NANO LETTERS
The Effect of Morphology on Electron Field-Effect Mobility in Disordered C60 Thin Films
2009 Vol. 9, No. 3 1085-1090
Joe J. Kwiatkowski,* Jarvist M. Frost, and Jenny Nelson Department of Physics, Imperial College London, London, SW7 2AZ, U.K. Received November 19, 2008; Revised Manuscript Received January 28, 2009
ABSTRACT We present a model of polycrystalline C60 field-effect transistors (FETs) that incorporates the microscopic structural and electronic details of the C60 films. We generate disordered polycrystalline thin films by simulating the physical-vapor deposition process. We simulate electron hopping transport using a Monte Carlo method and electronic structure calculations. Our model reproduces experimentally observed FET characteristics, including electrical characteristics, electrochemical potentials, and charge mobilities. Our results suggest that even relatively disordered films have charge mobilities that are only a factor of 2 smaller than mobilities in single crystals.
A critical factor that limits the efficiencies of organic electronic devices is poor charge mobility in disordered molecular films. Low mobilities are attributed variously to packing disorder,1 electronic interactions,2 chemical defects, and interface effects,3 but the precise factors that limit charge transport in these materials remain poorly quantified. An accurate description of these factors must incorporate phenomena that occur over widely varying length scales, from intermolecular charge transfer2 to device-scale variations of the morphology.4 While methods have been reported for simulation of charge transport in molecular crystals5,6 and highly disordered films,7 the intermediate regime of disordered polycrystalline molecular films has never been studied. Moreover, disordered films are often treated with simplified models which do not incorporate any microscopic detail.8 The few attempts that have been made to accurately model charge transport in disordered films have been restricted to small or highly stylized assemblies of molecules.7,9-11 In this paper we present a model of charge transport that incorporates both the details of intermolecular charge transfer and large scale variations in morphology. We use this model to consider the effect of morphology on charge mobilities in polycrystalline fullerene (C60) films. C60 is of interest because its mobility is very high for a molecular film12 and because the effect of morphology on mobility has been studied before experimentally.1,13 Furthermore, the high symmetry and rotational freedom (at room temperature) of C60 means that interactions between molecules can be treated as isotropic.14 This allows us to study the effects of packing density without complications arising * Corresponding author,
[email protected]. 10.1021/nl803504q CCC: $40.75 Published on Web 02/10/2009
2009 American Chemical Society
from large variations in rates due to disorder in molecular orientation. What is more, the isotropy makes it exceptionally computationally efficient to simulate C60 films, thus allowing experimentally relevant dimensions to be studied. We first present methods for modeling the growth of polycrystalline C60 films and show the effect of substrate temperature on film morphology. Then, in order to make comparison with experimental measurements of field-effect mobilities,1,13 we develop a model of hopping transport in field-effect transistors (FETs) using a Monte Carlo method together with electronic structure calculations of the hopping rates. In addition to linking molecular properties with device characteristics, our method allows the effects of film morphology to be decoupled from other important factors which affect FETs, such as the contacts3,15 and type of dielectric.16,17 We show that the calculated electrochemical potential, output and transfer curves, and field-effect mobilities are all consistent with experiment. Our simulations suggest that mobilities depend weakly on film morphology, varying by only a factor of 2 between the most disordered and crystalline films. Film Morphologies. Morphologies are generated by simulating the physical vapor deposition (PVD) process with a custom Monte Carlo code. The interaction between C60 models is modeled with the one-dimensional Girifalco potential14 (Figure 1a) which has been shown to accurately reproduce a variety of properties of C60.18-20 We treat the substrate as an impenetrable flat plane and assume a surface binding potential that scales linearly from -2 eV when the center of the C60 molecule is at the surface to 0 when the center is 0.7 nm from the surface. The film grows perpen-
Figure 1. (a) The Girifalco potential and intermolecular electron transfer integral JRMS against intermolecular separation. The error bars correspond to twice the standard deviation in J. The diameter of C60 is 0.71 nm. (b) PVD films grown at different temperatures viewed from the C60 source. Crystalline regions have been enlarged and colored red. (c) The radial distribution function for PVD slabs grown at 298, 523, and 748 K. The corresponding function for a crystal, scaled by a factor of 20, is represented by dashed lines. (d) Films modeled with GROMACS. Molecules on the bottom monolayer are shown colored according to the local orientation of crystal axes.
dicular to the {111} plane in which the separation between layers is 0.82 nm. Therefore, this surface potential means that only molecules in the first monolayer interact with the substrate. This approximate potential is broadly consistent with more detailed studies of C60 adsorption.21-23 Individual C60 molecules are released one at a time from the source at a random position above the substrate and a temperature of 773 K.24,25 Each molecule moves toward the substrate in steps of 0.05 nm, while all the other molecules on the substrate are held frozen. Each step is accepted or rejected according to the standard Metropolis algorithm; when first rejected the molecule stops its descent and assumes the temperature of the substrate. The newly arrived molecule and all others within a 1.85 nm radius are then each moved 5 × 104 times before the next molecule is deposited; molecules are moved consecutively by one Monte Carlo step at a time. As the new molecule moves across the substrate, the list of neighbors is updated. Periodic boundary conditions are applied in the plane of the substrate. This method is found to be a good compromise between equilibration and computational speed. Using this method, we generated molecular films measuring 50 × 20 nm2 on substrates held at temperatures of 298, 523, and 748 K. In each case, 104 molecules were deposited, resulting in rough, porous films with a maximum depth of about 15 nm. Just the bottom 5 nm was used for our FET simulations; these slabs had densities of 1.27, 1.44, and 1.59 1086
g/cm3 for the films grown at 298, 523, and 748 K respectively; this compares with a density of 1.67 g/cm3 for crystalline C60. This compares with a density of 1.67 g/cm3 for crystalline C60, assuming nearest neighbor spacing of 1.005 nm. Figure 1b shows examples of slabs grown by PVD simulation at substrate temperatures of 298, 523, and 748 K. Molecules lying in crystalline domains are defined as those that have 12 neighbors within a 1.075 nm radius (or eight when at the top or bottom of the film); all such molecules have been colored red and enlarged. The crystalline regions are estimated by eye to be 2, 4, and 6 nm in length for the 298, 523, and 748 K morphologies, respectively. This trend is consistent with experimental measurements,26,27 but the simulated grain lengths are orders of magnitude smaller. This is because the computational cost of simulating the PVD process prohibits equilibration on the scale that occurs naturally. Despite the very different grain lengths, the radial distribution functions (RDFs) of the three PVD morphologies are actually very similar to each other (Figure 1c), and indeed to the single crystal RDF. These similarities are due to the fact that it is easy for spheres to pack efficiently, even when they appear disordered to the eye. For comparison, we also generated C60 films with molecular dynamic simulations. Though not strictly modeling Nano Lett., Vol. 9, No. 3, 2009
the PVD process, this method allows us to generate films with larger crystalline grains. Two examples measuring 200 × 20 nm2 are shown in Figure 1d. The Girifalco potential14 was used in GROMACS 4.28 The substrate was represented as a bulk volume of strengthened C60 interaction potentials equivalent to a density of 6 nm-3. Simulations were started with evenly spaced C60 molecules at a density of 0.125 nm-3 and run for 20 ns using time steps of 20 fs, while connected to a Berendsen thermostat with a time constant of 1 ps. Though the resulting films were very rough, only the bottom 5 nm was used. Intermolecular Charge Transfer. We consider electrons to hop between neighboring molecules at a rate Γ defined by semiclassical Marcus theory2,29 Γ)
|J|2 p
(
π (∆G + λ)2 exp λkBT 4λkBT
)
(1)
where λ is the reorganization energy, J is the electronic coupling between initial and final states, and ∆G is the difference in free energy between initial and final states. The use of semiclassical Marcus theory is justified because λ . J for the vast majority of neighboring molecules. All quantum chemical calculations use the hybrid B3LYP functional with the 6-31G* basis set, as implemented in Gaussian30 with tight convergence criteria. We calculate the intramolecular contribution to λ using the potential energy surface method2 and obtain 0.132 eV for electron transfer. The contribution of the surrounding medium to λ is difficult to determine but has been shown to be small in oligoacenes31 and is neglected for the purpose of this study. For electron transfer, the relevant J is between lowest unoccupied molecular orbitals (LUMOs) and is calculated as described in ref 32. We treat the 3-fold degeneracy of the LUMOs by taking the root-mean-square (rms) of the nine relevant off-diagonals in the Hamiltonian. Figure 1a shows the variation in the rms of the transfer integral JRMS against intermolecular separation; each point is the rms of 100 randomly oriented C60 pairs at the same separation. The error bars are twice the standard deviation in J among these hundred calculations. The transfer integrals between HOMOs are found to be almost identical. Above 260 K, C60 molecules are reported to be completely rotationally disordered.33 Since these rotations occur on the nanosecond time scale,34 which is fast compared to the time scale of charge transport through the film, we use JRMS to calculate Γ. We justify this approximation by noting that the variation in J from orientational disorder is small compared to the variation that is caused by the range of intermolecular separations in the simulated films. We assume that the dominant contribution to ∆G is the Coulomb interaction between the mobile charges and neglect contributions from other electrostatic and inductive effects. This is reasonable because the symmetry of the C60 molecule means it has very small dipole and quadrupole moments (3 × 10-4 D and ∼ 10-3 DÅ, respectively, for the neutral molecule). The largest inductive interaction would be between the anion (dipole of 17.1 D) and the neutral molecule (polarizability volume of 69 Å3), giving an interacNano Lett., Vol. 9, No. 3, 2009
tion of roughly 5 meV at 1 nm separation. However, because of symmetry, this energy is essentially the same before and after charge transfer and therefore does not contribute to ∆G. Given these approximations, ∆G for electron transfer between a negatively charged molecule m and a neutral neighbor n is simply the sum of all Coulomb energies ∆Gmn )
∑(
1 1 e2 4π0r i*m |rm - ri| |rn - ri|
)
(2)
where the sum is over all charged molecules, e is the elementary charge, and 0 is the permittivity of free space. The dielectric constant r is taken as 4.4.35 If the neighbor n were occupied, ∆Gnm would be infinite, which ensures that molecules are never occupied by more than one charge. Because we only simulate electrons, the Coulomb interaction is always positive. During the simulation Γ is calculated from each charged molecule to all of its neutral neighbors. Neighboring molecules are defined as those whose centers are separated by less than 1.4 nm. Both ∆G and Γ are updated after every Monte Carlo move. Simulating FETs. Charges move between molecules according to a kinetic Monte Carlo algorithm.36 In the interest of computational efficiency we simulate transport in slabs that measure only 200 × 20 × 5 nm3. However, these dimensions are found to be sufficient for the field-effect mobilities to be independent of size. In the case of PVD morphologies, these slabs are obtained by stacking four identical copies of a generated slab along their long axis, taking advantage of the periodic boundary conditions used. The FET channel is the long axis of the slab. The potential difference between the source and drain is VDS. Molecules within 1 nm of each end of the slab are held in electrochemical equilibrium with the local contact: at each Monte Carlo step the Fermi-Dirac electron occupation probability of these molecules is adjusted to maintain equilibrium. There is thus no barrier to charge injection. In every other respect the charges on these contacted molecules are treated in the same way as all other charges. We assume that the FET is a bottom contact device with perfect gating efficiency so the gate voltage VG simply shifts the potential of both source and drain contacts with respect to the chemical potential of the C60 film at equilibrium. Once the simulation has reached steady state, the source-drain current IDS is calculated from the difference between the number of charges that are collected and injected at the drain per unit time. Figure 2a shows the variation in the average electrochemical potential along the length of a FET channel (PVD 523 K, 200 × 20 × 5 nm3) when VDS is fixed at 0.4 V and VG is varied from 0 to 0.8 V. This behavior is in qualitative agreement with scanning Kelvin probe microscopy studies in poly(3-hexylthiophene) FETs.37 Figure 2b shows the concomitant variation in the electron density. At high VG, the charge density is high throughout the channel length and the FET operates in the linear regime (IDS ∼ VDS). At low VG, charges are depleted from the region near the drain and the FET operates in the saturation regime (IDS ∼ constant). The transition between linear and saturated regimes is at the pinch-off where VG ) VDS.38 The applied voltages relative 1087
Figure 2. (a) Average electrochemical potential and (b) average electron density as a function of distance from the source. Both show the transition from saturated to linear regime via “pinch-off” when VG ) VDS. Simulations run at 300 K with the film deposited by PVD at 523 K. In (a), successive curves have been shifted by -0.05 eV to aid the eye.
Figure 3. (a) Output curves. (b) IDS,sat1/2 and average number of electrons in FET as a function of VG. Both simulated at 300 K with the film deposited by PVD at 523 K.
to the channel length, VG/L and VDS/L, are comparable to those in real FETs. In keeping with experimental practice, we calculate the saturated charge mobility µsat from the variation in the saturated source-drain current IDS,sat with VG IDS,sat )
WCµsat (VG - Vth)2 2L
(3)
where W and L are the width and length of the FET, C is the capacitance per unit area of the gate dielectric, and Vth is the threshold voltage.38 We extract µsat by plotting IDS,sat1/2 against VG. Equation 3 assumes that µsat is independent of VG, but we show that this is valid below. Figure 3a shows typical output curves at 300 K for FETs based on PVD films grown at 523 K. The current densities are similar or slightly higher than those seen in typical devices;39 the small difference may result from our neglect of contact resistance. Figure 3b shows IDS1/2 and the average number of mobile electrons in the FET as functions of VG; the fact that the two quantities are proportional means that we are justified in assuming that µsat is independent of VG in eq 3. The threshold voltage Vth is -0.26 V. The saturated mobilities µsat are shown in Figure 4 as a function of the estimated grain length in the simulated films. The single crystal is a face-centered cubic structure33 with nearest neighbor spacing of 10.05 Å, consistent with the 1088
Figure 4. µsat at 300 K as a function of the estimated grain size for PVD (O) and molecular dynamics grown films (b). Also shown are µsat from Singh et al.1 and Kobayashi et al.13 and time-of-flight mobilities for single crystals from Frankevich et al.40
Girifalco potential.14 We also estimate the electron mobility in a single crystal by simulating the time-of-flight method.7 For comparison, electron µsat measured in FETs1,13 are shown for a range of grain lengths, as estimated by atomic force microscopy. The electron mobility measured by the timeof-flight method40 is shown for the single crystal. Despite the various approximations in our model, our calculated µsat Nano Lett., Vol. 9, No. 3, 2009
are on the same order of magnitude as those measured experimentally. Our calculated mobilities suggest that, although film morphology does affect µsat, the variation is only a factor of 2 between the most disordered morphology and the single crystal. As discussed above, this is expected from the efficiency with which spheres pack, even in apparently disordered phases. This efficiency of packing manifests itself in the distribution of total hopping rates away from each C60 molecule: even in the 298 K PVD slab, 99.6% of molecules have total rates that are within an order of magnitude of the total rate hopping of a crystalline molecule (9 × 1013 s-1). For less symmetric molecules, J can vary widely, depending on the relative orientation of the coupled molecules. Furthermore, lower symmetry can lead to strong dipole moments which, in turn, cause large ∆G terms. In the case of disordered tris(8-hydroxyquinoline) aluminum (Alq3), these factors lead to a distribution of total hopping rates that spans 6 orders of magnitude.7 As a result, mobilities in disordered Alq3 are over 3 orders of magnitude lower than those in the crystal. The distribution of total hopping rates for C60 and Alq3 can be found in the Supporting Information. Our results are broadly in agreement with the work of Kobayashi et al.,13 who find that the morphology has no effect on µsat, but in contrast with the work of Singh et al.1 A critical difference between Kobayashi and Singh is that the former deposits C60 films by molecular beam epitaxy onto silica dielectrics, whereas the latter deposit films by hot-wall epitaxy onto organic dielectrics. In the light of our simulations and Kobayashi’s results, we propose that the variations in µsat seen by Singh are the result of some factor other than the disorder in packing of C60 molecules. For example, the mobilities may be limited by morphological phenomena that are not considered in this paper, such as large gaps between grains or the roughness induced by the dielectric.41 Alternatively, it may be that the electronic properties of the interface between the dielectric and the C60 (shown to be important in rubrene FETs)17 vary with substrate temperature. In conclusion, we have demonstrated a model of FETs that incorporates the microscopic structural and electronic details of disordered molecular films and applied it to polycrystalline C60 films. By simulating PVD over a range of substrate temperatures, we have investigated the effect of film morphology on FET characteristics. We have accurately reproduced FET characteristics that are measured experimentally, including output curves, transfer curves, electrochemical potentials, and charge mobilities. We find that even relatively disordered films have charge mobilities that are only a factor of 2 smaller than single crystal mobilities. We rationalize this result by noting that C60 molecules pack very well because of their symmetry so that, even in the most disordered films, molecules are well connected with each other. We suggest that other factors, such as the dielectric/C60 interface, might be responsible for previous indications of stronger effects of morphology on mobility. Nano Lett., Vol. 9, No. 3, 2009
Acknowledgment. Our thanks to Dr. Thomas Anthopoulos and his group for very helpful discussions. We acknowledge the support of Imperial College High Performance Computing Service. Supporting Information Available: Total hopping rates for C60 and Alq3, crystalline and disordered. This material is available free of charge via the Internet at http:// pubs.acs.org. References (1) Singh, T. B.; Sariciftci, N. S.; Yang, H.; Yang, L.; Plochberger, B.; Sitter, H. Appl. Phys. Lett. 2007, 90 (21), 213512. (2) Bredas, J.; Beljonne, D.; Coropceanu, V.; Cornil, J. Chem. ReV. 2004, 104 (11), 4971–5003. (3) Hulea, I. N.; Russo, S.; Molinari, A.; Morpurgo, A. F. Appl. Phys. Lett. 2006, 88, 113512. (4) Salleo, A. Mater. Today 2007, 10 (3), 38–45. (5) Troisi, A.; Orlandi, G. Phys. ReV. Lett. 2006, 96 (8), 086601. (6) Wang, L. J.; Peng, Q.; Li, Q. K.; Shuai, Z. J. Chem. Phys. 2007, 127, 044506. (7) Kwiatkowski, J. J.; Nelson, J.; Li, H.; Bredas, J. L.; Wenzel, W.; Lennartz, C. Phys. Chem. Chem. Phys. 2008, 10 (14), 1852–1858. (8) Bassler, H. Phys. Status Solidi B 1993, 175 (1), 15–56. (9) Athanasopoulos, S.; Kirkpatrick, J.; Martinez, D.; Frost, J. M.; Foden, C. M.; Walker, A. B.; Nelson, J. Nano Lett. 2007, 7 (6), 1785–1788. (10) Olivier, Y.; Lemaur, V.; Bredas, J. L.; Cornil, J. J. Phys. Chem. A 2006, 110 (19), 6356. (11) Kirkpatrick, J.; Marcon, V.; Nelson, J.; Kremer, K.; Andrienko, D. Phys. ReV. Lett. 2007, 98 (22), 227402. (12) Anthopoulos, T. D.; Singh, B.; Marjanovic, N.; Sariciftci, N. S.; Montaigne Ramil, A.; Sitter, H.; Co¨lle, M.; de Leeuw, D. M. Appl. Phys. Lett. 2006, 89 (21), 213504. (13) Kobayashi, S.; Takenobu, T.; Mori, S.; Fujiwara, A.; Iwasa, Y. Appl. Phys. Lett. 2003, 82 (25), 4581–4583. (14) Girifalco, L. A. J. Phys. Chem. 1992, 96 (2), 858–861. (15) Klauk, H.; Schmid, G.; Radlik, W.; Weber, W.; Zhou, L.; Sheraw, C. D.; Nichols, J. A.; Jackson, T. N. Solid-State Electron. 2003, 47 (2), 297–301. (16) Veres, J.; Ogier, S.; Lloyd, G.; de Leeuw, D. Chem. Mater. 2004, 16, 4543–4555. (17) Stassen, A. F.; de Boer, R. W. I.; Iosad, N. N.; Morpurgo, A. F. Appl. Phys. Lett. 2004, 85, 3899. (18) Fernandes, F.; Freitas, F. F. M.; Fartaria, R. P. S. J. Phys. Chem. B 2004, 108, 9251–9255. (19) Abramo, M. C.; Caccamo, C.; Costa, D.; Pellicane, G.; Ruberto, R. Phys. ReV. E 2004, 69 (3), 31112. (20) Hasegawa, M.; Ohno, K. J. Chem. Phys. 1999, 111, 5955–5963. (21) Teobaldi, G.; Zerbetto, F. Small 2007, 3, 1694. (22) Hobbs, C.; Kantorovich, L.; Gale, J. D. Surf. Sci. 2005, 591 (1-3), 45–55. (23) Hamza, A. V.; Balooch, M. Chem. Phys. Lett. 1993, 201, 404–408. (24) Bhuiyan, K. H.; Mieno, T. Thin Solid Films 2006, 506, 239–243. (25) Manaila, R.; Geru, I. I.; Fratiloiu, D.; Spoiala, D. M.; Dihor, I. T.; Devenyi, A. Fullerenes, Nanotubes, Carbon Nanostruct. 1999, 7 (1), 59–75. (26) Chen, R. S.; Lin, Y. J.; Su, Y. C.; Chiu, K. C. Thin Solid Films 2001, 396 (1-2), 103–108. (27) Cheng, W. R.; Tang, S. J.; Su, Y. C.; Lin, Y. J.; Chiu, K. C. J. Cryst. Growth 2003, 247 (3-4), 401–407. (28) Hess, B.; Kutzner, C.; van der Spoel, D.; Lindahl, E. J. Chem. Theory Comput. 2008, 4 (2), 435–447. (29) Freed, K.; Jortner, J. J. Chem. Phys. 1970, 52 (12), 6272. (30) Frisch, M. J. Trucks, G. W. Schlegel, H. B. Robb, M. A. Cheeseman, J. R. Montgomery, J. A. Kudin, K. N. Burant, J. C. Millam, J. M. Barone, V. Mennucci, B. Cossi, M. Petersson, G. A. Nakatsuji, H. Hada, M. Fukuda, R. Hasegawa, J. Ishida, M. Nakai, H. Klene, M. Li, X. Adamo, C. Jaramillo, J. Gomperts, R. Austin, A. J. Cammi, R. Pomelli, C. Morokuma, K. Voth, G. A. Salvador, P. Zakrzewski, V. G. Dapprich, S. Daniels, A. D. Farkas, O. Malick, D. K. Rabuck, A. D. Foresman, J. B. Ortiz, J. V. Cui, Q. Cioslowski, J. Stefanov, B. B. Liu, G. Komaromi, I. Martin, R. L. Fox, D. J. Peng, C. Y. Nanayakkara, A. Challacombe, M. Johnson, B. Chen, W. and Wong. M. W. Gaussian 03, ReVision B.05; Gaussian Inc.: Wallingford, CT, 2003. 1089
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NL803504Q
Nano Lett., Vol. 9, No. 3, 2009
