Optical parametric oscillators are widely-used pulsed and continuous-wave tunable sources for innumerable applications, as in quantum technologies, imaging and biophysics. A key drawback is material dispersion imposing the phase-matching condition that generally entails a complex setup design, thus hindering tunability and miniaturization. Here we show that the burden of phase-matching is surprisingly absent in parametric micro-resonators adopting monolayer transition-metal dichalcogenides as quadratic nonlinear materials. By the exact solution of nonlinear Maxwell equations and first-principle calculation of the semiconductor nonlinear response, we devise a novel kind of phase-matching-free miniaturized parametric oscillator operating at conventional pump intensities. We find that different two-dimensional semiconductors yield degenerate and non-degenerate emission at various spectral regions thanks to doubly-resonant mode excitation, which can be tuned through the incidence angle of the external pump laser. In addition we show that high-frequency electrical modulation can be achieved by doping through electrical gating that efficiently shifts the parametric oscillation threshold. Our results pave the way for new ultra-fast tunable micron-sized sources of entangled photons, a key device underpinning any quantum protocol. Highly-miniaturized optical parametric oscillators may also be employed in lab-on-chip technologies for biophysics, environmental pollution detection and security.
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