Dynamics of surface-localised electronic excitations studied with the scanning tunnelling microscope

Progress in Surface Science 82 (2007) 293–312 www.elsevier.com/locate/progsurf

Review

Dynamics of surface-localised electronic excitations studied with the scanning tunnelling microscope J. Kro¨ger a,*, M. Becker a, H. Jensen a, Th. von Hofe a, N. Ne´el a, L. Limot a, R. Berndt a, S. Crampin b, E. Pehlke c, C. Corriol d, V.M. Silkin d, D. Sa´nchez-Portal e, A. Arnau f, E.V. Chulkov f, P.M. Echenique f a

Institut fu¨r Experimentelle und Angewandte Physik, Christian-Albrechts-Universita¨t zu Kiel, Leibnizstraße 19, D-24098 Kiel, Germany b Department of Physics, University of Bath, Bath BA2 7AY, United Kingdom c Institut fu¨r Theoretische Physik und Astrophysik, Christian-Albrechts-Universita¨t zu Kiel, Leibnizstraße 15, D-24098 Kiel, Germany d Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, E-20018 San Sebastia´n, Spain e Unidad de Fı´sica de Materiales, Centro Mixto CSIC-UPV/EHU, Apartado 1072, E-20080 San Sebastia´n, Spain f Departamento de Fı´sica de Materiales, Facultad de Ciencas Quı´micas, Universidad del Paı´s Vasco UPV/EHU, Apartado 1072, E-20080 San Sebastia´n, Spain

Abstract The decay rates of electron and hole excitations at metal surfaces as determined by a scanning tunnelling microscope are presented and discussed. Surface-localised electron states as diverse as Shockley-type surface states and quantum well states confined to ultrathin alkali metal adsorption layers are covered. Recent developments in the analysis of the experimental procedures that are used to determine decay rates with the scanning tunnelling microscope, namely the analysis of line shapes and the spatial decay of standing wave patterns, are discussed.  2007 Elsevier Ltd. All rights reserved. PACS: 68.37.Ef; 72.10.Fk; 72.15.Qm; 73.20.At; 73.20.Fz; 73.21.b; 79.60.Jv; 73.61.Ph

*

Corresponding author. Tel.: +49 431 8803966; fax: +49 431 8801685. E-mail address: [email protected] (J. Kro¨ger).

0079-6816/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.progsurf.2007.03.003

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Keywords: Scanning tunnelling microscopy and spectroscopy; Electronic surface states; Decay rates; Electron– electron interaction; Quantum well states; Confinement

Contents 1. 2. 3. 4.

5. 6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historic remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lineshape analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Shockley surface states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Confinement of Shockley surface states . . . . . . . . . . . . . . . . . . . . . 4.3. Quantum well states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Defect scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase coherence length approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction The investigation of electronic excitations at surfaces is of great interest since they are involved in many important processes including catalytic reactions, photosynthesis, and light conversion in solar cells. The lifetime of these excitations determines the dynamics of charge and energy transfer and thus governs the effectiveness of the above mentioned processes. Consequently, substantial research effort has gone into elucidating the mechanisms which limit the lifetime of surface-localised electronic excitations. Scanning tunnelling microscopy and spectroscopy have contributed to the determination and the understanding of the finite lifetimes of surface-localised states as diverse as Shockley-type surface states and quantum well states. Two experimental methods are used to determine the lifetime from scanning tunnelling microscopy and spectroscopy data, namely the analysis of line shapes of spectroscopic signatures and the spatial decay of standing electron wave patterns – this latter method is often referred to as the phase coherence length approach. This article focuses on recent results based on the line shape analysis. However, we briefly touch upon some key results using the phase coherence length approach. The article is organised as follows: In Section 2 we give a historical overview of the investigation of surface-localised electronic excitations with focus on scanning tunnelling microscopy and spectroscopy experiments. Experimental details are summarised in Section 3. Section 4 is devoted to the line shape analysis method. Here, Shockley surface states of the (1 1 1) surfaces of noble metals, confinement of surface states to artificially fabricated quantum boxes, and quantum well states hosted by ultrathin alkali metal adsorption layers are discussed, and also scattering by adatoms. The phase coherence length approach and recent developments that improve this method are presented in Section 5, and we summarise in Section 6.

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2. Historic remarks The basic models of intrinsic surface states were introduced by Tamm [1] and Shockley [2]. Shockley surface states result from the truncation of the crystal potential at the surface, and are likely to exist when the electron orbitals on the atoms of the lattice interact strongly, and band crossing occurs. Tamm states are typically associated with a subsequent change in the potential in the outermost cell of the crystal. An overview on surface states is given by Davison and Steßs´licka [3]. The first experimental indication of surface states was reported by Meyerhof [4] who studied the difference of contact potential and work function in metal-silicon junctions. The surface state picture was further corroborated by Brattain and Shockley [5], Brattain [6], and Bardeen [7]. With the invention of ultrahigh vacuum, surface states became accessible via optical detection techniques [8], field emission [9], and photoemission [10,11]. Evidence for surface states in tunnelling spectroscopy experiments were reported by Jaklevic and Lambe [12,13] using planar metal–oxide–metal tunnel junctions. Kaiser and Jaklevic [14] used for the first time the scanning tunnelling microscope to report on electronic surface states on Au(1 1 1) and Pd(1 1 1). These observations demonstrated the instrument’s unique applicability to the investigation of the surface electronic structure. Subsequent real-space imaging of electron standing waves at metal surfaces belong to the iconic experiments performed with the scanning tunnelling microscope [15,16]. The existence of another class of surface states, so-called image potential states, was first conceived for isolators on theoretical grounds by Cole and Cohen [17]. Later, Echenique and Pendry showed that image potential states may also occur on metal surfaces, despite inelastic electron–electron scattering [18,19]. While Tamm and Shockley states may be characterised as intrinsic surface states, image potential states describe electrons which are bound to their own image in the solid. A key difference is that the probability amplitude of image potential states is maximum outside of the crystal surface. Inverse photoemission experiments were the first to report on such unoccupied metal surface states [20,21]. Becker et al. [22] and Binnig et al. [23] were able to measure image potential states with the scanning tunnelling microscope. In this case the states may alternatively be viewed as standing waves in the tunnelling gap as first discussed by Becker and coworkers [22]. This point of view emphasises the significant effect of the instrument’s tip on these states. Studies of image potential states using two-photon photoemission were also available at that time [24]. At present experiment and theory tend to separate Tamm and Shockley surface states on one hand and image potential states on the other hand due to their differing origins and characteristics, namely: (i) intrinsic or crystal-induced states which originate from the crystal termination and are localised on the crystal-side of the surface, (ii), imageinduced states which are localised outside the crystal and whose origin is the long-range image potential that results in a hydrogenic-like series of states. In Fig. 1 we show the probability amplitudes of Shockley surface states and image potential states as calculated for Cu(1 1 1). The calculation uses the one-dimensional pseudopotential introduced by Chulkov and coworkers [25]. Inside the solid (z < 0) the sinusoidal potential curve reveals the spatial periodicity of the substrate lattice (the atomic locations are indicated by circles). In the vacuum region (z > 0) the potential decays asymptotically like V = EV  1/ (4z) modelling the long-range image potential. The shaded areas reveal surface-projected

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Fig. 1. Schematic presentation of probability amplitudes of the Cu(1 1 1) Shockley surface state (curve close to EF) and the first image potential state (curve close to EV). The base lines of the curves are offset vertically to illustrate the typical energies of these states. The substrate region is located at z < 0 while the vacuum region is found at z > 0. The electron potential (grey curve) oscillates in the bulk and decays into vacuum [25]. Shaded areas correspond to surface-projected bulk electronic structure.

bulk electronic states, which around the Fermi level (EF) exhibit an energy gap (this being a metal, the gap is not absolute, but varies with the lateral momentum kk , in some cases vanishing). The lower of the two probability distributions that are plotted is that of the Cu(1 1 1) Shockley surface state. The maximum probability is located in the topmost layer of the substrate. The upper distribution is that of the first image potential state, whose energy is close to the vacuum energy (EV). Its probability maximum lies well outside the surface. We close this section by mentioning another type of surface-localised states which arises from confining electrons to thin metal layers on surfaces. The first direct observation of socalled quantum well states was reported by Jaklevic et al. [26] in planar Al–(Al oxide)–Pb tunnel junctions. Likewise the adsorption of ultrathin metal films on metal surfaces can lead to a trapping of electrons in quantum well states if the hosting substrate exhibits a local band gap normal to the surface. These quantum well states, which form two-dimensional electron gases with metal-like density of states, have also been observed with photoemission [27] and inverse photoemission [28]. The article focuses on the influence of electron–electron scattering on the lifetime of surface-localised states. The effect of electron–phonon coupling is discussed in, for instance, Refs. [29–31].

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3. Experiment The data presented below were acquired using home-made scanning tunnelling microscopes operated in ultrahigh vacuum (base pressure of the recipient better than 1 · 109 Pa) and at low temperatures (5 K and 8 K). Ag(1 1 1) and Cu(1 1 1) samples were prepared by repeated cycles of argon ion bombardment with subsequent annealing. Crystalline order was checked with low-energy electron diffraction while cleanliness was monitored with the scanning tunnelling microscope. Following a recipe by Everson et al. [32] and Michely et al. [33] monatomic deep hexagonal vacancy islands of various sizes on Ag(1 1 1) were fabricated by exposing the clean sample surface briefly to a low-flux argon ion beam. Triangular vacancy islands were produced by controlled tip-sample contacts. Single cobalt atoms were adsorbed on the cold surface by heating a cobalt-covered filament resistively. Individual silver and copper atoms were deposited from the tip by controlled contacts of the tip with the surface [34]. Cesium and sodium were evaporated at room temperature onto the Cu(1 1 1) surface from a commercial dispenser [35] keeping the pressure below 5 · 108 Pa. Spectroscopy of the differential conductivity was performed by superimposing a sinusoidal voltage (amplitude 1–5 mVrms, frequency 4– 10 kHz) on the tunnelling voltage and detecting the current response with a lock-in amplifier. 4. Lineshape analysis 4.1. Shockley surface states The first scanning tunnelling microscopy (STM) experiment addressing surface state lifetimes was reported by Li and coworkers [36]. They studied the Ag(1 1 1) Shockley-type surface state whose spectroscopic signature is shown in Fig. 2. Here, the differential conductivity, dI/dV, is measured as a function of the sample voltage. At the voltage which corresponds to the binding energy of the surface state a sharp rise of the dI/dV signal occurs. The width of this onset is determined by many-body interactions of the excited

dI / dV (nS)

4 3

Δ

2 1 0 -100

-80

-60

-40

-20

Sample voltage (mV)

Fig. 2. Constant-height tunnelling spectrum of the Ag(1 1 1) surface state at 4 K. The sharp rise of the differential conductivity (dI/dV) at the sample voltage V  67 meV is due to the onset of tunnelling into the Ag(1 1 1) surface state band. The width D of the onset is related to the lifetime of the band edge state, as discussed in the text. The constant offset of the spectrum of 1.2 nS results from tunnelling into bulk states whose density of states is assumed constant within the energy range of the spectrum.

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hole left behind in the surface state band by the tunnelling electron and reflects its lifetime. Electron–electron and electron–phonon scattering are the dominating processes leading to the observed width of the spectrum. Electron-defect scattering can be safely excluded since with the scanning tunnelling microscope minute amounts of contamination can be detected and thus an effectively defect-free area for the spectroscopy experiments can be chosen. The measured width, D, of the surface state spectroscopic signature, which is defined by extrapolating the slope at the midpoint of the rise to the continuation above and below the onset, is related to the imaginary part of the self-energy via 1 D ¼ bp Im R ¼ bpC; 2

ð1Þ

where Im R = C/2 denotes the imaginary part of self-energy which is related to the half width at half maximum, C/2, of the surface state; b is a number reflecting the probability of finding the electron in the tunnelling barrier region where Im R = 0. The self-energy concept allows to describe many-body physics and to incorporate lifetime effects. In good approximation the total line width can be written as the sum of independent contributions by electron–electron scattering (Cee), electron–phonon scattering (Cph), and electron-impurity scattering (Cim), i.e., C ¼ Cee þ Cph þ Cim :

ð2Þ

As discussed in Ref. [37], in the experiment thermal broadening of the Fermi edge and the modulation technique often used to measure differential conductivities contribute to the line width [38]. Comparison with lifetime data originating from, for instance, angleresolved photoelectron spectroscopy can now be performed via the line width: s¼

h  h  ¼ C 2Im R

ð3Þ

with s denoting the lifetime. As shown in Table 1, agreement has been achieved concerning the line widths of surface states as measured by scanning tunnelling microscopy [39], angle-resolved photoelectron spectroscopy [40,41], and calculated within the GW approximation of many-body theory [39], leading to the following picture of hole-state relaxation at metal surfaces (see Fig. 3). An excited hole in the surface state band, which results from an electron tunnelling out of the bottom of the band, relaxes via electron–electron and electron–phonon scattering. Both electron–electron scattering within the surfacestate band (intraband scattering) and electron–electron scattering with surface-projected bulk states (interband scattering) occur, with the former dominating due to the greater overlap between the hole and surface-band electron wave functions. The paramount

Table 1 Comparison of surface state lifetimes for the (1 1 1) surfaces of Ag, Au, and Cu as determined by scanning tunnelling microscopy (STM), angle-resolved photoelectron spectroscopy (ARPES), and theory (theo) Metal

CSTM (meV)

CARPES (meV)

Ctheo (meV)

Ag Au Cu

6 18 24

6 ± 0.5 21 ± 1 23 ± 1

7.2 18.9 21.7

Tunnelling spectroscopy and calculated data were adapted from Ref. [30,39], while photoemission data were taken from Refs. [40,41].

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Fig. 3. Current picture of the relaxation of an excited hole state by electron–electron scattering: the excited hole state (open circle at the bottom of the surface state band) can be occupied by electrons within the surface state band (filled circles) or by electrons from adjacent bulk states (shaded area).

importance of the availability of final states into which to scatter also plays an important role in understanding quantum well state decay rates, discussed below. Since for surface state excitations the excited hole state is localised in the topmost layer of the substrate the Coulomb interaction responsible for the scattering of electrons into the hole is not as effectively screened as in the case of a hole state in the bulk of the metal. As a consequence, the lifetime of holes at the surface is shorter than the lifetime of a hole in the bulk (at comparable energies). Another important lesson from the quantitative line shape analysis of surface states at the bottom of the surface state band is the important role played by the d-electrons in screening the Coulomb interaction. The binding energies of the Ag(1 1 1), Au(1 1 1), and Cu(1 1 1) surface states are perturbed in scanning tunnelling microscopy by the presence of the tip [42,43]. The potential importance of this to the analysis of scanning tunnelling spectroscopy of surface states has recently been addressed [44]. This follows the recognition that the analysis of image state lifetimes deduced with the scanning tunnelling microscope requires that the influence of the tip be taken into account [45]. In particular, at Cu(1 0 0) the first image potential state exists at a binding energy of 0.57 eV relative to the vacuum energy and has a lifetime of 37 fs. Using typical tunnelling currents of 0.1–1 nA the electric field of the microscope’s tip shifts the binding energy to +0.1 eV [46], and decreases the lifetime to 15 fs [45], due to a combination of changes to the image state wave function that increase the efficiency of inelastic scattering processes, and the greater number of available decay channels that accompany the Stark shift to higher image state energy. For the lifetimes of the n = 2 and n = 3 image states the perturbing influence of the scanning tunnelling microscope tip is even greater [47]. Using GW calculations, Becker et al. [44] have shown that although the rate of change of the decay rate with binding energy (dC/dEn) is even greater for Shockley states than for image potential states, the significantly smaller Stark shifts that result under normal tunnelling conditions when probing the dynamics of Shockley states means that the absolute change in decay rates is less than 1 meV, and within experimental uncertainties.

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Fig. 4. Constant-current map of dI/dV acquired at 20 mV on a triangular monatomic deep vacancy island on Ag(1 1 1). Inside the quantum box an interference pattern of the surface state wave function with six pronounced maxima is observed.

In the following section we describe the experimental methods which extend the measurements presented in this section to the case of excited surface-state and surfaceresonance electrons of different energies below and above the Fermi level. 4.2. Confinement of Shockley surface states One way to probe the dynamics of surface states at various energies is to confine them to quantum boxes. Quantum boxes may be fabricated by, e.g., single-atom manipulation [48] or by the preparation of nanoscale islands [49]. In Fig. 4 we show an experimental realisation of a quantum box which has been artificially fabricated with the tip on a Ag(1 1 1) surface, namely a triangular monatomic deep vacancy island. The standing wave pattern inside the island arises from confinement of the surface state wave function and is clearly visible. A dI/dV spectrum recorded above the centre of a hexagonal monatomic deep vacancy island on Ag(1 1 1) [50] is depicted in Fig. 5. As expected, a series of peaks reflecting discrete confined states is observed. Similar results are obtained for adatom islands [51], and atomic corrals [48,52]. The full width at half maximum of these peaks, C, decreases as the energy approaches the Fermi level (V = 0) which might be interpreted as being due to the vanishing of electron–electron and electron–phonon scattering as jE  EFj ! 0 [30]. However, the peaks at each energy are broader than inelastic processes like electron–electron or electron–phonon scattering would give alone. This reflects additional broadening of the quantised states due to a reflectivity coefficient at the confining steps which is less than one [53]. We have calculated [50,54] the influence of partial reflection within an analytic model of electrons confined within vacancy or adatom islands and atomic corrals. From a pole analysis of the single-particle Green function that satisfies the inhomogeneous Schro¨dinger equation including the electron self-energy, we find that for electrons in well-defined resonant levels the total decay rate s1 is given by 1 1 1 ¼ þ s sI sR

ð4Þ

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301

dI / dV (nS)

6 4 2 0 0

250 V (mV)

500

Fig. 5. Constant-height tunnelling spectrum acquired in the centre of a hexagonal vacancy island with an edge length of 7.4 nm on Ag(1 1 1). The peaks of the dI/dV signal correspond to the confined states of the quantum box. The feedback loop was opened at 1 nA and 300 mV.

with s1 I the decay rate due to inelastic (I) processes (electron–electron and electron–phonon scattering) and s1 R the decay rate resulting from partial reflection (R) at the confining barrier. For a circular island, which is a good approximation to the hexagonal vacancy islands, we get [50] sR ¼ 

S ; vg ln jRj

ð5Þ

where S is the island radius, vg denotes the group velocity of the surface state, and R = R(E) is the energy-dependent reflectivity coefficient as determined experimentally for ascending (vacancy islands) and descending (adatom islands) steps on Ag(1 1 1) [53]. A different expression for sR involving the adatom scattering phaseshift is obtained for resonant levels within atomic corrals [54]. What is the optimum size of nanostructures to be used for lifetime studies? Eq. (5) tends to suggest that the larger the size of the confining structure the lower the lifetime-limiting contribution of lossy scattering. However, simultaneously with increasing the size of the quantum box the level spacing, which scales as S2, is reduced. With the level spacing decreasing more rapidly than the level widths, the spectral features increasingly overlap, making the line shape analysis difficult. Based on this reasoning Crampin et al. [54] suggested that decay rate measurements for Ag(1 1 1) are best performed for high-symmetry resonators with radii in the range of 20 nm. 4.3. Quantum well states Quantum well states which are confined to ultrathin alkali metal adsorption layers on metal surfaces with a surface-projected band gap exhibit the appealing property that their binding energies depend on the alkali metal coverage. Consequently, decay rate analyses at a variety of energies can be performed conveniently by changing the coverage. In Fig. 6 we show the dependence of the binding energy of quantum well states confined to thin Cs layers adsorbed on Cu(1 1 1) upon the coverage. The p(2 · 2)Cs superstructure corresponds to a coverage of 0.25 ML, where one monolayer (ML) is defined as one Cs atom per Cu

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STS (von Hofe et al.) 2PPE (Bauer et al.) IPES (Arena et al.)

3.0

Binding energy (eV)

2.5 2.0

Cu(111)-p(2×2)Cs 1.5 1.0 0.5 0.0 0.00

0.05

0.10 0.15 0.20 Coverage (ML)

0.25

Fig. 6. Plot of the binding energy (E  EF) of quantum well states at C on Cu(1 1 1)-Cs as a function of coverage as measured by different techniques. Data from two-photon photoemission (2PPE) [55] are depicted as squares, data from inverse photoemission (IPES) [56] are presented by triangles, while data from scanning tunnelling spectroscopy (STS) [57] are depicted as circles.

b

Fig. 7. (a) (1 1 1)-projected bulk electronic structure of Cu along the CM direction of the surface Brillouin zone (shaded areas). The dashed line depicts the dispersion curve of the quantum well state of Cu(1 1 1)–p(2 · 2)Cs. (b) Like (a) for Cu(1 1 1)–p(2 · 2)Cs. The superstructure leads to a backfolding of the surface-projected band structure. (c) Surface Brillouin zone of Cu(1 1 1) and Cu(1 1 1)–p(2 · 2)Cs containing the CM direction.

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303

atom. Fig. 6 combines data acquired by two-photon photoemission (squares) [55], inverse photoemission (triangles) [56], and tunnelling spectroscopy (circles) [57]. The data at very low coverage (<0.05 ML) are interpreted in terms of excited states which are derived from unoccupied Cs orbitals. Here, we focus on the quantum well state of the p(2 · 2)Cs superstructure at a binding energy of 40 meV above the Fermi level [58]. The width of the spectroscopic signature of this quantum well state as measured using tunnelling spectroscopy is C = (18 ± 2)meV [58]. Interestingly, a first theoretical estimation of the total line width is C 6 7.6 meV [58] where the contribution of electron–electron scattering to the total line width is less than 0.1 meV, while electron–phonon scattering contributes with 7.5 meV. From Fig. 7 we can see that such a low value for Cee is not surprising: The quantum well state dispersion curve (dashed curve) lies within the surface-projected energy gap. For an injected electron there are no final states nearby to scatter into. The ordered p(2 · 2)Cs superstructure turns the M point of the Cu(1 1 1) surface Brillouin zone (see Fig. 7c) into a new C point. As a consequence, the quantum well state band disperses within backfolded unoccupied Cu states (see Fig. 7b). These backfolded states couple to the quantum well state and, therefore, introduce an elastic channel for broadening. Calculations taking into account this surface Brillouin zone backfolding show that this additional channel contributes Cel = 9.4 meV to the total line width C = Cee + Cph + Cel = 17.0 meV in good agreement  with  the experimental data. Interestingly, for Cuð1 1 1Þ  32  32 Na this additional elastic scattering into backfolded final states seems negligible. The width of the quantum well state of this adsorbate system is described quantitatively by electron–electron and electron–phonon scattering without need the  to invoke a significant contribution from backfolded Cu states [59]. The ordered 3 3 Na superstructure does lead to a backfolding of Cu states. However, compared to  2 2 the Cu(1 1 1)–p(2 · 2)Cs case these backfolded states, which originate from another region of the surface Brillouin zone, possess a significantly lower density of states [60]. Consequently, these states do not act as an efficient scattering channel that significantly reduces the lifetime of the quantum well state.

4.4. Defect scattering In this subsection we present tunnelling spectroscopy data acquired on individual adsorbed atoms on Ag(1 1 1) and Cu(1 1 1). Our preliminary line shape analysis of spectroscopic signatures indicates that the decay rate of the surface states may be influenced by the presence of the adsorbed atoms. Fig. 8 compares spectra of the differential conductivity as measured on a clean and wide Cu(1 1 1) terrace (lower panel) with spectra acquired on a single adsorbed Cu atom on Cu(1 1 1) (upper panel). The shape of the spectra differs appreciably: While on the Cu(1 1 1) terrace we observe the expected sharp increase of the dI/dV signal at the binding energy of the Cu(1 1 1) surface state, the spectra on the single Cu atom exhibit a drop of the dI/dV signal. Interestingly, this drop occurs at roughly the surface state binding energy (see the dashed line in Fig. 8). For different adsorbed atoms on different (1 1 1) noble metal surfaces this electronic excitation always occurs close to the binding energy of the corresponding surface state [61]. In order to model the data we use an extended Newns–Anderson model [62] to incorporate the coupling of a discrete adsorbate orbital to surface state electrons. The corresponding Hamiltonian reads schematically

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dI / dV (arb. units)

Cu(111)-Cu

Cu(111)

-600

-550

-500 -450 Sample voltage (mV)

-400

-350

Fig. 8. Upper panel: Spectrum of the differential conductivity as measured on a single Cu atom adsorbed on a Cu(1 1 1) surface. Lower panel: Spectrum of the differential conductivity as measured on a wide and defect-free Cu(1 1 1) terrace. The feedback loop was opened at 200 MX in both cases.

0

a BV H ¼ @ aq Ve 

ak

V aq q 0

1 Ve ak 0 C A;

ð6Þ

~k

where jai, jqi, jki denote respectively the adsorbate level, bulk states and surface states with corresponding energies a, q, ~k ; Vaq and Ve ak are the coupling matrix elements between the adsorbate level and the bulk and surface states of the substrate (for details the reader is referred to Refs. [37,61]). The density of states at the adsorbed atom can then be calculated analytically. It is given by na ðEÞ ¼

1 DðEÞ : p ½E  a  KðEÞ2 þ DðEÞ2

ð7Þ

Here, D(E) = Db(E) + Ds(E) describes the coupling of the adsorbate level to the bulk (b) states and the surface (s) state. In our model we set Db(E) = const and write Ds(E) = Dsn(E) with   1 1 2ðE  E0 Þ nðEÞ ¼ þ arctan ; ð8Þ 2 p C where E0 is the binding energy of the surface state and C its inverse lifetime. K(E) can be calculated from D(E) via a Hilbert transform [37,61]. The results of fitting na(E) to experimental data are shown in Fig. 9. Good agreement with experimental data is obtained. Consequently, the observed electronic excitation is interpreted in terms of an adsorbateinduced bound state which is split off from the surface state band. This result is plausible in connection with a theorem by Simon, who showed that an attractive potential with certain properties has always a bound state in two dimensions and in one dimension [63]. One could expect that a two-dimensional continuum perturbed by an attractive adsorbate will lead to a bound state. Zaremba et al. confirmed this idea demonstrating that the potential of an ion screened by a two-dimensional electron gas still supports bound states in spite of

dI / dV (arb. units)

J. Kro¨ger et al. / Progress in Surface Science 82 (2007) 293–312

a

Cu(111) - Cu

c

Ag(111) - Ag

b

Cu(111) - Co

d

Ag(111) - Co

-600

-500

-400

-300 -200

Sample voltage (mV)

-100

0

305

100

Sample voltage (mV)

Fig. 9. Spectra of dI/dV over the centre of a Cu (a) and a Co (b) atom on Cu(1 1 1) and over the centre of individual Ag (c) and Co (d) atoms on Ag(1 1 1). Full lines present fitting curves according to Eq. (7).

not being attractive in some regions of space [64,65]. However, the surface problem is a two-dimensional problem imbedded in three dimensions. The bound state resulting from the adsorbate-induced localisation is not stationary, but a resonance that eventually decays into the three-dimensional continuum bulk states. Our findings are in agreement with a combined theoretical and experimental tunnelling spectroscopy study by Olsson et al. [66]. Further, in a recent two-photon photoemission experiment Boger et al. [67] have shown that the decay rate of image potential states on Cu(1 0 0) is increased by quasi-elastic scattering at single adsorbed Cu atoms. A recent experiment using angle-resolved photoelectron spectroscopy revealed likewise the existence of an electronic state induced by gold atoms adsorbed on a silver film on Si(1 1 1) [68]. The model contains the inverse lifetime, C, as a parameter. Interestingly, for native adsorbate atoms – like Cu on Cu(1 1 1) or Ag on Ag(1 1 1) – C is increased from the clean-surface value (see Table 2), while for Co atoms C does not differ. Consequently, within the limits of our model, native adsorbate atoms decrease the surface state lifetime. While an increased value of C is consistent with the two-photon photoemission experiment by Boger et al. [67], the intriguing difference between Co, Cu, and Ag atoms deserves further investigation. Finally, we notice that the line shape analysis performed for the split-off state is of preliminary character because it relies on an assumption of the surface state density of states. In a recent theoretical analysis Lounis et al. [69] carried out ab initio calculations for single 3d and 4sp impurities on Cu(1 1 1). They found that over Ca and all 3d adsorbed atoms a bound state appears in the density of states of the surface state band. Moreover, although the bound state has s, p, and d-state contributions it arises mainly from the s-orbital of the adatom. Another theoretical study by Lazarovits et al. [70] dealt Table 2 Inverse lifetime (C) for the given adsorbate systems compared with inverse lifetime of the clean surfaces (C0) System

C (meV)

C0 (meV)

Ag(1 1 1)–Ag Ag(1 1 1)–Co Cu(1 1 1)–Cu Cu(1 1 1)–Co

12 ± 1 7±1 35 ± 5 24 ± 2

7±1 7±1 24 ± 1 24 ± 1

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with 3d impurities adsorbed on Cu(1 1 1). Again adsorbate-induced s-like resonances were found and attributed to a surface state localisation at the adatom. A refined analysis of the experimentally observed line shape of the resonance should be performed within these models in order to address the surface state decay rate. 5. Phase coherence length approach The decay rate of surface-localised states can also be extracted using the scanning tunnelling microscope by analysing the spatial decay of quantum interference patterns. This spatial decay is partly governed by the so-called phase coherence length (see below). Standing wave formation is observed, for instance, at step edges: Incoming and reflected surface state waves are superimposed giving rise to the characteristic interference pattern as observed by Hasegawa and Avouris [16] and Crommie et al. [15]. The phase coherence length method has been developed by Bu¨rgi et al. [71] and Jeandupeux et al. [72] to identify the energy dependence [71] and the temperature dependence [72] of the phase coherence length. Wahl et al. [46] applied the phase coherence length method to describe dynamical properties of electrons in two-dimensional image potential states. The interference pattern exhibits a spatial decay which has two components. First, scattering at steps in two dimensions leads to an interference pattern well described by the zeroth-orderpffiffiBessel function ffi [73]. This function incorporates an intrinsic decay proportional to 1= x with x being the distance from the step. Second, an additional decay of the interference amplitude is provided by inelastic scattering processes. These scatter electrons out of the coherent quantum state and so cause a reduction in the number of electrons that can contribute to the interference pattern. This latter effect can be described by calculations of the quan-

surface state: ab initio 2.5

Binding energy (eV)

2.0

bulk band edge: ab initio ∗ bulk band edge: m = 0.24 ∗ surface state: m = 0.4

∗

1.5 1.0 0.5 0.0

0.0

0.1

0.2 0.3 -1 Wave vector (Å )

0.4

0.5

Fig. 10. Surface band structure of Ag(1 1 1). The shaded area and the dashed line are the surface-projected bulk electronic structure and the surface state band as determined from ab initio calculations, respectively. The full line is the parabolic surface state dispersion curve using an effective mass of m* = 0.40me, while the dotted line depicts the parabolic shape of the bulk band edge with an effective mass of m* = 0.24me.

J. Kro¨ger et al. / Progress in Surface Science 82 (2007) 293–312

a

em

1.1 1.0 0.9

ai

vg / vg

307

0.8 0.7

em

vg

0.6

ai

vg (a. u.)

vg 0.4

0.2

0.0 0.0

b 0.1

0.2

0.3

0.4

0.5

-1

k|| (Å )

Fig. 11. (a) Ratio of group velocities for the Ag(1 1 1) surface state as calculated with a constant effective mass of ai em ai m* = 0.40me ðvem g Þ and as determined from ab initio calculations ðvg Þ. (b) Comparison of vg with vg as a function of the wave vector.

tum interference pattern in the vicinity of a step edge that are performed using an imaginary self-energy to describe the inelastic processes [74]. The result is   x ð9Þ .ðxÞ / 1  jRj exp  J 0 ð2kxÞ LU for the oscillations of the local density of states with distance x. The step reflection coefficient is jRjexp  ip, and J0 is the zeroth-order Bessel function containing the wave vector, k, as an argument. Fitting .(x) to experimental data then allows the phase coherence length, LU, to be extracted. A derivation of this expression can be found in Ref. [37]. We notice that Eq. (9) has the form used in Refs. [46,71,72] but with LU replaced by 2LU. As a result, linewidths and thus decay rates in Refs. [46,71,72] should be doubled. Fig. 12 shows a collection of decay rate data for the Ag(1 1 1) surface state. Data obtained by angle-resolved photoelectron spectroscopy (crosses, from Ref. [75]) as well as acquired by scanning tunnelling microscopy and spectroscopy using the line shape analysis method (circles, from Ref. [50]) and the phase coherence length approach (diamonds, from Ref. [76], triangles, from Ref. [77]) are included. Calculated results appear as full [75] and dashed [76] lines in Fig. 12. Correcting confinement data (circles) for the lossy scattering effect and introducing a factor of two in the phase coherence length data (diamonds, triangles) leads to a consistent picture of the Ag(1 1 1) surface state decay rate as measured by various techniques and modelled by theory. The lifetime is related to LU via the group velocity vg s¼

LU vg

ð10Þ

308

J. Kro¨ger et al. / Progress in Surface Science 82 (2007) 293–312 30 12

25

8 4

Γ (meV)

20

0 -60

-40

-20

0

20

15 10 5 0 0

100

200 300 E - EF (meV)

400

500

Fig. 12. Line width of the Ag(1 1 1) Shockley-type surface state as a function of the binding energy E  EF with EF the Fermi energy. The plot shows a collection of experimental and theoretical data: Photoemission data are depicted as crosses [75], data obtained by scanning tunnelling spectroscopy are presented as triangles [77], diamonds [76], and circles [50], theoretical data are shown as a full [75] and a broken [76] line. Inset: Decay rate data for 65 meV 6 E  EF 6 35 meV.

with vg ¼  h1

oEðkÞ ; ok

ð11Þ

where E(k) denotes the dispersion relation of the corresponding surface state. As a consequence, apart from extracting the phase coherence length from experimental data using Eq. (9), the electron dispersion E(k) is required for determining the lifetime. Based on angle-resolved photoelectron experiments, which can accurately measure the occupied part of surface state dispersion curves, the dispersion relation of Shockley states is normally assumed to be parabolic, for example for Ag(1 1 1) with an effective mass m* = 0.40me (me denotes the mass of the free electron). However, ab initio calculations which are able to describe the dispersion over the full range of surface state energies indicate that deviations from this parabolic variation occur, particularly for energies of unoccupied portions of the surface state band [44,78]. Fig. 10 compares surface state and surface-projected bulk band edge dispersion curves for the case of Ag(1 1 1). The parabolic dispersion relations for the surface state (m* = 0.40me, full line) and the surface-projected bulk band edge (m* = 0.24me, dotted line) deviate from the corresponding ab initio results (dashed line and shaded area). Fig. 11a and b demonstrate that some deviations of the group velocities ˚ 1 of the order of 10%. Taking this effect into account will enable an occur for k > 0.2 A even more accurate determination of lifetimes within the phase coherence length approach. More significant deviations are calculated for Cu(1 1 1) [79,80]. 6. Summary The scanning tunnelling microscope is a suitable tool to address the lifetime of surfacelocalised states on a femtosecond timescale. Line shape analysis of spectroscopic signatures and the phase coherence length approach allow one to determine the decay rates of both hole-like and electron-like excitations, enabling the study of electron–electron,

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electron–phonon scattering rates, and the effects of surface Brillouin zone backfolding in overlayer superstructures. In the case of confined electron states lossy boundary scattering contributes an additional increase of the overall decay rate, and must be allowed for when extracting intrinsic decay rates. The phase coherence length approach depends on a correct extraction of the phase coherence length, and on the precise surface state dispersion relation to determine accurate lifetimes. Finally, individual adsorbed atoms on noble metal (1 1 1) surfaces induce a resonance, which can be interpreted as a bound state split off from the surface state band. Modelling of the line shape could yield adatom-induced changes to the decay rate. Acknowledgements Financial support by the Deutsche Forschungsgemeinschaft and the British Council is gratefully acknowledged. This work has been supported by the Basque Departamento de Educacio´n, the UPV/EHU, the Spanish Ministerio de Educacio´n y Ciencia, the European Network of Excellence FP6-NoE NANOQUANTA, and the projects NANOMATERIALES and NANOTRON funded by the Basque Departamento de Industria, Comercio y Turismo within the ETORTEK programme and the Departamento para la Innovacio´n y la Sociedad del Conocimiento from the Diputacio´n Foral de Guipu´zcoa. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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