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 (T{}TD/3) is not simply thermally activated but described by a rate ωᵢⱼ{≈}({Δ}/{})²exp[-Eₐ/kBT+E_a'²/(kBT)²] where {Δ} is the renormalized tunneling matrix element, Eₐ is the conventional activation energy caused by linear adatom-phonon couplings, and E_a' 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.