  We investigate remote surface scattering (RSR) by the SiO$_2$/HfO$_2$
interface in Fully-Depleted Silicon-on-Insulator (FDSOI) devices using
Non-Equilibrium Green's Functions. We show that the RSR mobility is controlled
by cross-correlations between the surface roughness profiles at the Si/SiO$_2$
and SiO$_2$/HfO$_2$ interfaces. Therefore, surface roughness and remote surface
roughness can not be modeled as two independent mechanisms. RSR tends to
enhance the total mobility when the Si/SiO$_2$ interface and SiO$_2$ thickness
profiles are correlated, and to decrease the total mobility when they are
anti-correlated. We discuss the implications for the high-$\kappa$/Metal gate
technologies.
