The electron transport properties of ultra-scaled amorphous phase change material (PCM) GeTe are studied using non-equilibrium Green's function (NEGF). The inelastic electron-phonon scattering is included using Born approximation. It is shown that, in ultra-scaled PCM device with 6 nm channel length, less than 4% of the energy carried by the incident electrons from the source is transferred to the atomic lattice before reaching the drain, indicating that the electron transport is largely elastic. Our simulation results show that the inelastic electron-phonon scattering, which plays an important role to excite trapped electrons in bulk PCM devices, exerts very limited influence on the current density value and the shape of current-voltage curve of ultra-scaled PCM devices. The analysis reveals that the Poole-Frenkel law and the Ohm's law, which are the governing physical mechanisms of the bulk PCM devices, cease to be valid in the ultra-scaled PCM devices.