Enhanced conductance of molecular states at interstitial sites

Arrays of phthalocyanine molecules on Pb(100) are investigated with scanning tunneling microscopy. Maps of the differential conductance exhibit drastic changes as the sample voltage is being varied. Maximal conductances are observed at positions between the molecules mimicking bonding states. However, the maxima are shown to result from a superposition of non-interacting states. We expect that this effect may be observed from many other molecules.

The scanning tunneling microscopes (STMs) measures current values I as functions of the sample voltage V and the spatial coordinates x, y, and z, where the z-direction is perpendicular to the sample surface. Neglecting complications such as tip-sample interactions, the transition to a point contact, inelastic tunneling, or dynamic processes the interpretation of the data seems fairly straightforward. Assuming an s-wave character of the tip, Tersoff and Hamann showed that the data represent the local density of states (LDOS) [1,2], integrated over energy ranges determined by V and the temperature broadening of the Fermi levels [3,4].
To model the sample states, two-dimensional Bloch waves are typically used. For a given energy, their decay in the z-direction is slowest for vanishing lateral wave vector and increases for states closer to the Brillouin zone boundary. In other words, the tunneling gap acts as a low-pass filter for spatial frequencies. As the tip-sample distance is increased, the cut-off frequency of this filter is reduced.
The influence of the lateral wave vector on the tunneling current has long been known [5]. More recently, it has been experimentally shown that momentum filtering can result in a partial suppression of elastic tunneling in graphene, where no low-momentum states are available at the Fermi level [6]. A similar effect on the scale of a single molecule has been reported from pentacene [7]. Vibration-assisted tunneling was observed to depend on the symmetries of the lowest unoccupied molecular orbital (LUMO) and the tip. The spatial sign-changes of a molecular wave function were also shown to affect the interferences between different tunneling paths on an Fe-porphyrin [8] and on graphene nanoribbons [9]. Much earlier, interference effects in STM images of have been theoretically addressed [10,11].
The fact that the filtered LDOS high above a surface is not identical (except for a scaling factor) to the LDOS at the surface seems trivial. However, it can lead to unintuitive image contrast from three-dimensional structures like metal nano-particles [12]. Additional complication arises when more complex tip wave functions are taken into account. Chen analytically extended the Tersoff-Hamann model to higher angular momentum states, which leads to an interpretation of the data in terms of spatial derivatives of the LDOS [13]. Schemes have been designed to extract the LDOS from subsets of I(V, x, y, z), e.g. 'spectra' of I(V) or its derivative dI(V)/dV [14][15][16][17][18][19][20][21][22][23][24][25]. Laterally resolved maps recorded under experimental conditions such as constant height, constant current, or constant conductance have also been evaluated [17,18,26,27].
Here we present STM data from artificial molecular clusters. As the sample voltage is being varied, drastic changes of the image contrast occur in constant-height maps of the differential conductance. In particular, we find increased conductance values at interstitial positions. The corresponding image features in this case do not reflect some sort of chemical bond. They rather show local maxima of the three-dimensional LDOS that are located high above the molecular plane and, therefore, appear prominently in STM images. We

Experimental details
Pb(100) substrate surfaces were prepared by repeated bombardement with Ar + ions and annealing up to 530 K. H 2 phthalocyanine (H 2 Pc) was deposited in vacuo from a Knudsen cell onto the Pb substrate held at room temperature. All STM experiments were performed in ultrahigh vacuum at temperatures below 4.5 K. To measure the differential conductance dI/dV we used a lock-in amplifier with a modulation voltage of 14.1 mV PP at a frequency of 828.3 Hz. Figure 1 displays data from a trimer of H 2 Pc molecules, which was prepared by laterally moving molecules with the STM tip. The model in (a) indicates the geometry of the cluster that was derived from the STM topograph in (b). Each molecule gives rise to a clover-leaf pattern. One of the main observations of the present work is shown in figure 1(c). This constant-height dI/dV map of the same area reveals prominent maxima of the conductance (red spots) between the molecules. Comparison with the model and the topograph shows that the maxima occur in the areas where neighboring molecules are closest to each other. For further characterization, dI/dV spectra have been measured on the trimer (figure 2). Red and blue lines show data from an intersitial position (i.e. a protrusion in figure 1(c)) and an isoindole group at the periphery of the trimer. The lowest unoccupied orbital appears as a resonance peaking at ≈200 mV. Its intensity is clearly enhanced in the spectrum taken between two molecules.

Experimental results
At first glance, the protrusions and the enhanced conductance seem to indicate some sort of bonding state. In addition, the constant-height mode of measurement used is generally expected to be free of the instrumental artifacts that complicate the analysis of constant-current measurements [17,26]. However, previous analysis showed that no relevant bonding occurs in the H 2 Pc clusters, similar to clusters of Al-and Pb-phthalocyanine [29,30]. It turns out that the main features in the experimental dI/dV data may be reproduced fairly well by a superposition of the unoccupied orbitals of isolated molecules, which is displayed in figure 1(d).
Next we present data from artificial clusters of 3 × 3 molecules (figure 3). These structures are particularly useful because a detailed analysis of their geometric and electronic structure is available [31]. In brief, the clusters are comprised of molecules with two slightly different azimuthal orientations ( figure 3(a)). Using the axis defined by opposite lobes of the H 2 Pc molecules, we find that the molecules at the corners and at the center of the cluster in figure 3(b) are rotated by 50 • with respect to a ⟨001⟩-direction of the lead substrate while the edge molecules are rotated by 45 • .
At low V the central molecule exhibits prominent maxima where Yu-Shiba-Rusinov states are observed. As detailed in [31], the LUMO of the central molecule is pulled to the Fermi level by the electrostatic fields created by the adjacent molecules, the orbital becomes singly occupied, and a localized spin is present. Modeling of the electronic structure revealed that hardly any hybridization occurs between neighboring molecules. The corresponding energies are of the order of 1 meV. However, the electrostatic shifts of the molecular orbitals, partially due to the quadrupole field of neighboring molecules, were shown to be relevant. Figure 3(c) shows a constant-height map of the differential conductance dI/dV recorded at V = 100 mV. Like in the trimer case above, we observe distinct conductance maxima at interstitial positions where the phenyl rings of two adjacent molecules are particularly close to each other. The main features of the experimental map are reproduced by a superposition of calculated orbitals of the enneamer ( figure 3(d)).
The contrasts in constant-height dI/dV maps depend critically on the sample voltage ( figure 4). The low-bias data in figure 4(a) essentially show the LUMO of the central molecule. At 65 mV ( figure 4(b)), the image contrast is drastically different. A number of new bright features is observed. Closer inspection shows that the six most prominent features are located between molecules. A further increase of the voltage to 115 mV (figure 4(c)) leads to additional features at other interstitial positions. At 155 mV ( figure 4(d)), the intensity of these features is reduced.

One-dimensional model
The observations presented above may be qualitatively understood from a simple one-dimensional model. The molecules are represented by model wavefunctions built from six s-waves, Ψ ∝ ∑ 6 i=1 (−1) n i exp(−κr i ), where κ is an inverse decay length, r i is the distance from atom i, and n i ∈ {0, 1} determines the sign of wave centered at i. The corresponding probability densities evaluated at different distances from the molecule are  displayed in figure 5. The signs of the waves are chosen to obtain a node at the center of the molecule ( figure 5(a)) to mimick the experimental data. For completeness, we also consider a symmetric sign pattern in figure 5(b). As the distance is increased, high spatial frequencies are progressively suppressed in both cases. For the even wave function Ψ g in (a), however, the inner maxima vanish while a new central maximum along with secondary maxima develops in (b), Ψ u . At large distances, maxima in the propability density appear at positions located outside the molecule.
Next, figure 6 shows the probability density for chains of three model molecules. Neglecting interaction between the molecules the density is the incoherent sum of the individual densities. As the distance increases spatial filtering reduces the number of features. Ψ g ( figure 6(b)) results in predominant peaks at the molecular positions along with minor maxima between the molecules. In the case of Ψ u , maxima occur only at interstitial positions as observed in the experimental data ( figure 6(a)).
In the H 2 Pc cluster of the experiments, the orbital energies of corner, edge, and center molecules differ because the molecules in these positions have different environments. As discussed in [31], the quadrupole  moments of the neighbors result in different electrostatic shifts. The contributions of the respective orbitals to the conductance therefore are different as well. As a result, the dI/dV image contrast changes rapidly on the energy scale defined by the variation of the electrostatic shifts.
The uneven nodal pattern used above (Ψ u ) bears some similarity with cross-sections of the frontier orbitals of H 2 Pc. However, similar filtering effects are to be expected from many planar molecules.
So far, the incoherent sum ∑ i |Ψ i | 2 of the wave functions has been evaluated. This is appropriate when the hybridization between the molecular orbitals is negligible and bonding and antibonding supramolecular orbitals are degenerate. In this case, the probability density between the molecules can reach up to twice the value at the ends of a molecular chain, the value we approximately observe in the present data from H 2 Pc. If,  however, the degeneracy is lifted an enhancement up to a factor of four may be expected as long as the wave function of the supramolecular state Φ = ∑ i c i Ψ i does not change sign between the molecules.

Conclusion
The orbitals carrying the tunneling current between a STM tip and H 2 Pc feature several sign changes in the directions parallel to a on Pb(100) surface. When this orbital pattern is probed with an s-wave tip, unexpected image contrast occurs in maps of the differential conductance. This effect is caused by the low-pass filtering of spatial frequencies in the tunneling junction. Most likely this effect may be observed from many other molecules with suitable orbital structure.