title: '*Ab-initio* friction forces on the nanoscale: A DFT study of fcc Cu(111).'
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Introduction {#sec:Int}
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In the $17^{th}$ and $18^{th}$ century Amontons and Coulomb formulated the basic laws of friction, which state that the macroscopic friction force is independent of the apparent contact area and varies linearly with the external load at moderate constant sliding velocities [@amonton:1699; @coulomb:1773]. These friction laws have been further extended by Derjaguin for adhesive surfaces, so that a non-vanishing friction force can occur even for zero load [@derjaguin:34b]. Although generally very successful, on the nano-scale deviations from this classical behavior have sometimes been observed depending on the sliding conditions. If the contacting surfaces are aligned incommensurately, friction can be reduced by several orders of magnitude and structural superlubricity may occur [@hirano:90; @hirano:97; @dienwiebel:04; @dietzel:13; @zhang:13], though this phenomenon is suspected to be unstable at least for graphene flakes on graphite [@filippov:08; @feng:13]. It has also been proposed that the drastic reduction of frictional forces may not stem from incommensurability but from thermal effects and the low effective mass of the nanocontact [@krylov:05; @maier:05]. Inclusion of third bodies between incommensurate contacts hinders structual superlubricity and leads to a linear behavior of friction on load 