Decay mechanisms of excited electrons in quantum-well states of ultrathin Pb islands grown on Si(111): Scanning tunneling spectroscopy and theory

Using low-temperature scanning tunneling spectroscopy at 5 and 50 K, we studied the linewidth of unoccupied quantum-well states in ultrathin Pb islands, grown on Si(111) on two different Pb/Si interfaces. A quantitative analysis of the differential conductance spectra allowed us to determine the electron-electron $\left(e\text{−}e\right)$, electron-phonon $\left(e\text{-ph}\right)$ and the interface and defect contributions to the lifetime. Layer-dependent ab initio calculations of the $e\text{-ph}$ linewidth contribution are in excellent agreement with the data. Importantly, the sum of the calculated $e\text{−}e$ and $e\text{-ph}$ lifetime broadening follows the experimentally observed quadratic energy dependence.

Figure 1

(Color) STM images showing typical features of Pb islands grown on $\text{Si}\left(111\right)\text{−}7\times 7$ (a)–(c) and on $\text{Pb-}\surd 3\times \surd 3/\text{Si}\left(111\right)$ (d)–(f). (a) and (d) Large scale overview. The indicated island thickness includes the wetting layer (WL). (b) Disordered WL $\simeq 1\text{ML}$ high. (c) Atomic resolution of the surface Pb lattice. Buried $7\times 7$ interface seen through a 8ML island. (e) High resolution of the crystalline Pb WL, a saturated Pb ML. (f) Moiré pattern on a 8 ML island.

Figure 2

(Color online) Experimental (dots) and calculated (full line) $dI/dV$ spectra measured at 5 K by tunneling to a large atomically flat Pb island of selected thickness grown on (a) $\text{Si}\left(111\right)\text{−}7\times 7$ and (b) $\text{Pb-}\surd 3\times \surd 3/\text{Si}\left(111\right)$. The arrow indicates negative differential conductance. (c) Schematic energy diagram of the tunnel junction used to model the experiment. CBM: conductance band minimum, VBM: valence-band maximum, $d$: film thickness, $z_0$: vacuum gap. $E_{F,s}\left(E_{F,t}\right)$: sample (tip) Fermi level. $E_{vac,s}\left(E_{vac,t}\right)$: sample (tip) vacuum level. $\Phi \left(z\right)$: vacuum potential drop between tip and sample. $V$: tip-sample bias voltage.

Figure 3

(Color online) (a) Calculated dispersion of the electronic band crossing $E_F$ along $\Gamma \text{−}L$ for bulk Pb without spin-orbit coupling (with SOC): dashed (solid) line. Dots: computed QWS energies. (b) Calculated ${\Gamma }_{e\text{−}e}$ and ${\Gamma }_{e\text{-ph}}$ for unoccupied QWS as a function of energy. ${\Gamma }_{e-e}$ without SOC (with SOC): open (full) triangles. ${\Gamma }_{e\text{-ph}}$ at 5 K (50 K): dots (squares) with their fit. In $n_m\left(n=4,\dots ,10;m=1,2\right)n$ is the film number of monolayers, and $m$ is the QWS number counted from $E_F$.

Figure 4

(Color online) Linewidth versus energy of unoccupied QWS in Pb islands grown on $\text{Pb-}\surd 3\times \surd 3/\text{Si}\left(111\right)$ measured at 5 (50) K: full dots (full squares). The data are fitted according to 3D Fermi-liquid theory (continuous lines). Theoretical linewidth ${\Gamma }_{e\text{−}e}+{\Gamma }_{e\text{-ph}}$ at 5 (50) K: open dots (open squares) with corresponding fits (dashed lines). For easier comparison the theoretical data have been shifted up so that the theoretical fits coincide with the experimental ones at low energy. The linewidth difference $\Delta {\Gamma }_{e\text{-ph}}\simeq 25\text{meV}$ between the 50 and 5 K fit to the experimental data agrees very well with the corresponding calculated difference, yielding the QWS $e\text{-ph}$ coupling constant $\lambda \simeq 1.45–1.60$. Silicon conduction band minimum is indicated.