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 (cid:1) 111 (cid:2) on two different Pb/Si interfaces. A quantitative analysis of the differential conductance spectra allowed us to determine the electron-electron (cid:1) e - e (cid:2) , electron-phonon (cid:1) e -ph (cid:2) and the interface and defect contributions to the lifetime. Layer-dependent ab initio calculations of the e -ph linewidth contribution are in excellent agreement with the data. Importantly, the sum of the calculated e - e and e -ph

Understanding the basic processes governing the decay of elementary electronic excitations in metals and at metal surfaces is important because these excitations play a major role in a large variety of chemical and physical phenomena, including chemical reactions or catalysis at surfaces, moleculesurface interactions and transport properties. A clear picture of the decay mechanisms occurring in several types of bulk metals ͑simple, noble, paramagnetic and some ferromagnetic transition metals͒ has been obtained. 1 The analysis of the dynamics of surface and image potential states ͑SS and IPS͒ also clarified the decay processes at the surface of various metals. 1 Thin metal films are interesting from a fundamental point of view and for technological applications. In a thin metal film electrons occupy discrete eigenstates with a quantized wave vector perpendicular to the surface, known as quantum-well states ͑QWS͒. [2][3][4] These states, forming twodimensional ͑2D͒ bands, are intermediate between bulk states and SS. Due to technical limitations, few studies have reported so far detailed contributions to the QWS lifetime. For example, photoemission ͑PES͒, two-photon PES ͑2PPE͒ and time-resolved 2PPE ͑TR-2PPE͒ require homogeneous films over macroscopic areas. Nevertheless, the electronelectron ͑e-e͒ contribution ⌫ e-e was determined in Ag/ Fe͑100͒ by PES and TR-2PPE ͑Refs. 5 and 6͒ and in Pb/Si͑111͒. 7 The electron-phonon ͑e-ph͒ contribution ⌫ e-ph was extracted by PES in Ag/Fe͑100͒ ͑Ref. 5͒ and in Ag/Cu͑111͒. 8 Layer-dependent or electronic structure dependent e-ph contributions were also reported. [9][10][11][12] Scanning tunneling spectroscopy ͑STS͒ benefits from being a local probe but suffers from the lack of k resolution. However, detailed quantitative lifetime studies were achieved for SS ͑Refs. 13-16͒ and IPS. 17 Up to now only one STS study reported a quantitative linewidth analysis of a QWS metal system, Yb͑111͒/W͑110͒. 18 A quadratic energy dependence of the linewidth was found, in agreement with three-dimensional ͑3D͒ Fermi-liquid ͑FL͒ theory, and a large e-ph coupling constant. Both results were subsequently questioned by a TR-2PPE study on bulk Yb. 19 These controversial results illustrate the difficulties and limits encountered in STS experiments to retrieve reliable quantitative QWS lifetime data.
In this Rapid Communication we present a detailed lowtemperature STS study of the linewidth of unoccupied QWS in Pb islands of thicknesses 7-22 monolayers ͑MLs͒ grown on Si͑111͒. Using a simple model with tunneling allowed through a trapezoidal barrier for a set of discrete QWS, a quantitative analysis of the differential conductance dI / dV spectra allows us to determine the QWS lifetime broadening as a function of energy, and the e-ph contribution between 5 and 50 K. The interface and defect scattering contribution to the QWS linewidth from the disordered Pb/ Si͑111͒-7 ϫ 7 ͑hereafter 7 ϫ 7͒ interface is 90 meV larger than the one from the crystalline Pb-ͱ 3 ϫ ͱ 3 / Si͑111͒ ͑in short Pbͱ 3͒ interface. Layer-dependent ab initio calculations of ⌫ e-ph were performed for 4-10 ML free-standing Pb͑111͒ films, taking full account of the quantum-size effects on the electron and phonon band structures and on the e-ph coupling. 20,21 The theoretical results are in very good agreement with the experimental findings. ⌫ e-e was estimated from ab initio calculation of ⌫ e-e for the parent bulk band dispersing along ⌫-L. 20 The calculated ⌫ e-ph + ⌫ e-e is convincingly fitted by a quadratic equation in agreement with the experimental results. The effect of spin-orbit coupling ͑SOC͒ on the electronic band energies and on ⌫ e-e is small in the probed energy range. The e-ph coupling constant calculated for the unoccupied QWS, Ӎ 1.45-1.60, is generally larger than calculated at the Fermi energy ͑E F ͒ for the corresponding films. 21 The measurements were performed in a homebuilt scanning tunneling microscope ͑STM͒ operated at 50 and 5 K in ultrahigh vacuum using cut PtIr tips. 22 dI / dV spectra were measured using currents of 200Յ I Յ 500 pA, with open feedback loop via lock-in technique with a modulation amplitude of 10 mV pp at a frequency of 1.4 kHz. Pb was thermally evaporated on the Si͑111͒-7 ϫ 7 or on the Pbͱ 3 substrate kept at room temperature favoring the growth of Pb single crystals with their ͑111͒ axis perpendicular to the surface. 23,24 All dI / dV measurements were performed on large Pb islands far from steps or island boundaries to avoid additional broadening of the QWS linewidths. Figure 1 shows Pb islands grown on 7 ϫ 7 ͑a͒-͑c͒ and Pbͱ 3 ͑d͒-͑f͒. Thicknesses given in ML include the WL. Large islands of several hundreds of nm are formed. Figure  1͑c͒ reveals the buried Si-7 ϫ 7 interface superimposed with the atomic resolution of the Pb lattice indicating that the island surface is atomically flat. 25,26 As shown in Fig. 1͑b͒ the Pb WL formed on Si-7 ϫ 7 is disordered. In contrast, a crystalline topography is observed both on the island and on the WL on Pbͱ 3 ͓see Figs. 1͑d͒-1͑f͔͒. The WL displays a striped-incommensurate superstructure ͓see Fig. 1͑e͔͒ corresponding to a saturated Pb phase. 27,28 Figure 1͑f͒ shows a Moiré pattern on a 8 ML island, caused by interfacial strain due to the difference between the Si and Pb lattice constant. 29 Figure 2 presents single dI / dV spectra obtained at 5 K on Pb islands of selected thickness. Remarkably, the spectra consist of prominent maxima located at the QWS energies. Previous STS studies of QWS in Pb/Si͑111͒, 25 in Yb/W͑110͒, 18 and of lanthanide SS ͑Ref. 30͒ suggested that this line shape results from a high effective mass near the 2D subbands onset, which was confirmed by PES for Pb/Si͑111͒. 31 The measured QWS energies are similar for both interfaces, with larger dispersion on the disordered one. 32 If the WL thickness is assumed to be 1 ML, agreement occurs between calculated QWS energies for free-standing films 33 and our experimental data for thicknesses Ն17 ML. 32 At smaller thickness a systematic deviation exists, increasing with decreasing thickness. A comparison between the spectra shown in Figs. 2͑a͒ and 2͑b͒ reveals a considerable narrowing of the QWS linewidths on the Pbͱ 3 interface with respect to the ones on 7 ϫ 7.
To extract the intrinsic QWS linewidth, dI / dV is modeled based on a 1D WKB approach with a trapezoidal potential barrier. 25,34 Figure 2͑c͒ depicts the schematic energy diagram of the junction. The Pb island density of states ͑DOS͒ s is simulated as a series of Lorentzian peaks, whereas the tip DOS t is assumed to be constant. As a function of bias voltage V, I is written 35 The nonzero conductance observed between the QWS is modeled by an additional exponential term. This analysis describes convincingly the STS data ͓see Figs. 2͑a͒ and 2͑b͔͒.
Possible causes of extrinsic broadening are the transmission to the substrate, the QWS lateral dispersion and the ac voltage modulation. The latter contributes a few mV. Following Ref. 5, the reflectivity of both Pb/Si interfaces was found to be very close to one in the studied voltage range, contributing to negligible broadening. As symmetric QWS peaks are observed on both interfaces, tunneling of electrons with finite k ʈ should contribute less than 10 meV to the linewidth. 18,30 Consequently, extrinsic linewidth contributions were neglected in the following analysis. ⌫͑T , E͒ was further decomposed as follows: where T is the temperature and E is the QWS energy. ⌫ e-e ͑E͒ is the e-e interaction term and ⌫ e-ph ͑E , T͒ reflects the e-ph scattering. ⌫ 0 , independent of T and E, describes interface and defects scattering. Figure 3͑a͒ shows that our calculated QWS energies are almost lying on the parent bulk band. The calculated effective masses are very close to the free electron mass. Figure  3͑b͒ shows ⌫ e-e ͑E͒ computed for bulk band energy equal to the QWS energy with and without SOC. A quadratic dependence ⌫ e-e = ␣͑E − E F ͒ 2 is found, leading to ␣ = 0.023 eV −1 with SOC ͑0.021 without͒, which are very close to ␣ = 0.02 eV −1 obtained when treating bulk Pb as a free electron gas ͑r s = 2.30 a.u͒. Hence, in the probed energy range, the SOC effect on band ͑QWS͒ energies and on ⌫ e-e is small. Figure 3͑b͒ shows ⌫ e-ph versus QWS energy calculated for 5 and 50 K. It varies with QWS energy, but the difference between the averaged ⌫ e-ph 's ͑dashed lines͒, ⌬⌫ e-ph = ⌫ e-ph ͑50K͒ − ⌫ e-ph ͑5K͒, remains nearly constant, increasing from 23 meV close to E F to 26 meV at higher energies. Since the energy dependence of ⌫ e-e is much stronger than that of ⌫ e-ph , their sum ⌫ e-e + ⌫ e-ph , is fitted reasonably well by a quadratic equation with ␣ = 0.025 eV −1 ͑0.026͒ at 50 ͑5͒ K ͑see Fig. 4͒. Figure 4 shows ⌫ versus energy measured at 50 and 5 K on Pbͱ 3 with the theoretical ⌫ e-e + ⌫ e-ph . Both experimental data sets are well fitted by 3D FL theory: ⌫͑E͒ = ␣͑E − E F ͒ 2 , yielding the same value ␣ = 0.033 eV −1 . The difference ⌬⌫ e-ph Ӎ 25 meV between the 50 and 5 K fit to the experimental data yields an estimate of the average e-ph contribution to the QWS lifetime in excellent agreement with the theoretical ⌬⌫ e-ph Ӎ 23-26 meV. A similar analysis was conducted on the 7 ϫ 7 interface, which showed a larger linewidth dispersion due to disorder at this interface. In contrast to the crystalline Pbͱ 3, the linewidths increase with decreasing thickness ͑Ӎ20 meV from 22 to 7 ML͒. This linewidth variation was taken into account before ⌫ e-e , ⌫ e-ph , and ⌫ 0 were extracted. ⌫ 0 is found to be about 90 meV larger on 7 ϫ 7 ͑see Fig. 2͒. ␣ = 0.028 eV −1 at 50 K ͑0.037 at 5 K͒, ⌬⌫ e-ph Ӎ 26 meV, which is consistent with the values obtained on Pbͱ 3 and with the theoretical results.
The large ⌬⌫ e-ph measured on both interfaces reflect a strong e-ph coupling of the QWS in Pb thin films. A Debye model 36 with bulk = 1.55 yields ⌬⌫ e-ph = 23 meV, which is close to the measured averaged ⌬⌫ e-ph . The present ab initio calculations yield for most QWS 1.45ՅՅ1.6. These values are larger than those computed for Pb thin films at E F ͑Ref. 21͒ but close to bulk at E F . 36 The excellent agreement between theoretical and experimental e-ph coupling terms allows us to discriminate among the three contributions of Eq. ͑2͒. For 7-22 ML films the resulting electronic mean free path at E F , v F 0 ͑⌫ 0 = ប / 0 ͒ can be estimated for both interfaces, yielding 3-4 nm for 7 ϫ 7 and 11 nm for Pbͱ 3 ͓Fermi velocities v F are determined from the reconstructed band dispersion along ⌫-L ͑Ref. 32͔͒.
In a previous Yb/W͑110͒ QWS linewidth study by STS, the neglect of the interface and defect scattering term in the low-energy residual linewidth and a lack of temperaturedependent measurements, led to a strong e-ph coupling constant Ӎ 1.6-2.8. 18 In contrast, TR-2PPE measurements of the parent d band in bulk Yb found Ӎ 0.4. 19 Moreover TR-2PPE results together with ab initio calculations reported a linewidth energy dependence far from being quadratic. 19 In conclusion, the combination of high-accuracy temperature-dependent STS experiments with ab initio cal-culations allowed us to identify individual QWS in single ultrathin metal islands, to separate consistently the different decay mechanisms of these electronic excitations and to determine the QWS electron-phonon coupling strength. These achievements open up an avenue toward detailed investigations of the decay processes of electronic excitations on a local scale, e.g., of individual supported molecules, clusters or other nanostructures.