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Keywords:

  • C60;
  • charging energy;
  • density functional theory;
  • metal–organic interfaces

Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model and method of calculation
  5. 3 DFT calculations, density of states (DOS), the IDIS-model, and the molecule charging energy
  6. 4 Pillow effects and the transport energy gap of the molecule
  7. 5 Conclusions
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

The C60/Au(111) metal/organic (MO) interface barrier formation is analyzed from the single molecule limit to the full monolayer case by means of density functional theory (DFT) calculations taking into account charging energy effects to properly describe the electronic structure of the interface and van der Waals (vdW) interactions to obtain the molecule-surface adsorption distance. In our calculations we obtain the density of states (DOS) projected on the C60-molecule, the energy position of its charge neutrality level (CNL), its highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) levels, as well as the interface Fermi level. From these calculations we also obtain the charge transfer, the induced potential on the molecule and the interface screening parameter, S. We find a significant evolution from the single molecule to the full monolayer, with the energy barrier for electrons decreasing with the molecule deposition: this reflects an increasing interface screening effect for larger molecule coverages. We also analyze the relationship between S and the molecule charging energy, U, and deduce an effective interaction between electrons in different C60 molecules.


1 Introduction

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model and method of calculation
  5. 3 DFT calculations, density of states (DOS), the IDIS-model, and the molecule charging energy
  6. 4 Pillow effects and the transport energy gap of the molecule
  7. 5 Conclusions
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

The growing field of organic electronics relies on the use of organic conjugated molecules as components of multilayer devices. The performance of these devices depends critically on the energy barriers that control the carrier transport between layers, energy barriers that are determined by the relative alignment of the molecular levels at metal–organic (MO) or organic–organic (OO) interfaces 1, 2. This problem, which is typically analyzed by depositing an organic monolayer on a metal or an organic compound, is closely related to the problem of understanding the barrier formation for a single organic molecule sandwiched between two metals 3–5, a case of growing importance within the field of molecular electronics.

In this paper, we consider the paradigmatic case of a C60/Au(111) interface, and analyze how its interface barrier evolves when going from the isolated molecule to the full monolayer case. This approach will allow us to understand how these two limits are interconnected. The advantage of working with C60 is its chemical simplicity, which makes the density functional theory (DFT) calculations used in this paper and its corresponding analysis much simpler. Besides we should mention that these molecules have been widely studied deposited on metals 6, forming interfaces 7, 8, encapsulated as a film between two metals as a part of a solar cell 9, or in nanogap organic molecular junctions 10–12.

On the other hand, molecular level alignment at organic junctions has been widely investigated in the last decade 13–15. Since the Schottky–Mott limit (where use of the vacuum level alignment is made) was disproved 16, 17, many different mechanisms have been proposed to explain the barrier formation at MO interfaces: chemical reactions and the formation of gap states in the organic gap 15, 18–20; orientation of molecular dipoles 21, 22; or compression of the metal electron tails at the MO interface due to the Pauli exclusion principle 20, 23–25. It has also been suggested that the tendency of the charge neutrality level (CNL) of the organic material to align with the interface Fermi level 26, 27 plays also an important role; this mechanism is associated with the induced density of interface states (IDIS) and the charge transfer between the organic and the metal. More recently, this approach has been extended to the unified-IDIS model by inclusion of the dipole due to Pauli exclusion principle and intrinsic molecular dipoles 25, 28.

An extra problem appearing when analyzing organic interfaces is related to the standard DFT calculation, that do not yield an appropriate description of the organic transport energy gap 29. In particular, the Kohn–Sham (KS) eigenvalues calculated using the local density approximation (LDA) or generalized gradient approximation (GGA) exchange-correlation functionals yield a transport gap that is too small. It has been argued elsewhere 12, 30 that the effective charging energy of the molecule, U, can be used to correct the KS LDA energy gap, ELDA, to yield the following transport gap

  • equation image((1))

an equation that will be used below to determine Et self-consistently. Another problem is that many MO systems cannot be characterized accurately in a standard (e.g., LDA, GGA) DFT formalism due to the poor description of van der Waals (vdW) interactions in these methods 31–36 (the vdW interaction is nonlocal and long-range, while exchange-correlation functionals in standard DFT methods are local and short range, with a typical exponential decay).

In this paper the C60/Au(111) MO interface barrier formation is analyzed for a single molecule and for the equation image interfaces (m = 2, 3, 4, 5), the equation image corresponding to the full monolayer case. For this purpose, DFT calculations are combined with a calculation of the charging energy U of the C60 molecule on the metal surface to properly describe the transport gap and the energy level alignment and the interface. In these calculations vdW interactions are also taken into account to obtain a reliable molecule-surface adsorption distance. Our calculations yield a C60[BOND]Au(111) distance of ≃2.4 Å, a value in between the covalent (≃2 Å) and weak interacting (≃3 Å) regimes. We find a significant evolution of the position of the interface Fermi level with respect to the molecular levels as a function of the C60 coverage, the energy barrier for electrons decreasing as the coverage is increased from the single molecule to the monolayer. This evolution shows the important role played by neighboring adsorbed molecules in the screening properties of the interface. Regarding the charging energy U, we obtain U = 1.5 eV and a transport gap of 3.1 eV for C60 on Au(111).

The paper is organized as follows: in Section 2, we discuss in detail our approach, while in Sections 3 and 4 present our results. Finally, in Section 5 we present our conclusions.

2 Model and method of calculation

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model and method of calculation
  5. 3 DFT calculations, density of states (DOS), the IDIS-model, and the molecule charging energy
  6. 4 Pillow effects and the transport energy gap of the molecule
  7. 5 Conclusions
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

Figure 1 shows the systems and geometries we are interested in: an isolated C60-molecule deposited on a 8 × 8-Au(111) cluster; and a C60-monolayer on a Au(111)-surface. We have represented only the equation image periodic geometry, corresponding to a monolayer deposition; in order to analyze how the interface properties depend on the layer coverage, in our calculations we have also considered the equation image, equation image, and the equation image periodic geometries.

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Figure 1. (online color at: www.pss-a.com) Two of the C60/Au(111) geometries analyzed in this work. (left) A single C60 molecule on a 8 × 8-Au(111) cluster (side view); (right) the monolayer case, with equation image periodic geometry (unit cell is highlighted).

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We analyze all these cases using the local-orbital DFT code (FIREBALL) 37 as our basic computational tool. As commented above and discussed in detail below, these DFT calculations are combined with a calculation of the charging energy U of the molecule on the metal surface to properly obtain the transport gap, and vdW interactions are also taken into account. We stress that a standard (LDA or GGA) DFT calculation for MO systems like the one analyzed in this work, might be problematic due to the poor description of the organic energy gap and/or of the vdW interactions.

FIREBALL is a real-space DFT technique that uses basis sets of numerical atomic-like orbitals (NAOs) that are strictly zero beyond a given cut-off radius (the Fireball orbitals 38) and that is based in a self-consistent implementation 39 of the Harris–Foulkes 40, 41 functional. In this way, the DFT calculation is cast in a tight-binding-like form 42 where all the required integrals (orbital interactions) are at most three-center, and can be calculated beforehand and placed in data-tables, speeding up the calculations. In these particular calculations we use the LDA exchange-correlation functional and an optimized basis set of sp3d5 NAOs for Au, and sp3 for C 43, with the following cut-off radii (in a.u.): s = 4.5, p = 4.9, d = 4.3 (Au) and s = 4.5, p = 4.5 (C). For the C60 molecule this calculational approach yields C[BOND]C nearest neighbors distances of 1.40 and 1.47 Å (to be compared with 1.39 and 1.44 obtained in DFT in a plane wave basis 8 and experimental values 1.40, 1.45 Å 44), and a highest occupied molecular orbital (HOMO)/lowest unoccupied molecular orbital (LUMO) energy gap of 2.1 eV, to be compared with 1.6 eV for converged basis set LDA or GGA calculations 30. It is also worth mentioning that the experimental HOMO/LUMO gap (difference between affinity and ionization levels) for the gas-phase C60 is 4.9 eV 45, 46. For Au we obtain a bulk lattice parameter of 4.12 Å (experiment 4.07 Å 47). We have checked that with four layers we obtain converged results, that means that we are dealing with systems with a number of atoms between 108 (for the equation image case) and 360 (for the equation image one); that force us to use a very efficient local basis code. We have used 32 k-points to map the Brillouin zone at the equation image case, while 8, 8, and 2 k-points have been used for the equation image, equation image, and equation image, respectively. Regarding the cluster, an 8 × 8 × 4 size is enough to avoid border effects.

The main approximations of this calculational method are:

  • (i)
    The use of the LDA exchange-correlation functional. It is well-known that the LDA KS eigenvalues do not provide a good approximation for the transport gap of the organic material. On the other hand, as LDA–DFT exchange-correlation functionals do not take into account properly weak London dispersive forces, in physisorbed systems LDA adsorption energies and distances are not fully reliable.
  • (ii)
    In the Fireball approach a self-consistent version 39 of the so-called Harris–Foulkes 40, 41 functional is used; in this approximation the Hartree potential is calculated by approximating, in a self-consistent fashion, the total input charge by a superposition of spherical charges around each atom.
  • (iii)
    Use of a non-fully converged basis set of NAOs.

In the present calculations we correct the deficiencies introduced by these approximations in the following way:

(1) We correct the transport gap using Eq. (1). The point to stress here is that in order to determine U, we have first analyzed the case of an isolated molecule on the surface; from our DFT–LDA calculation, U can be related to the potential induced in the molecule, VIDIS (see below), and to the number of electrons transferred to it, δn, by the equation: equation image Then, Et is calculated by introducing in our Hamiltonian the following scissor operator, equation image, equation image (equation image) being the empty (occupied) orbitals of the isolated, but deformed, molecule (with the actual geometry of the molecule on the surface). Obviously U depends on Et, and this forces us to calculate Et and U self-consistently.

This approximation is justified as follows. For an isolated molecule the transport gap can be calculated as the difference between the electron affinity, equation image, and ionization potential, equation image, where E[Ni] is the total (LDA) energy for a neutral (N) or charged (N ± 1) molecule with Ni electrons

  • equation image((2))

Using this result together with Eq. (1) we can calculate the charging energy for the isolated molecule, U0.

When the molecule is deposited on a metal surface, image potential effects (or screening) at the MO interface induce on the metal side an opposite charge to the one appearing in the molecule, shifting the empty and filled molecular levels in opposite directions and reducing the value of the charging energy (U < U0) 48, 49. Considering that the exact total energy is expected to be a piecewise linear function of occupancy and using Janak's theorem 50, Sau et al. 30 have shown that, for a molecule weakly coupled to an environment, the new IP and EA are given by

  • equation image((3))

where equation image, equation image are the LDA KS eigenvalues and

  • equation image((4))

The charging energy is equation image. In the case of C60, equation image30. Notice that Eqs. (3), (4) are valid for both an isolated molecule, or a molecule weakly interacting with a metal surface, the different value of U in these two cases being related to the different dependence of the KS LDA eigenvalues with occupancy. In our case, the change in the LDA levels for an electron transfer δn is precisely the induced IDIS potential at the interface, eVIDIS

  • equation image((5))

which practically coincides in our calculations with equation image. A more detailed description of the IDIS model and the U calculation will be given in Section 3.

Once we have determined U from the single molecule case, we calculate the other adlayer cases introducing the same scissor operator which, in this way, takes into account the molecule charging effects. We also use equation image to correct the error in the LDA gap due to the basis set.

(2) The NAOs basis set has been optimized for each subsystem, so that it yields a reasonably good description of both the Au(111) surface and C60 molecule 43 independently. In order to have the Au and C60 levels correctly aligned at the experimental value before the contact is established we shift by ε0 the molecular levels of C60, using this Hamiltonian equation image (equation image being the eigenstates of the isolated molecule). This procedure has been checked previously for the case of benzene on Au(111) 51 (see also below). In this work, due to the error bar in the experimental data, we have chosen to use the alignment provided by the DFT-calculations of Ref. 8. Accordingly we have assumed the Fermi energy of the unmodified surface, ΦM, to be located 0.2 eV above the C60-mid gap. In this work we have also used equation image to change the Au workfunction fictitiously (e.g., see Fig. 4). This shift operator can be considered a kind of pseudopotential, that needs to be added in order to get a correct level alignment.

(3) The Harris–Foulkes functional neglects off-diagonal contributions of the induced charge (dipolar contributions) whose effects, although not important for the self-consistent calculation itself, introduces non-negligible contributions to the Hartree potential. One of these effects is due to the induced pillow-dipole (or Pauli-exclusion dipole), which is created by the compression of the electron metal tails due to their overlap with the organic molecule wavefunctions. The second effect we consider in this paper is associated with the charge induced on the metal surface and the accompanying induced “metal surface” dipole, that tends to shift that surface charge from practically the last metal layer to the image plane located outwards. Regarding the pillow-dipole, we calculate it following Ref. 25, and analyze how that dipole can be associated with the overlap between the MO wavefunctions. In particular, we calculate the charge rearrangement due to the orthogonalization of the organic/metal orbitals, by expanding the organic/metal many-body interactions up to second order in the wavefunction overlap using the Löwdin symmetric orthogonalization procedure; this rearrangement corresponds to the “push-back” effect experienced by the tail of the metal wavefunctions, and creates the following dipole, Dp, per molecule

  • equation image((6))

in this equation, ϕj and equation image are the wavefunctions of the molecule and the metal local orbitals, respectively, equation image is its overlap; and n and equation image are their occupancies; d is the vector joining the atomic positions of orbitals j and j′ and Δr′ joins d/2 with the integration variable r′. In our FIREBALL calculation, however, part of this dipole (in particular, the second term) is already taken into account through the potential created by the diagonal charges equation image. Thus we only need to consider here the following off-diagonal contribution

  • equation image((7))

We stress that this “pillow”-dipole is only the off-diagonal contribution of the total induced charge associated with the Pauli exclusion principle. Adding this off-diagonal term to the induced diagonal charge yields the total “push-back” contribution [Eq. (6)] that has been analyzed in detail by other researchers 24, 52.

On the other hand, the induced “metal surface” dipole is calculated by means of the off-diagonal induced charge: equation image; e being the electron charge and equation image being the Green function of our system, defined by the Hamiltonian, as obtained from our DFT–LDA calculation in the local orbital basis. From enij we calculate straightforwardly the corresponding induced Hartree potential.

(4) vdW forces are analyzed using an extension of the LCAO-S2-vdW formalism, developed specifically for weakly interacting systems, like graphitic materials 53, 54. In this approach, we consider two different contributions to the interaction energy, a short-range interaction associated with the overlap, S, between the wavefunctions of both systems, and the long range fluctuating dipole-dipole interaction leading to the vdW forces. In the present calculations we have used a simplified version of this approach, adding a semiempirical atom–atom vdW term to the short range interaction energy (see below). This way of including vdW interactions is a practical alternative to more ab initio approaches to include vdW in DFT (e.g., see Refs. 31, 55, 56), that are computationally very demanding for MO systems.

In the case of weakly interacting systems, the short range interaction can be calculated in an approximate way by means of a corrected LDA calculation, as follows: first, define the electron density for each subsystem (say, Au and C60); then, approximate the exchange-correlation energy for the complete system as the sum of the exchange-correlation energies for each subsystem taken each one independently, neglecting in this way the effect of the overlapping densities in the exchange-correlation energy. In the following we refer to the short-range interaction energy calculated in this way as the weak chemical interaction (WCI). This way of proceeding tries to avoid the double counting that would appear including both, the exchange-correlation energy provided by a conventional LDA and the correlation energy associated with the long-range vdW potential discussed below: in particular, Lang 52 has clearly shown how the LDA exchange-correlation hole for rare-gas atoms on metal surfaces mimics partially the vdW polarization hole induced in the metal. We should also mention that other authors 35 have analyzed how to avoid this double counting by means of an approach similar to the one presented in this paper, using a short-range exchange-correlation energy. Regarding the C[BOND]Au long-range (vdW) part, we use a typical atom–atom interaction with the standard form, equation image (R is the distance between atoms), where the factor equation image eliminates the vdW contribution for short distances 34. Following Refs. 34, 57, we take for C[BOND]Au RvdW = 3.3 Å 34 or RvdW = 4.1 Å 57 (see below) and d = 20, while the C6 parameter (C6 = 36 eV6) is taken from Ref. 58. The C60[BOND]C60 interaction is a constant that we calculate from Ref. 54, including the damping factor fD(R) (with RvdW(C[BOND]C) = 2.9 34 or 3.8 57, and d = 20).

3 DFT calculations, density of states (DOS), the IDIS-model, and the molecule charging energy

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model and method of calculation
  5. 3 DFT calculations, density of states (DOS), the IDIS-model, and the molecule charging energy
  6. 4 Pillow effects and the transport energy gap of the molecule
  7. 5 Conclusions
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

In a first step, we calculate the relaxed geometries and the C60/Au distance using both the standard LDA and the WCI approaches described above. Regarding the C60 geometry, we have found a very small relaxation, with C bonds changing less than 5% with respect to the gas-phase geometry, similar to the results by Wang and Cheng 8. Figure 2 shows our results for the adsorption energy of a C60 ML on Au(111): the black line represents our LDA-DFT energy, while the blue line corresponds to the WCI result. These two curves can be considered as two limiting cases for the short-range C60[BOND]Au interaction (basically the interaction of Au and the six nearest C atoms of C60): the WCI corresponds to weakly interacting systems, and the LDA to covalently bonded systems. In order to obtain the total adsorption energy we must add the long-range vdW interactions to this short-range energy. The C60[BOND]Au(111) vdW interaction is the result of many distant C[BOND]Au atom–atom interactions (the damping function fD(R) eliminates the contributions for C[BOND]Au atom pairs at closer distances). The purple and red curves in Fig. 2 show the energies calculated adding the vdW energy to both short-range curves. In this figure we have used RvdW(Au[BOND]C) = 4.1 Å and RvdW(C[BOND]C) = 3.8 Å 57. The minimum energy for the LDA + vdW result (purple curve) is located at a distance z (between the upper Au-layer and lower C-atoms of C60) of 2.25 Å, and at 2.67 Å for the WCI + vdW one (red curve). The adsorption energy is 2.17 and 1.89 eV, respectively; for comparison, the experimental value is 1.87 eV 7. This good agreement is probably fortuitous because of the minimal basis set used in our calculations; in particular, our LDA-adsorption energy is around 0.8 eV smaller than the one calculated using a LDA plane-wave converged basis 8, while an LDA-approach yields adsorption energies typically a few tenths of eVs larger than a GGA-calculation. Moreover, taking RvdW(Au[BOND]C) = 3.3 Å and RvdW(Au[BOND]C) = 2.9 Å 34 in the long range vdW interaction yields z = 2.20 Å (LDA + vdW) and 2.45 Å (WCI + vdW), with adsorption energies of 2.9 and 2.5 eV, respectively (see Table 1). Notice that the adsorption energies change significantly when changing the value of RvdW, while the equilibrium distance is much less affected. This is in line with a recent analysis of the semiempirical vdW forces for azobenzene on Ag(111) 59: these authors have concluded that, in the Tkatchenko–Scheffler approach 57, even if the adsorption energy is overestimated, the molecule/metal distance seems to be accurately described. Notice also that the sum of C and Au covalent radii is 2.1 Å, not too far from the values obtained above for the C60[BOND]Au(111) equilibrium distances. This suggests that the actual C60[BOND]Au(111) equilibrium distance should be in between the z values obtained in the LDA + vdW and WCI + vdW calculations. We conclude that 2.4 Å is a good guess to the distance between C60 and Au(111). Incidentally, this distance is similar to the minimum energy distance obtained in the LDA calculation (black curve).

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Figure 2. (online color at: www.pss-a.com) Adsorption energy for a C60 monolayer on Au(111) as a function of the distance between both subsystems, for several approximations (in all cases initial energy levels are correctly aligned). Black line: LDA Fireball. Blue line: Fireball with the LDA exchange correlation considered as the sum of exchange-correlation energies for each subsystem. Green line: vdW C6/R6-like correction. Purple line: LDA Fireball + C6/R6 vdW correction (good approximation at small distances). Red line: “Corrected” LDA + C6/R6 vdW correction (good approximation at large distances).

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Table 1. Distance, z, between the upper Au-layer and lower C-atoms of C60, and adsorption energy, Eads, for the different cases discussed in the text. Lengths are in Å and energies in eV.
techniquezEadsRvdW(Au[BOND]C)RvdW(C[BOND]C)
LDA2.450.39
WCI2.800.23
vdW + LDA2.252.174.1 573.8 57
vdW + WCI2.671.894.1 573.8 57
vdW + LDA2.202.823.3 342.9 34
vdW + WCI2.452.403.3 342.9 34

Once we have obtained the interface geometries, we calculate the interface electronic properties of: (i) the single molecule; and (ii) the other C60-layers, as well as the corresponding charging energy effects. The single molecule case allows us to determine U (Eq. (1)), while the other cases can be related to a kind of effective U

  • equation image((8))

where Ji is the effective intersite Coulomb interaction between charges in different molecules (see below for more details).

3.1 C60 molecule on Au(111)

Figure 3 shows the electron DOS projected on the C60 orbitals for the case of a single molecule adsorbed on Au(111) (Fig. 1, left). In the same figure we also show the molecule energy levels of the isolated (but deformed) molecule; the energy window around the energy gap is enlarged in the inset. The initial (ΦM) and the final (EF) Fermi energy (this is the metal Fermi level after the contact is established), the HOMO and LUMO levels, as well as the CNL, are shown. This last quantity is calculated integrating the molecule DOS up to charge neutrality conditions. The main effect of the contact is to broaden the molecular levels (also shown in Fig. 3), and to shift the relative position of the metal Fermi energy with the molecular levels from ΦM to EF, the final Fermi level. In this figure the vacuum level for the adsorbed molecule is kept aligned to the vacuum level for the isolated molecule. Also, in this figure the molecule transport gap (for both the isolated and the molecule on the surface) has been corrected using Eq. (1) with U = 1.5 eV. Notice that the first few peaks below the HOMO level are in good agreement with the ones observed in angle-resolved valence-band photoemission spectroscopy for a monolayer of C60 on Au; see also the case of a monolayer below 7.

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Figure 3. (online color at: www.pss-a.com) DOS projected on the C60 orbitals for the case of a single molecule over a Au(111) surface. Molecular levels of the isolated (but deformed) molecule are indicated by red lines. Right inset: enlarged image for the area around the Fermi level, showing HOMO, LUMO and CNL of the molecule, the initial (ΦM) and the final (EF) Fermi energy of the system. Note that EF − ΦM is the IDIS potential VIDIS. In this figure the vacuum level for the adsorbed molecule is kept aligned to the vacuum level for the isolated molecule. Also, the molecule transport gap (for both the isolated and the molecule on the surface) has been corrected using Eq. (1) with U = 1.5 eV.

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As in a typical MO interface, the C60/Au contact reacts creating a mean potential, Vt, between the molecule and the metal that tries to align the organic CNL and the metal workfunction; notice that ΦM and EF define respectively the initial and the final Fermi energy with respect to the molecular levels (which are defined with respect to the vacuum level of the isolated molecule), so that equation image. Following the Unified IDIS model 25, eVt has two components, the “pillow” and the IDIS potentials, Vp and VIDIS, respectively: equation image; in the results presented in this section we neglect Vp, so that Vt = VIDIS, this potential being associated with the charge transfer between the molecule and the metal, and is given by

  • equation image((9))

where S is a screening parameter related to the interface properties (see below); Eq. (9) implies that

  • equation image((10))

Notice that 0 < S < 1, with S[RIGHTWARDS ARROW] 0 or 1 for high or low screening, respectively 29. In the limit of large screening the CNL and EF tend to be aligned, while for S[RIGHTWARDS ARROW] 1, we recover the Schottky–Mott limit with no interface IDIS-potential. In order to calculate S with good accuracy, we have calculated (CNL − EF) versus (−ΦM) for the different interfaces analyzed in this work by changing fictitiously ΦM using the shift operator mentioned in Section 2, see Fig. 4. In particular, from the slope of the curve for the single C60 molecule on Au(111) we obtain S = 0.53 for this case.

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Figure 4. Values of (CNL − EF), VIDIS and δn versus (−ΦM), for the different interfaces (corresponding to different C60 coverages between the single molecule limit and the equation image monolayer) analyzed in this work. The values of CNL − EF and VIDIS have an error bar of ±0.03 eV; for δn the error bar is ±0.003.

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In order to obtain a convenient description of the screening parameter, S, we first consider the equation relating the molecule charging energy, U, to the potential induced by the charge transfer, eδn

  • equation image((11))

On the other hand, one can define a mean DOS (spin included), D, between CNL and EF so that δn is given by

  • equation image((12))

Combining Eqs. (9)–(12), one can easily find

  • equation image((13))

Equation (11), or equivalently equation image, has been used to obtain the molecule charging energy (see Fig. 4): our calculations yield U = 1.65 eV, with an error bar of ±0.1 eV, coming from the error bars in the slopes of eVIDIS and δn. We should mention, as a comment to this result, that introducing the “metal surface” dipole (see Section 2) reduces this value to U = 1.5 eV.

3.2 Monolayer and fraction of monolayer cases

Figure 5 shows the molecule local DOS for the equation image geometry (the full monolayer case); we show here the same quantities as in Fig. 3: ΦM, EF, the CNL and the HOMO and LUMO levels. As in the previous case, we also see how the quantity (CNL − ΦM), the initial misalignment between the energy levels of the metal surface and the molecule, has been screened to (CNL − EF), so that Eq. (10) is also valid for the monolayer case, but with a different screening parameter, which we have found in this case to be S = 0.19; see Fig. 4. It is interesting to realize that this parameter, S, is now significantly smaller than in the case of the single molecule on the surface, reflecting a larger screening effect. Similar effects are found for other coverages, the equation image geometries, that show an intermediate screening between the monolayer and the single molecule limits: Fig. 6 shows their DOS projected on the C60-orbitals, around the energy gap; notice how (CNL − EF) changes monotonously, being smaller for the compact monolayer and larger for the equation image structure. The values of S for the different equation image adlayers are 0.19, 0.23, 0.29, and 0.31 for m = 2, 3, 4, and 5, respectively. Notice also that the results shown in Figs. 3–6 indicate that the position of the CNL is practically the same, 0.8 eV above ΦM (−5.2 eV), in all the cases studied, with an error bar of ±0.02 eV.

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Figure 5. (online color at: www.pss-a.com) As Fig. 3 for the equation image monolayer.

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Figure 6. (online color at: www.pss-a.com) DOS projected on the C60 near Fermi level (EF) for the equation image coverages (m = 2, 3, 4, 5) discussed in the text. HOMO, LUMO, EF, CNL, and ΦM are shown. In this figure the vacuum level for the adsorbed molecules is kept aligned to the vacuum level for the isolated molecule. Also, in this figure the molecule transport gap (for both the isolated and the molecule on the surface) has been corrected using Eq. (1) with U = 1.5 eV.

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We can also relate S to Ueff, defined for the adlayer case as equation image, by replacing U by Ueff in Eqs. (11) and (13); this yields

  • equation image((14))

Notice that for these cases VIDIS depends also on the interaction between molecules, such that

  • equation image((15))

this equation indicating that the effective charging energy of the isolated molecule, U, is modified for an adlayer, by the interaction, Ji, between charges in different molecules 60. From our calculations we obtain the following values of Ueff (without including the “metal surface” dipole correction): 4.8, 3.3, 2.7, and 2.35 eV for m = 2, 3, 4, and 5, respectively; “metal surface” dipole reduces these values to 4.4, 3.0, 2.4, and 2.1 eV.

Figure 7 shows Ueff as a function of the distance, d, between molecules, as well as a fitting to these values using the following curve equation image. The A/d3 term is suggested by the dipole–dipole interaction between charges, and B/d4 is the next order correction to that term. We should stress that one could extract from that curve the particular values of Ji for a given adlayer, since the total sum equation image, the distance dependence of Ji (dipole–dipole) and the geometric arrangement of C60 molecules are known. In this way we can define the following many-body contribution equation image, to the effective Hamiltonian of the adlayer 60. Notice also the error bars in Fig. 7, associated with the error bars in our calculations of eVIDIS and δn.

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Figure 7. Ueff values versus distance between centers of nearest C60 for different coverages. The values have been adjusted to a equation image curve suggested by the multipole electrostatic interaction between molecules; a fitting including only a d−3 term has also been shown. A 6 × 6 geometry has been added in order to get a better fitting of the curve.

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4 Pillow effects and the transport energy gap of the molecule

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model and method of calculation
  5. 3 DFT calculations, density of states (DOS), the IDIS-model, and the molecule charging energy
  6. 4 Pillow effects and the transport energy gap of the molecule
  7. 5 Conclusions
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

In this section, we consider two effects that have not been included in our LDA calculation, which are associated with off-diagonal terms of the electron DOS. One is related to the “pillow” effect, the second one to our calculation of the transport energy gap of the molecule. Both effects are, however, small and it is reasonable to introduce them as a perturbation to our zeroth-order LDA-calculation.

4.1 Pillow effects

Up to this point, we have neglected the “pillow” potential, Vp, created by the compression of the electron metal tails due to their overlap with the organic molecule wavefunctions. As mentioned above, the C60/Au contact reacts creating a mean potential, equation image, so that equation image. The “pillow” potential, Vp, is associated with the “pillow”-dipole given by Eq. (6), which creates a mean potential equation image between the molecule and the metal. Because of the surface screening, S, we find that 25

  • equation image((16))
  • equation image((17))

and

  • equation image((18))

These equations show that, in general, we can define an effective, equation image, such that

  • equation image((19))
  • equation image((20))

which represents the initial ΦM shifted by the bare “pillow” potential, equation image. In this paper, we have calculated equation image for each interface using Eq. (6), and obtained the screened “pillow” potential, equation image, by means of Eq. (16). In our calculations, equation image is 0.13 eV for the isolated molecule, and 0.43, 0.30, 0.24, and 0.21 eV for the different adlayers with m = 2, 3, 4, and 5; this yields the following values of equation image, 0.07 eV for the molecule, and 0.08 eV for m = 2 and 0.07 eV for m = 3, 4, 5. These “pillow” potentials tend to reduce slightly the values of (CNL − EF) given above.

4.2 Transport energy gap

We again stress that the transport energy gap has been fitted self-consistently to the value given by Eq. (1), by means of the scissor operator mentioned above: the molecular orbitals equation image and equation image introduced in that Hamiltonian correspond to the case of an isolated, but deformed, molecule, so that equation image and equation image are pure molecular orbitals, written in terms of the C atomic-like orbitals. Neglecting C60 molecular orbital mixing and the corresponding contribution of the metal orbitals to the molecular wavefunctions is an approximation validated by the results shown in Figs. 3, 5, and 6, where the energy gap given by Eq. (1) is close to the HOMO–LUMO energy difference.

The use of the molecule charging energy, equation image, as the correction to the KS energy gap, equation image, deserves some comments. The point to realize is that, as shown by Sau et al. 30, one can define U as equation image, where equation image and equation image (assuming the metal Fermi level to be constant). In our particular case, the LUMO and HOMO levels, as well as the CNL, are practically shifted rigidly by the IDIS potential, VIDIS (as can be appreciated in Figs. 3, 5, and 6). This suggests that with a very good accuracy

  • equation image((21))

an equation that can be expected to be a good approximation even if equation image, because the CNL takes an average of the LUMO and HOMO shifts.

A comment on Eq. (21) should be done. When the molecule is interacting with the metal surface the molecular states are no longer well-defined, becoming instead mixed interface states. This means that the induced interface states around the CNL have a mixed character of molecular states coming from above the LUMO and below the HOMO levels. Thus, one could argue that equation image reflects a value of U associated with that mixture of states, instead of the pure HOMO or LUMO states. Nevertheless, we have found that the U-values associated with those levels (above the HOMO or below the LUMO) are very similar to the ones calculated for the HOMO and LUMO levels 29, 61. This justifies our use of Eq. (21) and rationalizes the good results shown by our calculations.

We pass now to make some comments about the “metal surface” dipole correction and its effect on our calculations for U and Ueff. As discussed in Section 2, we first calculate the “metal surface” dipole using the off-diagonal elements of the Green function as afforded by the tight-binding Hamiltonian obtained from our DFT–LDA calculation; in the next step, we calculate the screened potential associated with those off-diagonal terms. This is performed in a simplified calculation by considering only the metal first layer dipoles (all the other are strongly screened) and the mean potential these dipoles create between the molecule and the metal; then, this potential is screened by the screening parameter, S, in the same way that the “pillow” potential was screened in Eq. (16). This is a small effect that tends to reduce VIDIS; in the isolated molecule case (EF − ΦM) is reduced by 0.03 eV, while for the equation image structures, the reduction is around 0.02 eV. These effects also reduce U and Ueff in the way explained above.

A final comment should be made about the atomic orbital basis set used in our calculations. The point to stress is that our basis set is not a converged one; this is reflected in two deficiencies of our calculations: one is about the initial alignment of the MO levels; the second one is related to our calculated LDA energy gap which is a little too large (2.1 eV instead of 1.6 eV). However, as commented in Section 2, this has been corrected in two ways: the misalignment between the initial metal and organic electronic spectra is corrected introducing a rigid shift in the organic levels; on the other hand, the organic energy gap is corrected by means of a operator. These two corrections have mended the most important problems associated with using our restricted basis set. In order to test this procedure, we have performed test calculations using a sp3d5 basis set for C60: in this basis set the energy levels of C60 present a shift of several eV to lower energies as compared to the C60 energy levels for the sp3 basis set. Also, the C60 energy gap is reduced from 2.1 to 1.6 eV (the value obtained in PW LDA/GGA calculations 8). Nevertheless, using the corresponding shift and scissor operators we find only small changes for the values of Et, S, δn, and VIDIS.

However, our pillow potentials are increased in the new basis set by about 20%, so that equation image takes now the values: 0.08 eV for the molecule, 0.10 eV for the m = 2 adlayer case, and 0.08 eV for m = 3, 4, and 5. Preliminary results using a more extended basis for Au, suggest that these pillow potentials can increase up to 0.2 eV for the monolayer case, and 0.16 for the isolated molecule. Notice that for the full monolayer case (m = 2), this dipole potential can reverse the sign of (CNL − EF), so that charge is transferred from the Au surface to the C60 molecules, in agreement with previous theoretical and experimental results 7, 8; this, however does not change our main conclusions.

5 Conclusions

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model and method of calculation
  5. 3 DFT calculations, density of states (DOS), the IDIS-model, and the molecule charging energy
  6. 4 Pillow effects and the transport energy gap of the molecule
  7. 5 Conclusions
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

In this paper we have studied the C60/Au(111) MO interface formation for different coverages, going from the single C60 molecule adsorbed on the Au(111) surface to the monolayer case (the equation image structure). In our analysis we determine the atomic geometry for the C60/Au(111) interface by means of a local orbital DFT technique, taking into account vdW forces. We analyze the MO barrier formation for the following coverages of C60 molecules: θ = 1, 4/9, 1/4, and 4/25, corresponding to the adsorption of C60 molecules with equation image (m = 2, 3, 4, 5) adlayer structures. We have also calculated the single molecule limit, adsorbing one C60 molecule on a 8 × 8-Au(111)-cluster. From these calculations we obtain the DOS on the C60 molecules, the position of the interface Fermi level, and the position of HOMO and LUMO levels. An important step in our calculations is the self-consistent correction of the transport gap of the organic material, using Eq. (1). The MO interface barrier formation is analyzed in terms of the IDIS model. For this purpose we also calculate the charge transfer at the interface, the CNL of the organic layer, the interface dipole potential and the interface screening parameter S. In all the cases analyzed in this paper the C60 CNL is found at the same position, within ±0.02 eV. Regarding this screening parameter, we obtain an important change from the isolated molecule on the Au(111) surface, S = 0.53, to the monolayer case, S = 0.19 (notice that S = 1 corresponds to no screening while S = 0 corresponds to perfect screening). This indicates the important role played by other adsorbed molecules in the screening properties of the interface and in the interface barrier height formation.

The analysis in terms of the IDIS model allows us to calculate equation image; this quantity, for the single molecule case, corresponds to the charging energy of the molecule U used in Eq. (1) to correct the KS LDA gap to obtain the transport gap: equation image. This correction is introduced in our calculations using a scissor operator, and is calculated self-consistently. We obtain U = 1.5 eV, and a transport gap for C60 of 3.1 eV in good agreement with scanning tunneling spectroscopy experiments 6 and other theoretical calculations 30 based on a GW-approach. In the adlayer cases, Ueff is a combination of the intra-site charging energy U and intersite electron–electron interaction Ji for electrons in different C60 molecules. The values obtained for Ueff (4.4, 3.0, 2.4, and 2.1 eV for equation image, m = 2, 3, 4, 5) can be used to calculate Ji and thus obtain an effective many-body Hamiltonian for the adlayers 60.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model and method of calculation
  5. 3 DFT calculations, density of states (DOS), the IDIS-model, and the molecule charging energy
  6. 4 Pillow effects and the transport energy gap of the molecule
  7. 5 Conclusions
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

This work is supported by Spanish MICIIN under contracts MAT2007-60966 and FIS2010-16046, the Comunidad de Madrid under contract S2009/MAT-1467 and the European Project MINOTOR (FP7-NMP3-SL-2009-228424). E. A. gratefully acknowledges financial support by the Consejería de Educación de la Comunidad de Madrid, the Fondo Social Europeo and the European Project MINOTOR. We are grateful for stimulating discussions with J. I. Martínez.

Biographical Information

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model and method of calculation
  5. 3 DFT calculations, density of states (DOS), the IDIS-model, and the molecule charging energy
  6. 4 Pillow effects and the transport energy gap of the molecule
  7. 5 Conclusions
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

Enrique Abad received his PhD in 2011 from the Universidad Autónoma de Madrid. His thesis entitled “Energy level alignment and electron transport through metal/organic contacts: from interfaces to molecular electronics” was supervised by Fernando Flores and José Ortega. He has co-authored nine papers in the field of metal/organic interactions focusing on interfaces and molecular electronics. In 2012 he moves to the group of Johannes Kästner at Universität Stuttgart, to work on quantum tunneling in enzymes. His research interests include metal–organic interactions and application of mixed quantum/classical calculations to biological molecules.

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Biographical Information

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Model and method of calculation
  5. 3 DFT calculations, density of states (DOS), the IDIS-model, and the molecule charging energy
  6. 4 Pillow effects and the transport energy gap of the molecule
  7. 5 Conclusions
  8. Acknowledgements
  9. Biographical Information
  10. Biographical Information

José Ortega received his PhD in 1991 from the Universidad Autónoma de Madrid (UAM). In 1992–1993 he was a postdoctoral researcher at Arizona State University, and in 1993–1994 a postdoctoral researcher at the University of Cambridge. From 1995 up to now he has been an Associate Professor at UAM. His research interests include the development and application of local-orbital first-principles techniques for the atomistic simulation of complex materials, such as biological molecules, non-adiabatic molecular dynamics, the molecule/metal interface, semiconductor surfaces, nanowires, nanocontacts, and the theoretical simulation of STM.

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