Magnetoelectric composite materials are promising candidates for highly sensitive magnetic-field sensors. However, the composites showing the highest reported magnetoelectric coefficients require the presence of external d.c. magnetic bias fields, which is detrimental to their use as sensitive high-resolution magnetic-field sensors. Here, we report magnetoelectric composite materials that instead rely on intrinsic magnetic fields arising from exchange bias in the device. Thin-film magnetoelectric two-two composites were fabricated by magnetron sputtering on silicon-cantilever substrates. The composites consist of piezoelectric AlN and multilayers with the sequence Ta/Cu/Mn70Ir30/Fe50Co50 or Ta/Cu/Mn70Ir30/Fe70.2Co7.8Si12B10 serving as the magnetostrictive component. The thickness of the ferromagnetic layers and angle dependency of the exchange bias field are used to adjust the shift of the magnetostriction curve in such a way that the maximum piezomagnetic coefficient occurs at zero magnetic bias field. These self-biased composites show high sensitivity to a.c. magnetic fields with a maximum magnetoelectric coefficient of 96 V cm(-1) Oe(-1) at mechanical resonance.