Yade Wang(王亚德) Zijian Lin(林子荐) Siwei Xue(薛思玮) Jiade Li(李佳德)Yi Li(李毅) Xuetao Zhu(朱学涛) and Jiandong Guo(郭建东)

1Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,Chinese Academy of Sciences,Beijing 100190,China

2School of Physical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China

3Songshan Lake Materials Laboratory,Dongguan 523808,China

Keywords: Pb films,plasmons,quantum size effects,high-resolution electron energy loss spectroscopy

1. Introduction

The collective excitations of electrons,i.e., plasmons,in metal films are important not only for the understanding of electron-electron interactions in low-dimensional systems[1,2]but also promising in plasmonic electro-optic applications,[3-11]The properties of plasmons in metal films are mainly determined by the electronic band structure of the metal as well as the dielectric properties of the substrate.[1]The spatial confinement of electrons in the metal films may also generate an essential influence on the plasmon behavior.[12,13]The spatial confinement of metal films in the normal direction will generate the quantization of the electronic states,which can be depicted as the quantum well states(QWS).Due to the discrete energy levels and the electron confinement in the QWS, the properties of metal films, such as the work functions[14]and the transport coefficients,[15]may strongly depend on the thickness,known as the quantum size effect(QSE).The plasmons in metal films sometimes also exhibit obvious QSE, mainly resulting from the inherent QWS.For example, the surface plasmon of Ag films shows negative dispersion in the small momentum range due to enhanced screening effect,[16]and the wave of the QWS can reach the substrate and give rise to the hybridized interaction between the film surface and substrate.[12]Consequently, the damping of the surface plasmon in Ag films displays significant thickness dependence.[17]QSE is a crucial issue to take into account in the research of the surface plasmons of thin films.

Fig.1.A classic model of the surface plasmons in thin films.(a)A schematic diagram of the thin film epitaxy on the substrate,showing two interfaces. (b)A schematic plot of the surface plasmon in the film split into ω± branches.The green background illustrates the change of the width of the plasmon branch,with the qc defined at the position of the splitting or at the position of the minimum width. The dashed lines of ω± represent that the splitting may not be experimentally measured.

The most striking metal film system showing the QSE is the Pb(111) film, where the superconducting critical temperature, work function, and the Kondo effect exhibit strong thickness-dependent oscillations.[22-27]Different from other metal films in which the QSE vanishes when the thickness is beyond～10 monolayers (ML),[28-30]the QSE in Pb(111)films is so much stronger that it persists over 30 ML.[24]The plasmons in Pb(111)films are expected to show much stronger QSE than other metal films. However,the manifestation of the strong QSE in the plasmons of Pb(111)films has not been observed so far.

In this article, using high-resolution electron energy loss spectroscopy (HREELS) with the capability of twodimensional (2D) energy-momentum mapping,[31]we measured and analyzed the plasmon dispersions on the surface of Pb(111) films with different film thicknesses. We discovered that the QSE in Pb(111)films has a very strong effect on one of the surface plasmons,manifested as strong damping in the small momentum range. The damping is still noticeable even in 40-ML-thick Pb(111)films,clearly demonstrating the strong QSE in the perspective of the collective excitations.

2. Methods

3. Results and discussion

As one of the heavy metal elements in the periodical table,Pb possesses large spin-orbit coupling,which may induce topological phases in systems associated with Pb films.[34-37]From theoretical calculations,the spin-orbit effect is also predicted to have a great influence on the plasmons of Pb,e.g.,resulting in anisotropy of the plasmons or generating new excitation modes.[38,39]Previous HREELS experiment[40]of Pb(111) films grown on Si(111) focused on a plasmon mode around 2 eV(q →0),which was assigned to be a surface plasmon. While there was a huge disagreement with the theoretical calculations,where the mode around 2 eV was proved to be a bulk mode related to the spin-orbit coupling of Pb.[38]To clarify the issue, we performed systematical HREELS measurements to show the full dispersions of the plasmons in Pb(111) films and analyzed the possible manifestation of the QSE in the collective excitations.

Fig.2. The momentum-dependent energy loss curves of Pb(111)films with different thicknesses. (a)-(f)corresponding to film thickness of 40,30,20,13,7,and 4/3 ML,respectively. The dispersions of different branches are represented by dotted lines as guides to the eye.

Figure 2 shows the momentum-dependent energy loss curves(ELCs)of Pb(111)films with different thicknesses,extracted from the 2D HREELS mapping (see Fig. B1 in Appendix B). In the 4/3-ML sample, only two loss peaks can be discerned due to the strong effect of the substrate, since it is actually the Pb-induced reconstruction of the Si(111)surface, with details discussed in the appendix. Except for the 4/3-ML sample, all the other samples show four peaks in the ELCs, marked by the dashed lines and labeled as TE,α1,α2,α3, respectively. In order to check the details about the thickness-dependence,we plot the stacking ELCs of different thicknesses atq=0.07 ˚A-1andq=0.2 ˚A-1,in Figs.3(a)and 3(b),respectively. The line profiles of these modes at different thicknesses are clearly demonstrated. The exact energies of the peaks can be obtained by fitting the ELCs using Lorentz functions. Two typical fitting cases are shown in Figs. 3(c)and 3(d)as examples. The fitting results are plotted in Fig.4 to show the dispersions of the observed features. The assignments of these features are obtained by comparison with theoretical calculations,[38,41]with the results summarized in Table 1.

Table 1. Summary of the four observed features in the HREELS measurement of the Pb(111)films.

The TE branch represents the photoemission threshold excitation, usually manifested as a single-particle excitation peak in HREELS.[42,43]In Pb(111) films, the TE branch is located at about 4 eV, very close to the work function of Pb(4.25 eV).

Theα1,α2,α3branches are the collective excitations of Pb(111) films.α1is a bulk plasmon, with the energy of～1.8 eV atq=0 and dispersing up to～2.1 eV atq～0.1 ˚A-1.The dispersion matches well(shown in Fig.C1(a)in Appendix C)with the calculated bulk plasmon,[38]which is strongly related to the spin-orbit coupling effect of Pb.α2is a surface plasmon,with the energy of～7.0 eV atq=0 and being dispersionless up toq=0.4 ˚A-1. The comparison with the calculation (shown in Fig. C1(b) in Appendix C) indicates thatα2is closely related to the interband transitions.[41]The predicted acoustic plasmons in theoretical calculations[38]were not observed in our measurements,possibly because its crosssection,i.e.,the intensity in loss functions is too low to show obvious peaks in the ELCs.

α3is another surface plasmon,strongly related to the film thickness,which will be the focus in this study. The comparison betweenα3and the calculated surface plasmon dispersion(shown in Fig. C1(b) in Appendix C) indicates that the overall energy ofα3is slightly lower than the calculated results.This difference should be resulting from the screening effect of the d-electrons,which is difficult to be fully considered in the calculations due to the strong electron-electron interactions.It can be roughly understood by a phenomenological model,where the reduction of the surface plasmon energy due to the additional screening from the d-electrons can be described by Liebsch’s theory.[44]

Fig. 3. Comparison of the energy loss curves with different thicknesses and the curve fitting process: (a) the comparison at q=0.7 ˚A-1; (b) the comparison at q=0.2 ˚A-1. The dispersion of different branches is represented by dotted lines for guides to the eye; (c) and (d) typical peak fitting process of the 40-ML sample at q=0.07 ˚A-1 and q=0.2 ˚A-1,respectively.

A prominent feature ofα3is the damping in the small momentum range,as shown in Figs.4 and C1(b). In all the films,regardless of the thickness,α3cannot be measured aroundq=0.With increasingq,α3gradually appears after the critical momentum valueqc. This phenomenon has not been reported in previous experimental studies of Pb(111) films. There are two possible reasons for the damping of surface plasmons at the small momentum range: (i)the interaction with other collective excitations or (ii) the interaction between the top and bottom interfaces of the thin films.

The first scenario has been reported in two metal film systems, Cs films on Si(111) substrate[20]and Ag films on Cu(111) substrate.[16]In the case of ultrathin Cs films, there is a crossover between the multipole plasmon and the regular surface plasmon.[20]The regular surface plasmon cannot be measured whenq ＜0.1 ˚A-1due to the influence of the multipole plasmon.[20]In the case of thin Ag films, the surface plasmon cannot be measured as well whenq ＜0.1 ˚A-1,where only bulk plasmon can be measured.[16]In both cases,two plasmon modes cannot coexist,i.e., only one mode can be observed at each specificq. As an analogy, in our measurement, it seemsα3in Pb(111) films at small momentum range could be damped due to the influence ofα2. However,different from the cases of Cs and Ag films, it is clear from Fig. 2 thatα3andα2always coexist whenq ＞qc. These results rule out the possibility that the damping ofα3at the small momentum range is due to the interaction with other collective excitations.

Fig.4. Dispersions of the measured plasmons obtained from the fitting of loss curves: (a)-(f)corresponding to film thickness of 40,30,20,13,7,and 4/3 ML,respectively.

The second scenario,i.e.,the interaction between the top and bottom interfaces,is essentially a size effect due to the finite film thickness, as illustrated in Fig. 1. This scenario for Pb(111) films has been theoretically calculated in Ref. [41]and the overall damping feature ofα3observed in our experiment agrees well with this picture. The main difference is that the expected spitting ofα3in the small momentum range was not clear in the experiment, similar to the case of the surface plasmon in Ag films.[17]Instead,the damping was reflected by the change of the FWHM as a function ofq. As described in the scheme of Fig.1(b),qccan be determined as the momentum position of the minimum FWHM.To obtain the quantitative value of theqcfor each Pb(111)film with different thickness, the FWHMs ofα3at differentqare obtained from the fitting method shown in Figs.3(c)and 3(d). The fitting results are plotted in Fig.5(a). For each thickness,the FWHM gradually decreases with increasingquntil reaching the minimum,after which the FWHM starts increasing withq.

As shown in Fig.5(b),the experimentalqcof Pb(111)is almost maintained around 0.2 ˚A-1with the film thickness less than 30 ML; and it decays slowly to about 0.15 ˚A-1when the film thickness is up to 40 ML. The weak and slow decay of the experimentally measuredqcin Pb(111)is significantly different from the case of Ag(111) as well as the calculated of Pb(111), both showing an exponential decay with the increasing film thickness. The observed phenomenon provides a manifestation of the strong QSE in Pb(111)films in the perspective of collective excitations.

The origin of the surprisingly strong QSE in Pb(111)films has been theoretically investigated in Ref. [45]. Compared with other metals, the QSE in Pb(111) films are more prominent due to the slow decay of Friedel oscillations in the electron density from the Pb(111) surfaces, which is related to the strong nesting of the Fermi surface along the Pb(111) direction.[45]The Friedel oscillations at the Pb(111)surface decay as 1/xwith the distancexfrom the surface,different from the conventional 1/x2power law at other metal surfaces.[45]Rather than the mere presence of QWS, the interference in the electron density by the strong Friedel oscillations associated with the strong nesting of the Fermi surface along the Pb(111) direction,[45]would inevitably affect the plasmon behaviors. These effects are usually not considered in theab initiocalculations, resulting in the possible overestimation of the decay ofqcupon film thickness.[41]The indepth mechanism of the enhanced interface interactions in the surface plasmons by QSE is still not clear. More studies including both experiments and theories are needed in the future investigations.

Fig.5. The critical momentum qc of different thicknesses: (a)the variation of the FWHW with different film thicknesses of Pb(111). The stars on the horizontal axis mark the positions of the qc. (b) Comparison of the decay of qc between different systems. The experimental values of Pb(111) are obtained from panel(a). The theoretical values of Pb(111),[41] experimental values of Ag(111),[17] and theoretical values of Ag(111)[18] are extracted from previous studies.

4. Summary

We have measured the electronic collective excitations in Pb(111) films with different thicknesses. The dispersions of three different plasmons modes have been observed and analyzed. We discovered that one of the surface plasmons shows strong damping in the small momentum range whenq ＜qc,manifesting the strong QSE effect in Pb(111) films. Different from other metal films in which the critical momentumqcdecays exponentially with increasing film thickness,theqcin Pb(111)films decays much slower,and the strong damping is still observable even in 40-ML-thick Pb(111)films. These observations indicate that the interactions between the surface and interface of the Pb(111) films can be enhanced by the strong Friedel oscillations in the electron density and significantly affect the behaviors of the collective excitations. This work further proves that the QSE is an important issue that should be considered in the analysis of the surface plasmons of thin metal films. Moreover,the thickness-dependent damping behavior originated from the QSE may have potential applications in plasmonics based on metal films.

Acknowledgment

The authors would like to thank Prof. E V Chulkov and Prof. V M Silkin for discussions about the plasmon assignments.

Appendix A:Characterization of the films

Fig.A1. (a)-(f)The LEED patterns of Pb(111)films with different thicknesses,with the incident electron beam energy of 90 eV.(g)and(h)The STM images(V =-2 V,I=100 pA)of 30-ML-thick Pb(111)films and 4/3-ML-thick Pb(111)films,respectively.

Appendix B:Original 2D HREELS data

The original data obtained from our 2D HREELS system are the energy-momentum mappings, as shown in Fig.B1. The momentum-dependent energy loss curves(ELCs),i.e.,the scattering intensity as a function of energy loss for a given momentum value,shown in Fig.2 of the main manuscript,are extracted from these 2D HREELS mappings.

Fig.B1. (a)-(f)2D HREELS mappings showing the relationship between the energy loss and the momentum of Pb(111)films with different thicknesses.

Appendix C:Comparison of the measured plasmons with theoretical calculations

The assignments of the observed HREELS features are obtained by comparison with theoretical calculations.The colored background in Fig.C1(a)is adopted from the calculated loss functions of bulk Pb,[38]while the colored background in Fig. C1(b) is adopted from the calculated loss functions of Pb(111) films.[41]With our measured energy-momentum points superimposed on the theoretical backgrounds, we can obtain the assignments of the experimentally observed plasmon branches. The results are summarized in Table 1 in the main manuscript. The dispersions ofα1andα2matches well with the calculated results, while the overall energy ofα3is slightly lower than the calculated results. This difference should be resulting from the screening effect of the delectrons,which is difficult to be fully captured in the calculations.

In Fig.C1(c),we plot the dispersions ofα3for the films with different thicknesses in one panel,to show that the overall energy is consistent with the dispersion of the calculated surface plasmon.

Fig. C1. (a) Comparison of the experimental dispersion (dots) of α1 (40 ML) with the calculated loss functions (colored background, reprinted by permission from Ref.[38]. Copyright by the American Physical Society.) of bulk Pb. (b)Comparison of the experimental dispersion(dots)of α2 and α3(20 ML)with the calculated loss functions(colored background,reprinted by permission from Ref.[41]. Copyright by the American Physical Society.)of the 21-ML Pb(111)film. (c)The experimental dispersions of the α3 branch with different thicknesses.

Appendix D: Comparison of the HREELS results between the 4/3-ML-Pb(111) film and Si substrate

As shown in Fig. 2, different from the thicker films in which four energy loss peaks are clearly observed, the 4/3-ML-Pb(111) films can only roughly show two peaks. Especially,the bulk plasmonα1can no longer be observed in the ultrathin 4/3-ML-Pb(111)films. Here,in Fig.D1,the HREELS results of the 4/3-ML-Pb(111) films and the Si substrate are compared to show the possible influence from the substrate. It is clear that the ELCs of the 4/3-ML-Pb(111) films are more similar to the substrate than the thicker films,indicating strong substrate effects in the ultrathin films. Similar substrate effects have been previously reported in ultrathin Al films on Si(111).[48]Consequently,the dispersion of theα3branch and the correspondingqcof the 4/3 ML-Pb(111) are not as clear as those of the thick films. Theqcof the 4/3-ML-Pb(111) is roughly estimated from the momentum-dependent ELCs, as shown in Fig.D1(c). All other films with larger thicknesses in our study do not show obvious substrate effects, as shown in Fig.2.

Fig. D1. (a) and (b) The 2D HREELS mappings of 4/3-ML-Pb(111) films and the Si substrate, respectively. (c) and (d) The momentum-dependent ELCs of the 4/3-ML-Pb(111)films and the Si substrate extracted from the 2D HREELS mappings.

Appendix E: HREELS measurements at low temperature

In order to study the influence of temperature on the plasmons of Pb(111)films,we also performed the HREELS measurements on the 30-ML-Pb(111) films at 35 K, with the results shown in Fig.E1. The plasmons of Pb(111)films at 35 K marked in Fig.E1(b)do not show an obvious difference with the results obtained at room temperature.

Fig.E1. (a)The 2D HREELS of 30-ML-Pb(111)films measured at 35 K.(b)The corresponding momentum-dependent ELCs.