Impact of thermodynamic fluctuations and pattern size on the nucleation behavior during area selective deposition Free

Area selective deposition is a promising route for bottom-up synthesis that could help overcome the challenges of lithographic methods at the nanometer scale.^1,2 One of the most common approaches is the use of selective surface passivation using functionalization to hinder nucleation and growth in a specific material. Methods under this approach can be broadly divided into two categories: a first type of passivation strategy relies on functional groups that are tightly bonded to the surface.³ The challenge with this type of methods is how to ensure that during the passivation step the functional groups can block all available reactive sites on the surface: if the adsorption kinetics is too fast, the system risks getting stuck on a local minimum that leaves out a fraction of surface sites accessible to incoming precursors. Moreover, the intrinsic catalytic activity of many surfaces at growth temperature can also cause a degradation of the passivating layer.

A second group of processes comprises adsorbates that are dynamic and can relax toward the formation of tightly packed monolayers on the patterned surface. Self-assembled monolayers (SAMs) are the most common example of this second kind.^4–7 While the strong binding energy of thiolate, siloxane, and phosphonate-based SAMs confers them excellent thermal stability, experimental evidence of the dynamic nature of these layers is well established, particularly at higher temperatures. For instance, the coexistence and evolution of multiple crystalline phases, the ability to remove vacancies and to alter the concentration of gauche defects using annealing processes, and the presence of order-disorder phase transitions with negligible desorption are all hallmarks of systems capable of such dynamic evolution.^8–11 Broadly speaking, there are two types of relaxation mechanisms in SAMs: one is related to the dynamics and alignment of the monomer chains, and the second is the mobility of the monomer head itself, which is related to how corrugated is the potential energy surface for monomer adsorption. For instance, in the case of alkane thiols, density functional theory calculations predict barrier heights for the lateral translocation of SAMs between 5 and 25 kJ/mol depending of the direction on the Au surface and the chain length.¹² These values are in agreement with earlier calculations of $CH3S$ adsorption on Au 111 surfaces, where the binding energy surface showed barriers of the order of 5 kJ/mol.¹³ Even in the case of monoalkylsilanes, where cross-linking is thought to stabilize the monolayer, experiments show inconsistencies between the minimum monomer-monomer distance (4.2 Å) and the largest possible distance between two Si atoms connected through siloxane bridges (3.3 Å). This has led to the suggestion that alkylsilane monolayers exhibit a dynamic equilibrium involving the formation and breaking of siloxane bonds so that, at any given time, the layer is only partially cross-linked.¹⁴ 

Recently, analytic approaches building on the traditional Avrami’s nucleation model have provided a way of quantifying and fitting the evolution of growth selectivity in area selective deposition and to discriminate between the static and dynamic generation of new nucleation sites.¹⁵ One example of a system exhibiting such a continuous nucleation rate is Cu passivated with an octadecylphosphonic acid self-assembled monolayer.^7,15 While the underlying mechanistic aspects leading to a dynamic nucleation rate cannot be ascertained using this approach, the presence of dynamic nucleation rates emphasizes the need of understanding the dynamics of passivating layers during area selective deposition.

The fundamental premise of this work is that, if the passivating monolayer is in local thermodynamic equilibrium, thermodynamic fluctuations can lead to the spontaneous generation of surface defects that can affect growth selectivity even in the case of ideal surfaces that are fully covered with monomers or functional groups. These fluctuations would provide a lower boundary for the number of spontaneous defects in passivated surfaces and consequently for the growth selectivity that could be attainable. The presence of thermodynamic fluctuations would also make the density of defects on patterned surfaces sensitive to the pattern size due to the lower cohesive and binding energy of monomers at the perimeter of patterned areas compared to those in the bulk. We expect these effects to play a more dominant role as pitch sizes become smaller.

The goal of this work is to study the link between the equilibrium thermodynamics of such systems and the dynamic nucleation of defects during area selective deposition. In particular, we focus on the impact that pattern size has on the probability of finding coverage defects at equilibrium. This question is central to establish a connection between lab-scale experimental results, typically carried out on homogeneous surfaces or large pattern sizes, and patterned surfaces that are of interest for semiconductor industry, usually with linewidths in the nanometer scale.

Let us first start with the nucleation rate during growth due to spontaneous defect formation on a perfectly passivated homogeneous surface. In a fully covered surface, under thermal equilibrium, the formation of surface vacancies has to take place through the displacement of a monomer to either a grain boundary or an interstitial site. This is akin to the Frenkel defects present in crystals. Under thermal equilibrium, the surface density of defects $n$ will be given by

Here, $εd$ corresponds to the energy of formation (normalized to $kBT$⁠) of one of such vacancies and $N$ is the total number of monomers. This expression is derived in the canonical ensemble, where the number of monomers $N$ and also the total surface area remain constant. Equation (1) allows us to express the probability of having a surface defect $Pd$ at any given site as

Consequently, the number of precursor molecules that will be able to react in our passivated surface per unit surface area will be

where $β0$ is the reaction probability and $Jwall$ is the flux of particles per unit surface area. This provides a simple way of establishing a correlation between the dynamic nucleation rate $N˙0$⁠, the precursor pressure $p$⁠, and the energy of the surface defect $εd$⁠, as shown in Fig. 1.

In patterned surfaces, though, we can postulate the presence of two additional types of defects: first, as in conventional solid state physics, vacancies can also be formed when monomers migrate to the edge of the patterned area. Again, under the canonical ensemble the density of such defects can be expressed as

where $Ns$ is the number of edge sites. A key difference with defects in solid is that $Es$ will be affected by the lower binding energy of adsorbed species outside the pattern. If we consider a patterned line that is $W$ monomers wide, the probability $Ps$ of finding a defect in our patterned surface will be given by

A second type of defect comes from edge roughness effects: at equilibrium, we expect the edge of the patterned surface to become rough, introducing a higher concentration of defects near the interface. In contrast to surface vacancy defects, there is no simple model that can provide an estimate of the contribution of edge effects to the defect density: the presence of a patterned area where monomers preferentially adsorb removes the translational invariance that is assumed in the most common statistical models of surface and step roughening.¹⁶ One reasonable approximation is to assume that the probability of finding a defect due to edge effects $Pw$ depends on the linewidth $W$ and the difference in energy between adsorbates inside and at the outer edge of the pattern $εw$

where the $1/W$ comes from the ratio of edge to fully coordinated sites. Consequently, if our ansatz is correct, should these defects be dominant, we would expect a defect probability $Pw$ that is inversely proportional to the linewidth $W$⁠.

In general, the density of defects should be the sum of the three contributions, $n=nd+ns+nw$⁠, which means that the probability of finding a defect on a given site will be

In order to further explore the impact that pattern size has on the spontaneous formation of defects through thermodynamic fluctuations, we have developed a simple atomistic model of dynamic adsorbates. Since we are interested in a subset of conditions where the patterned surface is fully covered with monomers, we can sidestep the complex interactions arising between monomers at low surface coverages in systems such as SAMs. Instead, we can work with a simplified nonbonding interaction potential that captures the main features of the nonbonding potentials between monomers and the fluctuations around the equilibrium state.¹⁷ 

In particular, we have considered a lattice gas model, since it allows us to precisely define and track coverage defects on the patterned surface. A key difference between our model and most surface models in the literature is that in our model the equilibrium distance between monomers corresponds to second nearest neighbors in our lattice: we consider that the energy of a monomer is given by a repulsive term proportional to the occupancy of the first neighbors in the lattice $ε1$⁠, and an attractive term for the second neighbors $−ε2$⁠, where $ε1$ and $ε2$ are defined in $kBT$ units, so that

Consequently, under full coverage, the lowest energy configuration is a close packed structure occupying the second neighbors in the lattice. The main benefit of this approach is that it allows us to incorporate the contribution of interstitial defects. Our model also captures the structure of nonbonded potentials used in the simulation of SAMs, allowing us to incorporate the attractive and repulsive component of the interaction potential.^17,18 More specifically, $ε1$ is related to the compressibility of the monolayer, whereas $ε2$ represents the cohesive energy due to monomer-monomer interaction.

In addition to the interaction potential, we introduce the presence of two distinct surfaces by providing a spatially dependent binding energy. If we define $ε0$ as the difference in binding energies between the two surfaces, under equilibrium, there will be a nonzero probability of a small fraction of adsorbates moving outside the pattern. The selectivity of the surface functionalization can be directly taken into account by adjusting the value of $ε0$⁠. We also need to define how steep the transition is between the two surfaces, a parameter that is related to both the quality of the patterning and the size of the monomers covering the surface with respect to the characteristic lattice size of the underlying surfaces. We have introduced this parameter by allowing the energy to evolve over $rp$ number of surface sites, so that the binding energy progressively changes from 0 to $ε0$ in $εE$ increments given by

Consequently, if $W$ is the width of a patterned line, the probability of having a defect within our patterned surfaces will be a function of the following parameters:

This model is substantially simpler than atomistic models used in the literature to model self-assembled monolayers. Still, all these parameters can be traced back to experimental observables and, as discussed in Sec. V, we can establish a connection between the values of $ε1$ and $ε2$ and the chemical composition of the monomers.

In order to sample equilibrium states, we have relaxed an ideal compact monolayer of adsorbates within our patterned area [i.e., Fig. 2(a)] using a Metropolis algorithm where an adsorbate and one of its six nearest neighbors are selected at random. Whenever the nearest neighbor is not occupied, the switch between the two sites is either allowed if the change in energy $Δε≤0$ or otherwise accepted with a probability $p=e−Δε$⁠. The total number of monomers is preserved during the relaxation process, that is, the algorithm allows the exploration of configurations within the canonical ensemble. This corresponds to the ideal case where no desorption of adsorbates takes place during the selective area deposition process. In addition to sampling the configuration space, this algorithm also provides the kinetic evolution of the surface for the specific case where all reaction rates are given by $k−1=k1e−Δε$⁠, where $k1$ is the rate for energetically favorable transitions, assumed to be the same for all adsorbates.¹⁹ Under this approximation, every step advances the system a time $Δt=1/(6Nk1)$⁠, where $N$ is the number of adsorbates in our simulation domain.

In Fig. 2, we show snapshots of a monolayer of adsorbates adsorbed on a patterned line for different values of the interaction parameters. Figure 2(a) shows the starting configuration, where adsorbates form a compact layer that is confined to the patterned surface. All results shown in Fig. 2 were obtained for a highly selective adsorption case where $ε0=10$ and $rp=5$⁠, that is, there are three intermediate steps when transitioning between the two surfaces.

Figures 2(b)–2(f) show configuration states for different values of $ε1$ and $ε2$⁠. These have been taken after letting the system relax for 100 000 attempts per monomer in order to ensure that the snapshots are representative of the equilibrium conditions. The frequency of accepted moves typically reaches steady state values in fewer than 5000 attempts per adsorbate. From these snapshots, three regions are apparent: low values of cohesive energy $ε2$ are consistent with the presence of a highly disordered, almost fluidlike layer. This is consistent with observations of SAMs composed as short chain alkyl thiols, where the nonbonding interactions between monomers are weaker. As we increase $ε1$ and $ε2$ progressively, a more ordered self-assembled monolayer is obtained. Low values of $ε1$ promote the presence of interstitial states [Figs. 2(c) and 2(d)], whereas at sufficiently large values of $ε1$ and $ε2$⁠, only a few isolated vacancies and edge roughening are observed. Finally, as expected from an equilibrium behavior, in a few cases, monomers have drifted into the second surface, albeit at much smaller values of fractional coverage, as expected due to the large difference in binding energy (⁠$ε0=10$⁠).

To track the formation of spontaneous defects during the simulation, we have quantified the probability of finding vacancies in the self-assembled monolayer, defined as unoccupied sites in the patterned surface that do not have any other monomer in their first coordination sphere [i.e., Fig. 2(e)].

In Figs. 3(a) and 3(b), we show the evolution of the defect probability with linewidth $W$ for 12 different combinations of input parameters. Assuming a monomer-monomer distance of 4 Å, the range explored would correspond to linewidths between 7 and 35 nm. All datapoints are the average of five independent runs where the system, comprising $40×W$ monomers, has been allowed to evolve for 100 000 steps per monomer. The first 20% of the initial evolution is discarded in order to ensure that the model is sampling fluctuations around its equilibrium state. The defect probability values in Figs. 3(a) and 3(b) are obtained from the sampling of the remaining 80%. In all cases, the results are consistent with a $1/W$ dependence (dotted line in Fig. 3). Consequently, at these scales, defects are dominated by surface roughness effects. In Fig. 3(c), we have explored the validity of our ansatz [Eq. (6)] on the expected dependence of defect probability with the energy parameters in our model. The results show that, starting with defect probability values spanning three orders of magnitude, it is possible to coalesce the data points into a cloud of points clustered around a value of $C=6±2$⁠. Still, there is substantial case-to-case variability, indicating that Eq. (6) is a good approximation only within an order of magnitude.

Furthermore, we can discriminate between edge and bulk defects, where an edge defect is defined as any vacancy that has at least one site in its second coordination sphere outside the patterned line (regions in our simulation domain with a binding energy equal or greater than $εE$⁠), and a bulk defect (in the 2D sense) as a defect located deeper inside the patterned surface. In Fig. 4, we show the correlation between bulk and edge defects for $ε0=10$ and selected $(ε1,ε2,rp)$ values. The results in Fig. 4 are consistent with edge defects being the dominant contribution when the cohesive energy $ε2$ is high. Note that our definition of edge and bulk defect is purely geometrical and that we do not track the mechanism of formation of subsurface defects; some will be proper vacancies while in other cases, they may be due to fluctuations in the edge position due to roughening.

Finally, the simulations allow us to quantify the impact of patterning through the comparison of the defect probabilities obtained for the pattenred case and the predicted values for unpatterned surfaces, $P0$⁠. In Fig. 5, we show the ratio between the defect probability of our patterned model with that of the unpatterned surface. In order to generate this figure, we have chosen conditions that conduce to defect probabilities on an unpatterned surface greater than $10−6$ per monomer site according to Eq. (1). The results obtained show that defect probability can be up to three orders of magnitude higher in patterned surfaces than in the unpatterned reference. However, pattern quality, and in particular the degree of confinement of the monolayer in the pattern, introduced in our model via $rp$⁠, seems to be a key factor in determining this ratio, with highly confined monolayers leading to a lower number of defects.

In this work, we have explored the impact of thermodynamic fluctuations on spontaneous defect generation during area selective deposition on patterned surfaces. The results obtained suggest that edge effects and interfacial quality can play a critical role in ensuring a low density of defects in adsorbed monolayers at local thermodynamic equilibrium. More specifically, the model provides a number of predictions that can be experimentally validated:

1. If thermodynamic fluctuations play an important role in spontaneous nucleation during area selective deposition, Eq. (3) predicts that nucleation rate should be proportional to the total exposure.

2. If roughening dominates, we should expect a defect nucleation rate that is inversely proportional to the width of patterned lines. This is true both for edge defects and also for bulk nucleation defects.

3. We expect both cohesive energy between monomers and the quality of the pattern to have important effect in nucleation rates: highly confined monomer layers forming highly cohesive layers lead to a lower spontaneous nucleation rate. Equation (6) can also be used to determine the critical width below which we expect surface defects to dominate.

In order to explore the configuration space, we have chosen the simplest model that can capture the main physics, and in particular the phase transition from solid 2D crystals to a liquid state. While our model greatly simplifies the interaction between monomers in the passivating layer, we still can establish a connection between the values of these parameters and the chemical composition of the monomers for systems for which either experimental or computational data are available. For the case of alkanethiols on Au, the interaction energy seems to be one order of magnitude smaller than the binding energy, of the order of tens of kJ/mol versus the $∼180kJ/mol$⁠. For alkylthiolates, it is well known that cohesion energy increases with the length of the alkyl chain: pair potentials modeling the interaction between short alkanethiol monomers have also been extracted from ab initio calculations, resulting on binding energy potential curves with minima ranging between 0.05 and 0.15 eV.²⁰ This would correspond to values of $ε2≈2$ when the two chains are perfectly aligned at the optimal distance.

A second key parameter in our model is the sharpness of the transition between the two surfaces, broadly referred to in Sec. IV as the quality of the pattern. There are various factors that can affect this transition: one of them is atomic scale roughness, which can cause monomers to see mixed environments over a larger area. Another consideration is whether there is a thickness gradient at the interface: this can affect the local binding energy with respect to that expected from the surface of a bulk, crystalline sample. The results in this work point out that engineering that transition to be as sharp as possible helps mitigating these edge effects.

It is important to note, though, that this work has focused on a situation where local thermodynamic equilibrium can be established within the patterned surface. Although increasing disorder of the chains at higher temperatures in self-assembled monolayers is well established, the reversibility of the bonding of the head groups at higher temperatures is not fully understood, nor is the presence of intermediate states in many of the technologically relevant surfaces. Likewise, experimental studies on the stability of monolayers under hydrating conditions have been limited thus far to low-temperature, solution-based environments,²¹ so not much information is available regarding the impact of small molecules such as water on the bonding of the monomer. The predictions of simple models such as the one presented in this work could help rule out the role of fluctuations versus other degradation pathways.

In this work, we have not explored either how the interaction of the precursor with the monomers affects its reactivity toward the surface in defect sites: we have simply assumed that the presence of a point defect is enough to give the precursor access to the surface site, and that it reacts with the same sticking probability as in an unpassivated surface. While that could be a good approximation for small molecules, bulky precursors may require the formation of much larger defects. This is especially likely when the defects represent regions with disordered chains.

Finally, by focusing on the fully covered conditions, we have not explored how the different interaction parameters affect the formation of a compact monolayer in the first place. One simple generalization of the lattice model presented here would be to extend the simulation conditions to the grand canonical ensemble, where the adsorption and desorption of the monomer is driven by an excess chemical potential. This would allow us to explore how experimental conditions and the thermodynamic properties of the monolayer impact the quality of the passivating layer obtained during the surface passivation step. Another possible generalization is to formulate the dynamics of the monolayer as a Potts model to account for the orientational dependence of chain-chain interaction. Even though these models have been explored in the literature, the presence of a finite size domain with gradients on the binding energy can have an important impact on the system properties. These models may help uncover new behavior on patterned surfaces that is not apparent from the study of these systems under periodic boundary conditions.

This research is based upon work supported by the Laboratory Directed Research and Development (LDRD) funding from the Argonne National Laboratory, provided by the Director, Office of Science, of the U.S. Department of Energy (DOE) under Contract No. DE-AC02-06CH11357.