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⩾${\mathrm{T}}_{\mathrm{D}}$/3) is not simply thermally activated but described by a rate ${\mathrm{\omega }}_{\mathrm{ij}}$≈(Δ/ħ${)}^2$exp[-${\mathrm{E}}_{\mathrm{a}}$/${\mathrm{k}}_{\mathrm{B}}$T+${\mathrm{E}}_{\mathrm{a}\prime }^2$/(${\mathrm{k}}_{\mathrm{B}}$T${)}^2$] where Δ is the renormalized tunneling matrix element, ${\mathrm{E}}_{\mathrm{a}}$ is the conventional activation energy caused by linear adatom-phonon couplings, and ${\mathrm{E}}_{\mathrm{a}\prime }$ 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.