Main

We begin the discussion by analysing the quantized electronic structure in epitaxially grown Pb films. Figure 1a shows the photoemission intensity in a colour map as a function of Pb film thickness. We probe occupied states at energies below the Fermi level EF by one-photon photoemission (1PPE) with a photon energy h ν larger than the work function EF−Evac and unoccupied states above EF by two-photon photoemission (2PPE), where the sum of both photon energies h ν1+h ν2 is larger than the work function. These linear and nonlinear methods of photoemission are illustrated in Fig. 1, left. Symbols denote the peak positions of Lorentzian line fits. In Fig. 1b the fitting results are shown for three exemplary Pb thicknesses; details are given in the Supplementary Information. The data exhibit three branches of occupied quantum-well states (QWSs) and five branches of unoccupied QWSs that disperse towards EF with increasing coverage. The binding energies (E−EF) of the lowest unoccupied QWS (luQWS) oscillate between 1.65 and 0.6 eV by 45–165% on variation by one monolayer (1 ML) in thickness from N to N+1 ML and form the two lowest branches of unoccupied QWSs. Simultaneously, the QWS peaks exhibit a high intensity contrast owing to an efficient electron confinement to the film at energies of the Si bandgap. Hence, the study of electron dynamics of the luQWS is especially interesting because this state resides in the Si bandgap for even N and is confined to the film. In contrast, for odd N the luQWS is degenerate with Si bands. The thickness-dependent 1PPE results probe the quantized occupied electronic states21,23, confirm our earlier laser photoemission observations24 and are used as a thickness reference.

Figure 1: Quantized band structure of Pb/Si(111) for the occupied and unoccupied density of states at the  point observed by photoemission.
figure 1

a, Photoemission intensity in a colour map as a function of Pb thickness N in ML for the occupied and unoccupied states analysed by 1PPE and 2PPE, respectively. The 2PPE intensity is shown on a logarithmic scale, whereas the 1PPE intensity is plotted on a linear scale. The requirement that the sample work function is smaller than h ν in 2PPE prevented us from accessing energies E−EF<300 meV in these particular experiments. Red (blue) circles indicate the binding energies of the luQWS for even (odd) N; yellow circles indicate higher-lying QWSs. The binding energies are determined from a Lorentzian line fit as explained in b and the Supplementary Information. The horizontal dashed lines indicate the positions of the valence- and conduction-band edges (VBE/CBE). The 2 ML oscillation period of the binding energy of the luQWS is emphasized by a dotted line. The image is composed from three overlapping data sets; in total, 109 spectra have been analysed for this figure. b, 2PPE spectra for three exemplary values of N. The data are fitted by Lorentzian peaks and an exponentially decaying background (BG) and convoluted with a Gaussian instrument function. Background subtraction enhances the contrast between QWSs for odd and even N; see Supplementary Information.

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Delaying the two laser pulses at energies h ν1 and h ν2 employed in 2PPE with respect to each other enables us to study the decay of hot quasiparticles directly in the time domain in a pump–probe experiment. Figure 2 shows the electron population decay in the time domain by cross-correlation traces of the luQWS as a function of N. For odd N a fast population decay is observed, which slows down with increasing thickness. In contrast, for even N a slower and thickness-independent decay is found. The quasiparticle lifetimes are evaluated from a fit of a rate-equation model that is convoluted with the laser-pulse envelope30. This model enables us to determine the hot-electron lifetimes independently from a delayed population build-up due to secondary electrons; see also Supplementary Information. Essentially, the lifetimes are given by the inverse slope of the initial population decay, τ=ℏ/Γ, as indicated in Fig. 2. At later delays (odd N: >150–200 fs/even N: >800 fs) a second slower decay component is observed, which, however, does not contribute significantly to the initial decay dynamics at earlier delays30.

Figure 2: Decay of the transient quasiparticle population in the luQWS for various N.
figure 2

a,b, Cross-correlation traces for odd and even N, respectively, on a logarithmic intensity scale. Symbols indicate the 2PPE intensity obtained at h ν1=1.88 eV and h ν2=3.77 eV pump and probe photon energy by integrating the transient 2PPE intensity over an energy window of 100 meV width. The fast initial population decay time for odd N decreases with coverage, whereas the decay time for even N is slower and coverage independent. Solid lines are fits of a rate-equation model30 (see also the Supplementary Information), which yields the decay times that are discussed in Figs 3 and 4.

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Figure 3 summarizes the coverage-dependent quasiparticle lifetimes τ in Pb/Si(111). As evident from Fig. 2, τ oscillates with a 2 ML period, which is correlated to the characteristic binding-energy variation19; see Fig. 1. For odd N, the lifetimes depend on the coverage and increase from τ=5(5) fs for 1 ML to 37(5) fs for 15 ML. This is in contrast to even N, where the lifetimes are constant at τ=115(10) fs within the experimental accuracy. Hence, the lifetimes change by 200–2,000% when the film thickness is changed by 1 ML from odd to even N. On the one hand, the lifetimes seem to be correlated to the energetic position of the respective luQWS with respect to the Si bandgap: when the luQWS is degenerate with the Si bands we might expect a faster decay owing to extra decay and transport channels that open up towards the substrate. On the other hand, the binding energy of the luQWS oscillates with the characteristic 2 ML period (Fig. 1). In turn, the oscillating binding energies of the luQWS alone can introduce periodic changes of the phase space for electron–electron (e–e) scattering processes and thereby can lead to the observed lifetime oscillations.

Figure 3: Lifetime of the luQWS as a function of Pb thickness.
figure 3

The solid lines are linear fits to guide the eye; the shaded areas indicate a 90% confidence interval of the fits. The error bars of the analysed electron lifetimes amount to ±5 fs and are determined by the statistical error of the data and the fitting procedure.

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To distinguish these two limiting cases the quasiparticle lifetimes are plotted versus the binding energy E−EF of the luQWS in Fig. 4a. Fermi-liquid theory (FLT) for bulk materials with three-dimensional bands predicts a quadratic scaling law of the inelastic e–e scattering rate Γe–e=ℏ/τe–e of the excited quasiparticle with respect to the excess energy above the Fermi level ξ=E−EF

Here, n, e0, me and ε0 denote the bulk electron density, the elementary charge, the free-electron mass and the vacuum permittivity, respectively.

Figure 4: Comparison of the measured quasiparticle lifetimes with theoretical results from FLT and the G W approximation.
figure 4

a, Analysis of the quasiparticle lifetimes of the luQWS as a function of binding energy. The error bars of the analysed electron lifetimes amount to ±5 fs and are determined by the statistical error of the data and the fitting procedure. The dashed and dotted lines indicate the expectation from FLT (ref. 1) and the G W approximation4,7 for bulk Pb, respectively. The Si CBE is indicated. b, Scheme of the quantized band structure of a 4 ML freestanding Pb film. The solid lines indicate weakly dispersing bands near with mostly pz symmetry, whereas the dashed lines indicate strongly dispersing bands near with partial px and py orbital character. The dashed arrows depict interband scattering of hot electrons into bands with partial px and py character. The solid arrow represents intraband scattering within the same band of pz symmetry.

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Clearly, the measured data points are reproduced on an absolute scale using FLT (ref. 1) and employing the electron density of bulk Pb, nPb=13.2×1028 m−3. As n is given only by the size of the unit cell and the number of valence electrons, the FLT curve may be regarded as a parameter-free description. It is surprising that, despite the two-dimensional character of the film band structure20,21,22,24,25,27, the lifetimes scale quadratically with the excess energy ξ and hence exhibit a clear FLT behaviour. This can be justified by the strong screening of the Coulomb interaction in Pb films. If the screening length λs were larger than the typical thickness of the metal film we could expect finite-size effects reflected in the intensity of the e–e scattering. In this two-dimensional limit the screening would be reduced and could promote more intense e–e scattering, leading to shorter lifetimes than in the bulk1. To compare the electronic screening in the metal film with the typical layer thickness, the three-dimensional Thomas–Fermi screening length1 in Pb

where a0 denotes the Bohr radius, is compared with the interlayer spacing of d[111]=2.86 Å normal to the film. As λs<d[111] even a free-standing Pb(111) film of 1 ML thickness should exhibit three-dimensional screening. In addition, the energy-dependent quasiparticle lifetime in bulk Pb from an ab initio calculation employing the G W approximation4,7 is shown in Fig. 4a and is very similar to the FLT dependence. Owing to the agreement of the experimental and theoretical results, we conclude that the quasiparticle lifetimes are (1) dominated by e–e scattering that (2) follows the expectation from simple FLT for bulk materials.

This behaviour is consistent with the response function we have calculated for bulk Pb, which does not show any sharp feature in the energy interval of interest up to ξ=1.6 eV. Additionally, the binding energies of the luQWS in the Pb film are small compared with the energy of EF≈10 eV with respect to the band bottom. In this situation the imaginary part of the screened Coulomb interaction, being a linear function of energy, can be decomposed into a product of one energy-dependent and one energy-independent function7. Within the G W approximation7 this results directly in the quadratic dependence of Γe–e on ξ.

The Fermi-liquid character of Pb films accounts for a constant lifetime for the luQWS with even N (Fig. 3) because the binding energy of the state does not vary with thickness (Fig. 1). Further conclusions with respect to this constant lifetime can be drawn from the symmetry of the states contributing to the decay, as shown in Fig. 4b. In the vicinity of the QWSs are weakly dispersing (solid lines) and of pz symmetry. Towards they acquire px,py character (dashed lines) and disperse more strongly. This leads to two different decay channels at : (1) intraband scattering with a final state of dominant pz character (solid arrow) and (2) interband scattering into final states with partial px and py character (dashed arrows). The latter would lead to thickness-dependent decay rates as the respective final-state density in the interband process changes as a function of N. However, this process might have a low scattering rate owing to the different orbital character of initial and final states leading to a small scattering cross-section7. From the observed absence of a thickness dependence in the lifetime for even N within the investigated range from 1 to 15 ML (see Methods), we conclude that intraband scattering dominates.

Having explained the lifetimes of even N, we address the increasing lifetime with thickness for odd N. This behaviour can also be understood by FLT, because with increasing thickness the binding energy shifts towards EF, leading to a decreasing phase space for e–e scattering. A closer inspection reveals a systematic deviation from FLT at ξ>1.2 eV. We attribute this to an extra decay channel, namely electron transfer into the substrate, because at these energies the states are degenerate with the Si conduction band.

Methods

The experimental set-up and procedures have been described in detail elsewhere17,24. In brief, wedges31,32 of epitaxial Pb films are grown on the Si(111)--R30°-Pb reconstruction by evaporation of Pb under ultrahigh-vacuum conditions23,24 (base pressure <1×10−10 mbar). During preparation the temperature was kept at 100 K to avoid island formation and surface diffusion. The evaporation rate of 0.5–1 ML min−1 was monitored by a quartz crystal microbalance and cross-checked after deposition by comparison of the observed binding energies of occupied QWSs (ref. 24) with density functional calculations19. Furthermore, the discrete QWSs only appear at integer Pb thicknesses, providing an intrinsic coverage calibration. For characterization of the unoccupied electronic structure and the analysis of the electron dynamics, Pb wedges are grown with a thickness gradient of 1–2 ML mm−1. Together with the laser spot size of ∼100 μm, this corresponds to a coverage resolution of 0.1–0.2 ML.

For the 2PPE experiments, which address the unoccupied part of the band structure, the output of an amplified Ti:sapphire laser system (Coherent RegA 9050) operating at 300 kHz repetition rate is used to drive a tunable optical parametric amplifier. The signal at h ν1=2.7–1.7 eV is frequency doubled in a β-barium borate crystal into the ultraviolet wavelength range, h ν2=5.4–3.4 eV. These time-correlated pairs of femtosecond pump and probe laser pulses, h ν1 and h ν2, respectively, are focused into the ultrahigh-vacuum chamber and spatially overlapped on the sample surface with typical focus diameters of 100 μm. Typical pump fluences are of the order of 50–100 μJ cm−2 and the overall time resolution is 60 fs. Alternatively, the RegA output is frequency quadrupled to yield h ν=6.2 eV for the characterization of the occupied electronic structure with 1PPE.

The binding energy of the photoemitted electrons is determined from their kinetic energy Ekin analysed in a time-of-flight spectrometer: E−EF=Ekin+Φ−h ν. Here, Φ denotes the spectrometer work function and h ν the photon energy of the probing laser pulse. All measurements were made at a sample temperature of 100 K and in normal emission geometry at the  point.

The relativistic ab initio calculations of the quasiparticle lifetimes were carried out in the framework of the G W approximation4,7. The energy-dependent quasiparticle lifetimes τe–e(E)=1/Γe–e(E) are estimated for the 6pz parent bulk band dispersing along Γ–L.