A 3-D full-band particle Monte Carlo (MC) simulator, with full electron and phonon dispersion and a 2-D quantum correction is self-consistently coupled to a phonon MC simulator. The coupling entails feeding the phonon data obtained from the 3-D electrical MC to the phonon MC. The phonon MC reciprocates by providing the resulting spatial temperature map, which is used in the electron MC, with temperature-dependent scattering table, in a self-consistent manner. A key feature of our model is its ability to delineate the influence of the various phonon modes on the electronic transport through the application of anharmonic phonon decay and full phonon dispersion. The electrothermal simulator developed is utilized to assess the performance of silicon-on-insulator (SoI) multigate (MG) MOSFET with nanoscale cross sections. This paper shows that the hotspot in inversion mode SoI MG MOSFET with 20-nm gate length permeates into the channel as the cross section is reduced (covering similar to 50% of the channel for the 5 nm x 5 nm cross section). Furthermore, cross-sectional scaling, a key design rule to mitigate short-channel effects, degenerates device performance well beyond the ideal current gain limits of MG MOSFET architecture of double-gate, trigate, and gate-all-around MOSFET. Consequently, at the sub-20-nm scale adding more gate does not necessarily improve performance.