Interplay between electronic states and structure during Au faceting

F Schiller¹, M Corso², J Cordón¹, F J García de Abajo³ and J E Ortega^1,2,3,4

¹ Dpto. Física Aplicada I, Universidad del País Vasco, Plaza Oñate 2, E-20018 San Sebastián, Spain and Unidad de Física de Materiales CSIC/UPV, Manuel Lardizábal 3, E-20018 San Sebastián, Spain
² DIPC, Manuel Lardizábal 4, E-20018 San Sebastián, Spain
³ Instituto de Óptica-CSIC, Serrano 121, E-28006 Madrid, Spain

⁴ Author to whom any correspondence should be addressed.

E-mail: enrique.ortega@ehu.es

Received 21 July 2008
Published 14 November 2008

Contents

• 1. Introduction
• 2. Experimental details
• 3. Experimental results
• 4. Summary
• Acknowledgments
• References

1. Introduction

Surface electrons in metals have become widely popular thanks to scanning tunneling microscopy and spectroscopy (STM/STS), which allows the creation of electron-confining nanostructures of varied shape, such as atom corrals, and the ability to map the standing wave patterns inside [1]. Beyond the striking visualization of the wave nature of the electron, STM/STS experiments show that surface electrons undergo significant scattering at atoms, defects or steps on metal surfaces. This results in a strong spatial modulation of the density of states at E_F (period ~λ_F/2) that affects the effective surface potential. In fact, Cu adatoms on Cu(111) [2], and, more strikingly, H and Ce atoms on W(110) and Ag(111), respectively  [3, 4] self-assemble forming ordered structures with characteristic λ_F/2 spacings. In metallic nano-object arrays, such as quantum dots and steps, the periodic perturbation of the electronic density of states originated by surface state scattering extends over the entire two-dimensional (2D) surface plane [5]. In such systems the question that arises is how the structure responds when the lattice constant d matches λ_F/2, i.e. what happens if we force the nesting of the Fermi surface with 2k_F superlattice vectors.

Noble metal surfaces display a variety of periodic self-assembled structures that are stable at 300 K. These comprise 1D step arrays and surface reconstructions in vicinal surfaces  [6, 7], as well as 2D dislocation patterns in monolayers and bilayers [8, 9]. All of them exhibit superlattice bands, as a clear signature of coherent surface electron scattering in the array. Vicinal (111) surfaces are particularly attractive to tune 1D Fermi surface nesting. This occurs when the radius of the ring-like Fermi surface κ_F matches the superlattice wavevector π/d [10], where d is the terrace size. The latter can be selected via the surface misorientation (or miscut angle α) with respect to the (111)-direction with sin α = h/d, where h is the step height. Macroscopic angles α~1°–15° turn into nanoscopic lattice constants d~100–10 Å, covering the variation range of π/κ_F, i.e. from 15 Å in Cu(111), to 19 Å in Au(111) and 39 Å in Ag(111) surfaces [11], and intermediate values in overlayer systems [8, 12]. In copper, for example, 1D nesting occurs near the (443) surface orientation, where a Fermi gap opening has been observed [10]. In this case, the step lattice disorder has been claimed as the structural instability associated with nesting, although clear proof of such structural/electronic interplay has not been reported.

In this work, we use a curved crystal surface to study the connection between structure and electronic states in 1D step superlattices. The curved surface allows one to smoothly vary the miscut α, and hence the step lattice constant d, as sketched in the lower panel of figure 1. The local structure is determined by STM (middle panels in figure 1), whereas locally resolved surface bands are measured by angle resolved photoemission spectroscopy (ARPES, top panel in figure 1) using a very small synchrotron light spot. The system of choice is an Au crystal that covers a Δα = ±15° range of miscut angles around the [111]-direction (α = 0). As depicted in figure 1, Au(111) vicinals are not stable within the 4°–9.5° miscut range, exhibiting faceting in two phases of large (d_w) and small (d_n) (111) terraces [13]. The ARPES data indicate the presence of distinct d_w and d_n bands in the faceted range, whose intensities evolve with the miscut angle. At the onset of faceting (~4°) a superlattice gap opens at the Fermi energy, suggesting that the faceting is triggered by electronic energy gain in the d_w phase. It is also found that d_n corresponds to the critical size for surface state occupation of a single terrace, which may have an influence in both surface energy and faceting kinetics.

2. Experimental details

The curved Au crystal (Mateck GmbH, Germany) was mechanically polished defining a 30° cylindrical section (11 mm radius) around the (111)-direction. The surface was prepared in vacuum following standard sputter-annealing cycles. The procedure ends after the complete removal of point defects and contaminants, which give rise to step pinning and bunching. The middle panels in figure 1 display two STM images from the curved surface taken at the right side (positive α) and the left side (negative α) of the sample. At a given α, the structures are analogous to those observed in regular vicinal Au(111) crystals with flat surfaces [13]. The advantage of the new approach is that the curved surface delivers all miscut angles in a single sample. Left and right sides of the crystal correspond to surfaces with A- ({001}-oriented microfacet) and B-type steps ( microfacet), respectively. Spontaneous faceting occurs for both step types within the same ~4°–9.5° miscut range. At the lower faceting onset (~4°), one observes monatomic step arrays made of relatively wide d_w = 41±1 Å and d_w = 36±1 Å terraces, for A and B steps, respectively, whereas at the upper α~9.5° onset both sides exhibit a single phase with d_n = 14±1 Å lattice constant. For A-type steps, the d_w phase is defined by a single terrace and the d_n phase by bunches [13], whereas in the B-type both d_n and d_w form bunches. The STM image of the A-type steps of figure 1 corresponds to α = –7.5° with a d_n phase containing six terraces. In contrast, the B side at α = 8.3° displays large, alternating bunches of d_w and d_n terraces. This difference between A and B faceting is connected to their distinct atomic packing at step edges, which results in significant step energy variations [14]. However, there is no apparent structural reason for the system to prefer specific d_n and d_w values. These are explained on the basis of electronic surface states, as discussed below.

STM experiments were carried out using a variable temperature Omicron STM in San Sebastián (Spain), while the photoemission data are taken at the PGM beamline of the Synchrotron Radiation Center (SRC) of the University of Wisconsin in Stoughton (USA). For ARPES measurements, we used a hemispherical Scienta SES200 spectrometer with energy and angular resolutions set to 25 meV and 0.1°, respectively, and p-polarized light with the polarization plane set parallel to the steps on the sample, i.e. along the -direction in figure 1. Characteristic surface bands within the faceted regions are shown on top of figure 1. The miscut α is selected by scanning the 100 μm synchrotron light beam across the curved surface. The spot size defines an effective miscut broadening Δα < 0.5°. In the spectra of figure 1, the wavevector scale is referred to the center of the second Brillouin zone of the local (nominal) surface plane, i.e. , where ( ) is the electron kinetic energy, m the electron mass, θ is the emission angle with respect to the (111)-direction, and d is the average terrace size d = h/sin α, with h = 2.35 Å being the monatomic Au step height. To better observe different superlattice umklapps we choose  [6]. Measurements have been performed at 150 K.

3. Experimental results

The surface bands in figure 1 nicely reflect the structural differences between A-type and B-type faceting. In both, the free-electron-like, Shockley surface state splits into two different contributions, one for each phase. The d_n phase with narrow terraces leads to folded bands of lower binding energy E_B = 0.29 eV at A and B sides. The multi-terrace d_w phase at the B side exhibits zone folding of a higher binding energy E_B = 0.45 eV dispersing band. In contrast, the single d_w terrace at the A side leads to a 1D quantum well level at E_B = 0.4 eV. In the A-side one can observe the additional umklapp folding of d_n bands by the periodic faceted structure, in the same way as in flat samples [6]. Such an umklapp emission makes it difficult to analyze separate d_w and d_n bands across the faceted region in the A-side. This analysis is thus limited to the B-side.

In figure 2, we compare photoemission intensity images (ω  =  21.2 eV) for a regular flat Au(788) surface [6] and for the curved crystal at α = + 3.5° (d_w = 38 Å), i.e. very close to the faceting onset in the B side (d_w = 36 Å). To better analyze the band topology, the second derivative of the photoemission intensity is displayed. The close similarity between top and bottom panels in figure 2 demonstrates the validity of the curved crystal approach for ARPES experiments. Surface bands deviate significantly from the free-electron-like behavior of the Au(111) surface in both cases. They show the signatures of superlattice scattering, namely band folding with 2π/d vectors and small ~0.1–0.15 eV energy gaps, namely the first superlattice gap at ~E_F–0.3 eV and a second one around E_F. The existence of such superlattice gaps is further proven in figure 3(a), where we plot the photoemission intensity in a constant energy surface in the middle of the first gap (~E_F–0.28 eV). The data correspond to the faceting onset at the A side (α = –4.4°). The intensity drop at is clear. The magnitude of the gap can be determined from a fit to the photoemission spectrum at this point. This is shown in figure 3(b), together with a gapless spectrum, for the sake of comparison. The fit in figure 3(b) uses Lorentzian lines, giving a gap width of 124 meV. The second superlattice gap at the Fermi energy is poorly resolved in individual spectra. In order to have an estimation of such a gap we can fit the overall band structure with a Kronig–Penney model, as done in figure 2 (red solid lines) [6]. For the spectra in figure 2, the Kronig–Penney fit gives a second superlattice gap of ~100 meV right below the Fermi energy.

Figure 3. (a) Constant energy surface at EF–0.28 eV showing the depleted intensity at the first superlattice gap for α = –4.4°. (b) Energy distribution curves at the (kx, ky) points marked in (a) with the same colors. At (spectrum at the bottom) the two peak features delimit (124 meV) the first superlattice gap.

As demonstrated for Cu(443) [10], a Fermi gap leads to a significant reduction in the electronic energy per surface area. This can be estimated from the total energy difference Δγ between gapped and nongapped 2D band structures with the same occupation [10]. For the 0.1 eV gap of figure 2 we obtain Δγ~0.25 meV atom^–1, a number that falls within the range of the surface stress energy oscillations originated by the periodic fcc/hcp terrace reconstruction in vicinal Au(111) [14]. The latter was, up to now, thought of as the reason for the faceting transition in Au(111). Since reconstructed terraces have lower surface energy [13], one may naively expect that beyond 4° the system locks in the smallest terrace size (d_w) that allows the fcc/hcp reconstruction [15]. However, fcc/hcp disconmensuration lines run perpendicular to the steps in the B-side [14], and hence there is no a straightforward restriction to limit the size of a reconstructed terrace with B steps. In contrast, the data in figure 2 suggest that d_w is a magic terrace size that provides a significant reduction in electronic energy, and hence can trigger surface faceting.

Figure 4 illustrates the detailed surface state evolution within the faceted region at the B side of the sample, where one can follow the changing contribution from each phase. To better identify the nature of the different bands, the solid lines represent parabolic envelopes (circles in figure 3(a)) that fit d_w and d_n bands, respectively, at faceting onsets. The data at 4.4° and 9.9° are shown as direct intensity maps, and the rest as second derivative plots. The latter show that the gapped topology of the bands in figures 2 and 3, and in particular the E_F gap, is also present in the d_w phase. Between 4.4° and 9.9°, the d_n emission builds up, whereas the d_w bands are progressively quenched. d_w bands exhibit -folding with g_w = 0.175 Å^–1 vectors, i.e. d_w = 36 Å, whereas d_n bands are folded with g_n = 2π/d_n = 0.45 Å^–1, in agreement with the value d_n = 14 Å found in STM. Within the error limits, both g_n and g_w are constant values across the faceting regime, reflecting the stability of the d_w and d_n lattice constants in each phase. In figure 4, the arrows join folded parabolas at the Fermi energy defining the nesting vector 2k_F for each phase. For the d_w bands, the parabolic fit gives 2k_F = 0.177±0.005 Å^–1, i.e. 2k_F~g_w, as expected for Fermi surface nesting and gap opening with 4π/d umklapp vectors (dotted lines). Furthermore, for d_n bands, the Fermi level crossing defines 2k_F = 0.180±0.005 Å^–1. Thus, we may conclude that, within error bars, 2k_F~g_w = 2π/d_w is the same for both d_w and d_n phases.

To better analyze the electronic states in the B-like faceted surface, we have simulated a photoemission intensity map in figure 5(a). In detail, the model has the following key elements: (i) we solve Schrödinger's equation in 1D, with the spatial coordinate representing the distance measured along the -direction; (ii) the potential is fixed as zero inside d_w terraces, whereas a small offset for d_n terraces (~0.1 eV) and the potential step barrier (width = 2 Å and height = 1 eV) are needed to fit both d_w and d_n band minima; (iii) the surface electron wavefunction is then assumed to be separable into the product of a function that depends on the coordinate along the 111-direction and a function depending on the coordinates parallel to the surface [16]; (iv) this separation leads to a factorization of the photoemission matrix element into a factor that is the Fourier transform of the parallel component of the wavefunction times the matrix element of the normal wavefunction going to an emitted plane wave; (v) each terrace contributes with a term to the first of these factors (i.e. the Fourier transform), and each term is multiplied by a phase that takes into account the difference in path-length of the emitted photoelectron for each emitting terrace; (vi) finally, the matrix element of the normal wavefunction is calculated from rigorous solutions for surface-state and free electron-wavefunctions corresponding to a 1D model potential along the 111-direction that is adjusted to yield the main characteristics of the measured surface and bulk electronic structure of the Au(111) surface [17]. In practice, a finite set of terraces is considered in each calculation, and in particular, we have considered a system consisting of a periodic succession of 30 small d_n = 14 Å terraces, embedded between two 30×d_w (d_w = 36 Å) facets, as sketched on top of figure 5. The similarity of the model calculation with the data in figure 4 is remarkable. Both d_w and d_n phases are indeed large enough to develop separate bands (blue and red, respectively). Interestingly, the simulation shows that d_n states change their nature within the second d_w superlattice gap, i.e. at the Fermi energy. Inside the gap, we observe discrete levels due to confinement within the d_n phase. Out of the gap, d_n electrons are transmitted, displaying continuous band dispersion.

It is interesting to note that the d_n = 14 Å size is sharply defined in the faceted region of the curved surface for both A and B sides. This suggests a local energy minimum also for this terrace size, but this cannot be readily proven. On the one hand, the elastic energy curve for Au(111) vicinals does not show any free energy minimum at d_n = 14 Å [13]. There is only a shallow inflection point for A-type facets, resulting from the cross over of the dominating terrace energy at low miscuts and the steadily increasing step energy at high miscuts. From electronic energy arguments, one may rather expect faceting with d_n~17 Å, such that the first superlattice gap would straddle E_F for d_n bands [10]. Instead we observe the first superlattice gap well above the Fermi energy for the d_n phase. Yet figures 4 and 5 contain clues that allow us to consider the role of the electronic structure in a different way.

Note that the unique k_F observed in figure 4 implies crossing of d_w and d_n bands near E_F. As shown in figure 5, such a band overlap permits nesting between Fermi surface sheets that belong to bonding d_n states and antibonding d_w states at . We can find the same topology in systems with coexisting periodic potentials, where such an indirect nesting leads to the Fermi gap opening and electron energy reduction [18]. However, the large size of laterally coupled phases in the present case suggests a weak electronic interaction between d_n and d_w facets, and hence an irrelevant energy gain. On the other hand, the presence of d_n discrete states within the Fermi energy (d_w) gap can lead to a different phenomenon, namely the change in surface state occupation of d_n terraces by quantum size shift, which may influence not only system energetics but also faceting kinetics. It is known that surface band depletion eliminates the Ehrlich–Schwoebel barrier for atom diffusion across surface steps [19], thereby speeding interlayer mass transport and favoring kinetics. Since band occupation lowers the electron energy, we may expect a critical d_n size that balances fast kinetics (favoring depletion) and surface energetics (favoring occupation). Using the same parameters of figure 5(a), we calculate in figure 5(b) surface states for one d_n terrace inserted between two 30×d_w arrays. The d_n size is varied in ±1 atomic rows around d_n = 14 Å, i.e. we consider d_n terraces containing 4(1/3), 5(1/3) and 6(1/3) atomic rows (11 Å, 13.5 Å and 16 Å). The single d_n terrace leads to a single quantum well level inside the d_w Fermi gap, which becomes occupied for d_n~13 Å, i.e. within the error bars of the measured d_n = 14 Å size. Thus, at the onset of the faceting transition, when the d_n phase segregates, d_n~14 Å appears as a critical terrace size that may prompt fast kinetics, while reducing surface energy.

4. Summary

In summary, we have explored the rich behavior exhibited by vicinal Au surfaces probed continuously in a cylindrical crystal with a wide range of miscut angles. Faceting extends over a large miscut range, although, remarkably, the facets are defined by terraces of fixed widths d_w and d_n. In light of the measured band structures and the photoemission simulations, the d_w width appears clearly as a magic size that lowers the electronic energy. In contrast, d_n is explained as a critical size at which the system may improve the faceting kinetics. Further experiments to clarify this later point are encouraged.

Acknowledgments

Fruitful discussions with E Tosatti are acknowledged. The work is supported through projects of the Basque Government (GI1058) and the Spanish Ministerio de Educacion y Ciencia (MAT2007-63083, MAT2007-66050). The SRC is funded by the National Science Foundation (Award no. DMR-0084402).

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