  Giant radio relics in the outskirts of galaxy clusters are known to be lit up
by the relativistic electrons produced via diffusive shock acceleration (DSA)
in shocks with low sonic Mach numbers, $M_{\rm s}\lesssim3$. The particle
acceleration at these collisionless shocks critically depends on the kinetic
plasma processes that govern the injection to DSA. Here, we study the
preacceleration of suprathermal electrons in weak, quasi-perpendicular
($Q_\perp$) shocks in the hot, high-$\beta$ ($\beta = P_{\rm gas}/P_{\rm B}$)
intracluster medium (ICM) through two-dimensional particle-in-cell simulations.
\citet{guo2014a,guo2014b} showed that in high-$\beta$ $Q_\perp$-shocks, some of
incoming electrons could be reflected upstream and gain energy via shock drift
acceleration (SDA). The temperature anisotropy due to the SDA-energized
electrons then induces the electron firehose instability (EFI), and oblique
waves are generated, leading to a Fermi-like process and multiple cycles of SDA
in the preshock region. We find that such electron preacceleration is effective
only in shocks above a critical Mach number $M_{\rm ef}^*\approx2.3$. This
means that in ICM plasmas, $Q_\perp$-shocks with $M_{\rm s}\lesssim2.3$ may not
efficiently accelerate electrons. We also find that even in $Q_\perp$-shocks
with $M_{\rm s}\gtrsim2.3$, electrons may not reach high enough energies to be
injected to the full Fermi-I process of DSA, because long-wavelength waves are
not developed via the EFI alone. Our results indicate that additional electron
preaccelerations are required for DSA in ICM shocks, and the presence of fossil
relativistic electrons in the shock upstream region may be necessary to explain
observed radio relics.
