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Content. Electron-cyclotron maser emission (ECME) is the favored mechanismfor solar radio spikes and has been investigated extensively since the 1980s.Most studies relevant to solar spikes employ a loss-cone-type distribution ofenergetic 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 madein this theory, one major problem is how the fundamental emissions pass throughthe second-harmonic absorption layer in the corona and escape. This isgenerally known as the escaping difficulty of the theory. Aims. We study theharmonic emissions generated by ECME driven by energetic electrons with thehorseshoe distribution to solve the escaping difficulty of ECME for solarspikes. Methods. We performed a fully kinetic electromagnetic PIC simulationwith ${\omega}_ {pe}/{\Omega}_{ce}$ = 0.1, corresponding to the stronglymagnetized plasma conditions in the flare region, with energetic electronscharacterized by the horseshoe distribution. We also varied the density ratioof 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 arelatively weak growth of O1 and X3. With a higher-density ratio, the X2emission becomes more intense, and the rate of energy conversion from energeticelectrons into X2 modes can reach $\sim$0.06% and 0.17%, with $n_e/n_0$= 5% and10%, respectively. Conclusions. We find that the horseshoe-driven ECME can leadto an efficient excitation of X2 and X3 with a low value of ${\omega}_{pe}/{\Omega}_{ce}$, providing novel means for resolving the escapingdifficulty of ECME when applied to solar radio spikes. The simultaneous growthof X2 and X3 can be used to explain some harmonic structures observed in solarspikes.

Harmonic elctron-cyclotron maser emissions driven by energetic electrons of the horseshoe distribution with application to solar radio spikes