Field-Emission Resonances in Thin Metallic Films: Nonexponential Decrease of the Tunneling Current as a Function of the Sample-to-Tip Distance

Field-emission resonances (FERs) for two-dimensional Pb(111) islands grown on Si(111)7x7 surfaces were studied by low-temperature scanning tunneling microscopy and spectroscopy (STM/STS) in a broad range of tunneling conditions with both an active and disabled feedback loop. These FERs exist at quantized sample-to-tip distances Z(n) above the sample surface, where n is the serial number of the FER state. By recording the trajectory of the STM tip during ramping of the bias voltage U (while keeping the tunneling current I fixed), we obtain the set of the Zn values corresponding to local maxima in the derived dZ/ dU(U) spectra. This way, the continuous evolution of Zn as a function of U for all FERs was investigated by STS experiments with an active feedback loop for different I. Complementing these measurements by current-distance spectroscopy at a fixed U, we could construct a 4-dimensional I-U-Z-dZ/dU diagram that allows us to investigate the geometric localization of the FERs above the surface. We demonstrate (i) that the difference (5Z(n) = Z(n+1) - Zn between neighboring FER lines in the Z-U diagram is independent of n for higher resonances; (ii) that the (5Z(n) value decreases as U increases; (iii) that the quantized FER states lead to the periodic variations of ln I as a function of Z with periodicity (5Z; and (iv) that the periodic variations in the ln I-Z spectra allow an estimation of the absolute height of the tip above the sample surface. Our findings contribute to a deeper understanding on how the FER states affect various types of tunneling spectroscopy experiments and how they lead to a nonexponential decay of the tunneling current as a function of Z at high bias voltages in the regime of quantized electron emission.