A test of the method of images at the surface of molecular materials

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 120, NUMBER 8 22 FEBRUARY 2004 A test of the method of images at the surface of molecular materials A. Eilmes Faculty of Chemistry, Jagiellonian University, Ingardena 3, Kraków, Poland and Department of Chemistry, UMIST, Manchester M60 1QD, United Kingdom R. W. Munn Department of Chemistry, UMIST, Manchester M60 1QD, United Kingdom Received 7 October 2003; accepted 24 November 2003 The method of images is tested by comparing two ways of calculating the polarization energy in crystalline fullerene C 60 and in bulk amorphous polyethylene PE : i treating the whole molecular material microscopically, and ii replacing part of the material by a uniform dielectric continuum of the same relative permittivity. The method of images is accurate to within 5% once the distance of the charge from the surface of the dielectric continuum exceeds about twice the average spacing between the polarizable units in the molecular material. For C 60 crystals the method of images always overestimates the magnitude of the polarization energy, partly because it ignores the reduction in the relative permittivity of the dielectric continuum near its surface. For amorphous PE the method of images can overestimate or underestimate the true result, depending on the local density around the charge American Institute of Physics. DOI: / I. INTRODUCTION Electronic and electrochemical devices often involve charge transport through a surface or interface of an organic molecular crystal or polymer. Photovoltaic cells, thin-film transistors, light-emitting diodes, and electrochromic devices are based on thin films or layers of dielectric material deposited on a metallic surface or a different dielectric medium. With increasing interest in applying organic materials in such devices, understanding of charge transport and other electronic processes at interfaces has become more important. Electronic polarization of an organic material by a charge carrier significantly affects the carrier energy and hence significantly affects transport of the charge carrier. The value of the polarization energy at a surface or interface may be significantly different from that in the bulk. 1 4 Therefore calculations of the polarization energy at the surfaces are an important part of a complete description of the energetics and mechanism of charge injection and charge transport through the interface. The method usually used to calculate polarization energies at the surface of molecular crystals is the self-consistent polarization field SCPF method. 5 At the interface between the dielectric and a metal or another dielectric one needs to account properly for the polarization induced in the metal or in the other dielectric by charges located in the dielectric. The standard approach is the method of image charges, where such polarization is accounted for by placing charges of opposite sign in the other medium. 6 However, the image charge method is based on a macroscopic treatment of the dielectric medium. In a microscopic description the polarization results from shifts of the electrons, which in a molecular material are confined to the volumes of the molecules if we neglect intermolecular overlap. This raises the question as to how well the field of the image charges mimics the field of these polarized molecules once one needs to deal with lengths on a scale comparable with intermolecular distances or the sizes of the molecules. The limitations of the method of images could be estimated if we were able to compare its predictions to exact results of SCPF calculations that treated both sides of the interface on an equal footing. Calculations for a real interface between different materials are complicated, since they require a knowledge of the polarizability and structure of the two materials, as well as a proper simulation of the structure of the interface. We have therefore sought to obtain information on the limitations of the method of images by calculating polarization energies for bulk materials in two ways. Imagine dividing the infinite bulk medium into two parts with the interface between them close to a polarizing charge. This is illustrated in Fig. 1 a for the bulk crystal, treated as an array of discrete polarizable molecules or, more realistically, submolecules. This is just a notional division of the real bulk crystal, for which the polarization energy can be obtained by standard means. Alternatively, the part containing the external charge can be treated explicitly in this way, but with the other part treated as a dielectric continuum with its polarization described using the image-charge method, as illustrated in Fig. 1 b. Ideally, the results for the two treatments should be the same, and so the difference between them serves to indicate the error attributable to using the method of images. This paper presents comparative calculations of this sort that draw upon our previous SCPF results for polarization energies in fullerene C 60 crystals 7 and for bulk amorphous polyethylene PE. 8 Being of cubic and isotropic symmetry, respectively, these materials have an isotropic dielectric tensor with a single independent nonzero component. II. METHOD The SCPF procedure calculates iteratively the dipole moments induced by an external charge. In the initial step /2004/120(8)/3887/6/$ American Institute of Physics

2 3888 J. Chem. Phys., Vol. 120, No. 8, 22 February 2004 A. Eilmes and R. W. Munn FIG. 1. Two equivalent representation of a crystal with an external charge: a the whole crystal treated as a discrete lattice of polarizable molecules circles ; b part of the crystal treated as a dielectric continuum gray in which image charges and image dipoles are located. the local fields at the positions of the submolecules are assumed to be equal the fields of the external charge. Given the polarizability of the submolecules and the local fields, the induced dipole moments are calculated. Then the local fields are corrected for the field of the induced dipoles and the next approximation for the induced dipoles is obtained. The procedure is repeated until convergence is achieved to within the desired accuracy. The polarization energy P is then calculated as P 1 E 0 2 i p i, 1 i where the index i runs over all polarizable points, E 0 i is the field of the external charge at point i, and p i is the dipole moment induced on submolecule i. In the bulk this procedure is carried out for a spherical region of radius R, and at a surface the procedure is carried out similarly for a hemispherical region of radius R. The above procedure applies for the discrete array of polarizable points. For a proper description of the interface it should also take into account the dielectric continuum on the other side of the interface. This is done by the method of images. Assume that the interface is defined by the plane x 0 with the discrete medium on the positive side x 0 and the dielectric continuum on the negative side x 0 later it proves convenient in one material to choose a different location of the interface, in which case the definitions are changed accordingly. Associated with each charge q at position r (x,y,z) is an image charge q I q positioned at r I ( x,y,z), where ( 1)/( 1) is the dielectric contrast, with the relative permittivity of the dielectric continuum. 9,10 Analogously, associated with each dipole p at position r (x,y,z) is an image dipole p I R"p positioned at r I ( x,y,z), where the matrix R is given by R The image charge and image dipoles contribute to the field experienced by the real dipoles and have to be included during the calculation of the local fields. After each SCPF step, the values of the image dipoles are updated. Otherwise the procedure including images is the same as described above for the bulk medium. After the induced dipole moments and their images have converged, one has all the ingredients to calculate the polarization energy, but exactly how to do so needs elucidation. Equation 1 includes not only the interaction energy between the permanent charge and the induced dipoles, and the mutual interactions between the induced dipoles, but also the polarization work, i.e., the energy required to induce the dipole moments by displacing electrons against the internal forces of molecules. 11 To extract the polarization energy within the method of images we use the following argument. Consider a surface of a dielectric medium and an external charge outside the medium. Using the SCPF method one can find the induced dipole moments and calculate P from Eq. 1. However, since the interaction between a charge and a dipole may be calculated either as the energy of the dipole in the field of the charge or as the energy of the charge in the potential of the dipole, we may alternatively express P as P 1 qv 2 i. 3 i Here q is the external charge and V i is the potential produced at the position of the charge by the dipole on submolecule i. On the other hand, in the method of images, the polarization of the medium is represented by an image charge, which is chosen so that its field and the potential V I outside the dielectric resemble those caused by the real charge distribution, i.e., V I i V i. The polarization energy may be therefore calculated as P 1 2 qvi. In our case the field acting on the dielectric continuum is caused not only by the external charge but also by the induced real dipoles, but, nevertheless, the image charge and image dipoles are still intended to mimic the resulting polarization and to reproduce the resulting potential outside the continuum i.e., in the part treated as discrete. Therefore after convergence has been achieved in the SCPF procedure, the polarization energy may be calculated as P 1 E 0 2 i p i 1 i 2 qvim, 5 where the first term is defined as in Eq. 1 and V IM is the potential produced at the external charge q by the image charge and the image dipoles. Alternatively, with the help of Eq. 3 the polarization energy may be expressed as P 1 2 qvr IM, where V R IM is the potential at the charge position caused by the image charge and all dipoles, both real and image. Within this result, all the real dipoles increase the stabilization of the system, i.e., shift the polarization energy to more negative values, as expected for any polarization. The net potential of the image charge and of all the image dipoles together also stabilizes the system. However, within this net effect, although the effect of the image charge is always stabilizing, the effect of an individual image dipole depends on its position. An example is shown in Fig. 2. If the dipole is located between the charge and the interface position S in 4 6

3 J. Chem. Phys., Vol. 120, No. 8, 22 February 2004 Test of the method of images 3889 FIG. 2. A crystal divided into a discrete and a continuous part as in Fig. 1 b. Broken symbols show the positions of the image charge and the image dipoles. The image S of the dipole S increases the stabilization of the external charge, whereas the image D of the dipole D decreases it. Fig. 2 its image stabilizes the charge position S ). The effect is different for the position marked in Fig. 2 as D, in which case the image dipole (D ) leads to destabilization. The overall effect of all image dipoles is destabilizing, but its absolute value is three to four times smaller than the stabilizing effect of the image charge, and hence the latter prevails. III. RESULTS FIG. 3. Dependence of the polarization energy P on the number N of real dipoles included in the calculations, for different distances d of the external charge to the interface. The lines are quadratic regression fits to the data. Using the method described above we have calculated the polarization energies for an external charge in crystalline fullerene C 60 and amorphous PE. These were chosen as model systems for which results were already available for the bulk material. 7,8 Fullerene adopts a cubic crystal structure, 12 while amorphous PE is isotropic, so that each is characterized by a scalar relative permittivity. The C 60 crystal was divided into two parts by the plane x 0.25a, where a Å is the period of the fcc structure; 12 as explained below, this choice of interface facilitates convergence. One part was treated as a discrete lattice of fullerene molecules described as single polarizable points using the crystal structure and molecular polarizability as in Ref. 7, and the other as a dielectric continuum with permittivity 4.4. The charge was positioned on one of the molecules in the discrete part of the dielectric at a distance to the interface varying from 0.25a to 3.25a. In this case the shape of the discrete part of the crystal was a cuboid rather than a hemisphere, to make the calculations compatible with those performed for C 60 microcrystals in Ref. 7. The calculations for amorphous PE were performed for a structure used in our previous work 8 containing four PE chains each 400 CH 2 units long. We used local segment polarizabilities, 13 with the permittivity of the dielectric continuum calculated as Since there is a distribution of the polarization energies in the disordered polymer, we picked five positions for the charge that gave a range of P values across the whole distribution. As the system is locally anisotropic, for each charge position we repeated the calculations for three perpendicular planes dividing the sample into the two parts treated as discrete and as a dielectric continuum. The distance of the charge to the interface was 3 6 Å. For the polymer material an additional difficulty arises because there is nonzero probability of finding a polarizable CH 2 unit arbitrarily close to the interface and therefore to its image. In such a situation the field of the image dipole acting on the real dipole is so high that it causes the SCPF procedure to diverge. For C 60 molecules treated as single points this difficulty was readily avoided by choosing the interface plane so that the closest distance to the interface was 0.25a. For PE the difficulty was avoided by arbitrarily neglecting image dipoles for all CH 2 units closer than 1 Å to the interface. The calculations were performed for increasing system size, i.e., increasing size of the cuboid or hemisphere within the discrete part of the material. A typical dependence of the polarization energy on the system size N number of molecules in the discrete region is shown in Fig. 3 for the fullerene crystal. As is readily seen, the curves for different values of the distance d of the external charge from the interface become close, and hence the polarization energy saturates, when the charge is about two periods away from the surface. Quadratic regression was used to extrapolate the data to N, yielding estimates of the converged P values for different distances d of the external charge from the interface. The extrapolated values of the polarization energy for C 60 are given in Table I. Linear and quadratic regression yield and ev, respectively, so that the accuracy of the extrapolated value obtained here can be taken as about ev. Nevertheless, we display four significant figures in Table I to make the convergence towards the bulk value more clearly visible. The exact value obtained in Ref. 7 for the bulk crystal was P ev although it was quoted to only three figures there to match the precision of the input polarizability. As expected, the deviations from the bulk value are larger for smaller distances of the charge to the interface. However, once d reaches about one lattice spacing, the deviation from the exact value is about 5% and for larger separations it becomes numerically insignificant. This means that the method of images works well in C 60 for distances to the surface above one lattice spacing, and even

4 3890 J. Chem. Phys., Vol. 120, No. 8, 22 February 2004 A. Eilmes and R. W. Munn TABLE I. Values of the polarization energy P for C 60 crystals calculated for increasing distance d of the charge to the interface in units of the crystallographic period a. The last decimal digit is displayed in order to show the convergence towards the exact bulk value. d/a P/eV exact below this limit it provides a reasonable estimate of the polarization energy. The data extrapolated for the amorphous PE matrix are given in Table II for the three orientations of the interface for each charge position. Because of fluctuations in the polymer density and the different orientations, the convergence towards the exact value is not as well defined as for C 60, but nevertheless it is clear that the error of the method of images decreases for larger charge interface distances. For d 5 Å, the relative errors do not exceed 5% and in most cases they are smaller. This distance can be compared with the C C bond length in polyethylene, which is about 1.5 Å, and with the minimum intermolecular distances of Å, so that relative to the spacing of the polarizable points it corresponds roughly to d 2a for fullerene. It can be seen that the method of images overestimates the polarization for small absolute values of the polarization energy P, and underestimates it for larger absolute values. This observation may be rationalized by taking into account the fact that the value of the relative permittivity used for the TABLE II. Values of the polarization energy calculated for different positions of the charge within a polyethylene sample and for different orientations and distances of the interface relative to the charge. The exact values were calculated as in Ref. 8. Charge position Normal to the interface P/eV d 3Å d 5Å d 6 Å Exact x y z x y z x y z x y z x y z dielectric contrast was an average for the whole sample. Therefore it is too large in regions of lower density of the polymer, where the polarization energy is small in magnitude, and too small in regions of higher density, where the polarization energy is large in magnitude. Calculations of the spatial variation of the relative permittivity in PE are presented elsewhere. 13 IV. LIMITATIONS OF THE METHOD OF IMAGES The preceding results have shown how far the method of images is valid, and have given some indication of its limitations. Here we explore these limitations more systematically in order to understand them and to establish how far they could usefully be overcome. Interpretation is clearer for a crystal than it is for an amorphous material, where the local density fluctuations can mask the surface effects, as we have seen for PE. For a crystal, two main factors can be recognized: loss of symmetry once the two parts of the material are treated differently, as a discrete material and as an equivalent continuum, and spatial variations in dielectric response in the crystal, which the continuum cannot reproduce. A further factor associated with the latter point is the actual definition of the interface in the crystal. First, the method of images uses a uniform relative permittivity for the dielectric continuum, whereas the continuum is only semi-infinite and hence by symmetry its relative permittivity must be a function of the distance to the interface. Outside the interface, dielectric response is missing, and so near the interface the relative permittivity is lower than in the bulk, which would make the image charge and dipoles smaller than those given by the bulk relative permittivity. Indeed, the data in Table I show that for the value of the bulk relative permittivity of C 60 used in our calculations the stabilization energy is always overestimated. On the other hand, if one uses a relative permittivity of unity, corresponding to a vacuum, the image charge and image dipoles are all zero and the problem reduces to a charge close to a free surface of the crystal. In this case the stabilization energy due to polarization is smaller than that for the bulk crystal, as shown in Ref. 1, for example. Hence there exists a value of the dielectric permittivity between 1 vacuum and 4.4 bulk C 60 ) that when used in the method of images will reproduce the correct polarization energy, and this value will depend on the distance of the charge from the interface. This argument suggests that the method of images could be improved by modeling this variation in some way, perhaps within a dipolium model 14 parameterized to reproduce the relative permittivity of the bulk dielectric. This is of conceptual value, and could also be of practical value in providing a way of characterizing the surface relative permittivity. The loss of symmetry also enhances the dipole moment induced at the site of the charge itself. The polarization energy for fullerene ions in the C 60 crystal 7 with which we compare the present calculations does not depend on the polarizability of the fullerene ion, as the fields of the induced dipoles cancel by symmetry at the position of the charge. However, in the present calculations the polarizability of the fullerene ion results in a dipole moment being induced on the

5 J. Chem. Phys., Vol. 120, No. 8, 22 February 2004 Test of the method of images 3891 FIG. 4. Two positions of the charge near the crystal surface used to test the potential of the image charge see text. As in Fig. 1, the crystal is represented either as a discrete lattice a or as a continuous polarizable medium b. ion because the potential is no longer symmetric about the ion. The precise value of the induced dipole moment also depends on the polarizability, which here as in Ref. 7 is assumed to be the same as for the neutral molecule. Although the field or potential of this dipole is excluded when the polarization energy is calculated according to Eqs. 5 or 6, its image has an additional stabilizing effect. For a chargeinterface distance of 0.25a, we find that the polarization energy changes from 1.02 ev for a polarizable ion Table I to about 0.96 ev for a nonpolarizable ion. For larger separations, the effect is smaller, as the total field of the image charge and the image dipoles is a better approximation to the real field experienced by the charge. Second, even if we admit the use of a relative permittivity that varies as a function of distance to the interface, in practice the surface permittivity does not vary monotonically as a function of distance from the interface. A planewise treatment of molecular crystal surfaces allows their surface linear response to be calculated. 15 Deviations from the bulk behavior depend on the range of dipolar interactions between planes, which is typically a couple of lattice spacings, consistent with present results. Detailed numerical calculations for different surfaces of para-nitroaniline 16 showed that the surface refractive index depended not only on the crystallographic surface chosen since the crystal is noncubic but also on the specific termination chosen. Different terminations for the same surface in general correspond to different molecular orientations relative to the surface plane and hence to different surface properties. This is not relevant to the isotropically polarizable C 60 molecule, but it is a reminder that the true interface with a molecular material is not a mathematical plane, and that even if it were, the position of that plane would not be uniquely defined. Our choice of an interface halving the distances in the crystal structure cf. Fig. 1 is arbitrary but plausible, and works well for sufficiently large charge interface distances. A slight shift of the interface would change the results for small d without significantly affecting them for larger distances to the interface. In general, the proper choice of the interface plane is likely to be more difficult than in C 60, and hence this will be an additional source of uncertainty and errors for charges close to the boundary. The use of a continuous dielectric medium in the method of images also disregards the role of the microscopic crystal structure in causing fluctuations of the relative permittivity for different positions in the plane parallel to the interface, as well as perpendicular to it. This is illustrated in Fig. 4 a, where we show two positions of the charge close to the crystal surface. The stabilization energy due to the polarization for charge at position A will be smaller than at position B because of the larger distance to the nearest neighbor. On the other hand, when we replace the discrete crystal below the broken line representing the interface by the continuous medium and use the method of images to calculate the potentials Fig. 4 b, the result will obviously be the same for both positions of the charge. We have confirmed this by placing a nonpolarizable charge or dipole at position A or B, performing SCPF calculations and obtaining the potential of the induced dipoles at the position of the charge or dipole. The result was compared with the potential resulting from the image charge or image dipole. As expected, at position A the method of images yielded too large an absolute value of the potential compared with the SCPF method, while at position B it yielded too small an absolute value. Furthermore, it happens that because of the crystal structure of C 60, the molecules closest to the interface are all at positions equivalent to A in Fig. 4. Therefore the method of images overestimates the stabilizing effect of a charge located at this position the result for d 0.25a in Table I or the stabilizing effects of the dipoles closest to the interface in the case of a charge at d 0.75a). This also helps to explain why the method of images overestimates the stabilization energy for a small charge-interface distance. The conclusion is that for the method of images one has to expect deviations from the true polarization energy for small charge interface separations because the crystal structure causes the polarization energy to depend on the charge position. As the description of the discrete structure by an averaged continuous medium is expected to suffer from increasing errors at length scales of the order of the lattice period of the crystal, such deviations seem to be inevitable. It seems unlikely that a simple approach could be found to estimate such errors, because, as shown above, they depend on the position of the charge and interface relative to the discrete structure, which in the general case is unknown if they were known, one could obtain exact results by performing SCPF calculations without using the method of images. This analysis is less relevant for an amorphous medium where the major deviations from the exact value are due to the density fluctuations in the real medium that is approximated by the continuous dielectric, and these fluctuations are in practice unknown. Their effects are enhanced for polymers, as in that case they are determined not only by the number of polarizable units per volume but also by the number of polymer chains and their orientation relative to the interface. V. CONCLUSIONS We have estimated the error involved in using the method of images to represent a molecular material by a dielectric continuum by dividing an infinite molecular material into two semi-infinite parts, one of which was treated as a discrete molecular material and the other as a dielectric continuum. Using the SCPF approach we have calculated the polarization energies for charges placed in the discrete part

6 3892 J. Chem. Phys., Vol. 120, No. 8, 22 February 2004 A. Eilmes and R. W. Munn of the medium close to the interface and compared the values with the exact results of the discrete SCPF treatment for the whole system. The results suggest that the accuracy of the method of images increases rapidly the farther the charge is from the interface, with errors smaller than 5% for charge surface distances some two to three times larger than the average spacing of the polarizable units in the molecular material. The results were similar for crystalline C 60 and amorphous PE, but in C 60 the polarization energy found by the method of images was always larger in magnitude than the exact value, whereas in the amorphous PE samples the convergence towards the exact values was not monotonic because of the density fluctuations, and the sign of the difference was not obvious a priori. However, subject to these differences between crystalline and amorphous materials, we expect essentially the same findings for other materials. As explained in the preceding section, errors are inevitable for charge interface distances that are small compared with the lattice period of a crystal or characteristic distance of an amorphous medium. The origin of the errors can be traced to neglecting the loss of symmetry and to neglecting the real microscopic structure when part of the molecular material is replaced by the dielectric continuum. Moreover, the errors seem rather difficult to quantify, as the real position of the charge relative to the real discrete structure of the dielectric is generally unknown. Nevertheless, the errors should fall off quite rapidly with increasing distance from the interface. Our analysis of the energetics of a charge near the surface of the molecular material shows that the magnitude of the polarization energy, i.e., the stabilization energy, increases as the relative permittivity increases. This means that the distribution of energies near the surface of an amorphous material broadens as the relative permittivity of the dielectric outside the surface increases. A model of charge transport by hopping between localized states then implies that, all things being equal, the mobility increases as the relative permittivity decreases. This is consistent with the suggested interpretation 17 of measurements showing that dielectrics of low relative permittivity afford higher mobilities and hence are preferable for use in organic field-effect transistors; in that interpretation the broadening of the distribution was attributed explicitly to the effect of dipoles in the amorphous polymer dielectric, but the dipoles also determine the magnitude of the relative permittivity. Overall, these results confirm the usefulness of the method of images for microscopic calculations of polarization energies at the interface of a molecular material and another dielectric medium. Calculations using the method for a polymer/inorganic glass interface will be the subject of future work. ACKNOWLEDGMENT This work was partly supported under the EU FP5 Project ELEVAG, reference ENK6-CT E. A. Silinsh, Organic Molecular Crystals: Their Electronic States Springer, Berlin, A. J. Twarowski, J. Chem. Phys. 77, E. V. Tsiper, Z. G. Soos, W. Gao, and A. Kahn, Chem. Phys. Lett. 360, E. V. Tsiper and Z. G. Soos, Phys. Rev. B 68, D. B. Knowles and R. W. Munn, J. Mater. Sci.: Mater. Electron. 5, L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media Pergamon, Oxford, A. Eilmes, Synth. Met. 109, A. Eilmes, R. W. Munn, and A. Góra, J. Chem. Phys. 119, R. W. Munn, J. Chem. Phys. 101, M. in het Panhuis and R. W. Munn, J. Chem. Phys. 113, C. J. F. Böttcher, Theory of Electric Polarization Elsevier Scientific, Amsterdam, 1973, Vol D. André, A. Dworkin, H. Szwarc, R. Céolin, V. Agafonov, C. Fabre, A. Rassat, L. Stravet, P. Bernier, and A. Zahab, Mol. Phys. 76, A. Eilmes, R. W. Munn, V. G. Mavrantzas, D. N. Theodorou, and A. Góra, J. Chem. Phys. 119, W. L. Schaich and B. S. Mendoza, Phys. Rev. B 45, R. W. Munn, J. Chem. Phys. 97, M. Malagoli and R. W. Munn, J. Chem. Phys. 112, J. Veres, S. D. Ogier, S. W. Leeming, D. C. Cupertino, and S. M. Khaffaf, Adv. Funct. Mater. 13,