Thermal damping of quantum interference patterns of surface-state electrons

The temperature-dependent damping of quantum-mechanical interference patterns from surface-state electrons scattering off steps on Ag(111) and Cu(111) has been studied using scanning tunneling microscopy (STM) and spectroscopy in the temperature range 3.5–178 K. The thermal damping of the electron standing waves is described quantitatively within a simple plane-wave model accounting for thermal broadening due to the broadening of the Fermi-Dirac distributions of sample and tip, for beating effects between electrons with different k_|| vectors, and for inelastic collisions of the electrons, e.g., with phonons. Our measurements reveal that Fermi-Dirac broadening fully explains the observed damping for Ag and Cu. From the analysis of our data, lower limits of the phase-relaxation lengths at the Fermi energy E_F of the two-dimensional electron gas of L_φ(E_F)≳600 Å at 3.5 K and ≳250 Å at 77 K for Ag(111), and of L_φ(E_F)≳660 Å at 77 K and ≳160 Å at 178 K for Cu(111) are deduced. In contrast to integral measurements such as photoemission we measure L_φ close to E_F and also locally. The latter eliminates residual line widths due to surface defect scattering found in the integrating techniques. Our STM results, therefore, currently provide a very good absolute estimate of L_φ and the inelastic lifetime τ=L_φ/v_F, respectively. Our values can be combined with photoemission results on dL_φ/dT to derive the inelastic lifetime of surface state electrons at any T.