with large efficiency [@Miron2011; @Ryu2013; @Emori2013]. Also, it has been predicted that isolated skyrmions injected in nanotracks can be moved with very low current density and are moreover insensitive to defects [@Sampaio2013]. Engineering materials with large DMI has therefore become an important issue both for domain wall and skyrmion physics.

So far *ab-initio* calculations of interfacial DMI are rare and concern perfect interfaces difficult to compare with the mixed interfaces found in real samples [@Freimuth2014; @Yang2015]. The information presently available on the DMI strengths relies on experimental work. A large input has been given by Spin-polarised Scanning Tunneling Microscopy measurements that show the presence of chiral magnetic textures or skyrmions in systems consisting of one monolayer of Fe (or Mn) on heavy metal substrates [@Bode2007; @Ferriani2008; @Meckler2009; @Heinze2011] in ultra-high vacuum and at low temperature. In the last few years, domain wall dynamics and nucleation measurements at room temperature have revealed the presence of DMI in less ordered, non centrosymmetric ultrathin magnetic layers with PMA, made by magnetron sputtering [@Ryu2013; @Emori2013; @Haazen2013; @Pizzini2014]. More recently, Brillouin light scattering experiments have also highlighted the presence of DMI in similar PMA samples [@DiPRL2015; @Belmeguenai2015].

It has been shown recently that when, in 