  Correlated photon-pair sources are key components for quantum computing,
networking, and sensing applications. Integrated photonics has enabled
chip-scale sources using nonlinear processes, producing high-rate entanglement
with sub-100 microwatt power at telecom wavelengths. Many quantum systems
operate in the visible or near-infrared ranges, necessitating broadband
visible-telecom entangled-pair sources for connecting remote systems via
entanglement swapping and teleportation. This study evaluates broadband
entanglement generation through spontaneous four-wave mixing in various
nonlinear integrated photonic materials, including silicon nitride, lithium
niobate, aluminum gallium arsenide, indium gallium phosphide, and gallium
nitride. We demonstrate how geometric dispersion engineering facilitates
phase-matching for each platform and reveals unexpected results, such as robust
designs to fabrication variations and a Type-1 cross-polarized phase-matching
condition for III-V materials that expands the operational bandwidth. With
experimentally attainable parameters, integrated photonic microresonators with
optimized designs can achieve pair generation rates greater than ~1 THz/mW$^2$.
