  The tunnelling anisotropic magnetoresistance (TAMR) effect describes the
sensitivity of spin-polarized electron transport to the orientation of the
magnetization with respect to the crystallographic axes. As the TAMR effect
requires only a single magnetic electrode, in contrast to the tunnelling
magnetoresistance effect, it offers an attractive route towards alternative
spintronics applications. In this work we consider the TAMR effect at the
single-atom limit by investigating the anisotropy of the local density of
states in the vacuum above transition-metal adatoms adsorbed on a noncollinear
magnetic surface, the monolayer of Mn on W(110). This surface presents a
cycloidal spin spiral ground state with an angle of 173$^\circ$ between
neighbouring spins and thus allows a quasi-continuous exploration of the
angular dependence of the TAMR of adsorbed adatoms using scanning tunnelling
microscopy. Using first-principles calculations, we investigate the TAMR of Co,
Rh and Ir adatoms on Mn/W(110) and relate our results to magnetization
direction dependent changes in the local density of states. The anisotropic
effect is found to be enhanced dramatically on the adsorption of heavy
transition-metal atoms, with values of up to 50% predicted from our
calculations. This effect will be measurable even with a non-magnetic STM tip.
