in SHG and OPO.

It is worth repeating that the work here provides theoretical guidelines on performance expected if the layering within superlattice waveguides is nearly perfect. Of course, in developmental experiments, there will be fabrication errors including surface roughness, etching errors, variation in layer thicknesses, point defects, threading dislocations and perhaps unwanted strain as well as unintentional doping. Such errors will induce propagation loss within the passive SLOI waveguides and will decrease the performance metrics for the SLOI EFIM devices.

The paper is organized as follows. Motivations for realizing PICs and OEICs based on SLOI are detailed in Section II. Numerical results about the EFIM processes are detailed in Section III. Efficient SHG, and OPO features are found, and they are deployed to demonstrate the potential of the proposed platform. Finally, Section IV summarizes the work.

In this paper, a physics-and-engineering procedure has been performed in order to investigate EFIM processes in a straight waveguide, and in bus-coupled integrated racetrack-resonators based upon ${\left(\text{ZnS}\right)}_N/{\left(\mathrm{S}{\mathrm{i}}_2\right)}_M$ short-period-superlattices. In this context, general physical aspects have been investigated by means of the empirical $s{p}^3{s}^{*}$ tight-binding method, by determining the features of the electronic structure and the influence of the monolayers number upon ${\chi }_{xxxx}^{\left(3\right)}$ and the SPSL refractive index dispersion. We have explored large third-order nonlinearity coefficients in undoped, unstrained multi-layered waveguided (ZnS)N/(Si2)M structures, demonstrating that the electric-field-induced second-order nonlinearity can be used to induce efficiently SHG and OPO in the wavelength range from 1000 nm to 2000 nm. Moreover, we have demonstrated that superlattices based on the [111] ${\left(\text{ZnS}\right)}_{\mathrm{N}}/{\left(\mathrm{S}{\mathrm{i}}_2\right)}_M$ SL structure are suitable for realizing Quasi-Phase-Matched waveguides, operating with the fundamental TE modes for all the wavelengths involved in the EFIM process, where segmented side-electrodes that are periodically cascaded along the strip waveguide allow periodically reversed applied voltage. As a result, the maximal confinement in the SPSL core, and the large overlap between pump and second-harmonic signals, lead to increasing the conversion efficiency with respect to the case of SPSL waveguides based on large inherent ${\chi }_{zzz}^{\left(2\right)}\left(2\omega ,\omega ,\omega \right)$, where the overlap between fundamental and higher-order TM modes is required. Using the large electric-field-induced ${\chi }_{eff}^{\left(2\right)}$ values provided by ${\left(\text{ZnS}\right)}_3/{\left(\mathrm{S}{\mathrm{i}}_2\right)}_3$ superlattices, we predict on-chip resonant SHG and OPO with performances comparable to the results experimentally found for the LNOI platform. For the above reasons, the present work offers a performance motivation for using the ${\left(\text{ZnS}\right)}_{\mathrm{N}}/{\left(\mathrm{S}{\mathrm{i}}_2\right)}_M$ SL platform in future experimental heterogeneous SLOI opto-electronc PIC chips for classical and quantum applications.

The multilayered waveguide proposed here, is a ZnS/Si short-period superlattice (SPSL). Generally speaking, the superlattice ${\left(\mathrm{A}\right)}_N/{\left(\mathrm{B}\right)}_M$ we consider has N monolayers (MLs) of A and M monolayers of B, repeated periodically along the [111] growth direction z. Note that the single ML is comprised of a two-atom-thick layer (i.e. cation and anion). In this context hereafter, we denote with ${\left(ca\right)}_N$/${\left(\mathrm{A}\right)}_N/{\left(\mathrm{B}\right)}_M$ the SL structure. All atoms are on the sites of a zinc-blende lattice with lattice constant aL (under the perfect AB-matched condition). The x and the y coordinate axes are chosen along [$\overline{2}$ 1 1] and [0 $\overline{1}$ 1]. In our investigations we assume $N+M=3q$ (q=1,2,3…), for which the Bravais lattice is hexagonal and the point group is C3v.

In our approach, the Hamiltonian matrix, H(k), related to the generic ${\left(ca\right)}_N$/${\left(CA\right)}_M$ SLs is obtained by means of the empirical $s{p}^3{s}^{*}$ tight binding (TB) method and is defined as⁴:

where the 10×10 matrices $H_{ca\left(CA\right)}$, $G_{ca\left(CA\right)}$ and $F_{ca\left(CA\right)}$ denote intramaterial interaction for ${\left(ca\right)}_N$ and ${\left(CA\right)}_M$, respectively. In these blocks, every element represents a 5×5 matrix taking into account the intrasite energies, as well as second and nearest neighbor interactions. Moreover, the matrices G_i and $F_i$ (i=0,1) denote the intermaterial interactions.

Within the density matrix calculation, both linear and nonlinear susceptibilities are directly dependent upon the dipole matrix elements $〈n,\mathit{k}|\mathit{u}·\mathit{r}|m,\mathit{k}〉$,$\mathit{u}·〈n,\mathit{k}\left|\mathit{r}\right|m,\mathit{k}〉$, where u is the polarization vector of the electric field and r is the vector position operator. Here n and m denote the band index, running in either VB or CB, and k indicates the wavevector. In k-space the position operator is proportional to the gradient with respect to k of the Hamiltonian H(k)⁴:

where Ei(k) represents the energy at the k point for the i-th band. It should be noted that both the Hamiltonian H(k) and the dipole matrix elements are calculated taking into account the band offset parameter (see Ref. 25 for the ${\left(\text{ZnS}\right)}_{\mathrm{N}}/{\left(\mathrm{S}{\mathrm{i}}_2\right)}_M$ SL).

We first calculate the electronic structure by diagonalizing the Hamiltonian H(k) and consequently we apply Eq. (12). The calculation for both the refractive index (${\chi }^{\left(1\right)}$) and the ${\chi }_{xxxx}^{\left(3\right)}$ susceptibility have been addressed by considering the transitions from VB to CB. According to the general definition given in²⁵ (not reported here for compactness reasons), we estimate the linear and third-order susceptibility by performing a summation over all VB and CB bands and integrating in k space. In this way, the refractive index and ${\chi }_{xxxx}^{\left(3\right)}$ take into account all the effects related to band non-parabolicity and related to the momentum-dependence of the dipole matrix elements (see Eq. (12)).

The method adopted here can be functionalized to our aim, guaranteeing the possibility of fitting optimization on the basis of future experimental measurements.

In turn, the coefficient ${\chi }_{xxxx}^{\left(3\right)}$ is used for evaluating the EFIM process. Indeed, the dynamics of the EFIM system are described by the following coupled-mode equations for the fundamental (${\tilde{A}}_m$) and second harmonic (${\tilde{B}}_0$) field amplitudes. The terms ${|{\tilde{A}}_m|}^2$ and ${|{\tilde{B}}_0|}^2$ correspond to the intracavity photon numbers for the fundamental and second harmonic modes.

The terms nSH, nm and nn indicate the SL refractive index at Ω0, ωm and ωn, respectively. All the other parameters are defined in the main text. In the case of SHG, we set G0 = 0, F0≠ 0; ωm = ω0 and n = m. To the contrary, for OPO application: F0 = 0, G0≠ 0; n = −m. Finally in Eq. (14), the degeneration factor K = 1 or 2, for SHG or OPO, respectively.

Author contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. R. S. conceived the idea, F. D. L. performed the simulations.

Acknowledgment The work of Richard Soref is supported by the Air Force Office of Scientific Research under Grant FA9550-21-1-0347.

Declaration of Competing Interest The authors declare no competing interests.