Non-Equilibrium Phonon Distributions in Sub-$100nm$ Silicon Transistors

Intense electron-phonon scattering near the peak electric field in a semiconductor device results in nanometer-scale phonon hotspots. Past studies have argued that ballistic phonon transport near such hotspots serves to restrict heat conduction. We reexamine this assertion by developing a new phonon transport model. In a departure from previous studies, we treat isotropic dispersion in all phonon branches and include a phonon emission spectrum from independent Monte Carlo simulations of electron-phonon scattering. We cast the model in terms of a non-equilibrium phonon distribution function and compare predictions from this model with data for ballistic transport in silicon. The solution to the steady-state transport equations for bulk silicon transistors shows that energy stagnation at the hotspot results in an excess equivalent temperature rise of about 13% in a $90nm$ gate-length device. Longitudinal optical phonons with non-zero group velocities dominate transport. We find that the resistance associated with ballistic transport does not overwhelm that from the package unless the peak power density approaches $50W∕μm3$⁠. A transient calculation shows negligible phonon accumulation and retardation between successive logic states. This work highlights and reduces the knowledge gaps in the electro-thermal simulation of transistors.