Chiral domain walls in ultrathin perpendicularly magnetised layers have a N{e}el structure stabilised by a Dzyaloshinskii-Moriya interaction (DMI) that is generated at the interface between the ferromagnet and a heavy metal. Different heavy metals are required above and below a ferromagnetic film in order to generate the structural inversion asymmetry needed to ensure that the DMI arising at the two interfaces does not cancel. Here we report on the magnetic properties of epitaxial Pt/Co/Au$_x$Pt$_{1-x}$ trilayers grown by sputtering onto sapphire substrates with 0.6 nm thick Co. As $x$ rises from 0 to 1 a structural inversion asymmetry is generated. We characterise the epilayer structure with x-ray diffraction and cross-sectional transmission electron microscopy, revealing (111) stacking. The saturation magnetization falls as the proximity magnetisation in Pt is reduced, whilst the perpendicular magnetic anisotropy $K_mathrm{u}$ rises. The micromagnetic DMI strength $D$ was determined using the bubble expansion technique and also rises from a negligible value when $x=0$ to $sim 1$ mJ/m$^2$ for $x = 1$. The depinning field at which field-driven domain wall motion crosses from the creep to the depinning regime rises from $sim 40$ to $sim 70$ mT, attributed to greater spatial fluctuations of the domain wall energy with increasing Au concentration. Meanwhile, the increase in DMI causes the Walker field to rise from $sim 10$ to $sim 280$ mT, meaning that only in the $x = 1$ sample is the steady flow regime accessible. The full dependence of domain wall velocity on driving field bears little resemblance to the prediction of a simple one-dimensional model, but can be described very well using micromagnetic simulations with a realistic model of disorder. These reveal a rise in Gilbert damping as $x$ increases.
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