Quantum Coherence of Image-Potential States

The quantum dynamics of the two-dimensional image-potential states in front of the Cu(100) surface is measured by scanning tunneling microscopy and spectroscopy. The dispersion relation and the momentum resolved phase-relaxation time of the first image-potential state are determined from the quantum interference patterns in the local density of states at step edges. It is demonstrated that the tip-induced Stark shift does not affect the motion of the electrons parallel to the surface.

Figure 1

(a) Upper half: STM topography of an artificially created step on Cu(100) taken at $5.2\mathrm{V}$ bias voltage. Lower half: $dI/dV$ map of the same place at the same bias voltage. The image-potential-state electrons reflected at the step edge create a density modulation which appears as a standing wave pattern in the $dI/dV$ map. (b) Measured $z(V)$ curve on a terrace of Cu(100) using two different tunneling currents of $0.1$ and $1.0\mathrm{n}\mathrm{A}$. We observe up to four image-potential states in the electric field of the STM tip. The measured increments in the tip-sample distance are labeled $\Delta z_n$.

Figure 2

(a) One-dimensional model potential used and probability density of the wave function of the first image-potential state obtained. (b) Schematic drawing of the energy levels of the resonances, the Fermi levels of tip and sample, the bulk band edges of the Cu(100) crystal [$E(X_{4^{\prime }})$ and $E(X_1)$] and the potential in the tunneling gap (dashed line). (c) Calculation (dotted lines) of the energies of the image-potential states in the electric field of the STM tip as a function of the applied voltage. Each of the dotted lines corresponds to one particular state ($n=1$ to $4$) calculated for a tip-sample distance $z=\Delta z_n+z_0$, where $\Delta z_n$ is taken from the measurement shown in Fig. 1b and $z_0=22.5\mathrm{\AA }$. Full circles are plotted at the voltages where the steps in Fig. 1b occur and at the corresponding energies according to Eq. (1).

Figure 3

(a) $dI/dV$ signal at 5.0 V sample bias (upper graph) and topographic signal at a step edge. Electron interference produces the oscillations in $dI/dV$. (b) $dI/dV(V,x)$ map. $dI/dV$ is plotted as a function of the lateral distance $x$ from a step edge and of the bias voltage $V$ [17]. Horizontal line cuts reveal the standing waves due to the interference of electrons in the image-potential state scattered at the step edge. The dashed line indicates the energetic position of the data shown in (a). (c) Points represent the $k$ values obtained from a fit to each line in (b); the solid line shows the parabolic dispersion fitted to these points.

Figure 4

Phase coherence lengths of the interference pattern at a step edge converted into linewidths $\Gamma (E)$. Solid symbols are the data taken by STM, and the solid line is a linear fit to them; open symbols represent linewidths determined by 2PPE measurements from Ref. 3.