  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.
