  Content. Electron-cyclotron maser emission (ECME) is the favored mechanism
for solar radio spikes and has been investigated extensively since the 1980s.
Most studies relevant to solar spikes employ a loss-cone-type distribution of
energetic electrons, generating waves mainly in the fundamental X/O mode
(X1/O1), with a ratio of plasma oscillation frequency to electron gyrofrequency
(${\omega}_ {pe}/{\Omega}_{ce}$) lower than 1. Despite the great progress made
in this theory, one major problem is how the fundamental emissions pass through
the second-harmonic absorption layer in the corona and escape. This is
generally known as the escaping difficulty of the theory. Aims. We study the
harmonic emissions generated by ECME driven by energetic electrons with the
horseshoe distribution to solve the escaping difficulty of ECME for solar
spikes. Methods. We performed a fully kinetic electromagnetic PIC simulation
with ${\omega}_ {pe}/{\Omega}_{ce}$ = 0.1, corresponding to the strongly
magnetized plasma conditions in the flare region, with energetic electrons
characterized by the horseshoe distribution. We also varied the density ratio
of energetic electrons to total electrons ($n_e/n_0$) in the simulation.
Results. We obtain efficient amplification of waves in Z and X2 modes, with a
relatively weak growth of O1 and X3. With a higher-density ratio, the X2
emission becomes more intense, and the rate of energy conversion from energetic
electrons into X2 modes can reach $\sim$0.06% and 0.17%, with $n_e/n_0$= 5% and
10%, respectively. Conclusions. We find that the horseshoe-driven ECME can lead
to an efficient excitation of X2 and X3 with a low value of ${\omega}_
{pe}/{\Omega}_{ce}$, providing novel means for resolving the escaping
difficulty of ECME when applied to solar radio spikes. The simultaneous growth
of X2 and X3 can be used to explain some harmonic structures observed in solar
spikes.
