We examine the current-induced dynamics of a skyrmion that is subject to both structural and bulk inversion asymmetry. There arises a hybrid type of Dzyaloshinskii-Moriya interaction (DMI) which is in the form of a mixture of interfacial and bulk DMIs. Examples include crystals with symmetry classes C$_n$ as well as magnetic multilayers composed of a ferromagnet with a noncentrosymmetric crystal and a nonmagnet with strong spin-orbit coupling. As a striking result, we find that, in systems with a hybrid DMI, the spin-orbit-torque-induced skyrmion Hall angle is asymmetric for the two different skyrmion polarities ($pm 1$ given by out-of-plane core magnetization), even allowing one of them to be tuned to zero. We propose several experimental ways to achieve the necessary straight skyrmion motion (with zero Hall angle) for racetrack memories, even without antiferromagnetic interactions or any interaction with another magnet. Our results can be understood within a simple picture by using a global spin rotation which maps the hybrid DMI model to an effective model containing purely interfacial DMI. The formalism directly reveals the effective spin torque and effective current that result in qualitatively different dynamics. Our work provides a way to utilize symmetry breaking to eliminate detrimental phenomena as hybrid DMI eliminates the skyrmion Hall angle.
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