PRL 106, 026802 (2011)
PHYSICAL REVIEW LETTERS
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Tunable Quantum Dot Arrays Formed from Self-Assembled Metal-Organic Networks F. Klappenberger,1,* D. Ku¨hne,1 W. Krenner,1 I. Silanes,2,3 A. Arnau,2,4,5 F. J. Garcı´a de Abajo,6 S. Klyatskaya,7,1 M. Ruben,7,8 and J. V. Barth1 1 Physik Department E20, TU Mu¨nchen, 85748 Garching, Germany Donostia International Physics Center (DIPC), 20018 San Sebastian, Spain 3 Instituto de Hidra´ulica Ambiental de Cantabria (IH), 39005 Santander, Spain 4 Centro de Fisica de Materiales CSIC-UPV/EHU, Materials Physics Center MPC, 20080 San Sebastian, Spain 5 Depto. Fisica de Materiales UPV/EHU, Facultad de Quimica, 20080 San Sebastian, Spain 6 ´ ptica—CSIC, Serrano 121, 28006 Madrid, Spain Instituto de O 7 Institut fu¨r Nanotechnologie, Karlsruher Institut fu¨r Technologie (KIT), 76344 Eggenstein-Leopoldshafen, Germany 8 IPCMS-CNRS, Universite´ de Strasbourg, F-67034 Strasbourg, France (Received 28 June 2010; published 13 January 2011) 2
The confinement of Ag(111) surface-state electrons by self-assembled, nanoporous metal-organic networks is studied using low-temperature scanning tunneling microscopy and spectroscopy as well as electronic structure calculations. The honeycomb networks of Co metal centers and dicarbonitrileoligophenyl linkers induce surface resonance states confined in the cavities with a tunable energy level alignment. We find that electron scattering is repulsive on the molecules and weakly attractive on Co. The tailored networks represent periodic arrays of uniform and coupled quantum dots. DOI: 10.1103/PhysRevLett.106.026802
PACS numbers: 73.20.At, 73.63.Kv, 31.15.ae
The controlled adjustment of materials properties by modification on the atomic scale is a major goal of nanoscale science. On the close-packed faces of noble metals the surface-state electrons are well suited to monitor and influence the properties of the surface [1]. In 1993 Crommie et al. demonstrated the engineering of quantum well states by the manipulation of individual adatoms [2]. Since then the quasiparticle excitations of the surface-state band have been studied with great interest both experimentally [3–8] and theoretically [9–11], especially because their lifetime is connected to elementary scattering processes [10,11]. The high level of control over local electronic properties achievable by atom manipulation was demonstrated in the quantum mirage effect [12], in the controlled modification of the electronic structure of an adsorbate [13], and in quantum holographic encoding [14]. However, for the design of a targeted electronic spectrum of an entire surface it is impractical to use atom manipulation due to the slow serial production process. Since not only metal adatoms but also organic molecules scatter the surface-state electron waves [15], the extended regular structures that can be produced with molecular self-assembly [16] allow the tuning of electronic properties not only locally but surface wide, for example, in between linear molecular lines [17]. An elaborate electronic structure has been bestowed to Ag(111) by a chiral kagome´ network providing a 2D periodic array of quantum dots [18]. With the reflectivity for the electron waves being finite, the leakage-induced electronic overlap between neighboring quantum dots results in dispersive bands [19]. Here, we use supra-molecular design to construct metalorganic networks providing regular hexagonal cavities on 0031-9007=11=106(2)=026802(4)
the Ag(111) metal surface [20,21]. The obtained roomtemperature stable, nanoporous arrays create a novel periodic structure to the surface-state electrons, in which the energy positions of the different resonances depend on the lateral dimensions of the confinement region. These dimensions are easily tunable by the length of the linking molecule. The electronic characteristics of the confined states have been reproduced both in energy and real space by Green’s functions based electronic structure calculations using a boundary element method. Our analysis indicates that the molecules produce a repulsive scattering potential whereas the Co potential is slightly attractive. Thus the coordination bond markedly alters the scattering properties of the Co atoms. The crystal-like quality of our networks provides a perfect template for the lateral engineering of surface states [22]. The experiments were carried out in a vacuum apparatus, where a clean Ag(111) surface was prepared by a standard procedure. Metal-organic networks of Co and dicarbonitrile-quaterphenyl (NC-Ph4 -CN), respectively, dicarbonitrile-sexiphenyl (NC-Ph6 -CN) molecules were produced as described earlier [20,21]. Posteriorly, the sample was transferred into our homemade [23] beetle-type low-temperature scanning tunneling microscope (STM) where data were recorded at 8 K. Scanning tunneling spectroscopy (STS) was carried out by open-feedback loop dI=dV point spectra with bias modulation at 1400 Hz, an amplitude of 5 mV rms, and a lock-in time constant of 20 to 50 ms. The dI=dV maps were extracted from a set of 84 84 normalized point spectra. The spectroscopic maps are displayed as measured without convolution or high pass filtering. The automated procedure for taking a set of
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Ó 2011 American Physical Society
spectra takes 10 to 20 h, therefore maps are slightly distorted due to drift. More experimental details can be found in the supporting information (SI) [24]. We reported previously [21] that after the evaporation of 3 parts NC-Ph6 -CN and 2 parts Co [Fig. 1(a)] large domains of RT stable metal-organic networks self-assemble as exemplified in Fig. 1(b). These supramolecular structures provide regular arrays of uniform hexagonal pores. The spectroscopic dI=dV map obtained at a bias value VB ¼ 101 mV [Fig. 1(c)] demonstrates a standing electron wave pattern indicating confinement [2,3,8,18,25] of the electrons of the surface state described by the energy @2 2 dispersion EðkÞ ¼ E0 þ 2m k , where E0 ¼ 65 meV is the onset energy, m ¼ 0:42me the effective mass [25,26], @ the reduced Planck constant, and k the electron wave vector. Thus each pore represents a quantum dot. Earlier work showed that for this network type the size of the unit cell is controlled by the length of the organic linker [20]. Consequenently, the confinement imposed by the networks can be tuned accordingly. As an example we chose NC-Ph4 -CN [Fig. 1(d)] to construct quantum dots with a reduced confinement area [Fig. 1(e)]. Now, the same standing-wave pattern is obtained at a higher VB ¼ 204 mV. In comparison to confinement in previous, sputtered or grown, nanostructures [4,7,8,27] the intensity distributions of the network-confined resonances exhibit more regular shapes directly reflecting the hexagonal symmetry of the networks. Furthermore, the identical spectroscopic features in the different pores [Figs. 1(c) and 1(f)] reveal the crystal-like quality that can be achieved by using a)
b)
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PHYSICAL REVIEW LETTERS
PRL 106, 026802 (2011)
c)
dI/dV(101 mV)
self-assembly protocols for the production of such arrays of quantum dots. For a more quantitative investigation of the confinement properties we focus on the NC-Ph6 -CN case in the following before discussing the tunability in more detail later. A set of representative points of the hexagonal unit cell [Fig. 2(a)] was chosen to conduct dI=dV point spectroscopy [Fig. 2(b)]. In contrast to the reference spectrum (gray dashed line), recorded over a pristine surface with the same tip that was used to obtain all the spectra of the set, a strong position dependent variation is present in the spectra distributed over the quantum dot. The c spectrum (black, plus symbols), i.e., the spectrum at the center of the hexagon, shows two clear maxima at VB ¼ 6 and 205 mV, with the second maximum being enclosed by a shoulder on the low as well as on the high energy side (VB 140 and 280 mV). The ‘‘halfway’’ spectrum (red, crosses), obtained at half of the distance between the center of the cavity and the center of a molecule, displays its most prominent peak at 68 mV. The ‘‘molecule’’ spectrum (blue squares), on top of the center of a molecule, indicates the smallest local density of states (LDOS) of all positions and appears without sharp features. The Co spectrum (green diamonds), on top of a threefold coordinated Co center, shows more intensity than the molecule spectrum for VB < 150 mV, and very similar intensity for higher bias values. Therefore, the difference between the Co and the molecule spectrum is a first indication that, with respect to the electron scattering close to the Fermi level, the two positions behave differently.
a)
b) D
x
Co
+
1nm
d)
e)
f)
c)
dI/dV(204 mV)
Co
d)
x
+
1nm
FIG. 1. (a) The self-assembly of NC-Ph6 -CN and Co produces (b) a crystal-quality metal-organic 2D network with honeycombshaped pores (STM image, VB ¼ 1 V, IT ¼ 0:1 nA). (c) The experimental dI=dV map (STS, VB ¼ 101 mV) shows a uniform electron standing-wave pattern in all pores qualifying them as quantum dots. The lateral confinement dimensions are tunable by choice of the length of the molecules. (d) The shorter linking species creates (e) a network of the same symmetry, but with a smaller unit cell (STM image, VB ¼ 0:8 V, IT ¼ 0:1 nA) resulting (f) in the same electron wave pattern as in (c), but at an increased bias voltage STS, vB ¼ 204 mV.
FIG. 2 (color online). (a) Topograph of a single quantum dot. The symbols depict the positions at which the dI=dV spectra of (b) were taken. D is the distance between parallel sides. (b) The experimental dI=dV spectra (symbols) demonstrate a strong position dependent modulation of the nearly constant DOS of the pristine surface (dashed line). (c) The 2D potential model with black, white and green areas corresponding to potential values of V0 ¼ 0 meV, Vmol ¼ 500 meV, and VCo ¼ 50 meV. (d) The spectra calculated with the BEM.
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PHYSICAL REVIEW LETTERS
PRL 106, 026802 (2011)
In the following we will analyze the confinement in a first approximation by comparing it to a quantum particle in a 2D hexagonal box [27]. The lowest eigenstates n of this system that exhibit intensity in the center of the hexagon are 1 and 4 . They are reflected in the two maxima of the c spectrum in Fig. 2(b), whereas the maximum of the halfway spectrum is connected to 2 . The numerically obtained eigenvalues n for the hexagonal box [27] scale with the inverse of the area of the hexagon A ¼ cosð6 ÞD2 and thus the eigenstate energies En can be calculated by En ¼ E0 þ mnA . For the 20 20 supercell obtained in Ref. [21] the distance between parallel hexagon With this D the energy sides amounts to D ¼ 57:78 A. values En deviate from the experimental ones ([24], Table I). However, by using an effective diameter Deff ¼ 1:05D good agreement is obtained. This indicates a substantial penetration of the wave functions into the confining potential, i.e., an appreciable overlap between neighboring quantum dots. The origin of this overlap will be further discussed after the presentation of a more comprehensive theoretical analysis. Next, we investigate the lateral intensity distributions of the confined resonances. The 1 state [Fig. 3(a)] exhibits a domelike structure inside the cavity. The molecules appear dark, the Co atoms at the coordination sites show medium brightness. A central depression surrounded by a bright ring leads to a donut shape for 2 [Fig. 3(b)]. In this case also the corners exhibit a higher electron density than the molecules. At 200 mV [Fig. 3(c)] the map is characterized by a protrusion in the center encompassed with a dark inner and a brighter outer ring. At the energy of the 5th eigenstate [Fig. 3(d)] the central maximum is very shallow and the outer ring is divided into six bright spots near to the centers of the molecules. The comparison with the theoretical state densities of Ref. [4] allows extracting two conclusions. First, the intensity in the outer ring of the 4 maps is too high compared to the central peak, but can well be explained if a mixing with 3 is taken into account. Thus the map at 200 mV is not an eigenmode, but a mixture with the 3rd state that should appear at VB ¼ 174 mV (see a)
b)
c)
d)
e)
f)
g)
h)
FIG. 3 (color online). (a)–(d) Measured dI=dV maps of a hexagonal quantum dot at indicated bias. (e)–(h) The electron wave patterns obtained with the BEM for the model potential of Fig. 2(c) reproduce well the characteristics of the experimental maps.
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SI [24]), but is not resolved because of the width of the resonances. Second, for n ¼ 2 to 5, all eigenstates are expected to display maxima inside the cavity near the corners. Since the experimental maps do not show these maxima, the scattering potential of the Co atoms must be less reflective than that of the molecules. We have gained further insight into the confinement properties of the network through simulations using a scalar version of the electromagnetic boundary element method (BEM) [18,22,28]. The Schro¨dinger equation is solved for a 2D potential consisting of regions of constant potential with abrupt boundaries. The solution is expressed by boundary sources, which are propagated in each region via the electron Green’s function. The sources are determined from the continuity of both the wave function and its gradient, thus defining a system of integral equations [24]. Using the already mentioned values for E0 and m for the surface-state electron band, we have defined the effective scattering potential landscape as depicted in Fig. 2(c), where the black regions have zero potential and in which the molecules are described by rectangles (white) with the same length (2.96 nm), widths (0.25 nm, C-C distance perpendicular to the long molecular axis), and effective scattering potential value (Vmol ¼ 500 meV) that were successfully employed earlier [18]. The value VCo ¼ 50 meV for the hexagonal Co regions (green) and the energy-independent phenomenological broadening (25 meV) were adjusted to obtain the best agreement with experiment. Using the BEM, we calculated the energy dependent LDOS at selected (x, y) positions and compared them to the measured dI=dV spectra. Despite the simplicity of the model, the simulated spectra [Fig. 2(d)] agree rather well with the measured dI=dV point spectra [Fig. 2(b)] regarding peak positions, intensities, and width. The only exception is the Co spectrum, which deviates from the experiment for E * 100 meV. Furthermore, conductance maps [LDOSðx; yÞ] of the resonances i [Figs. 3(e)–3(h)] were calculated at the energies Ei indicated by the maxima in the dI=dV spectra shown in Fig. 2(d). In this model description, the surfacestate electrons are restricted to a 2D plane and, therefore, there is not any z coordinate representing the height. Thus, we compare with experimental spectroscopic data taken under open-feedback-loop conditions, corresponding to constant height conditions. Note that all simulated maps are displayed using an identical color scale, and that the same is true for the experimental maps. Close examination of the simulated standing-wave patterns reveals a reproduction of the experimental data in great detail within the cavities. For the Co sites, the agreement is very good for energies E < 100 meV, but too much intensity is present at higher energies in the simulated maps, as anticipated from the dI=dV point spectra. The approximation of describing the actual three-dimensional scattering of surface-state electrons by an energy-independent 2D effective scattering
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PRL 106, 026802 (2011)
PHYSICAL REVIEW LETTERS
dI/dV (arb. units)
NC-Ph4-CN 6
4 NC-Ph4-CN scaled
2
NC-Ph6-CN 0
200 bias voltage(mV)
400
FIG. 4 (color online). The c spectrum for cavities made with NC-Ph6 -CN linkers (black) and with NC-Ph4 -CN linkers (red). The four local maxima are present in both spectra and are upshifted for the shorter molecules as clearly demonstrated by the scaled version (red, dashed) of the NC-Ph4 -CN spectrum, for which the bias values were scaled by the ratio of the confining areas RExp (for details see [24]). The bold arrows mark the energies of the maps presented in Figs. 1(c) and 1(f).
potential has some limitations and, therefore, one cannot expect a fully correct description of the real system. The overall agreement between the measured and the simulated shapes of the resonances substantiates that the difference between the eigenstates of a purely hexagonal quantum box and our system originates from the attractive scattering potential at the Co atoms. In contrast, an isolated Co adatom on the Ag(111) surface is characterized by an effective repulsive scattering potential for electrons at the Fermi level [29]. Thus, our study indicates that the metalligand interactions induce a significant change of the embedded Co centers and their interplay with the surface-state electrons. Even though in an extended system it is not trivial to assign a specific charge to an individual entity, we interpret this result using the following picture. The electron charge transfer from the Co centers to the CN-ligands changes the character of the electron-Co interaction from repulsive (isolated Co adatom) to attractive (Co bond to CNligands) for electron scattering close to the Fermi level. However, the HOMO-LUMO gap of the molecules represents a repulsive potential for these electrons, even after the formation of the metal-organic coordination. Now we discuss the tunability of the energetic positions of the quantum dot states in more detail. In Fig. 4 we compare the c spectra of the networks constructed from NC-Ph6 -CN and NC-Ph4 -CN. The characteristic features (highlighted by vertical lines) shift to higher energies for the shorter molecules. As shown in [24], we obtain a constant scaling ratio Rexp ¼ 1:74 for all four features marked in the Figure proving that the energy levels are controlled by the quantum dot dimensions. A version of the NC-Ph4 -CN spectrum with the bias values scaled by Rexp highlights that the shape of the spectrum remains nearly unchanged when reducing the length of the linkers. The ratio of the quantum dot areas for the two networks is R ¼ 1:83, thus compared to confinement in a hexagonal box the expected scaling factor is somewhat larger than Rexp . This difference is consistent with the enlargement of the
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confining area produced by the attractive Co potential that does not scale with the molecular length. In conclusion, we have self-assembled RT stable metalorganic networks behaving as arrays of coupled quantum dots. The preservation of the metal-organic binding motif upon a variation of the organic linker allows to control the confinement geometry and accordingly the electronic level alignment. A further functionalization of the molecules can provide an additional parameter to tailor the system. Thus, we introduced a versatile method to tune the interface electronic properties by employing an easy-to-control supra-molecular engineering protocol. This work has been supported by the ERC Advanced Grant MolArt, the TUM Institute of Advanced Studies, the Deutsche Forschungsgemeinschaft (BA 3395/2-1), the Spanish MICINN (MAT2007-66050 and FIS2010-19609C02-01), the EU (NMP4-SL-2008-213660-ENSEMBLE), and the Basque Departamento de Educacio´n, UPV/EHU (IT-366-07).
*
[email protected] [1] N. Memmel, Surf. Sci. Rep. 32, 91 (1998). [2] M. F. Crommie, C. P. Lutz, and D. M. Eigler, Science 262, 218 (1993). [3] J. T. Li et al., Phys. Rev. Lett. 81, 4464 (1998). [4] J. T. Li et al., Phys. Rev. Lett. 80, 3332 (1998). [5] J. Kliewer et al., Science 288, 1399 (2000). [6] K. F. Braun and K. H. Rieder, Phys. Rev. Lett. 88, 096801 (2002). [7] H. Jensen et al., Phys. Rev. B 71, 155417 (2005). [8] C. Tournier-Colletta et al., Phys. Rev. Lett. 104, 016802 (2010). [9] G. A. Fiete and E. J. Heller, Rev. Mod. Phys. 75, 933 (2003). [10] P. M. Echenique et al., Surf. Sci. Rep. 52, 219 (2004). [11] J. Kro¨ger et al., Prog. Surf. Sci. 80, 26 (2005). [12] H. C. Manoharan, C. P. Lutz, and D. M. Eigler, Nature (London) 403, 512 (2000). [13] J. Kliewer, R. Berndt, and S. Crampin, Phys. Rev. Lett. 85, 4936 (2000). [14] C. R. Moon et al., Nature Nanotech. 4, 167 (2009). [15] L. Gross et al., Phys. Rev. Lett. 93, 056103 (2004). [16] J. V. Barth, Annu. Rev. Phys. Chem. 58, 375 (2007). [17] Y. Pennec et al., Nature Nanotech. 2, 99 (2007). [18] F. Klappenberger et al., Nano Lett. 9, 3509 (2009). [19] J. Lobo-Checa et al., Science 325, 300 (2009). [20] U. Schlickum et al., Nano Lett. 7, 3813 (2007). [21] D. Ku¨hne et al., J. Am. Chem. Soc. 131, 3881 (2009). [22] F. J. Garcia de Abajo et al., Nanoscale 2, 717 (2010). [23] S. Clair, Ph.D. thesis, Ecole Polytechnique Fe´de´rale de Lausanne, 2004. [24] See supplementary material at http://link.aps.org/supplemental/10.1103/PhysRevLett.106.026802. [25] J. T. Li, W. D. Schneider, and R. Berndt, Phys. Rev. B 56, 7656 (1997). [26] L. Bu¨rgi et al., Phys. Rev. Lett. 81, 5370 (1998). [27] J. Li et al., Surf. Sci. 422, 95 (1999). [28] V. Myroshnychenko et al., Adv. Mater. 20, 4288 (2008). [29] M. A. Schneider et al., Appl. Phys. A 80, 937 (2005).
026802-4
Tunable Quantum Dots Arrays by Self-Assembled Metal-Organic Networks: Supporting Information F. Klappenberger,1 D. K¨ uhne,1 W. Krenner,1 I. Silanes,2, 3 A. Arnau,2, 4, 5 F. J. Garc´ıa de Abajo,6 S. Klyatskaya,7 M. Ruben,7, 8 and J. V. Barth1 1 Physik 2 Donostia 3 Instituto
Department E20, TU M¨ unchen, 85748 Garching, Germany
International Physics Center (DIPC), 20018 San Sebastian, Spain
de Hidr´ aulica Ambiental de Cantabria (IH), 39005 Santander, Spain 4 Centro
de Fisica de Materiales CSIC-UPV/EHU,
Materials Physics Center MPC, 20080 San Sebastian, Spain 5 Depto.
Fisica de Materiales UPV/EHU, Facultad de Quimica, 20080 San Sebastian, Spain 6 Instituto
´ de Optica – CSIC, Serrano 121, 28006 Madrid, Spain
7 Institute
f¨ ur Nanotechnologie, Karlsruher Institut f¨ ur
Technologie (KIT), 76344 Eggenstein-Leopoldshafen, Germany 8 IPCMS-CNRS,
Universit´e de Strasbourg, F-67034 Strasbourg, France (Dated: January 3, 2011)
EXPERIMENTAL DETAILS
The experiments were carried out in a vacuum apparatus with a base pressure of ∼ 3×10−11 mbar, where a clean Ag(111) surface was prepared by repeated cycles of Ar+ sputtering and annealing to 740 K. Dicarbonitrile-quaterphenyl (NC-Ph4 -CN) and dicarbonitrilesexiphenyl (NC-Ph6 -CN) molecules [1, 2] were sublimated from a quartz glass crucible inside a Knudsen cell, held at 483 K for NC-Ph4 -CN and at 572 K for NC-Ph6 -CN, onto the Ag substrate stabilized at room temperature (RT). The deposition time for a dense-packed monolayer [3] is approximately 10 mins. After the sublimation of the organic molecules, Co adatoms were dosed by e-beam evaporation while keeping the subtrate at RT. Posteriorly, the sample was transfered into our home-made [4] beetle-type low-temperature STM where data were recorded at ≈ 8 K. The indicated bias values VB refer to the sample voltage. Scanning tunneling spectroscopy (STS) was carried out by open-feedback loop dI/dV point spectra with set points as indicated, bias modulation with frequencies ∼1400 Hz, an am-
2 plitude of 5 mV rms, and a lock-in time constant of 20 to 50 ms. The dI/dV maps were extracted from a set of 84 x 84 point spectra equally distributed over the area of interest. All spectra were normalized such that the average value of the intensity between -250 mV and -150 mV was 1. The spectroscopic maps are displayed as measured without convolution or high pass filtering. The automated procedure for taking a set of spectra takes 10 to 20 h, therefore maps are slightly distorted due to drift.
DESCRIPTION OF THE BOUNDARY ELEMENT METHOD
In the boundary element method (BEM), surface electrons are considered to evolve in a 2D potential that represents the effect of corrugations such as surface steps or adsorbed atoms and molecules. The surface state wavelength is generally large compared to the atomic spacing, so we can consider planar surfaces supporting surface states, and in particular Ag(111), as described by flat effective potentials. Setting the zero of energy at the bottom of the surface-state band, the evolution of the electron wave function φ(x, y) along the surface is governed by Schr¨odinger equation, (∇2 + k 2 )φ = 0, where k =
√
(1)
2meff E/~ is the electron wave vector, E is the energy, and meff is the effective
mass [0.42 me ] of electrons in the Ag(111) surface state. In the presence of a metal organic network, the electrons are subject to an effective potential V as described in Fig. 2 of the main paper, so that Eq. (1) has to be changed to (∇2 + k 2 − 2meff V /~2)φ = 0.
(2)
We solve Eq. (2) by placing electron sources at the boundaries between different potential regions. These sources are propagated through each region j of potential Vj by means of the 2D Green function of Helmholz equation (1)
Gj (R) = (−imeff /2~2 )H0 (kj R), implicitly defined by the relation (∇2 + kj2 )Gj (R) = (2meff /~2 )δ(R),
3 where R = (x, y) and kj =
p 2meff (E − Vj )/~ is the wave vector inside region j. The sources
σj defined on the boundary of region j contribute to the scattered wave function inside it as Z scat φ (R) = ds Gj (|R − Rs ) σj (s), (3) Sj
where the integral extends over points Rs along the contour Sj defining region j. The total wave function is then imposed to be φsource + φscat , where φsource describes an external source (e.g., a point source, as considered below). The boundary sources are determined from the condition that both the total wave function and its derivatives are continuous across the boundaries, following similar procedures to those employed in the application of this method to the solution of Maxwell’s equations for the electromagnetic problem [5]. We are in particular interested in calculating the local density of states (LDOS). The LDOS at a position R0 within a given region j can be derived by considering a point source at that position defined by φsource (R) = Gj (|R − R0 |). The LDOS is simply given
by meff /π~2 − (2/π)Im{φscat (R0 )}, where the first term is the background LDOS of the unpatterned surface, the second term is computed using Eq. (3), and a factor of 2 acounting for electron spin has been included.
[1] U. Schlickum, et al., Nano Lett. 7, 3813 (2007). [2] D. K¨ uhne, et al., J. Am. Chem. Soc. 131, 3881 (2009). [3] D. K¨ uhne, et al., J. Phys. Chem. C 113, 17851 (2009). [4] S. Clair, Ph.D. thesis, Ecole Polytechnique F´ed´erale de Lausanne (2004). [5] F. J. G. de Abajo, and A. Howie, Phys. Rev. B 65, 115418 (2002).
4 EIGENSTATES OF CONFINED ELECTRONS
Table 1 shows the energies of the five lowest-lying eigenstates of a hexagonal box, i.e., a two-dimensional, infinitely strong confining potential surrounding a hexagonal area of zero potential. In the metal-organic network employing NC-Ph6 -CN linkers the distance D between parallel sides of the hexagons amounts to D = 57.78 ˚ A. The assumed effective distance Def f = 60.75 ˚ A results from fitting the theoretical values of the eigenstates En (Def f ) for n = 1 to 4 to the experimental energies En (Exp). The energy separation Esep,n = En+1 (Def f ) − En (Def f ) between the energies of Ψn and Ψn+1 is smallest for n = 3. TABLE I: For the five lowest-lying eigenstates of a hexagonal box the columns denote the number, the energy assuming a distance D, the energy assuming distance Def f , the energy separation Esep,n and the experimental energy En (Exp). Ψn
En (Dreal )
En (Def f )
Esep,n
En (Exp)
1
-6.7 meV
-12.2 meV
80.8 meV
-11 meV
2
82.7 meV
68.6 meV
105.6 meV
69 meV
3
199.5 meV
174.2 meV
37.2 meV
167 meV
4
240.5 meV
211.4 meV
74.7 meV
209 meV
5
323.1 meV
286.1 meV
36.9 meV
≈280 meV
5 TUNING OF EIGENSTATES
The eigenstate energies can be tuned by choosing the appropriate length of the linking molecule. The bias voltage VP h6,Exp of the characteristic features (n = 1 to 4) of the cspectrum in the network constructed with NC-Ph6 -CN is converted into energy and shifted by the surface state offset E0 = −65 mV to give the energetic separation EP h6,Sep from the surface state band minimum. The same conversion was used for the bias voltages VP h4,Exp of the corresponding spectrum of the network employing NC-Ph4 -CN. The shorter linkers result in higher-lying resonance energies, EP h4,Sep , as expected for smaller confinement dimensions. Scaling EP h6,Sep with the ratio of the unit cell areas of the two networks, R = 1.83, results in energies, EP h4,Scaled slightly higher than observed in the experiment. The fact that the constant experimental ratio Rexp = 1.75, obtained by dividing EP h4,Sep by EP h6,Sep, is slightly smaller than R is consistent with the attractive potential of the Co atoms, which enlarges the confining volume. The bias values of the scaled version of the c-spectrum in the NC-Ph4 -CN case (Fig. 4, red dashed line) were calculated using the formula: EB,scaled = ((EB − E0 )/Rexp ) + E0 .
TABLE II: For the four lowest-lying characteristic features, n = 1 to 4, the columns denote the bias voltage VP h6,Exp, the energetic separation EP h6,Sep, the bias voltages VP h4,Exp, the energetic separation EP h4,Sep, the scaled energies EP h4,Scaled , and the experimental ratio Rexp . n
VP h6,Exp
EP h6,Sep
VP h4,Exp
EP h4,Sep
EP h4,Scaled
Rexp
1
-13.7 mV
51.3 meV
23.5 mV
88.5 meV
93.9 meV
1.73
2
125.5 mV
190.5 meV
268.2 mV
333.2 meV
348.6 meV
1.75
3
210.9 mV
275.9 meV
414.6 mV
479.6 meV
504.9 meV
1.74
4
293.2 mV
358.2 meV
564.8 mV
629.8 meV
655.4 meV
1.76
