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We perform a combined experimental and theoretical study of a magnetic-field($B$) induced evolution of magnetic and ferroelectric properties in anantiferromagnetic material Pb(TiO)Cu$_4$(PO$_4$)$_4$, whose structure ischaracterized by a staggered array of Cu$_4$O$_{12}$ magnetic units with convexgeometry known as square cupola. Our experiments show a $B$-induced phasetransition from a previously reported low-$B$ linear magnetoelectric phase to anew high-$B$ magnetoelectric phase, which accompanies a 90$^\circ$ flop ofelectric polarization and gigantic magnetodielectric effect. Moreover, weobserve a $B$-induced sign reversal of ferroelectric polarization in thehigh-$B$ phase. Our model and first-principles calculations reveal that theobserved complex magnetoelectric behavior is well explained in terms of a$B$-dependent electric polarization generated in each Cu$_4$O$_{12}$ unit bythe so-called exchange striction mechanism. The present study demonstrates thatthe materials design based on the magnetic structural unit with convex geometrydeserves to be explored for developing strong magnetoelectric couplings.

Magnetic Structural Unit with Convex Geometry: a Building Block Hosting an Exchange-striction-driven Magnetoelectric Coupling