  This theoretical modeling and simulation paper presents designs and projected
performances of two non-volatile, broadband, on-chip 2-by-2 electro-optical
switches based upon the germanium-on-insulator (GeOI) photonic-electronic
platform operating at the 2.5 $\mu$m mid-infrared wavelength. These compact
devices facilitate large-scale integration on a monolithic wafer where all
components are made of group-IV semiconductors. The switches are the
two-waveguide directional coupler (DC) and the Mach-Zehnder interferometer
(MZI). A thin-film graphene Joule-effect micro-heater is assumed on the
planarized GeOI device to change the phase (reversably) of DC-slot-embedded
Sb$_2$Se$_3$ phase-change material (PCM) from crystalline to amorphous. The MZI
has this PCM within its slotted-arm waveguides. Simulations show
high-performance bistable or multi-stable cross-bar switching in both devices.
The 2-by-2 DC has an active coupling length of 17 $\mu$m, 130 nm gap, and a
footprint of 5 $\mu$m -by- 31 $\mu$m. The device bandwidth is 30 nm over a
wavelength range where cross and bar insertion losses IL are less than 0.3 dB,
and where optical crosstalk is less than -15 dB. Results for the 2-by-2 MZI
show crossbar switching attained with a 7.8 $\mu$m-length Sb$_2$Se$_3$ slot and
a 5 $\mu$m -by- 51 $\mu$m switch footprint. Stable, multi-level switching in
both devices is attained via partial amorphization. Thermal modeling shows that
careful control of the voltage-pulse amplitude V applied to graphene
(rectangular pulse duration of 500 ns) can give 32 levels, for example, using V
in the range from 6.18 to 7.75 Volts.
