Microscopic Description of Elementary Growth Processes and

J. Phys. Chem. B 2007, 111, 139-150

139

Microscopic Description of Elementary Growth Processes and Classification of Structural Defects in Pentacene Thin Films Stijn Verlaak,* Cedric Rolin, and Paul Heremans IMEC, Kapeldreef 75, B-3001 LeuVen, Belgium ReceiVed: August 16, 2006; In Final Form: NoVember 8, 2006

Elementary growth processes such as kink initiation, adding a molecule to a kink, and adding a molecule between two neighboring kinks and between two grains are theoretically studied in pentacene films by adding one molecule at a time to a predefined aggregate. For each molecule, the potential energy surface is calculated using the MM3 molecular mechanics force field, which allowed one to identify useful parameters like the energy barrier for diffusion and the energy to create kinks, as well as defect configurations. Depending on the properties of the potential energy surface and the resulting growth-condition-dependent probabilities of initiating defect configurations in the film, three types of pentacene defects are identified: a thermally activated defect, an intrinsic defect, and a kinetic defect. Upon film growth, most defects relax into the ideal crystal configuration. Bulk defects that resist relaxation have densities lower than 1016 defects/cm3 at typical growth conditions. Grain boundary defects, on the other hand, are very stable. Moreover, interstitial molecules at grain boundaries are identified as a source of compressive stress.

1. Introduction Organic molecular single-crystals have been studied for decades for their challenging electronic1-3 and structural4 properties. With the demonstration of organic electronic devices like light-emitting diodes,5 solar cells, and transistors,6,7 molecular thin films have come into focus. Epitaxy and quasiepitaxy of monolayers of organic small molecules on crystalline metallic, semiconductor, and insulating substrates have been studied as model cases for heterointerfaces that are inevitably present in such electronic devices.8-11 More recently, the growth of polycrystalline molecular semiconductor thin films of pentacene and oligothiophenes on less-defined and inert dielectric substrates like SiO2 is being explored.12-18 Besides scientifically challenging, the semiconductor-dielectric interface is technologically crucial for the properties of thin-film transistors, where current is concentrated in the first couple of monolayers near this interface.19-21 Electronic defects at this interface, within the bulk of crystalline semiconductor grains, or concentrated at grain boundaries, may severely hinder current flow and deteriorate a transistor’s performance.19,22-24 One important class of electronic defects is extrinsic to the film morphology and is related, for example, to moisture,25 chemical reactions,26 hydroxyl groups at the SiO2 surface trapping electrons,27 etc. Another class of electronic defects finds its origin in structural defects affecting the crystalline quality of the semiconductor film, having considerable impact on both molecular energy levels and intermolecular charge-transfer probability.2,3,28 While some reports suggest a clear relation between morphology and charge-carrier mobility and define grain boundaries as the dominant defects,29,30,24 other data are less conclusive.31-34 Moreover, a focus on geometric parameters such as (apparent) morphology (i.e., grain size and shape) as dominant factor determining macroscopic charge transport behavior in those organic films neglects the profound and omnipresent influence of the film-growth thermodynamics on the creation and type of * Corresponding author. E-mail: [email protected].

structural and electronic defects. For example, charge-carrier mobilities are sometimes related to (growth condition dependent) grain sizes,29,30 neglecting the fact that the trap distribution at grain boundaries is also expected to depend on growth conditions. That film-growth thermodynamics has an influence on the creation of bulk defect states has been demonstrated for parasexiphenyl grown by hot wall epitaxy,35 as well as for some perylene derivatives.36 In the former study, some photoluminescence features were found to depend on substrate temperature and were attributed to dimer-like structural defects. The classification of structural defects and their response to varying growth conditions will be the subject of this Article. A microscopic picture is given for defects in pentacene films with negligible film-substrate interactions, with the help of a molecular mechanics force field. This more quantitative and detailed information gives an idea about relevant energetics and processes and may act as input data for future macroscopic models of film growth. Only “thermodynamic” defects, which are incorporated in the film during growth, will be considered here. 2. Microscopic Model of Film Growth This section describes the elementary film-growth physics as a basis for the remainder of the Article. Next, the quasistatic methodology is proposed to study elementary film-growth processes and the resulting defects. Last, the computational methods and procedures are presented. 2.1. Relevant Growth Processes. Growth of a film or a crystal can be envisaged as adding molecules one after another to a kink site, see Figure 1. This process does not consume the kink position: upon addition/adsorption of a molecule to the kink, the kink site simply moves one molecule further. For that reason, a kink site is sometimes referred to as a “repetitive step”. Similarly, a crystal can be deconstructed by desorbing molecules from kink positions one after another. It comes as no surprise then that the interaction energy of a molecule at a kink site

10.1021/jp0653003 CCC: $37.00 © 2007 American Chemical Society Published on Web 12/14/2006

140 J. Phys. Chem. B, Vol. 111, No. 1, 2007

Verlaak et al. ∆µ ) 0 eV (where their desorption rate would have been larger than their adsorption rate). For this system consisting of a kink site that can either be not occupied or be occupied by a molecule in a perfect crystal configuration, the probability of occupancy is determined by statistical mechanics to be:39

( ) ( )

∆µ kTsub 1 ) pocc(E1/2, ∆µ) ) ∆µ -∆µ exp + 1 1 + exp kTsub kTsub exp

Figure 1. Adding a mobile molecule to a monolayer aggregate, viewed at its (001) surface as if the growth substrate would lay parallel to the paper: (a) to the (110) surface edge, (b) to a kink site, (c) interstitial between two kink sites, (d) interstitial between two grains that are displaced by 1 Å toward each other. The nomenclature is illustrated in (b). In-plane unit cell parameters are a, b, and γ; the principal molecular axes are declared long (L), medium (M), and normal (N); ω indicates the rotation of a molecule along its long axis; the two (inequivalent) molecules of the triclinic unit cell are named 1 and 2.

with its nearest neighbors is closely related to the enthalpy of sublimation. Last, when ignoring foreign substrate interactions, a molecule at a kink site binds to exactly one-half a monolayer or crystal. Kink sites are therefore also often referred to as “halfcrystal positions”.15,37,38 Kinks not only deserve their special status because they can build repetitive steps, but also because their half-crystal binding energy is representative for the energetics of the film or crystal growth.38 The rate of adding molecules one after another to a kink site is determined by externally applied growth conditions. Any growth condition can be uniquely defined by the substrate temperature Tsub and the supersaturation ∆µ. The supersaturation is a function that depends on Tsub, the deposition flux, and surface parameters. A crude example of an expression linking those growth parameters to the supersaturation can be found in ref 15, valid only in the early nucleation regime and for negligible surface interactions. The supersaturation determines the energy level of the configurations that have a ∼50% probability to be occupied by a molecule. The important point here is that a molecule at a kink position in a perfect crystal configuration, whose energy is calculated in one of the next sections to be E1/2 ) -1.18 eV, is considered as the reference level. In other words, when ∆µ ) 0 eV, any molecular state with an energy of -1.18 eV has a 50% chance of being occupied. Stated differently, the rates of adsorption and desorption are equal at those kink states under those growth conditions, and there will be no net growth of the film or crystal.38 When growth parameters are chosen such that the supersaturation increases, for example, to ∆µ ) 0.15 eV by decreasing the substrate temperature and/or increasing the deposition flux and/ or increasing the interaction with the substrate, ideal kink states will capture more molecules than there are molecules desorbing from those states: the film will grow. Under those same conditions with ∆µ ) 0.15 eV, states with an energy of -1.03 eV will now have equal rates for adsorbing and desorbing molecules and will now consequently be more stable than at

( )

(1)

where the denominator of the first right-hand side is the grand canonical ensemble of this system, summing over the occupied (exp(∆µ/kTsub)) and the unoccupied (1) state. Note that this probability is identical to the Fermi-Dirac probability function, yet has a different origin. When, besides the molecule in a perfect crystal configuration, also other stable (i.e., local energy minimum) defect states with configurations Qi and energies EQi are accessible to occupy the kink site, eq 1 has to be extended. The probability to occupy the kink site by a molecule in state Qi is now given by:

( (

exp pocc(EQi, ∆µ) )

∑ ∀j

exp -

) )

EQi - E1/2 - ∆µ kTsub

EQj - E1/2 - ∆µ kTsub

1 + exp

(

1

≈ +1

)

EQi - E1/2 - ∆µ kTsub

(2)

where the second right-hand side approximation is valid only if that particular state Qi is more stable than any other state by at least 0.1 eV. Equation 2 reduces to eq 1 when there is only one accessible state Qi that has the ideal crystal configuration with energy E1/2. In retrospect, it would be more accurate to restate the first sentence of this section as “growth of a film or a crystal at low supersaturations can be envisaged as adding molecules one after another to a kink site.” That is, at low supersaturations, molecules at some coincidental kink site have slightly higher rates of adsorption than desorption, just enough to have net growth at this coincidental kink site. Molecules attached to a step-free (110) surface are less bound to the aggregate than molecules at a kink, and therefore have higher rates of desorption, and will not contribute to net growth. At higher supersaturations, however, even molecules attached to a stepfree surface will be sufficiently stable to survive. In other words, the initiation of new kinks on a step-free surface becomes more probable under those growth conditions and might even compete with growth along kink sites at very high supersaturations. In addition, the more new kinks are initiated in parallel, the faster the film can grow by “regular” incorporation of molecules at kink sites because more such kink sites are available. If the appearance of kink sites increases at higher supersaturation, also the incorporation of a single molecule in the void separating two neighboring kinks might become an important process to consider. A last growth process is the incorporation of interstitial molecules at grain boundaries. Despite that the number of molecules involved in grain boundaries is relatively limited, they may have pronounced effects on charge transport in polycrystalline films.

Structural Defects in Pentacene Thin Films

J. Phys. Chem. B, Vol. 111, No. 1, 2007 141

Figure 2. Sketch of the equilibration of a molecule with an aggregate in a configurational coordinate diagram (energy versus configuration of the molecule relative to the aggregate). (1) The molecule hits the aggregate at a random position; (2) the molecule relaxes to the nearest local energy minimum; (3) the molecule jumps back and forth to other energy minima until they are equilibrated. In equilibrium, most molecules will adopt the lowest-energy configuration Q0, but a number of molecules will still adopt the configuration Q1.

To conclude, the following growth processes are illustrated in Figure 1 and are the subject of study in this work: (a) initiation of new kink sites, (b) incorporation of molecules at kink sites, (c) incorporation of molecules in the void between two neighboring kinks, and (d) incorporation of interstitial molecules between two grains. 2.2. The Quasi-Static Approach. Macroscopically, a film grows uniformly by simultaneous addition of molecules at various binding sites across the surface. When zooming in to one particular binding site or a small collection of binding sites on a small piece of an aggregate, film growth can microscopically be envisaged as adding one molecule after another to the aggregate. In the quasi-static limit, a molecule or an ensemble of molecules has enough time to equilibrate with its environment before another molecule is added. The equilibration process is sketched in the configurational coordinate diagram in Figure 2. When a molecule approaches an aggregate close enough to be attracted by it, it will soon adopt a binding configuration corresponding to a local energy minimum in the potential energy surface: any small change in configuration by translating or rotating the molecule relative to the aggregate will result in a higher potential energy, that is, a less bound state. The pace to find a local energy minimum is clocked by the frequency of intermolecular vibrations υ, which is characterized by typical optical phonon frequencies ∼1012 s-1 for organic molecular crystals, corresponding to optical phonon energies around 10 meV.40,41 Consequently, this is a picosecond process. Having found a local energy minimum configuration Q1 with energy EQ1, the molecule or ensemble of molecules might flip to another local energy configuration Q0 with energy EQ0 by crossing an activated configuration Qb with energy EQb, separating the initial and the final states. The characteristic time τrelax for any configuration consisting of one or more molecules to jump into another configuration is given by considering thermal activation over the barrier EQb - EQ1:

τrelax )

1 EQb - EQ1 ν‚exp kTsub

(

)

(3)

The attempt-to-jump frequency υ is the frequency at which two neighboring molecules vibrate relative to each other and is again characterized by typical optical phonon frequencies.

Backward jumps from final to initial states are also possible, but now the barrier to be overcome by thermal activation is given by EQb - EQ0. When the system is given enough time, any ensemble of molecules can jump back and forth between all its possible configurations until an equilibrium occupation of all configurations is reached. In equilibrium, the number of forward jumps between any two configurations equals the number of backward jumps. The probability to find an equilibrated molecule or ensemble of molecules in one or the other particular local energy minimum configuration can be deduced by imagining N identical systems composed of identical aggregates and identical molecules or ensemble of molecules to equilibrate in each system. Once the equilibrium condition is reached for all systems, there will be NQ1 systems in configuration Q1 with energy EQ1, and NQ0 systems in the configuration Q0 with energy EQ0. Expressing this equilibrium by equaling the number of forward and backward jumps yields:

(

)

(

) (

NQ0 EQ1 - EQb EQ0 - EQb ) NQ0 exp , or ) kTsub kTsub N NQ1 1 1 ) and (4) EQ0 - EQ1 N EQ1 - EQ0 1 + exp 1 + exp kTsub kTsub

NQ1 exp

(

)

)

irrespective of the energy EQb of the highest activated configuration Qb separating the two states, and irrespective of which configuration was created first. Alternatively, this same expression for the equilibrium distribution of states can also be obtained (and extended to more than two stable states) using statistical mechanics, by calculating the partition function and the classical probability to find the system in one or another state.39 A necessary condition to reach this equilibrium is that the system has enough time to equilibrate. This means that the time for relaxation/equilibration, represented by the characteristic time to make forward or backward jumps τrelax, should be smaller than the characteristic time to add yet another molecule to the system, τadd. The condition that τrelax , τadd represents the limit of validity of the quasi-static approach presented here. Note that, contrary to the equilibrium occupation of states, the time τrelax required to reach this equilibrium depends on EQb. The characteristic time scale τadd obviously depends on the rate at which molecules offer themselves to a kink site, hence on supersaturation and substrate temperature through a series of events including nucleation, creation of kink sites, and diffusion. While analytical expressions exist for those events, they do not often yield very quantitative results. To avoid those complexities, a more “top-down” estimate will be given here based on typical experimental values. Pentacene is typically grown at a net flux (this is the final film thickness measured after deposition, divided by the total deposition time) of 10-1100 Å/s,34,47 corresponding to 0.01-0.1 monolayers per second. Each monolayer contains about 1014 molecules per squared cm. On average, the deposition flux is then 1012-1013 molecules/ cm2 s. Typical pentacene grain sizes range from 10-1 to 101 µm diameter,13,34,42 which can be traced back to nucleation densities of at least 1010-106 nuclei/cm2. In general, the nucleation density is higher at higher fluxes, hence at higher supersaturations.37 Assuming each nucleus to have between 100 (very low ∆µ) and 103 (high ∆µ) kink sites per µm perimeter (this number in addition varies while the nucleus is growing), and distributing the total flux of molecules over all available kink sites at the surface, we end up with an average back-of-

142 J. Phys. Chem. B, Vol. 111, No. 1, 2007 the-envelope time τadd of 10 µs (at low flux/supersaturation!) to 100 ms (at high supersaturation!) between the addition of two molecules to one specific repetitive step. Note that those values and the following analysis are grafted to films grown by organic molecular beam deposition. When molecules have a high initial kinetic energy as in supersonic molecular-beam deposition, the impact of impinging molecules might induce a sort of local annealing affecting the processes described here.43,44 There is one more limitation implicit to the quasi-static approach presented above. So far, it has been assumed that the aggregate itself, to which molecules are added, is frozen in an equilibrium crystal configuration. Above a certain substrate temperature, however, this aggregate itself is subject to instantaneous thermal fluctuations that may produce a certain degree of intrinsic disorder, particularly at its surfaces. For tetrathiophene at room temperature, for example, the distribution of inter-ring torsion angles is much wider at the surface than in the bulk.45 Given that pentacene has a more rigid molecular structure than thiophenes, and that typical pentacene films are grown at substrate temperatures significantly smaller than the pentacene sublimation temperature, such effects will be neglected here. 2.3. Computational Methods. All computations are based on homemade Matlab routines unless specified differently. Freestanding monolayer aggregates of pentacene molecules adopting the crystal structure reported by Siegrist et al. were constructed,46 exposing one free (110) surface edge as in Figure 1. The monolayer has (001) top and bottom surfaces (i.e., the surface with the lowest energy) as is typical for thin films grown on inert substrates and also for charge transport layers in organic thin-film transistors.47-49 The (110) surface edge was chosen for adding molecules one by one to the aggregate, because it is one of the more pronounced surfaces in the equilibrium crystal shape.15 Each freshly arriving molecule was translated along the a and b unit cell directions in steps of 0.1 Å and was allowed to rotate along its long molecular axis in steps of 1° (relative to the nearest ideal crystal configuration).50 For each configuration of the molecule, intermolecular interactions with its nearest neighbors in the aggregate were computed using the van der Waals and electrostatic components of the MM3 force field.51-53 As such, the potential energy surface was constructed for each molecule, which allowed identifying local and global energy minima, as well as rigorously determining potential barriers between minima.54 Neglecting next-nearest neighbors introduces errors