  The magnetic properties of BaFe$_{2}$As$_{2}$(001) surface have been studied
by using first-principles electronic structure calculations. We find that for
As-terminated surface the magnetic ground state of the top-layer FeAs is in the
staggered dimer antiferromagnetic (AFM) order, while for Ba-terminated surface
the collinear (single stripe) AFM order is the most stable. When a certain
coverage of Ba or K atoms are deposited onto the As-terminated surface, the
calculated energy differences among different AFM orders for the top-layer FeAs
on BaFe$_{2}$As$_{2}$(001) can be much reduced, indicating enhanced spin
fluctuations. To identify the novel staggered dimer AFM order for the As
termination, we have simulated the scanning tunneling microscopy (STM) image
for this state, which shows a different $\sqrt{2}\times\sqrt{2}$ pattern from
the case of half Ba coverage. Our results suggest: i) the magnetic properties
of the top-layer FeAs on BaFe$_{2}$As$_{2}$(001) can be tuned effectively by
surface doping; ii) both the surface termination and the AFM order in the
top-layer FeAs can affect the STM image of BaFe$_{2}$As$_{2}$(001).
