Phonon-assisted tunneling of adatoms on a solid surface: The influence of quadratic adatom-phonon couplings

We investigate the influence of quadratic adatom-phonon couplings on the phonon-assisted adatom tunneling on solids based upon the early work of Tonks and Dick. We find that the incoherent tunneling at high temperatures (TTD/3) is not simply thermally activated but described by a rate ij(Latin small letter h with stroke)2exp[-Ea/kBT+Ea2/(kBT)2] where is the renormalized tunneling matrix element, Ea is the conventional activation energy caused by linear adatom-phonon couplings, and Ea is an effective energy arising from the quadratic couplings. We determine the conditions when the activated tunneling behavior is qualitatively altered by the quadratic couplings. As a result of reduced dimensionality, the latter play much more important roles in the tunneling diffusion of hydrogen on metal surfaces than they do in the interstitial tunneling in solids.