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 EF of the two-dimensional electron gas of L_φ(EF)600 at 3.5 K and 250 at 77 K for Ag(111), and of L_φ(EF)660 at 77 K and 160 at 178 K for Cu(111) are deduced. In contrast to integral measurements such as photoemission we measure L_φ close to EF 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/vF, 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.