  Electro-optic modulators are utilized ubiquitously ranging from applications
in data communication to photonic neural networks. While tremendous progress
has been made over the years, efficient phase-shifting modulators are
challenged with fundamental tradeoffs, such as voltage-length, index
change-losses or energy-bandwidth, and no single solution available checks all
boxes. While voltage-driven phase modulators, such as based on lithium niobate,
offer low loss and high speed operation, their footprint of 10's of cm-scale is
prohibitively large, especially for density-critical applications, for example
in photonic neural networks. Ignoring modulators for quantum applications,
where loss is critical, here we distinguish between current versus
voltage-driven modulators. We focus on the former, since current-based schemes
of emerging thin electro-optical materials have shown unity-strong index
modulation suitable for heterogeneous integration into foundry waveguides.
Here, we provide an in-depth ab-initio analysis of obtainable modulator
performance based on heterogeneously integrating low-dimensional materials,
i.e. graphene, thin films of indium tin oxide, and transition metal
dichalcogenide monolayers into a plurality of optical waveguide designs atop
silicon photonics. Using the fundamental modulator tradeoff of
energy-bandwidth-product as a design-quality quantifier, we show that a small
modal cross section, such as given by plasmonic modes, enables high-performance
operation, physically realized by arguments on charge-distribution and low
electrical resistance. An in-depth design understanding of phase-modulator
performance, beyond doped-junctions in silicon, offers opportunities for
micrometer-compact yet energy-bandwidth-ratio constrained modulators with
timely opportunities to hardware-accelerate applications beyond data
communication towards photonic machine intelligence.
