  Radio relics in the outskirts of galaxy clusters imply the diffusive shock
acceleration (DSA) of electrons at merger-driven shocks with Mach number
$M_{s}\lesssim3-4$ in the intracluster medium (ICM). Recent studies have
suggested that electron preacceleration and injection, prerequisite steps for
DSA, could occur at supercritical shocks with $M_{s}\gtrsim2.3$ in the ICM,
thanks to the generation of multiscale waves by microinstabilities such as the
Alfv\'en ion cyclotron (AIC) instability, the electron firehose instability
(EFI), and the whistler instability (WI). On the other hand, some relics are
observed to have subcritical shocks with $M_{s}\lesssim2.3$, leaving DSA at
such weak shocks as an outstanding problem. Reacceleration of preexisting
nonthermal electrons has been contemplated as one of possible solutions for
that puzzle. To explore this idea, we perform Particle-in-Cell (PIC)
simulations for weak quasi-perpendicular shocks in high-$\beta$ ($\beta=P_{\rm
gas}/P_{B}$) plasmas with power-law suprathermal electrons in addition to
Maxwellian thermal electrons. We find that suprathermal electrons enhance the
excitation of electron-scale waves via the EFI and WI. However, they do not
affect the ion reflection and the ensuing generation of ion-scale waves via the
AIC instability. The presence of ion-scale waves is the key for the
preacceleration of electrons up to the injection momentum, thus the shock
criticality condition for electron injection to DSA is preserved. Based on the
results, we conclude that preexisting nonthermal electrons in the preshock
region alone would not resolve the issue of electron preacceleration at
subcritical ICM shocks.
