Why EM waves having only x momentum transfers to electron z momentum? Electron begins oscillating along z, so will not radiate EM waves along z direction, to compensate its z momentum. It seems that it is violated conservation of momentum?
In the particle formulation of electromagnetism, light is carried by photons. Momentum is absolutely conserved. If a photon hits an electron, there will be momentum balance.
In the wave formulation of EM the electric field,( supposed it is a polarized beam), oscillates in the z direction. An electron that acquires a momentum will take part of the energy of the beam and the beam will acquire a small z component to conserve momentum. On the average,since the electron will be hit in both z directions by the incoming beam, the effect will be that the beam will spread/degrade, to conserve momentum and energy.