  Lack of nonreciprocity is one of the major drawbacks of solid-state acoustic
devices, which has hindered the development of microwave-frequency acoustic
isolators and circulators. Here we report giant nonreciprocal transmission of
shear-horizontal surface acoustic waves (SH-SAWs) on a LiTaO3 substrate coated
with a negative-positive magnetostrictive bilayer structure of Ni/Ti/FeCoSiB.
Although the static magnetic moments of two layers are parallel, SH-SAWs can
excite optical-mode spin waves much stronger than acoustic-mode ones at
relatively low frequencies via magnetoelastic coupling. The measured magnitude
nonreciprocity exceeds 40 dB (or 80 dB/mm) at 2.333 GHz. In addition, maximum
nonreciprocal phase accumulation reaches 188{\deg} (376{\deg}/mm), which is
desired for an effective SAW circulator. Our theoretical model and calculations
provide an insight into the observed phenomena and demonstrate a pathway for
further improvement of nonreciprocal acoustic devices.
