Thermal damping of quantum interference patterns of surface-state electrons

Abstract

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 ${\mathbf{k}}_{||}$ 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 ${L}_{\ensuremath{\varphi}}{(E}_{F})\ensuremath{\gtrsim}600 \mathrm{\AA{}}\mathrm{}$ at 3.5 K and $\ensuremath{\gtrsim}250 \mathrm{\AA{}}\mathrm{}$ at 77 K for Ag(111), and of ${L}_{\ensuremath{\varphi}}{(E}_{F})\ensuremath{\gtrsim}660 \mathrm{\AA{}}\mathrm{}$ at 77 K and $\ensuremath{\gtrsim}160 \mathrm{\AA{}}\mathrm{}$ at 178 K for Cu(111) are deduced. In contrast to integral measurements such as photoemission we measure ${L}_{\ensuremath{\varphi}}$ 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 ${L}_{\ensuremath{\varphi}}$ and the inelastic lifetime $\ensuremath{\tau}{=L}_{\ensuremath{\varphi}}{/v}_{F},$ respectively. Our values can be combined with photoemission results on ${\mathrm{dL}}_{\ensuremath{\varphi}}/dT$ to derive the inelastic lifetime of surface state electrons at any T.