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## Journal of the Chilean Chemical Society

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*On-line version* ISSN 0717-9707

### J. Chil. Chem. Soc. vol.51 no.2 Concepción June 2006

#### http://dx.doi.org/10.4067/S0717-97072006000200014

J. Chil. Chem. Soc., 51, Nº 2 (2006) , pags: 905-912
Programa de Doctorado en Fisicoquímica Molecular. Facultad de Ecología y Recursos Naturales, Universidad Andrés Bello, Santiago, Chile; and Universidad de Chile, Facultad de Ciencias, Departamento de Química. Santiago, Chile.
A Density Functional microscopic model of C Keywords: Buckminsterfullerene, ADF, band structure, eigenvalue spectrum, molecular electronics, solid band gap.
Molecular electronics has reached a point in which molecules are not the starting point for bulk electronic materials but the active device components within electronic circuitry [1]. In this sense molecular electronics can be defined as the technology using single molecules, small groups of molecules, molecular wires or similar nanostructures to perform electronic functions [2]. The work at this level needs a correct knowledge of the single molecule's geometric, optical, chemical, electronic, etc. properties. Due to the small size of the constituents, new physical and chemical effects are to be expected, or known properties may be improved substantially. The logical approach to the experimental study at this level is the single molecule spectroscopy [3]. The ideal situation is the experimental study of molecules in a highly diluted gas phase. Essential experimental tools are Surface-Enhanced Resonance Raman Scattering spectroscopy, Scanning Near-Field Optical Microscopy, Scanning Probe Microscopy (SPM), Total Internal Reflection Fluorescence Microscopy, etc. SPM is especially useful for the study of structural (topographic) and functional (electronic conduction) properties at the level of a single molecule, as a function of the environment conditions. Nevertheless, the quantity of data recorded up today with these techniques is still not enough for comparison with theoretical results for a great number of different molecular systems. The lack of enough single molecule studies makes it necessary to use, as a first approach, experimental results obtained for bulk materials. We must emphasize here that a comparison between experimental results obtained for the bulk material and the single- molecule theoretical results can be done only on a qualitative basis. At the level of solid molecular systems, one of the properties used to characterize them is their band structure. Two of the bands, the valence band (VB) and the conduction band (CB), together with their energy gap (VB-CB gap), are useful to classify a solid as an insulator, a semiconductor, or a metal. An associated concept is that of density of states (DOS), which describes the energy levels per unit energy increment. The band structure for bulk materials is obtained with several experimental techniques like optical reflectivity [4-7] and photoelectron spectroscopy (PES) [8]. One of the most useful implementations of PES is the angle-resolved photoemission spectroscopy technique. With this technique the most crucial low-energy excitations near the Fermi surface may be directly probed [9]. The analysis of the conduction band structure is carried out with the technique of inverse photoemission spectroscopy (IPES, the new name for bremsstrahlung isochromat spectroscopy), which yields the energy and momentum of a photon emitted when an electron makes the transition from the conduction band to the valence band [10, 11].Quantum chemistry represents an essential tool to assign photoelectron spectra, to analyze band structure and to rationalize experimental observations. We shall center ourselves hereafter only on band structures. The general procedure to obtain the band structure of a single molecule within any standard quantum chemical scheme (semiempirical, There are many conceptual problems remaining unsolved to date at the experimental and theoretical levels. The experimentalist must solve problems such as device addressability, heat dissipation, defined placement of molecules and prevention of diffusion, as well as the exchange of information with other devices and the macroscopic world. At the level of the theory, an analysis of the literature shows that some fundamental aspects of the broadening process have never been fully addressed. The first one is that, in contrast with experimental measurements, the convolution procedure is not limited by resolution. The second one is related to the energy distance between each calculated DOS value (called here scanning distance, SD) to generate the theoretical curve. The use of different resolution and SD values produces different spectra. The third, and most important one, is related to the following fact. In general, theoretical calculations are compared with experimental results obtained by separate for the valence and conduction bands for the bulk material. The "experimental" gap between the valence and conduction bands is determined in an indirect way. On the other hand, if we take the whole molecular eigenvalue spectra of a single molecule, a putative BV-CB gap is obtained. Nevertheless, this putative BV-CB gap is influenced by the convoluting function's half-width and by the scanning distance. The fourth point is related to the selection of the experimental results we pretend to compare with theory and it is resumed in the following question. What experimental results should be selected to compare with theoretical ones and for what purposes? This question is very important when molecules are the single bricks in electronic circuitry. In this paper we address the above points through the analysis of a simple molecular model and its comparison with some experimental results. Also we discuss the problem associated with the "experimental" determination of the VB-CB gap in solids and the role of theory in selecting experimental results.
The first step consists in building an appropriate molecular model to deduce some possible restrictions to be imposed on the broadening process for the theoretical calculations. The foregoing analysis will be based on the C For this task, a microscopy model of Buckminsterfullerene C With these MOs we shall build the theoretical DOS spectrum for the first two valence and first three conduction bands. The DOS spectrum can be obtained through the convolution of the MOs with a variety of mathematical functions. For our analysis we selected a Gaussian function [20], a 50:50 mixture of Gaussian and Lorentzian functions [21] and a purely Lorentzian function taken from Ref. 21. Other convoluting functions have been also used in the literature [22]. The mathematical forms of these functions are:
for a purely Gaussian function,
for a 50:50 combination of Gaussian and Lorentzian functions, and
for a purely Lorentzian function. In Eq. 1 to 3, DOS(E) is the density of states at energy E, E Examining the form of Eq. 1 to 3 we may see that the choice of s and SD will necessarily influence the resulting DOS curves shape. Therefore, we calculated the DOS curve for several values of both parameters. Also, for each couple of values of s and SD we obtained a VB-CB energy "gap". We used these theoretical results to find possible restrictions to apply for the selection of the correct theoretical spectrum. The second step consisted in comparing the theoretical DOS spectrum with some experimental spectra selected from the literature [23-26]. The last part is a short discussion about how experimentalists select their results.
The C
Let us begin by making some purely theoretical considerations. Table 1 shows the degeneracies and energies of the selected MOs. The HOMO is separated by about 1.15 eV from the HOMO-1 and it will form the first valence band (VB1).The HOMO-1 is separated only by about 0.11 eV from the HOMO-2. Therefore the HOMO-1 and HOMO-2 should both form the second valence band (VB2) which must have more states that VB1. This is the first formal qualitative restriction that we must impose on the DOS spectrum results.
The LUMO, LUMO+1 and LUMO+2 are separated by about 1 eV. The degeneracies are the same (3) for all. LUMO+3 (fivefold degenerate) is separated by only 0.16 eV from LUMO+2. Therefore the first two conduction bands of the DOS spectrum (the LUMO-derived and LUMO+1-derived respectively) should have almost the same number of states but the third band must have more states because it is formed by LUMO+2 and LUMO+3. Associating the above number of states with the value at the peak of each band in the theoretical DOS spectrum we obtain some relationships between the band's relative heights which are displayed in Table 2.
Let us analyze now the results of the various convolution results at the light of the above restrictions. Figure 1 shows the DOS curves for several values of s for the case of a Gaussian function with a fixed scanning distance of 0.1 eV. Figures 2 and 3 show the same curves for a Lorentzian function and a 50:50 Gaussian and Lorentzian functions mixture respectively. For the Gaussian case we may see that the only acceptable spectra is the one obtained with
Figures 4 to 6 show, for the same three functions, the influence of the variation of the scanning distance on the DOS curves when the broadening parameter is kept constant (s = 0.1 eV). The associated VB-CB band "gaps" are shown in Table 4. We may see immediately that a rising of the scanning distance SD= 0.3 and 0.5 eV) produces spectra which are not compatible with the theoretical considerations exposed above. The conclusion is that a scanning distance of 0.1 eV or less is to be used. We must note that diminishing the scanning distance could influence only a band's fine structure.
This spectra has a VB-CB band "gap" of about 1.3 eV and the relative intensities are I(CB2) H" 1.1 I(CB1), I(VB1) H" 1.3 I(CB1), I(VB2) H" 2.0 I(CB1) and I(CB3) H" 1.6 I(CB1). These intensities do not match very well the theoretical ones. The BV-CB band "gap" obtained here should be considered only as the lower bound of the HOMO-LUMO gap or an approximate value for the lowest p
As it has been stressed, the molecular property that corresponds to the band gap of the molecular solid is the difference between the ionization potential (IP) and the electron affinity (EA) of the isolated molecule and not the lowest excitation energy [26].
Using the experimental values IP = 7.6 eV and EA = 2.7 eV we get (IP-EA) = 4.9 eV. Some reported values for the solid band gap obtained from experimental results are: 1.85 eV [31], less that 3.5 eV [26], 1.5 eV [32], 2.3 eV [25] and less than 3.7 eV [24]. Why are these values so different? The answer is that the band gap is a property that is not determined in a direct way. Interestingly the above results below 2 eV seem to be closer to the HOMO-LUMO gap rather than to the solid band gap. Let us examine some experimental results to clarify this point. Lof et al. [25] employed 3 to 6 layers of C
Our calculated HOMO-LUMO gap is about 1.7 eV. High-resolution electron-energy-loss spectroscopy studies of C
Figures 9 and 10 show the experimental PES results for the valence band of condensed C
Regarding the valence band region features displayed in figures 9 to 12 it can be observed that the line shapes change dramatically as a function of the incident photon energy. It is expected that the photons with lower energies provide information mainly from the upper part of the valence band.
Let us center our attention on the experimentally resolvable features of the valence bands VB1 and VB2. We shall look for cases in which the experimental intensity of the VB2 band is higher that the intensity of the VB1 band (about 1.8 times, see Table 2). Figure 9 shows that this condition is fulfilled for hr = 65 and 170 eV, the latter being more acceptable. In Fig. 10, with hr = 50 eV, VB1 and VB2 have almost the same intensity. In Fig. 11, with hr = 70 eV, only in the case of C The empty states of C
With these criteria, the best experimental result for the valence band displayed in Fig. 9 should be selected. It corresponds to a photon energy of 170 eV. In the case of the conduction band the results displayed in Fig. 13, corresponding to an electron beam energy of 15.25 eV, should be selected. Ohno et al. made a different choice. They used the 65 eV spectrum of Fig. 9 and the spectrum with hr = 19.25 eV of Fig. 13 [24]. normalized them so that the total intensity of the LUMO-derived peak is about 60% of the HOMO-derived peak we follow a similar procedure than Ohno et al [24], procedure that is in agreement with the theoretical results presented in Table 2. Does this mean that other experimental results should be neglected? The answer is that these results are explainable in terms of more complex phenomena and theories but they are discarded for further discussion in these papers only because a philosophical choice was made. This is a clear case of dependence on theory [36]. Then, if we want to provide an "experimental" estimate of the band gap of the solid we must select spectroscopic results matching the theoretical DOS spectra.
Prof. Dr. John H. Weaver (U. of Illinois at Urbana-Champaign, USA) and the American Physical Society are thanked for their permission to reproduce some figures. This work has been partially funded by Departamento de Química, Facultad de Ciencias, Universidad de Chile. Dr. Ramiro Arratia-Pérez (UNAB) is thanked for granting access to the ADF program.
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