J. Phys. Chem. B 2001, 105, 8979-8988
8979
The Need for Quantum-Mechanical Treatment of Capacitance and Related Properties of Nanoelectrodes† Jeffrey R. Reimers*,‡ and Noel S. Hush‡,§ School of Chemistry, UniVersity of Sydney, NSW 2006, Australia, and Department of Biochemistry, UniVersity of Sydney, NSW 2006, Australia ReceiVed: March 19, 2001; In Final Form: June 26, 2001
Capacitance and other properties of large metal clusters proposed for use as nanoelectrodes in complex molecular-electronic devices, or as cores of the monolayer-passivated nanoparticles studied by Murray (J. Phys. Chem. B 1999, 103, 9996), are discussed using atomistic formalisms based on classical electrostatics as well as INDO electronic structure theory. Using classical electrostatics, both finite-size and atomicity effects are found to be important for properties such as the surface charge distribution but unimportant for other properties such as the electric field profile between electrodes. The INDO and classical atomistic charge distributions are found to be strikingly different, with both departing from textbook expectations based on theorems of classical continuum electrostatics such as Gauss’ law. For linear chains of metallic atoms, ab initio full configuration interaction as well as density-functional (DFT) calculations validate the INDO/S picture in which both positively and negatively charged atoms appear within a chain of net positive charge, contrary to the classical treatment that permits only distribution of the net charge. Examination of the form of the INDO/S Hamiltonian reveals that a key aspect of the failure of classical atomistic electrostatics arises from its treatment of self-energy (the energy required to store a finite charge in the finite atomic volume). Exchange operators present in the quantum approaches halve the classical self-energy contributions, facilitating charging. Even the requirement that atomic charges be distributed across the width of a surface atomic plane is found to significantly modify the classical self-energy and hence induce large short-range deviations from standard capacitance relationships. For large clusters, the INDO/S results are shown to depict qualitatively reasonable properties by comparison with published DFT calculations. INDO/S may prove an efficient computational scheme for the study of a wide range of nanoparticle electronic properties: here, we deduce the voltage differential arising from the cluster to cluster charge-transfer state.
1. Introduction The rapidly increasing interest in molecular electronics is resulting in a number of diverse approaches to fabrication of nanometer-sized structures that can in principle function as wires, electrodes, capacitors, and other elements of relevance to electrical response at the nanoscopic level; see, e.g., ref 1. Perhaps the most basic of these is the nanoscale metal cluster, which can function, inter alia, in each of the above manners. Until recently, no well-defined clusters of this type that can be prepared reproducibly have been produced. However, a number of workers have in the past few years isolated nanometer-scale metal clusters in solvent-free form that are stable can be reversibly redissolved and can be subjected to further synthetic manipulation: the metals include Au, Ag, Pt, and alloys, and their properties have been extensively discussed by Chen and Murray (see ref 2 and references therein). Cluster stabilization is accomplished through coverage by ligand monolayers of alkanethiolates and related thiolate ligands using the Brust3 reaction. Stabilized clusters of this kind exhibit quantized capacitance charging.2,4 We are concerned here with the metal nanoclusters that form the cores of such systems. Extensive †
Part of the special issue “Royce W. Murray Festschrift”. * To whom correspondence should be addressed: chem.usyd.edu.au. ‡ School of Chemistry. § Department of Biochemistry.
reimers@
theoretical treatments of these nanowires and their one-atom contacts have recently been advanced by Landman and coworkers (see, e.g., refs 5 and 6). We are particularly interested in the relationship between actual properties and those expected from atomistic classical electrostatics and from continuum classical electrostatics for macroscopic clusters, including in particular the applicability of Coulomb’s and Gauss’ law. Gauss’ law derives from continuum electrostatics and is not strictly applicable to atomistic problems; while we use Coulomb’s law to describe the interaction of charge densities within atoms with either themselves or those of other atoms, Coulomb’s law is found not to hold on an atom by atom basis. To test the reliability of atomistic classical electrostatics, quantum-mechanical methodologies are used. We employ ab initio full configuration interaction as well as density-functional methods to study charge distributions for linear chains of atoms, but to perform quantum-mechanical calculations on large clusters, we employ the INDO/S computational scheme.7-10 The simplest quantum-mechanical approach, the Hu¨ckel or tightbinding method, has been used with some success in a number of applications, e.g., characterization of the electrical characteristics of nanotubes (see, e.g., ref 11) and characterization of electrode-single molecule-electrode current-voltage relationships (see, e.g., refs 12-15). However, in general, the method is only qualitatively descriptive of basic molecular properties.
10.1021/jp011023i CCC: $20.00 © 2001 American Chemical Society Published on Web 08/10/2001
8980 J. Phys. Chem. B, Vol. 105, No. 37, 2001 This is well-known, being demonstrated, for example, by the widely different interatomic interactions required in order to fit data of different types (e.g., ionization energies and electronic transition energies) for series of closely analogous molecules.16 It also provides better descriptions for smaller molecules than for larger ones, eventually incorrectly yielding metallic-type zero band-gap behavior for extensive systems. Successful applications of this method always have, at their core, an external buildingin of the basic features of the properties being investigated into the specification of slightly modified Hamiltonian operators. Molecular properties are controlled by the fine details of Coulombic and exchange interactions, and the fact that these are treated only implicitly is a weakness of the method. INDO methodology overcomes these limitations. In section 2, the properties of classical atomistic charge descriptions are considered. In particular, the need to introduce self-energy expressions is discussed. This and other differences between classical atomistic and classical continuum electrostatics are discussed in section 3, which considers the atomistic parallelplate capacitor. The pronounced effects of quantum mechanics are demonstrated in section 4 through comparison of the classical and quantum (full configuration interaction, DFT, and INDO/ S) results for the linear chains H5+ and Au5+. Section 5 compares the classical and INDO energy expressions, highlighting that a major cause of the failure of classical electrostatics to produce realistic charge distributions is its neglect of electron exchange in evaluating the atomic self-energy. In section 6, classical and INDO/S results are presented for isolated neutral or charged clusters, and the quantitative performance of INDO/S for large metal clusters is examined in section 7 through comparison with DFT results. In section 8, the cluster to cluster charge-transfer state is investigated and properties such as the voltage between the two electrode-representing clusters is deduced. 2. Atomicity and Classical Electrostatics A metal is treated in classical electrostatics as a perfect uniform conductor. As such, only one type of charge (positive or negative) may reside on the metal at one time, and Gauss’ law requires that this charge is distributed continuously over the metallic surface with no net charge present in the bulk material. The introduction of atoms necessitates major changes to this physical picture. Both positive (nuclear) and negative (electronic) charges are simultaneously present within atoms, and the net atomic charge is determined by the balance between them. An atomistic classical electrostatics approach can be developed by considering each atom as a separate unit represented by its net atomic charge, and allowing these units to interact with each other according to the laws of classical electrostatics. Each atom therefore contains finite electron density. Assembling a finite charge on an object requires expenditure of energy; this is known as the charging self-energy and can be expressed as E ) q2/2C where q is the charge and C is in effect the capacitance of the atom.17 The nature of the self-energy is strongly dependent on the dimensionality of the system: if the charge distribution is constrained to a line (1D), a plane (2D), or an atomic volume (3D), the self-energy decreases as the dimensionality, and hence the degree of spreading-out of the charge, increases. Gauss’ law for a continuous metal requires that the charge resides on its infinitesimally thin surface; this result can be introduced as an assumption in an atomistic theory using a 2D formalism for the self-energy. However, a layer of atoms has finite thickness and the atomic charge per atom is actually distributed throughout it, demanding a 3D determination of the self-energy.
Reimers and Hush
Figure 1. 2D hexagonal unit cell used to surround a gold atom (marked O) on the (111) surface, as well as the 96 point integration grid (marked b) used to divide the unit cell into equal-area triangular sections.
Self-energy is extremely important in all quantum-mechanical theories of the electronic structure of atomistic systems, and the ability of any atomic electronic structure scheme to predict realistic charge distributions is closely related to the quality of its description of the self-energy. We also note that, in the context of chemisorption and electrode-molecule-electrode conduction, the self-energy is often generalized to be a complex quantity.18-20 This allows the electronic structure of the electrodes to be represented by an imaginary component of the self-energy of a reduced-dimensionality molecule-only system. In the isolated cluster systems we are concerned with here, there is no external bath and hence the self-energy remains a purely real quantity. We consider first the self-energy appropriate for the 2D situation in which the atoms are taken from the (111) surface layer of gold, a layer with hexagonal symmetry. We assume that the charge on an atom is uniformly distributed within an (infinitely thin) hexagon placed around its center, as shown in Figure 1. A charge of 1 electron (or, equivalently, hole) is introduced to the atom and distributed uniformly over an integration grid containing N internal points; a sample grid with N ) 96 is shown the figure. In this grid, each point is at the center of a fixed-area triangular subsection. The self-energy is then evaluated from the Coulomb repulsion energy of the charges on the integration grid. Using a total of N ) 96, 600, 5400, and 15 000 integration points, the self-energy γ evaluates to 15.953, 16.029, 16.087, and 16.101 eV, respectively. Numerical integration is thus shown to produce quite slowly convergent results. The net Coulomb attraction between an electron distributed over this 2D layer and a nearby nucleus, or alternatively the net Coulomb repulsion between two such electrons on atoms separated a distance r from each other, may also be obtained by numerical integration over the associated charge densities. Results for the two-electron repulsion integrals are shown in Figure 2 where they are compared to the repulsion expected for point charges according to Coulomb’s law. The atomistic 2D repulsion is seen to reduce to the Coulomb repulsion at distances r much greater than the size of the atom, but at short distances the interaction is bounded above by the self-energy γ. In summary, we see that while the infinitesimal elements of the charge densities of two atoms interact according to Coulomb’s law, the total energy can be expressed without approximation as the non-Coulomb interaction of two atoms. While the need to introduce explicit one-electron and two-
QM Treatment of Properties of Nanoelectrodes
J. Phys. Chem. B, Vol. 105, No. 37, 2001 8981 energy is significantly larger than the 3D experimental value; the value for a uniform sphere of charge whose volume is equal to that of an atom in a gold crystal is23 12q2/5r ) 10.86 eV 3. A Classical Atomistic Parallel-Plate Capacitor
Figure 2. Potential energy of two interacting charges of magnitude e separated a distance r along a line normal to one of the edges of the hexagonal unit cell used to represent the charge of a 2D sheet of gold (111) atoms compared to that for two classical point charges at the same separation and that for two 3D atoms obtained using the Mataga approximation.
electron Coulomb integrals into quantum treatments is wellknown, we see that even when classical electrostatics is used, Coulomb’s law as such is inappropriate, at atomic-scale distances, for the description of interatomic interactions. In Figure 2 are also shown results obtained using Mataga’s21,22 interpolation scheme to approximate the 3D Coulomb potential energy E for two interacting gold atoms; in atomic units, this is given by
E(r) ) q2Γ(r)
Γ(r) )
1 r + γ-1
(1)
At large r, this formula reduces to Coulomb’s law while E(0) is simply the self-energy γ. Evoking Koopmans’ theorem, the self-energy is obtained from the difference between the observed atomic ionization energy and electron affinity and is 6.8 eV for gold. As expected, the 2D uniform charge value of the self-
For a parallel-plate capacitor made of continuous material, the electrical properties of the system are not affected by the thickness of the plates. The simplest atomistic model for a parallel-plate capacitor is thus two facing planes of atoms. Figure 3 shows two charge distributions obtained by minimizing the classical potential energy for a model parallel-plate capacitor with two parallel sheets of charge, each containing 31 atoms from the gold (111) face in an approximately circular arrangement. The diameter of the plates is 1.6 nm, while the spacing between them is 0.84 nm; the planes are parallel, but there is a small offset of 0.022 nm in their locations, chosen24 such that the central atoms could be spanned by trans-C2H2S2 (this electrode-molecule-electrode problem we have considered in detail elsewhere24). As a result, the total system has C2h rather than D3h symmetry. In this and subsequent figures, the planes are viewed nearly edge-on at an angle of 82° to their normal. The total energy is evaluated using an integration grid containing 9600 points per atom, as described above, with each point on the integration grid of one atom interacting with all those of another in accordance with Coulomb’s law. For this system we find two distinct solutions, both shown in Figure 3, to the classical electrostatics problem; for all other systems that we have considered, various local minima are found, but these all differ insignificantly from each other. This is the smallest system studied, and it may be for small systems that significantly different local minima may arise. In this case the two solutions differ in energy by just 2.5%. In Figure 3 a net electric charge of q ) 1 e is transferred from one electrode to the other, with the net charge per layer being written above each of the layers in this and subsequent figures. The atomic charge on each atom is indicated by colored circles whose area is proportional to the charge, red for positive
Figure 3. Two solutions of energy E for the classical charge distribution for a model parallel-plate capacitor with two parallel sheets of charge, each containing 31 atoms from the gold (111) face in an approximately circular arrangement, viewed nearly edge-on. The diameter of the plates is 1.6 nm, while the spacing between them is 0.84 nm. For each atom, the magnitude of the charge is proportional to the area of the drawn circle, while red indicates positive and green indicates negative charge.
8982 J. Phys. Chem. B, Vol. 105, No. 37, 2001
Reimers and Hush TABLE 1: Comparison of Properties Calculated Using Classical Electrostatics for the Two 31-Atom Planes Shown in Figure 3 and Those Expected for Continuous Infinite Plates two 31-atom planes property
units
E F V CqV CqE CEV
eV V nm-1 V aF aF aF
soln 1
soln 2
infinite continuous plates
1.49 3.54 2.98 0.0538 0.0537 0.0539
1.53 4.02 3.38 .0474 .0523 .0429
3.42 8.13 6.84 0.0234 0.0234 0.0234
TABLE 2: Capacitance Ratios Obtained for Two Parallel Planes of 2029 Gold Atoms Each Using Mataga’s Formula for the Classical Coulomb Energies and Self-Energies l/nm CqV/C∞ CqV/CEV Figure 4. Electric field emanating from the lower-energy electronic structure shown in Figure 3 for a model atomistic parallel-plate capacitor, contoured on the (C2h) symmetry plane that bisects each plate. The contours are equally spaced in units of 0.3 V, and the extent of the clusters above and below the plane of section are indicated by thick vertical lines.
charge and green for negative. In the high-energy solution, 65% of the charge resides on the 38 boundary atoms that constitute 61% of the total number of atoms, and hence the charge is evenly distributed, while for the low-energy solution 75% of the charge is localized on the outside atoms. Localization of charge on the outside atoms is consistent with the well-known result that the electric field emanating from a metal is largest near atoms of low coordination. For the low-energy solution, this fraction is 57% if the total charge on the cluster is reduced to 0.5 e and 87% when increased to 2 e; the tendency to localize charge on the outside of the cluster is thus indicative of the relatively high atomic charge densities present in this 31-atom cluster. For a near-infinite continuous parallel-plate capacitor, essentially all of the charge is expected to be located on the face. The electric potential originating from the two charge distributions is very similar, and that for the lowest-energy solution is shown in Figure 4. It indicates that, in the central region at least 0.2 nm from each face and 0.3 nm from the edges, the electric field is quite uniform. Hence a cluster of this size has some properties that are reminiscent of an infinite surface but not others. In Table 1 is shown for each solution the total energy E, the electric field strength F at the midpoint between the plates, the voltage obtained from using as V ) lF where l ) 0.84 nm is the interplate separation, and the capacitances
CqV ) q/V CqV ) q2/2E
(2)
CEV ) 2E/V2 obtained using various expressions from continuum classical electrostatics. From either solution, the electric field strength and charging energy are less than half and the capacitance double that expected for continuous infinite plates. For the lower-energy solution, all three formulas return consistent values for the capacitance with CqV/CEV ) CqE/CqV ) 1.002; however, significant deviations occur for the slightly higher-energy solution for which this ratio is 1.105. Comparing the two
0.42 1.00 0.42
0.84 1.07 0.53
1.68 1.14 0.60
3.26 1.32 0.64
solutions, their energy difference is just 2.5% while their voltage difference is 12%, and hence, CqE provides the estimate of the capacitance which has least variance. Also, this capacitance is the most straightforward to determine as q and E are welldefined while V may be estimated in a number of ways. When the plate diameter is expanded to 13.6 nm (2029 atoms per plate), only one distinct solution is obtained for which the field at the midpoint of the plates is 0.116 V nm-1, only 8% less than that expected for infinite continuous plates. The ratio CqV/CEV becomes 0.97, closer to the expected value than that for the lowest-energy solution of the 31-atom per plate problem. However, if the interplate separation l is reduced from 0.84 to 0.105 nm, this ratio decreases to 0.911, and we see that the continuum infinite plate result is only obtained in the limit where the interplate separation is larger than the interatomic separation but much less than the plate diameter. In the previous calculations, the atoms were treated as forming an infinitely thin sheet of charge through the use of the 2D selfenergy expression to mimic the electronic structure predicted by application of Gauss’ law to a continuous 3D metal. We find replacing this with the 3D Mataga formula for the selfenergy has a dramatic effect on calculated properties, as this smears out the charge distribution over a thickness of atomic dimension rather that forcing it to reside “on the surface” of the cluster. Results calculated for the 2029 atom per layer system are shown in Table 2. They display a large variation in CqV/C∞ as the interplane separation l increases, despite the largest separation of 3.26 nm remaining small compared to the plate diameter of 13.6 nm. Further, CqV/CEV ranges from 0.42 to 0.64, indicating that the various possible definitions of capacitance in this microscopic system are no longer self-consistent. Clearly, interplate spacings l must be of a length scale larger than the length scale for atomistic deviations from Coulomb’s law (see Figure 2) for simple notions of capacitance to be applicable. The self-energy contribution is seen to be profoundly important by inspection of the various components contributing to the total energy; we consider in detail the 2029 atom per layer problem at l ) 0.84 nm by assuming a 2D uniform charge within each atom. The total energy of 0.006 29 eV is composed of an interplane attraction of -0.35236 eV, intraplate interatomic repulsions totaling 0.350 76 eV, and a total self-energy of 0.006 48 eV. It arises through near-complete cancellation of large repulsive and attractive contributions, with the total energy being of the same order as the self-energy! Hence the selfenergy, a term that might have been thought to be of minor importance, becomes in fact the central focus.
QM Treatment of Properties of Nanoelectrodes
Figure 5. Atomic charges for H5+ and Au5+ linear chains. For each atom, the magnitude of the charge is proportional to the area of the drawn circle, while red indicates positive and green indicates negative charge.
4. Quantum and Classical Charge Distributions for Linear Chains of Atoms One limit of the properties of nanoscopic electrodes is that of very small clusters, and here we focus on the properties of a linear chain of five (metallic) atoms. It is for systems such as this that Hu¨ckel (tight binding) Hamiltonians, often used in quantum-mechanical electronic-structure calculations, are most appropriate, and there is much interest in electrode-moleculeelectrode conduction through such systems.25,26 It is highly desirable that any computational method, whether classical or quantum, for the electronic charge distribution in nanoelectronic structures be able to produce realistic results in this limit. Figure 5 shows Mulliken charges and electrostatic-potential (ESP) charges obtained by a number of quantum electronic structure methods for linear H5+ and Au5+, as well as classical results obtained using the 2D uniform charge density per atom model. In each case, the atoms are taken to be equally spaced at distances of 0.0635 nm (H-H) and 0.0288 nm (Au-Au). For the quantum methods, the standard 6-311G basis set was used for H while LANL2DZ27 was used for Au; calculations were performed using full configuration interaction (FCI) using MOLPRO28 and the B3LYP density functional using GAUSSIAN-98.29 Also shown in the figure are self-consistent field (SCF) results obtained using the quantum semiempirical INDO/S method discussed in detail in the next section. The classical results, which are independent of atom type, show an almost uniform charge distribution along the chain, with the outside atoms bearing slightly more charge than the inside ones. This pattern is in stark contrast to those from the quantum methods, however, which predict large positive charge on the central atom, smaller positive charges on the end atoms, and significant negatiVe charges on the intermediary atoms. Indeed, the spontaneous charging of the system, which also occurs in quantum calculations on neutral clusters of metal atoms, is not permitted under classical electrostatics. Such effects are included in approximate quantum schemes such as INDO, however, and as shown in the next section, the problems of the classical approach are due to neglect of the effects of electron exchange on the self-energy. 5. The INDO/S Method and the Cause of the Failure of Classical Electrostatics INDO/S7-10 is a semiempirical quantum mechanical scheme for the determination of the electronic properties of molecules.
J. Phys. Chem. B, Vol. 105, No. 37, 2001 8983 It is based on the Hartree-Fock self-consistent-field (SCF) plus configuration-interaction (CI) approach, modified such that many of the resulting atomic orbital integrals are neglected while the remainder are approximated using simple analytical expressions. Only the valence electrons are usually included, and experimental data are used to derive values for the atomic parameters used in the integral expressions. One such expression is the Mataga equation (eq 1) used in describing the Coulomb, exchange, and self-energy interactions; its parameter γ is set to reproduce the observed difference between the atomic ionization energy and electron affinity, hence embodying a realistic description of the self-energy associated with adding or removing an electron from the atom. We use an INDO/S scheme in which only the valence s orbital per gold atom is explicitly included,30 although we have also generated variants in which the valence p and d orbitals are included.30 In the s-only scheme, four parameters are required in order to describe a gold atom: γ ) 6.910 eV, the exponent for the 7s Slater orbital that was fitted to ζ ) 1.986 au-1, the experimental atomic ionization energy Ip ) 9.226 eV, and one freely adjustable parameter that specifies interatomic resonance energies, β. We adjusted30 β to fit DFT-calculated ionization energies, dipole moments, and dissociation energies for a range of molecules. Sensible molecular properties are obtained for -1.5 eV > β > -4 eV, and we chose β ) -2.5 eV. INDO/S can be considered to be a generalization of, in increasing order of sophistication, the Hu¨ckel31 (tight binding), Hubbard,32 Pariser-Parr-Pople (PPP),23,33 and CNDO/S34-36 methods; it is the simplest method that describes in a qualitatively correct manner all of the electronic charge, spin, and excited states relevant to electrodesingle molecule-electrode conduction. Recently,37 the PPP scheme has been applied to study image electrostatics of finite clusters. For n equivalent atoms with one s orbital and one valence electron each, the INDO/S SCF energy7-10 for n even can be written succinctly as
EINDO ) -nIP +
nΓ11
n
+
4
∑ i)1
Γ11qi2
n
+
4
∑ qiqjΓij + i
