Using a two-dimensional frictional force microscope (2D-FFM), we investigated the two-dimensional nature of spatially quantized friction with a lattice periodicity on an atomic scale. As a result, in addition to the well-known stick-slip behavior along the scanning direction, we found the appearance of friction with square-wave behavior which works across the scanning direction. To explain the observed two-dimensional friction with the lattice periodicity, we proposed a two-dimensional stick-slip model which predicts spatially quantized adhesions and jumps with the lattice periodicity. By calibrating displacements due to frictional forces using the lateral force curve, we obtained quantitative agreements between predicted and experimentally deduced displacements on an atomic scale. We also observed the fluctuation of the spatially quantized friction. Further, the two-dimensional stick-slip model with an effective adhesive region enabled us to explain individual sawtooth and square-wave behaviors in more detail. This model also enabled us to interpret a normal load dependence of the frictional force hysteresis as a normal load dependence of the effective adhesive region. Besides, by comparing the normal load dependence of the frictional force hysteresis on layered materials such as MoS2 with that on the three-dimensional NaF(100) crystal surface, we made clear that the atomic-scale facial friction works on layered materials, in contrast to the near single-atom friction on the three-dimensional one.
