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Probing wave functions at step superlattices: confined versus propagating electrons
Materials Science and Engineering B96 (2002) 154 /158 www.elsevier.com/locate/mseb
Probing wave functions at step superlattices: confined versus propagating electrons J.E. Ortega a,*, A. Mugarza a, V. Pe´rez-Dieste b, V. Repain c, S. Rousset c, A. Mascaraque b a
Departamento de Fı´sica Aplicada I, Donostia International Physics Center (DIPC) and Centro Mixto CSIC/UPV, Universidad del Paı´s Vasco, Plaza On˜ati 2, E-20018 San Sebastian, Spain b LURE Centre Universitaire Paris-Sud, Baˆt 209D B.P. 34, 91898 Orsay Cedex, France c Groupe de Physique des Solides, CNRS, Universite´s Paris 6 et 7, 2 Place Jussieu, 75251 Paris Cedex 5, France
Abstract Electron wave functions at lateral nanostructures can be readily explored using photoemission with tunable synchrotron radiation. Such systems are ideal for applications in nanotechnology because they are easily accessed by read /write devices that scan across a surface. As model system here we use vicinal Au(111) surfaces, with an array of terraces separated by straight, monatomic steps, which exhibit atomically accurate periodicity due to an intrinsic surface reconstruction. We analyze two extreme cases where electrons are either localized into lateral quantum-well states or propagating across the step superlattice. We are able to probe the electron probability density at the lateral quantum-wells as well as the Fourier spectrum of the two-dimensional (2D) superlattice states. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Probing wave functions; Step superlattices; Confined versus propagating electrons; Stepped surfaces
The development of new devices based on nanostructured solids relies on the ability to tailor structural and electronic properties [1 /3]. As shown for vertical nanostructures [1], photoemission with synchrotron radiation is a powerful tool to investigate electronic states and probe their wave functions. Here, we use photoemission to study electronic states in lateral nanostructures. These have recently attracted broad interest either for fundamental reasons, since they are ideal to test the exotic electronic properties predicted in low dimensional solids [2] or for practical purposes, due to their potential use in optoelectronic devices [3]. Our model lateral nanostructure is a vicinal surface, i.e. a staircase of atomically flat, nanometer-wide terraces separated by atomic-height steps at which electrons scatter. The highly regular array required for photoemission can be naturally achieved at two vicinal Au(111) surfaces, namely Au(788) and Au(322). Unlike the other Au(111) vicinals [4], these are stable orienta-
* Corresponding author. Tel.: /34-943-018-403; fax: /34-943-219727 E-mail address: [email protected] (J.E. Ortega).
tions that display extraordinarily homogeneous step arrays. Fig. 1 shows the respective scanning tunneling microscopy (STM) images. Regular arrays of straight, ˚ ) steps characterize micron-size monoatomic (h /2.36 A surface areas. The side view of the vicinal surface is schematized in the inset. Since h is fixed, the macroscopic miscut angle a determines the terrace width L and the superlattice periodicity d . From the STM we obtain the average values of the superlattice period d / ˚ for Au(322) and d /38.7 A ˚ for Au(788). 12.8 A Flat noble metal (111) surfaces are characterized by a free-electron-like surface band that produces and intense photoemission peak [5]. At stepped Au(322) and Au(788) the free-electron-like character is maintained parallel to the steps (Fig. 1, y-direction). The band dispersion is only affected in the direction perpendicular to the step array (x-direction). The photoemission spectra and the resulting E (kx ) bands are shown in Fig. 2. The data have been taken at the S8 undulator beam line of the synchrotron LURE in Paris with photon energy hn/27 eV, angular resolution lower than 0.58, and total energy resolution of 50 meV. All the energies in Fig. 2 are referred to the Fermi level (EF).
0921-5107/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 0 2 ) 0 0 3 0 8 - 2
J.E. Ortega et al. / Materials Science and Engineering B96 (2002) 154 /158
155
Fig. 1. Step superlattices in Au(111) vicinal surfaces. The STM images display highly regular step arrays that extend over micron-size areas, thereby ˚ for Au(322) (left) and d/38.7 A ˚ for Au(788) (right). defining coherent lateral step superlattices of period d/12.8 A
Although both Au(788) and Au(322) only differ in terrace size, the electronic structure is completely different. The surface state is dispersing in Au(322), but appears broken up into two non-dispersing levels for Au(788). This means that the surface state is a twodimensional (2D) state for Au(322) and a 1D state for Au(788). In Au(322), the scattering at the steps is reflected in the topology of the surface band, which is folded along the kx direction with superlattice wave vectors g /2p /d . ˚ 1 The parabolas have their centers at kx /0.239/0.04 A 1 ˚ , consistent with g /2 /p /d / and kx /0.749/0.04 A ˚ 1 and 3g /2 /3p/d/ 0.74 A ˚ 1, respectively. The 0.25 A strength of the step potential can be estimated by assuming delta-function-like potential barriers at the steps in a 1D Kronig/Penney model [6], and then comparing the band minimum in Fig. 2(a) with that of the flat surface. The Au(322) band in the first zone displays an upwards shift of /0.1 eV with respect to the Au(111) surface state, which has been measured in a different crystal under the same experimental conditions. From this shift we deduce a small repulsive step ˚ 1, similar to the step barriers found barrier of 1.4 eV A for 2D surface states of Cu(111) vicinals [6]. In contrast, the flat levels of Fig. 1(b) indicate that in Au(788) the steps act as infinite potential barriers, such that the electrons are confined within 1D terraces. Indeed, the two discrete levels at /0.40 and /0.11 eV are consistent with the N /1 and 2 energy levels of the infinite ˚ , with EN /E0/ quantum-well of width L /38.7 A (’2p2/2m *L2)N2, E0 //0.5 eV and m * /0.27 me. Such 1D behavior agrees with STM observations at singled terraces in Au(111) and Ag(111) [7,8]. The changing nature of the surface state from Au(788) to Au(322) can be explained in the light of the bulk band structure projection on each vicinal surface, which leads to a different degree of coupling between surface and bulk states [9]. The key feature of
the band structure projection concerns the neck of the Fermi surface. The projection of the neck results in a gap at G¯ in flat Au(111) that shrinks in vicinal surfaces as the miscut angle increases. At small miscuts this G¯ gap supports surface states with terrace-like modulation [9], like 1D states of Au(788). At large miscuts, the gap has vanished, and surface states of Au(322) overlap with projected bulk states becoming surface resonances. Gap states are pure surface states, i.e. they are expected to be physically close to the surface plane. This makes surface electrons in Au(788) very sensitive to the step potential, leading to complete confinement within terraces. By becoming a resonance, the center of gravity of the surface state moves deeper in the bulk and hence the effective step potential felt by the electron is reduced. In this case, the surface electron can tunnel across the step array and form a 2D band, as in Au(322). 1D terrace quantum-well wave functions of Au(788) can be mapped in reciprocal space using angle-resolved photoemission [10], in a similar way as done in real space with STM [7,8]. This is shown in Fig. 3. Data points represent the angle-resolved photoemission intensity for the N /1 and 2 levels of Fig. 2(b). Such intensity can be theoretically calculated using Fermi’s golden rule for optical transitions: I 8jhci jApjcf ij2
(1)
Here, A and p, respectively, correspond to the light vector potential and the momentum operators, and ci and cf to the electron wave functions at the initial (quantum-well) and the final states. Assuming plane electron waves for cf and within our particular experimental conditions (light polarization, photon energy, and small emission angles) the matrix element of Eq. (1) is reduced to [9,10]: I8
jg
2
eiqx? x? ci (x?) dx?
j
(2)
156
J.E. Ortega et al. / Materials Science and Engineering B96 (2002) 154 /158
Fig. 2. Superlattice states vs. lateral quantum-well states in vicinal Au(111). Top, photoemission spectra from (a) Au(322) and (b) Au(788) and the corresponding E (kx ) bands perpendicular to the step array (x -direction in Fig. 1). The emission angle (u ) is measured with respect to the surface normal. At Au(322), the surface state forms superlattice bands (zone-folded by g/2p /d ), whereas at Au(788) it breaks up into 1D quantum-well levels.
Eq. (2) actually represents the probability density jci(qx ?)j2 in reciprocal space for the 1D terrace quantumwell, which is being mapped in Fig. 3(a). As shown in the inset in Fig. 3(a), x ? refers to the direction parallel to the terraces, i.e. the appropriate coordinate system in this case [11]. Using Eq. (2) we can calculate the
probability density for the infinite quantum-well of ˚ , which is included in Fig. 3(a). width L /38.7 A Furthermore, we can directly obtain the probability density in real space by Fourier transforming these curves, which can be in turn compared with the probability measured by STM [7,8]. This Fourier trans-
J.E. Ortega et al. / Materials Science and Engineering B96 (2002) 154 /158
157
Fig. 3. Probing wave functions of lateral quantum wells with angleresolved photoemission. (a) Reciprocal space mapping, the photoemission intensity reflects the probability density jc (q )j2 of the 1D quantum well wave function at terraces in Au(788). It fits to the probability density for the corresponding quantum well states of the 1D infinite potential well (dashed lines and shaded areas) calculated in the x ?y ?z ?-reference system of the terraces (right inset). (b) Fourier transform of the experimental curves shown in (a), which represents the probability density in real space for the lateral quantum well. The result is compared with the infinite quantum well of the same width. The agreement indicates that both infinite and terrace potential wells share similar wave functions.
form is shown in Fig. 3(b) for both levels, and again compared with the case of the infinite quantum well of the same width. The agreement indicates that both the terrace and the infinite quantum-well have similar wave functions. We can further explore the electron wave function properties at step superlattices by tuning the photon energy. This is shown for 2D superlattice states of Au(322) in Fig. 4. On the left, the photoemission spectra are displayed in a gray scale for increasing photon energies, such that superlattice bands are directly observable. The most remarkable feature is the intensity shift from the first to the second superlattice Brillouin zones as the photon energy increases. The situation resembles that of electron diffraction from a stepped surface, where split spots are only observed under outof-phase interference conditions [12]. In the present case, the relative intensity of the zone-folded (split) bands is explained by the spectral composition of the surface state perpendicular to the surface (z -direction). This is demonstrated in the wave vector plot shown in Fig. 4(b). To simplify the discussion we limit ourselves to the lowest electron energy, i.e. the surface band minima in Fig. 4(a). At the minima, we can determine the wave vector component perpendicular to the surface (kz ) from
Fig. 4. Fourier analysis of the 2D superlattice wave function by photon-energy scan. (a) 2D bands of Au(322) measured at different photon energies shown in a gray scale. (b) Corresponding (kz , kx ) plot for the band minima. The size of the dots is proportional to the peak intensity, which in turn is maximum near the L -point of the bulk band structure. This gives the spectral composition of the surface state wave function perpendicular to the surface, which is described in real space in (c). The evanescent Bloch wave with kz /kL along the z -direction is consistent with the kz -broadening shown in (b).
kx and the kinetic energy of the electron (E ) by following constant energy lines: E (’2 =2m)(kx2 kz2 )
(3)
Thus data points in Fig. 4(b) represent the so-called photoemission final state outside the crystal [13]. Using bulk band calculations, we can approximately find the corresponding final states inside the crystal and locate the bulk states that are being probed. In Fig. 4(b) this is done for the high-symmetry L-point of the bulk Brillouin zone. Note that the photoemission intensity (size of the data points) is maximum close to this Lpoint and decreases slowly away from it. As shown in flat crystal [14], such kz -dependent intensity reflects the spectral composition of the wave function in the initial
158
J.E. Ortega et al. / Materials Science and Engineering B96 (2002) 154 /158
(surface) state in the direction perpendicular to the surface. Thus Fig. 4(b) represents the 3D (xyz ) description of the superlattice wave function in the reciprocal space. The corresponding wave function in the real space is shown in Fig. 4(c). 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 kz /kL /(â3/2)(2p/a ), consistent with the kz -broadening away from the L -point. In summary, we observe that Au(788) and Au(322) represent two limiting cases in a lateral nanostructure, which can be generally defined as a regular array of nano-sized objects on a surface. In Au(788) we have the case of individual, non-interacting objects (terraces) that display quantum-well levels. In Au(322) the object-toobject 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 complete Fourier spectrum of the relevant electronic states, which is in turn necessary to properly describe the wave functions of the lateral nanostructures.
Acknowledgements J.E. Ortega and A. Mugarza are supported by the Universidad del Paı´s Vasco (1/UPV/EHU/ 00057.240EA-8078/2000) and the Max Planck Research Award Program. V. Repain and S. Rousset are supported by the CNRS-ULTIMATECH program, the CRIF and the Universite´ de Paris 7. A. Mascaraque is supported by a Marie Curie Fellowship of the European Union under contract No. HPMF-CT-2000-00565. V. Pe´rez-Dieste is supported by the Comunidad Auto´noma de Madrid
(Project No. 07N/0042/98) and the Spanish DGICYT (Grant No. PB-97-1199). The experiments performed at LURE were funded by the Large Scale Facilities program of the European Union. Technical support from the Spanish /French beam line staff is gratefully acknowledged.
References [1] R.K. Kawakami, E. Rotenberg, H.J. Choi, E.J. Escorcia-Aparicio, M.O. Bowen, J.H. Wolfe, E. Arenholz, Z.D. Zhang, N.V. Smith, Z.Q. Qiu, Nature 398 (1999) 132. [2] P. Segovia, D. Purdie, M. Hengsberger, Y. Baer, Nature 402 (1999) 504. [3] R. No¨tzel, Z. Niu, M. Ramsteimer, H.P. Scho¨nherr, A. Trampert, L. Da¨weritz, K.H. Ploog, Nature 392 (1998) 56. [4] V. Repain, J.M. Berroir, B. Croset, S. Rousset, Y. Garreau, V.H. Etgens, L. Lecoeur, Phys. Rev. Lett. 84 (2000) 5367. [5] F. Reinert, G. Nicolay, S. Schmidt, D. Ehm, F. Hu¨fner, Phys. Rev. B63 (2001) 115415. [6] O. Sa´nchez, J.M. Garcı´a, P. Segovia, J. Alvarez, A.L. Va´zquez de Parga, J.E. Ortega, M. Prietsch, R. Miranda, Phys. Rev. B52 (1995) 7894. [7] P. Avouris, I.-W. Lyo, Science 264 (1994) 942. [8] L. Bu¨rgi, O. Jeandupeux, A. Hirstein, H. Brune, K. Kern, Phys. Rev. Lett. 81 (1998) 5370 /5373. [9] J.E. Ortega, A. Mugarza, V. Repain, S. Rousset, V. Pe´rez-Dieste, A. Mascaraque, Phys. Rev. B, 65 (2002) 165413. [10] A. Mugarza, A. Mascaraque, V. Pe´rez-Dieste, V. Repain, S. Rousset, F.J. Garcı´a de Abajo, J.E. Ortega, Phys. Rev. Lett. 87 (2001) 107601. [11] In agreement with this assumption, note that the center of the probability density functions are shifted by 3.58 from the surface normal, i.e. along the (111) direction. [12] M. Henzler, Appl. Phys. A9 (1976) 11. [13] F.J. Himpsel, Adv. Phys. 32 (1983) 1. [14] S.G. Louie, P. Thiry, R. Pincahux, Y. Petroff, D. Chandesris, J. Lecante, Phys. Rev. Lett. 44 (1980) 549 /553.
Probing wave functions at step superlattices: confined versus propagating electrons J.E. Ortega a,*, A. Mugarza a, V. Pe´rez-Dieste b, V. Repain c, S. Rousset c, A. Mascaraque b a
Departamento de Fı´sica Aplicada I, Donostia International Physics Center (DIPC) and Centro Mixto CSIC/UPV, Universidad del Paı´s Vasco, Plaza On˜ati 2, E-20018 San Sebastian, Spain b LURE Centre Universitaire Paris-Sud, Baˆt 209D B.P. 34, 91898 Orsay Cedex, France c Groupe de Physique des Solides, CNRS, Universite´s Paris 6 et 7, 2 Place Jussieu, 75251 Paris Cedex 5, France
Abstract Electron wave functions at lateral nanostructures can be readily explored using photoemission with tunable synchrotron radiation. Such systems are ideal for applications in nanotechnology because they are easily accessed by read /write devices that scan across a surface. As model system here we use vicinal Au(111) surfaces, with an array of terraces separated by straight, monatomic steps, which exhibit atomically accurate periodicity due to an intrinsic surface reconstruction. We analyze two extreme cases where electrons are either localized into lateral quantum-well states or propagating across the step superlattice. We are able to probe the electron probability density at the lateral quantum-wells as well as the Fourier spectrum of the two-dimensional (2D) superlattice states. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Probing wave functions; Step superlattices; Confined versus propagating electrons; Stepped surfaces
The development of new devices based on nanostructured solids relies on the ability to tailor structural and electronic properties [1 /3]. As shown for vertical nanostructures [1], photoemission with synchrotron radiation is a powerful tool to investigate electronic states and probe their wave functions. Here, we use photoemission to study electronic states in lateral nanostructures. These have recently attracted broad interest either for fundamental reasons, since they are ideal to test the exotic electronic properties predicted in low dimensional solids [2] or for practical purposes, due to their potential use in optoelectronic devices [3]. Our model lateral nanostructure is a vicinal surface, i.e. a staircase of atomically flat, nanometer-wide terraces separated by atomic-height steps at which electrons scatter. The highly regular array required for photoemission can be naturally achieved at two vicinal Au(111) surfaces, namely Au(788) and Au(322). Unlike the other Au(111) vicinals [4], these are stable orienta-
* Corresponding author. Tel.: /34-943-018-403; fax: /34-943-219727 E-mail address: [email protected] (J.E. Ortega).
tions that display extraordinarily homogeneous step arrays. Fig. 1 shows the respective scanning tunneling microscopy (STM) images. Regular arrays of straight, ˚ ) steps characterize micron-size monoatomic (h /2.36 A surface areas. The side view of the vicinal surface is schematized in the inset. Since h is fixed, the macroscopic miscut angle a determines the terrace width L and the superlattice periodicity d . From the STM we obtain the average values of the superlattice period d / ˚ for Au(322) and d /38.7 A ˚ for Au(788). 12.8 A Flat noble metal (111) surfaces are characterized by a free-electron-like surface band that produces and intense photoemission peak [5]. At stepped Au(322) and Au(788) the free-electron-like character is maintained parallel to the steps (Fig. 1, y-direction). The band dispersion is only affected in the direction perpendicular to the step array (x-direction). The photoemission spectra and the resulting E (kx ) bands are shown in Fig. 2. The data have been taken at the S8 undulator beam line of the synchrotron LURE in Paris with photon energy hn/27 eV, angular resolution lower than 0.58, and total energy resolution of 50 meV. All the energies in Fig. 2 are referred to the Fermi level (EF).
0921-5107/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 0 2 ) 0 0 3 0 8 - 2
J.E. Ortega et al. / Materials Science and Engineering B96 (2002) 154 /158
155
Fig. 1. Step superlattices in Au(111) vicinal surfaces. The STM images display highly regular step arrays that extend over micron-size areas, thereby ˚ for Au(322) (left) and d/38.7 A ˚ for Au(788) (right). defining coherent lateral step superlattices of period d/12.8 A
Although both Au(788) and Au(322) only differ in terrace size, the electronic structure is completely different. The surface state is dispersing in Au(322), but appears broken up into two non-dispersing levels for Au(788). This means that the surface state is a twodimensional (2D) state for Au(322) and a 1D state for Au(788). In Au(322), the scattering at the steps is reflected in the topology of the surface band, which is folded along the kx direction with superlattice wave vectors g /2p /d . ˚ 1 The parabolas have their centers at kx /0.239/0.04 A 1 ˚ , consistent with g /2 /p /d / and kx /0.749/0.04 A ˚ 1 and 3g /2 /3p/d/ 0.74 A ˚ 1, respectively. The 0.25 A strength of the step potential can be estimated by assuming delta-function-like potential barriers at the steps in a 1D Kronig/Penney model [6], and then comparing the band minimum in Fig. 2(a) with that of the flat surface. The Au(322) band in the first zone displays an upwards shift of /0.1 eV with respect to the Au(111) surface state, which has been measured in a different crystal under the same experimental conditions. From this shift we deduce a small repulsive step ˚ 1, similar to the step barriers found barrier of 1.4 eV A for 2D surface states of Cu(111) vicinals [6]. In contrast, the flat levels of Fig. 1(b) indicate that in Au(788) the steps act as infinite potential barriers, such that the electrons are confined within 1D terraces. Indeed, the two discrete levels at /0.40 and /0.11 eV are consistent with the N /1 and 2 energy levels of the infinite ˚ , with EN /E0/ quantum-well of width L /38.7 A (’2p2/2m *L2)N2, E0 //0.5 eV and m * /0.27 me. Such 1D behavior agrees with STM observations at singled terraces in Au(111) and Ag(111) [7,8]. The changing nature of the surface state from Au(788) to Au(322) can be explained in the light of the bulk band structure projection on each vicinal surface, which leads to a different degree of coupling between surface and bulk states [9]. The key feature of
the band structure projection concerns the neck of the Fermi surface. The projection of the neck results in a gap at G¯ in flat Au(111) that shrinks in vicinal surfaces as the miscut angle increases. At small miscuts this G¯ gap supports surface states with terrace-like modulation [9], like 1D states of Au(788). At large miscuts, the gap has vanished, and surface states of Au(322) overlap with projected bulk states becoming surface resonances. Gap states are pure surface states, i.e. they are expected to be physically close to the surface plane. This makes surface electrons in Au(788) very sensitive to the step potential, leading to complete confinement within terraces. By becoming a resonance, the center of gravity of the surface state moves deeper in the bulk and hence the effective step potential felt by the electron is reduced. In this case, the surface electron can tunnel across the step array and form a 2D band, as in Au(322). 1D terrace quantum-well wave functions of Au(788) can be mapped in reciprocal space using angle-resolved photoemission [10], in a similar way as done in real space with STM [7,8]. This is shown in Fig. 3. Data points represent the angle-resolved photoemission intensity for the N /1 and 2 levels of Fig. 2(b). Such intensity can be theoretically calculated using Fermi’s golden rule for optical transitions: I 8jhci jApjcf ij2
(1)
Here, A and p, respectively, correspond to the light vector potential and the momentum operators, and ci and cf to the electron wave functions at the initial (quantum-well) and the final states. Assuming plane electron waves for cf and within our particular experimental conditions (light polarization, photon energy, and small emission angles) the matrix element of Eq. (1) is reduced to [9,10]: I8
jg
2
eiqx? x? ci (x?) dx?
j
(2)
156
J.E. Ortega et al. / Materials Science and Engineering B96 (2002) 154 /158
Fig. 2. Superlattice states vs. lateral quantum-well states in vicinal Au(111). Top, photoemission spectra from (a) Au(322) and (b) Au(788) and the corresponding E (kx ) bands perpendicular to the step array (x -direction in Fig. 1). The emission angle (u ) is measured with respect to the surface normal. At Au(322), the surface state forms superlattice bands (zone-folded by g/2p /d ), whereas at Au(788) it breaks up into 1D quantum-well levels.
Eq. (2) actually represents the probability density jci(qx ?)j2 in reciprocal space for the 1D terrace quantumwell, which is being mapped in Fig. 3(a). As shown in the inset in Fig. 3(a), x ? refers to the direction parallel to the terraces, i.e. the appropriate coordinate system in this case [11]. Using Eq. (2) we can calculate the
probability density for the infinite quantum-well of ˚ , which is included in Fig. 3(a). width L /38.7 A Furthermore, we can directly obtain the probability density in real space by Fourier transforming these curves, which can be in turn compared with the probability measured by STM [7,8]. This Fourier trans-
J.E. Ortega et al. / Materials Science and Engineering B96 (2002) 154 /158
157
Fig. 3. Probing wave functions of lateral quantum wells with angleresolved photoemission. (a) Reciprocal space mapping, the photoemission intensity reflects the probability density jc (q )j2 of the 1D quantum well wave function at terraces in Au(788). It fits to the probability density for the corresponding quantum well states of the 1D infinite potential well (dashed lines and shaded areas) calculated in the x ?y ?z ?-reference system of the terraces (right inset). (b) Fourier transform of the experimental curves shown in (a), which represents the probability density in real space for the lateral quantum well. The result is compared with the infinite quantum well of the same width. The agreement indicates that both infinite and terrace potential wells share similar wave functions.
form is shown in Fig. 3(b) for both levels, and again compared with the case of the infinite quantum well of the same width. The agreement indicates that both the terrace and the infinite quantum-well have similar wave functions. We can further explore the electron wave function properties at step superlattices by tuning the photon energy. This is shown for 2D superlattice states of Au(322) in Fig. 4. On the left, the photoemission spectra are displayed in a gray scale for increasing photon energies, such that superlattice bands are directly observable. The most remarkable feature is the intensity shift from the first to the second superlattice Brillouin zones as the photon energy increases. The situation resembles that of electron diffraction from a stepped surface, where split spots are only observed under outof-phase interference conditions [12]. In the present case, the relative intensity of the zone-folded (split) bands is explained by the spectral composition of the surface state perpendicular to the surface (z -direction). This is demonstrated in the wave vector plot shown in Fig. 4(b). To simplify the discussion we limit ourselves to the lowest electron energy, i.e. the surface band minima in Fig. 4(a). At the minima, we can determine the wave vector component perpendicular to the surface (kz ) from
Fig. 4. Fourier analysis of the 2D superlattice wave function by photon-energy scan. (a) 2D bands of Au(322) measured at different photon energies shown in a gray scale. (b) Corresponding (kz , kx ) plot for the band minima. The size of the dots is proportional to the peak intensity, which in turn is maximum near the L -point of the bulk band structure. This gives the spectral composition of the surface state wave function perpendicular to the surface, which is described in real space in (c). The evanescent Bloch wave with kz /kL along the z -direction is consistent with the kz -broadening shown in (b).
kx and the kinetic energy of the electron (E ) by following constant energy lines: E (’2 =2m)(kx2 kz2 )
(3)
Thus data points in Fig. 4(b) represent the so-called photoemission final state outside the crystal [13]. Using bulk band calculations, we can approximately find the corresponding final states inside the crystal and locate the bulk states that are being probed. In Fig. 4(b) this is done for the high-symmetry L-point of the bulk Brillouin zone. Note that the photoemission intensity (size of the data points) is maximum close to this Lpoint and decreases slowly away from it. As shown in flat crystal [14], such kz -dependent intensity reflects the spectral composition of the wave function in the initial
158
J.E. Ortega et al. / Materials Science and Engineering B96 (2002) 154 /158
(surface) state in the direction perpendicular to the surface. Thus Fig. 4(b) represents the 3D (xyz ) description of the superlattice wave function in the reciprocal space. The corresponding wave function in the real space is shown in Fig. 4(c). 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 kz /kL /(â3/2)(2p/a ), consistent with the kz -broadening away from the L -point. In summary, we observe that Au(788) and Au(322) represent two limiting cases in a lateral nanostructure, which can be generally defined as a regular array of nano-sized objects on a surface. In Au(788) we have the case of individual, non-interacting objects (terraces) that display quantum-well levels. In Au(322) the object-toobject 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 complete Fourier spectrum of the relevant electronic states, which is in turn necessary to properly describe the wave functions of the lateral nanostructures.
Acknowledgements J.E. Ortega and A. Mugarza are supported by the Universidad del Paı´s Vasco (1/UPV/EHU/ 00057.240EA-8078/2000) and the Max Planck Research Award Program. V. Repain and S. Rousset are supported by the CNRS-ULTIMATECH program, the CRIF and the Universite´ de Paris 7. A. Mascaraque is supported by a Marie Curie Fellowship of the European Union under contract No. HPMF-CT-2000-00565. V. Pe´rez-Dieste is supported by the Comunidad Auto´noma de Madrid
(Project No. 07N/0042/98) and the Spanish DGICYT (Grant No. PB-97-1199). The experiments performed at LURE were funded by the Large Scale Facilities program of the European Union. Technical support from the Spanish /French beam line staff is gratefully acknowledged.
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