Electron accelerations at high Mach number collision-less shocks are investigated by means of two-dimensional electromagnetic Particle-in-Cell simulations with various Alfven Mach numbers, ion-to-electron mass ratios, and the upstream electron beta_e (the ratio of the thermal pressure to the magnetic pressure). We found electrons are effectively accelerated at a super-high Mach number shock (MA~30) with a mass ratio of M/m=100 and beta_e=0.5. The electron shock surfing acceleration is an effective mechanism for accelerating the particles toward the relativistic regime even in two dimensions with the large mass ratio. Buneman instability excited at the leading edge of the foot in the super-high Mach number shock results in a coherent electrostatic potential structure. While multi-dimensionality allows the electrons to escape from the trapping region, they can interact with the strong electrostatic field several times. Simulation runs in various parameter regimes indicate that the electron shock surfing acceleration is an effective mechanism for producing relativistic particles in extremely-high Mach number shocks in supernova remnants, provided that the upstream electron temperature is reasonably low.
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