  The band structures of single-layer MoS$_2$ and WS$_2$ present a coupling
between spin and valley degrees of freedom that suppresses spin-flip scattering
and spin dephasing. Here we show that out-of-plane deformations, such as
corrugations or ripples, enhance spin-flip scattering in armchair MoS$_2$ and
WS$_2$ nanoribbons. Spin transport in the presence of surface roughness is
systematically investigated, employing the non-equilibrium Green's function
method along with the tight-binding approximation. Both transmission and
conductance have been calculated as a function of surface roughness. Our
results indicate that the spin-flip rate, usually neglected in flat pristine
samples, increases significantly with the surface roughness amplitude. These
results are important for the design and fabrication of transition metal
dichalcogenides based spintronic devices.
