One special interest for the industrial development of Hall thruster is characterizing the anomalous cross-field electron transport observed after the channel exit. Since the ionization efficiency is more than 90%, the neutral atom density in that domain is so low that the electron collisions cannot explain the high electron flux observed experimentally. Indeed this is 100 times higher than the collisional transport. In Hall thruster geometry, as ions are not magnetized the electric and magnetic field configuration creates a huge difference in drift velocity between electrons and ions, which generates electron cyclotron drift instability or $vec E times vec B$ electron drift instability. Here we are focusing on collision-less chaotic transport of electrons by those unstable modes generated by $vec E times vec B$ drift instability. We found that in presence of these electrostatic modes electron dynamics become chaotic. They gain energy from the background waves which increases electron temperature along perpendicular direction by a significant amount, $T_{rm perp}/T_{rm parallel}sim 4$, and a significant amount of crossfield electron transport is observed along the axial direction.
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