One-dimensional versus two-dimensional surface states on stepped Au „ 111 ...

Donostia International Physics Center and Centro Mixto de Materiales CSIC/UPV, Paseo Manuel Lardizabal 4, 20018-San Sebastian, Spain Universidad del Paı ́s Vasco, Departamento de Fı ́sica Aplicada I, Plaza de On ̃ate 2, E-20018 San Sebastian, Spain Groupe de Physique des Solides, CNRS, Universite ́s Paris 6 et 7, 2 Place Jussieu, 75251 Paris Cedex 5, France LURE Centre Universitaire Paris-Sud, Ba ˆt 209D Boı ˆte Postal 34, 91898 Orsay Cedex, France ~Received 19 November 2001; published 4 April 2002 !


I. INTRODUCTION
Vicinal surfaces are natural templates for self-assembling ordered arrays of nanowires and nanodots with tailored electronic states and transport properties. 1,2They can also be used as model systems for studying the electron wave functions and other basic properties of lateral nanostructures. 3,46][7][8] Here we study this surface state at Au͑322͒ and Au͑788͒ surfaces, which are vicinal to the ͑111͒ plane.Both are characterized by highly regular step arrays over micron-size areas, making them ideal for photoemission studies.Although the basic structural difference between Au͑322͒ and Au͑788͒ is the terrace width ͑i.e., the period of the step superlattice͒, they display completely different electronic structures.In Au͑322͒ we have relatively narrow ͑1.3 Å͒ terraces and the surface state is a two-dimensional band, folded at the center of the step superlattice Brillouin zone.In Au͑788͒ the terraces are wider ͑3.9 Å͒ and the surface state breaks up into one-dimensional lateral quantum-well states.Such different behavior can be explained in the light of the different bulk band structure projection on each surface plane.
The photoemission experiments have been performed at the SU8 undulator, Spanish-French beam line at LURE ͑Paris͒.The experiments were done at 300 K using p-polarized light.The experimental station is equipped with an angle-resolved photoemission setup that provides an angular resolution lower than 0.5°.The total energy resolution is 50 meV.Au͑322͒ is vicinal with respect to Au͑111͒ by a miscut angle of 11.4°towards ͓21 ¯1 ¯͔ and presents ͕100͖-like steps.Au͑788͒ has a miscut angle of 3.5°towards ͓2 ¯11͔ and ͕111͖-like steps.In situ surface preparation is done by extensive sputtering-annealing cycles.Most of Au͑111͒ vicinals undergo facetting, 9 but Au͑322͒ and Au͑788͒ are stable orientations.Figure 1 shows the respective scanning tunneling microscopy ͑STM͒ images.We observe micron-size surface areas with defect-free, regular arrays of straight, monatomic (hϭ2.35Å) steps.In Au͑322͒ terraces are 1ϫ1, ͑111͒ planes, whereas Au͑788͒ terraces display a characteristic reconstruction that is similar to the reconstruction of Au͑111͒. 9,10This is better observed in the closer view shown in the inset of Fig. 1͑b͒ where the corrugation due to the terrace levels has been subtracted.Discommensuration lines run perpendicular to the step edges indicating the presence of alternative fcc-and hcp-packed domains along a single terrace.In the bottom panel we show the terrace width distribution measured over 30 images and more than 200 terraces.The resulting average terrace width values are dϭ13Ϯ2 Å for Au͑322͒ and dϭ39Ϯ5 Å for Au͑788͒.In both cases the sharp terrace width distribution is expected from the strong elastic interaction between adjacent steps as well as from the high kink energy of gold.In Au͑788͒ it is also possible that the highly homogeneous step array is related to the terrace reconstruction, as observed on Si͑111͒. 11

II. 1D VERSUS 2D SURFACE STATES
The flat Au͑111͒ surface electronic structure is characterized by the presence of a well-known, free-electron-like, two-dimensional surface state close to the Fermi energy E F in the center of the surface Brillouin zone. 12At stepped Au͑322͒ and Au͑788͒ the free-electron-like character of the surface state is maintained parallel to the steps (y direction in Fig. 1͒, where we observe a parabolic dispersion with an effective mass m*ϭ0.27m e similar to the effective mass in Au͑111͒. 4 The band dispersion is only affected in the direction perpendicular to the step array (x direction͒.Figures 2͑a͒ and 2͑b͒, respectively, contain the photoemission spectra of Au͑322͒ and Au͑788͒ near the Fermi level showing the dispersion of the surface state peak in the x direction.The emission angle is measured with respect to the surface normal.The differences between both surfaces are remarkable.The surface state peak disperses in Au͑322͒, but it splits into two non-dispersing features in Au͑788͒.This means that the surface state is a two-dimensional ͑2D͒ state for Au͑322͒ and a one-dimensional ͑1D͒ state for Au͑788͒. 13he E(k x ) band dispersions in the direction perpendicular to the steps for both Au͑322͒ and Au͑788͒ are shown in Fig. 3. k x is obtained from the emission angle and the ͑measured͒ kinetic energy via the known equation k x ϭͱ(2m/ប 2 )E kin sin .All the energies are referred to the Fermi level E F .For Au͑322͒ data points in Fig. 3͑a͒ are determined directly from the peak maxima in the spectra in Fig. 2͑a͒ ͑tick marks͒.They are fitted with two parabolas.These appear symmetric with respect to the surface Brillouin zone center, suggesting band folding by step superlattice vectors.The band centers are located at k x ϭ0.23Ϯ0.04Å Ϫ1 and k x ϭ0.74Ϯ0.04Å Ϫ1 , consistent with g/2ϭ/d ϭ0.25 Å Ϫ1 and 3g/2ϭ3/dϭ0.74Å Ϫ1 , respectively.The shift of the p z -like surface band to the surface Brillouin zone edge is expected for vicinal surfaces, as explained in Sec.V.The band minimum is found at E min ϭϪ0.36 eV in the first Brillouin zone and slightly higher in the second zone.This appears to be an artifact related to the lower intensity of the folded band. 14Within the error bars, the effective mass deduced from the parabolic fit is m*ϭ0.28me , which is again similar to the effective mass in Au͑111͒, indicating that the scattering at the step edges is weak in Au͑322͒.Assuming a one-dimensional Kronig-Penney model with ␦-function-like step potentials, the strength of the step barrier can be estimated from the upwards shift of the superlattice band with respect to the flat surface band. 15Although the correct reference is the energy of the infinite fcc-packed terrace we can use the surface state energy of Au͑111͒, since its herringbone reconstruction only results in minor energy changes. 10The surface state of Au͑111͒ has been measured separately under the same experimental conditions, giving E F ϪE 0 ϭ Ϫ0.46 eV.Thus we obtain ⌬EϭE min ϪE 0 ϭ0.1 eV.Following the analysis of Ref. 15, we deduce a repulsive barrier of ϳ1.4 eV Å Ϫ1 and a transmission probability ͉T͉ϭ0.84 at monatomic steps in Au͑322͒.These values are similar to those found in Cu͑111͒ vicinals. 15he data points in Fig. 3͑b͒ for Au͑788͒ are obtained from a line fitting to the spectra in Fig. 2͑b͒, which gives the accurate energy position and the intensity of the two 1D features.The fitting uses asymmetric Lorentzian lines for the peaks, plus a Shirley and a polynomial background, all convoluted with a Gaussian to account for the overall energy resolution.We clearly observe two flat levels at Ϫ0.40 eV

FIG. 3. E(k x
) surface bands corresponding to the spectra of Fig. 2. In Au͑322͒, the surface state forms superlattice bands ͑zone folded by gϭ2/d), whereas in Au͑788͒ we observe two 1D quantum-well levels.The size of the dots in ͑b͒ is proportional to the corresponding peak intensity in Fig. 2͑b͒.The energy gap between quantum levels in Au͑788͒ is consistent with total electron confinement within terraces.and Ϫ0.11 eV, suggesting strong confinement along the k x direction.If we assume the terraces in Au͑788͒ as infinite quantum wells of size L, the energy of the N th level, referred to the ground state of the infinitely wide terrace E F ϪE 0 , is given by The energy gap between the two levels in Fig. 3͑b͒ agrees with the energy interval between the lowest two levels of this infinite one-dimensional quantum well, i.e., (2 2 Ϫ1 2 ) ϫ͓(ប 2 2 )/(2m*L 2 )͔ϭ0.28 eV, where m*ϭ0.27m e and L ϭ39 Å.We conclude that surface electrons are strongly confined within individual terraces at the Au͑788͒ step array, in the same way as they get confined within isolated terraces in Au͑111͒ and Ag͑111͒. 5,6sing Eq. ͑1͒ and the absolute energy of the quantum levels in Fig. 3͑b͒ we obtain E F ϪE 0 ϭϪ0.50 eV.This value is different from E F ϪE 0 ϭϪ0.46 eV measured in Au͑111͒.Assuming the latter as the reference energy of the infinitely wide terrace, such deviation could indicate that the actual terrace potential in Au͑788͒ is not the ideal infinite quantum well.In this sense note that the data points in Fig. 3͑b͒ display some random variation within the error bars that can be interpreted as a narrow band.Using the Kronig-Penney model, the width of the band allows to estimate an upper limit for the transmission probability across the step barrier.We obtain a maximum value of ͉T͉ϭ 0.1 and ͉T͉ϭ0.19 for the first and the second levels, which are still considerably smaller than those found for Au͑322͒.For this value of T and within the same Kronig-Penney model we obtain E F ϪE 0 ϭ Ϫ0.48 eV, i.e. closer to the energy of the Au͑111͒ surface state. 16

III. MAPPING THE WAVE FUNCTION AT LATERAL QUANTUM WELLS
The size of the data points in Fig. 3͑b͒ reflects the normalized photoemission intensity ͑area under the peak͒ of the 1D quantum-well levels in Au͑788͒.Such intensity is shown in Fig. 4͑a͒ as a function of the emission angle.It thus represents the experimental angle-resolved photoemission matrix element in the direction perpendicular to the steps for the two terrace quantum levels.We can estimate such matrix element using Fermi's golden rule for optical transitions,

͑2͒
Here A and p, respectively, correspond to the light vector potential and the momentum operators, and i and f to the electron wave functions in the initial ͑quantum-well͒ and the final states, respectively.At a terrace we can approach the initial state wave function as the product of a wave decaying into the bulk, a plane wave parallel to the terrace, and the wave function in the xЈ direction, i.e., i (xЈ,yЈ,zЈ) ϭe ϪzЈ e ik yЈ .yЈk xЈ (xЈ).The (xЈ,yЈ,zЈ) reference system is defined in the ͑111͒ terrace plane, as depicted in the inset of Fig. 4͑a͒.The matrix element can be simplified assuming the final-state wave function as a plane wave f ϭe iq•r .With p polarization AӍA z k ˆand making q y Ј ϭ0, Eq. ͑2͒ is reduced to At the photon energy used in the experiment and close to the surface normal q z Ј (q z Ј ӷq x Ј ,q y Ј ) is fixed by energy conservation.The first term in Eq. ͑3͒ is thus constant, and the matrix element is only a function of the second term, which is written as where q x Ј is the final-state wave vector along the xЈ direction that is fixed by the emission angle.Equation ͑4͒ is describing the 1D wave function probability density ͉(q x Ј )͉ 2 in reciprocal space, which is the analog of the charge density mapped in real space by STM. 5,6By chosing a plane wave for the final state we neglect the effect of the step superlattice, which results in a smooth modulation of Eq. ͑4͒.In our analysis we avoid this modulation by chosing a photon energy such that the interference effect of the superlattice is canceled. 17Within the limits of the approach made to the matrix element, we can conclude that the probability density in reciprocal space for Nϭ1 and Nϭ2 in the terrace quantum well is being obtained experimentally in Fig. 4͑a͒.It is thus interesting to compare this result with the infinite, one-FIG.4. ͑a͒ Photoemission intensity for the two quantum-well levels of Fig. 1͑b͒.Such intensity reflects the probability density of the one-dimensional terrace quantum-well states in Au͑788͒.It fits to the probability density for the corresponding quantum-well states of the 1D infinite potential well ͑dashed lines͒ calculated in the xЈyЈzЈ reference system of the terraces ͑inset͒, indicating that both share similar wave functions.͑b͒ Wave functions in real space for the terrace quantum well obtained from ͑a͒, compared with the infinite quantum-well wave functions.dimensional quantum well of size L. In this case we have the well-known wave functions N (xЈ)ϭsin(NxЈ/L) that allow us to simplify Eq. ͑4͒ to give where A is a normalization constant that can be chosen to make the total intensity ͐I QW ϱ (q x Ј )dxЈ equal to the area under the experimental intensity curve for Nϭ1.With this value of A we obtain the dotted lines shown in Fig. 4͑a͒ for Nϭ1 and Nϭ2 of the infinite quantum well.Peak positions are well reproduced, indicating that the terraces effectively behave as infinite potential wells with similar wave functions.Actually, from Eq. ͑4͒ we deduce that the wave function in real space can be obtained experimentally from I(q x Ј ) by Fourier transformation.The missing phase is determined by supposing a symmetric well, and therefore only even or odd wave functions.Using the data in Fig. 4͑a͒ we obtained the wave functions in real space for the terrace quantum well.The result is shown in Fig. 4͑b͒, together with the wave functions of the infinite quantum well.We observe small differences that can be due to the actual shape of the terrace potential.However, the differences lie within the error bars of the experiment and the approach used to simplify the matrix element in Eq. ͑2͒.A more refined, quantitative analysis requires a better theoretical approach of this matrix element, especially in order to account for photoemission final-state effects.This analysis is in progress. 17

IV. THREE-DIMENSIONAL FOURIER ANALYSIS BY PHOTON ENERGY TUNING
The center of the probability density functions in Fig. 4͑a͒ is shifted by 3.5°from the surface normal.This angle corresponds to the direction perpendicular to the ͑111͒ plane, proving that this plane is the appropriate reference of the 1D terrace potential in Au͑788͒.In contrast the 2D surface state in Au͑322͒ is modulated on the ͑322͒ average plane of the surface.This can be shown by probing the Fourier components of the surface wave function in the perpendicular direction, which in turn can only be done by tuning the photon energy in photoemission. 3 The analysis is presented in Figs. 5 and 6.In Fig. 5 we show the spectra from Au͑322͒ along the x direction for increasing photon energies.The photoemission intensity is now shown in a gray scale, such that surface bands are directly observed.The most remarkable feature is the intensity shift from the first to the second superlattice Brillouin zone as the photon energy increases.The situation resembles that of the electron diffraction from a stepped surface, where the split spots are only observed under out-of-phase interference conditions. 18In the present case the relative intensity of the zone-folded bands is explained by the spectral composition of the surface state perpendicular to the surface (z direction͒.This is demonstrated in the wave vector plot shown in Fig. 6.To simplify the discussion, we limit ourselves to the lowest electron energy, i.e. the surface band minima in Fig. 5.At the band minima we measure the photoemission intensity normalized to the photon flux ͑size of the dots͒ and calculate the wave vector component perpendicular to the surface for the excited photoelectron out of the crystal (k z out ).The latter is obtained from the electron kinetic energy E kin and k x by assuming constant energy lines (k y ϭ0) Therefore, data points in Fig. 6͑a͒ correspond to the photoemission final state outside the crystal.In the initial state ͑surface state͒ the wave vector components parallel to the surface k x and k y are the same as in the final state, since both are conserved in the photoemission process.k z is not strictly conserved but broadened due to the lack of symmetry of the crystal in the z direction.In order to find the approximate value of k z it is necessary to use bulk band calculations to find, first, the corresponding final state inside the crystal at the same energy, and then assume wave vector conservation in the reduced Brillouin zone.In Fig. 6 we have located the high-symmetry L point of the bulk Brillouin zone for the initial state, which is probed at ϳ26 eV and ϳ50 eV above E F . 19The photoemission intensity ͑size of the data points͒ is maximum close to this L point and decreases slowly away from it in the perpendicular direction.This k z -dependent intensity is similar to that found in flat crystals and reflects the spectral composition of the wave function in the surface state along the direction perpendicular to the surface. 20Such k z broadening around L, which is actually the consequence of the two-dimensional confinement in the surface plane, is depicted schematically in Fig. 6͑a͒ with the dashed cones.Thus Fig. 6͑a͒ provides a ͑qualitative͒ three-dimensional description of the superlattice wave function in the reciprocal space where all the Fourier components (k x ,k y ϭ0,k z ) are being probed and weighted. 21The corresponding wave function in the real space is depicted in Fig. 6͑b͒.Along the x direction, it is composed of Bloch waves with the periodicity of the superlattice.Along the z direction, it is an evanescent oscillation with k z ϭk L , consistent with the k z broadening away from the L point.

V. BULK BAND STRUCTURE PROJECTION ON THE VICINAL PLANE
In order to understand the different nature of the surface state in Au͑788͒ and Au͑322͒ we examine the bulk band structure projection on each surface plane and along the di-rection perpendicular to the steps.This is shown schematically in Fig. 7.The vicinal surface plane is represented by a thick solid line rotated by the corresponding miscut angle about the ͑111͒ plane.The circles describe the bulk Fermi surface in the first Brillouin zone with its characteristic necks of width k neck ϭ0.24 Å Ϫ1 . 12  The different degree of coupling between projected bulk states and ⌫ ¯and /d surface states explains their different dimensionality.Gap states are sharply confined to the surface plane and they feel a strong step barrier potential.Mixing of /d states with bulk states displaces the center of gravity of the electron wave function deeper in the bulk, and hence /d states sense a smoother effective potential at the step. 22For a low effective barrier the electron tunneling across the step can lead to significant terrace to terrace coupling, and hence to 2D superlattice band formation.The stronger surface-bulk mixing of /d surface resonances is also supported by the recent line shape analysis done by Baumberger et al. in Cu͑111͒ vicinals. 8igure 7 suggests a transition from ⌫ ¯states to /d states as a function of the miscut angle.Such transition has been indeed observed in Cu͑111͒ vicinals with about 7°miscut, i.e., when the ⌫ ¯gap gets closed at the energy range of the surface state. 3,23As deduced from Fig. 7͑b͒, the gap at E F closes in Au͑111͒ vicinals with ␣ c ϭarctan(k neck /k L )ϭ10.2°m iscut.Thus for larger angles, as in Au͑322͒, surface states at ⌫ ¯are not longer supported.However, the data for Cu͑111͒ vicinals indicate that significant tunneling across the step barrier occurs before the spectral weight is transferred from ⌫ ¯to /d.Indeed in Cu͑111͒ with 5°miscut the surface state displays ⌫ ¯-like character ͓k z Ј broadening along ͑111͒ ter- races͔, though it is already a 2D dispersing band. 3This appears to be again a consequence of the surface-bulk mixing that occurs during the ⌫ ¯gap closing.For 5°the ⌫ ¯gap in Fig. 7͑b͒ has shrunk, such that part of the ⌫ ¯state Fourier components overlap with bulk states. 23As a result, the ⌫ ¯state senses a lower effective step barrier and eventually becomes a 2D surface state.

VI. CONCLUSIONS
In summary, we observe that the p z -like surface state can have one-dimensional or two-dimensional character at vicinal Au͑111͒ surfaces depending on the terrace size.It is completely confined within wide terraces in Au͑788͒ and becomes a two-dimensional, step superlattice state at narrow terraces in Au͑322͒.Both surfaces represent the two limiting cases of a lateral nanostructure.This can be generally defined as a regular array of nano-sized objects on a surface.In Au͑788͒ we have the case of individual, non interacting objects ͑terraces͒ that display quantum-well levels.In Au͑322͒ effective barriers are lower and the object-to-object interaction is switched on, thereby leading to 2D superlattice bands.By means of angle-resolved photoemission and synchrotron radiation we can obtain, in both cases, the Fourier spectrum of the relevant electronic states, which is in turn necessary to properly describe the wave functions of the lateral nanostructure.

FIG. 1 .
FIG. 1. STM pictures from ͑a͒ Au͑322͒ and ͑b͒ Au͑788͒ showing highly regular one-dimensional step arrays.The right inset shows the detailed structure of Au͑788͒ terraces.Bright and dark areas indicate the presence of alternating fcc-and hcp-packed square areas within the same terrace ͑Ref.9͒.The terrace width distribution for each surface is shown in the bottom panels.The top inset schematically depicts the side view of the one-dimensional step array, where miscut angle, step height, and terrace width appear defined.

FIG. 5 .
FIG. 5. Photoemission spectra of Au͑322͒ measured at different photon energies are shown in a gray scale.The 2D, zone-folded surface bands along k x are directly obtained.The photon energy dependent intensity of each band is related to the spectral weight of the different surface state Fourier components, which are selectively probed at every energy ͑see Fig. 6͒.
Dark shaded regions indicate the projection of bulk states from the two halves of the Fermi surface, whereas light shaded areas correspond to bulk states projected from one side of the Fermi surface.Only in the case of Au͑788͒ the projection of the two necks overlaps, leading to a gap around ⌫ ¯.The dashed cones in Fig. 7͑a͒ represent the same Fourier spectrum of the 2D surface state in Au͑322͒ probed in Fig. 6͑a͒, which is characterized by the k z broadening along the surface normal.In Fig. 7͑b͒ the dashed cones indicate that 1D terrace quantum-well states in Au͑788͒ have k z Ј broadening along the ͑111͒ terrace normal.This is concluded from the probability density functions shown in Fig.4͑a͒, which are centered around the ͑111͒ direction.Note that all the Fourier components of the 2D surface state in Au͑322͒ lie at the L-point projection on the surface, i.e., at /d.In contrast, the Fourier components of the 1D surface states of Au͑788͒ with k z Ј broadening along the ͑111͒ direction lie at the ⌫ ¯gap.This means that these 1D terrace quantum-well states are truly gap states, whereas 2D surface states of Au͑322͒ are resonances that overlap with projected bulk states.

FIG. 6 .
FIG.6.͑a͒ k z out and k x wave vector components of photoelectrons out of the crystal.The data correspond to the band minima in Fig.5.The size of the dots is proportional to the peak intensity.This is maximum near the L point of the bulk band structure which is probed at two different energies in the first and the second Brillouin zones.The plot reflects the spectral composition of the surface state wave function, as qualitatively described by the dashed cone.The corresponding wave function is described in real space in ͑b͒.Along the x direction, we have Bloch waves of the step superlattice.Along the z direction, the superlattice wave function is an evanescent wave with k z ϭk L , consistent with the k z broadening shown in ͑a͒.