Microscopic Origin of the Apparent Activation Energy in Diffusion

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Microscopic Origin of the Apparent Activation Energy in Diffusionmediated Monolayer Growth of Two-Dimensional Materials Miguel Angel Angel Gosálvez, and Joseba Alberdi-Rodriguez J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b05794 • Publication Date (Web): 21 Aug 2017 Downloaded from http://pubs.acs.org on August 29, 2017

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The Journal of Physical Chemistry

Microscopic Origin of the Apparent Activation Energy in Diffusion-mediated Monolayer Growth of Two-Dimensional Materials Miguel A. Gosálvez∗,†,‡,¶ and Joseba Alberdi-Rodriguez,¶ Dept. of Materials Physics, University of the Basque Country UPV/EHU, 20018 Donostia-San Sebastian, Spain , and Centro de Física de Materiales CFM-Materials Physics Center MPC, centro mixto CSIC – UPV/EHU, 20018 Donostia-San Sebastian, Spain E-mail: [email protected]

Abstract

simple understanding of the temperature and coverage dependence of this observable. Most importantly, we describe the microscopic origin of the apparent diffusion barrier extracted from a typical Arrhenius plot at any given coverage.

Current trends indicate that mass production of high-quality, two-dimensional materials will likely be based on the use of On-SurfaceSynthesis (OSS) and/or Chemical Vapor Deposition (CVD) on selected substrates. However, the success of these techniques heavily relies on a deeper understanding of terrace and perimeter diffusion of the adspecies across and around many self-generated, mobile clusters and/or motionless islands with a variety of shapes (dendritic, compact, irregular, polygonal,...). We show that, for typical monolayer growth conditions at constant deposition rate, the total square distance traveled by all adsorbed particles departs from the total number of diffusion hops due to the onset of correlations between subsequent hops along the perimeters of a growing density of obstacles. As a result, we propose a new expression to determine the tracer diffusivity of the adparticles, directly providing a

Introduction Graphene and related compounds are ideal candidates to achieve superior energy efficiency by enhancing the storage and release of charge and fuel. 1,2 However, this requires the production of high-quality, large amounts of these twodimensional (2D) materials. On-surface synthesis (OSS) by evaporation of molecular precursors at moderate temperatures has recently received much attention as it enables atomic precision molecular assembly, generating defectless nanoribbons and complex 2D molecular networks on selected substrates. 3–7 Another alternative is chemical vapor deposition (CVD), which is considered suitable for batch production of large amounts of extended monolayer samples with sufficient quality at reasonable cost. 8–14 A common feature for OSS and CVD is the prominent role of diffusion. In a typical CVD process, a metal substrate is heated up to about 1000 ◦ C in the presence of a carbon source vapor. As a result, the vapor is thermally

∗

To whom correspondence should be addressed Dept. of Materials Physics, University of the Basque Country UPV/EHU, 20018 Donostia-San Sebastian, Spain ‡ Centro de Física de Materiales CFM-Materials Physics Center MPC, centro mixto CSIC – UPV/EHU, 20018 Donostia-San Sebastian, Spain ¶ Donostia International Physics Center (DIPC), 20018 Donostia-San Sebastian, Spain †

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decomposed and adparticles are adsorbed on the metal surface, where they diffuse randomly, eventually forming small clusters at random locations (nucleation). The clusters gradually evolve into larger islands by accretion of the diffusing adparticles (growth), and the islands eventually merge to form a single layer (coalescence). The quality of the resulting monolayer depends markedly on the density and structure of the grain boundaries formed during coalescence, which depends on the actual shape of the islands (dendritic, compact, irregular, polygonal,...), which is ultimately a function of the relative occurrence and strength of perimeter diffusion. 15–17 At lower temperatures and with larger diffusing particles, the same picture is valid for OSS. In fact, both dendritic and compact islands can be obtained in OSS, depending on the details of perimeter diffusion. 5 Thus, one cannot overstate the importance of diffusion for the future production of 2D materials. Previous work on surface diffusion has focused on such nucleation, growth and coalescence aspects, analyzing the time evolution of the monomer and island densities, characterizing the island size distributions, and describing the origin of the island capture numbers and capture zones. 18–22 Similarly, much effort has been invested on clarifying the differences and similarities between the tracer and collective diffusion coefficients, and describing them for various systems. 23–29 The tracer diffusivity, DT , which is the relevant transport coefficient when the particles can be followed, can be determined using three alternative formulations, e.g. Equations (1), (3) and (19) in Ref. 28 In this study we present a fourth expression, describing DT in terms of the total hop rate, Rh , defined as the sum of all distinct hop rates accessible to the adparticles, each multiplied by the corresponding hop multiplicity (the number of hops that share a given rate). The proposed formula accounts for small-cluster diffusion across the terraces (monomers, dimers, trimers,...) as well as perimeter diffusion at mobile clusters and motionless islands of any size. In particular, our approach provides an exact formulation of an empirical procedure, 23 where DT is described as the product of an average

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hop rate, Γ, and a correlation factor, fT , which accounts for all memory effects between consecutive hops at finite coverages. 23–27 Such correlations refer to the fact that an adparticle hopping from site i to site j leaves site i empty and, thus, at finite coverage the adparticle has a higher chance of returning to i. We show that Γ must be understood as a triple average of the total hop rate, Γ = hRh i/Na , Rwhere hXi is the Xdt k ∆tk is ensemble average of X, X = R dt = ΣΣk Xk ∆t k the time average of X, and X/Na is the ’perparticle’ average of X, with Na the number of adparticles. Similarly, we stress that fT should be understood as the proportionality factor between the total square distance travelled by all adparticles, hR2 i, and the total number of diffusion hops, hNh i, namely: fT =

hR2 i , l2 hNh i

(1)

with l the hop distance between adjacent adsorption sites. At very low coverage the particles perform completely independent random walks and hR2 i = l2 hNh i. 24,27 Thus, fT = 1. At finite coverages, however, each particle is affected by the presence of (and the interaction with) the other particles and, thus, their random walks become correlated. 26,27 As a result, a fT departs from 1. Here, R2 = ΣN i=1 |ri (t) − 2 ri (0)| , with ri (t) denoting the position of particle i at time t. Although some studies have been devoted to understanding the behavior of the prefactor and the apparent diffusion barrier extracted from an Arrhenius plot of the temperature dependence of the diffusivity, 23,24,26,30 the actual microscopic origin of these quantities is still poorly understood. As shown below, this issue becomes clear once DT is described in terms of the distinct hop rates and their multiplicities, as proposed here.

Theory Let us consider a two-dimensional lattice-gas in order to model a collection of adparticles on a substrate. The substrate is treated as a two-


2

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is to take all adsorption rates equal to a single deposition flux (Fi = F ). θ = LNx La y designates the total coverage, where Lx Ly the total number of adsorption sites (before adsorption of any particle). Since the increase in coverage with = F (1 − θ), direct time satisfies the relation dθ dt integration gives: θ = 1 − e−F t . Thus, initially θ increases linearly (θ ≈ F t) and later saturates at value 1. Defining θi = LxnLi y as the density of particles of type i, the total coverage satisfies the relation: θ = Σi θi . Similarly, Na = Σi ni . Regarding Equation 4, Gi is the desorption rate for a particle of type i. For completeness, the desorption term is shown in our equations, although it can be neglected in both OSS and CVD. Note that we consider perimeter detachment events ending at terrace sites. Now, we propose describing the tracer diffusivity as follows:

dimensional lattice of adsorption sites, where particles from the surrounding environment are deposited randomly (on empty sites) while the already existing adparticles are able to either hop diffusively to a neighboring (empty) site or desorb (back to the environment). Since adsorption, desorption and diffusion events may occur at any given time, let us consider the corresponding total event rate Re = Rh + Ra + Rd , with Rh , Ra and Rd the total hop rate, total adsorption rate and total desorption rate, respectively: X X Rh = Mij νij = ni mij νij , (2) i,j

Ra =

X

Rd =

X

i,j

Fi nφi

= F (1 − θ)Lx Ly ,

(3)

i

Gi ni .

(4)

i

hR2 i , t hNh i , = gl2 fT t = gl2 fT hRh i, X = gl2 fT hMij iνij ,

In Equation 2, i, j = 0, 1, 2, ..., S denote the particle/site type, taken as the coordination number, i.e. the number of in-plane occupied nearest neighbor sites; νij is the hop rate for a particle of type i to jump to a neighbor site of type j; and Mij = ni mij is the corresponding total multiplicity, i.e. the number of hops with rate νij that can be performed. Here, ni is the number of particles of type i and mij is the local multiplicity, i.e. the number of sites of type j surrounding a particle of type i. For triangular and square lattices the maximum coordination number S is 6 and 4 (7 and 5 particle types), respectively. In general, any suitable variable–or set of variables–may be used to uniquely describe the different particle types. Based on Transition State Theory (TST), 31 the hop rates in Equation 2 are written as νij = νa e−Eij /kB T , where the Boltzmann factor e−Eij /kB T indicates the probability to perform the hop at temperature T if the energy barrier is Eij , and the attempt-frequency νa (≈ 1×1013 Hz) describes the frequency with which the substrate phonons and the adparticle vibrations couple with each other. 27 Regarding Equation 3, Fi is the adsorption rate at a site of type i and nφi is the number of empty sites of type i. A typical assumption (adopted in the right-hand-side of Equation 3)

DT = g

(5) (6) (7) (8)

i,j

= gl2 fT

X

hni ihmij iνij .

(9)

i,j 1 , where d is the Here, t is time and g = 2dN a dimensionality (= 2). Equation 5 is based on previous definitions. 27,28,32 Equation 6 is equal to Equation 5 based on the fact that hR2 i = fT l2 hNh i (Equation 1). In turn, the equality between Equations 6 and 7 is based on the observation that dNh /dt is nothing else but the number of hops per unit time, i.e. the total hop Σ R ∆tk rate Rh . Correspondingly, Rh = kΣkh,k = ∆tk Σk

∆Nh,k ∆tk

∆tk

= Nth is an exact expression. Finally, the identity between Equations 7, 8 and 9 is based on the exact decomposition proposed in Equation 2. Note that, defining the correlationfree diffusivity as: t

ˆ T = gl2 hNh i , D t


3

(10)


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hop rates, να , and the corresponding hop multiplicities, Mα : X Ea = E f + ǫα , ǫα = ωα (Eα + EαM ),

then Equations 5 and 6 allow expressing the correlation factor (Equation 1) equivalently as: fT =

DT . ˆT D

(11)

α

To our best knowledge this is the first study where the tracer diffusivity (Equation 5) is exactly linked to the total hop rate (Eqs. 7-9). Mathematically, the apparent activation energy (Ea ) of DT is defined as: Ea = −d(log DT )/dβ = −(1/DT )dDT /dβ, where β = 1/kB T . Thus, using Equation 8 gives: Ea = −

fT

P

d(fT

1 i,j

P

hMij iνij

ωα =

hMij iνij ) , dβ

i,j

i,j

where the weights ωij describe the relative contribution of the hops with rate νij to the total hop rate, i.e. the probability of observing a hop with rate νij :

Σij hMij iνij

.

Σα hMα iνα

.

(16)

To illustrate our theoretical formulation we present computational simulations of monolayer growth using the Kinetic Monte Carlo (KMC) method. 16,17,21,27,33–39 We use a typical lattice-gas model and the rejection-free, timedependent implementation. 17,27,34,35 Although the algorithmic details are provided in the Supporting Information, here we stress the fact that the time increment is calculated as ∆t = − log(e)/Re , where e ∈ (0, 1] is a uniform random number and Re is the total event rate (see Equations 2-4). With some exceptions, 37 selflearning KMC studies usually indicate that only a few key barriers are important in order to describe surface diffusion, even if the KMC simulations automatically generate many dozens of barriers. 38,39 Thus, in this study the activation energies are chosen in order to generate realistic morphologies for both triangular and square lattices while keeping the resulting models (i) simple enough (to enable a foundational analysis of the apparent activation energy of the tracer diffusivity) and, (ii) disengaged from any specific material (to maintain a general discussion for both OSS and CVD). Although the particular activation energies are listed in Table 1S of the Supporting Information, we indicate here that (i) they fulfill detailed balance, 27,33 and (ii) a few barriers for the triangular lattice are so large that they can be regarded as ∞ for the discussions below. This way there are only four (six) different hop rates for the triangular (square) lattice, referred to as: να = νa e−Eα /kB T , with α = 0, 1, 2, 3 (α = 0, a, 1, b, c, 2). To keep the discussion general, we use a wide range of temperatures and deposition fluxes (triangular lat-

(12)

hMij iνij

hMα iνα

(15)

Computational method

where l and g = 1/2dNa in Equation 8 are considered independent of T (and β). [Na = Lx Ly θ = Lx Ly (1 − e−F t ) only depends on the deposition flux.] Then, applying the chain rule to Equation 12 and writing νij ∝ e−Eij β , M f hMij i ∝ e−Eij β , and fT ∝ e−E β easily leads to: X Ea = E f + ωij (Eij + EijM ), (13)

ωij =

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(14)

Note that ωij depends not only on the value of the hop rate itself, νij , but also on the number of hops with that rate at a given time, i.e. the hop multiplicity Mij . In this manner, Equation 13 includes the dependence on both coverage and morphology (through their effect on the actual values of the multiplicities), in addition to the temperature dependence (through the activation energies assigned to the correlation factor, E f , the multiplicities, EijM , and the hop rates, Eij ). For the triangular (square) lattice used here (see next section), the notation can be simplified in terms of the 4 (6) distinct


4

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mental formulation of fT , which is equivalent to the empirical definition 24 (fT = 4DT /l2 Γ), provided Γ is defined properly (e.g. as above). Should part of (or all) hop distances lij be different, Equation 9 is generalized as follows:

temperature, thus describing νij , hMij i and fT M f as proportional to e−Eij β , e−Eij β and e−E β , respectively, for the derivation of Equation 13 from Equation 12. The variation of Eij with temperature is not used here, but it might be useful in future studies. As a result, the only assumptions made in order to derive Equation 13 are that the lattice parameter of the subM strate and the prefactors of e−Eij β , e−Eij β and f e−E β remain essentially constant with temperature. Should any of these variables turn out to be strongly dependent on temperature, then their actual contribution can easily be added into Equation 13. In this study we simply regard their contribution to Equation 13 as a second order effect and, thus, negligible. The computational aspects of the study (i.e. the KMC simulations) are used to test the validity of Equation 13 for particular examples. For this purpose, the main approximation consists on considering a number of regions in the overall temperature range, so that EijM , E f and, especially, wij are estimated as accurately as possible. We conclude that, from a theoretical perspective, Equation 13 provides an excellent model to describe the origin of the apparent activation energy and, from a computational viewpoint, the estimates for EijM , E f and, especially, wij become progressively more accurate as the number of regions is increased. Overall, we conclude that, even when a single process dominates, the macroscopic activation energy is a modified version of the microscopic activation energy of that process, due to the temperaturedependence of the corresponding multiplicity. Regarding Equation 7, note that it can be l2 Γ where Γ = hRh i/Na . rewritten as DT = fT 2d Thus, this study establishes that the average single-particle hop rate, Γ, loosely introduced in Ref., 23 should be understood as the time average of the total hop rate per particle. Although Γ has been determined in various ways, 24–27 the exact formulation based on the time average has not been reported previously. In fact, our exact formulation provides a fundamental relation between the effective barrier Ea and the microscopic activation barriers (Equation 13), unavailable before. Regarding the correlation factor itself, Equation 1 provides a more funda-

DT = fT

1 X 2 hWi ihµij ilij , 2d i,j

(17)

where hWi i = h Nnia i = h θθi i is the occupancy, understood as the probability of occupying a site of type i, and hµij i = hmij iνij is the rateplicity (hop rate × local hop multiplicity). Both concepts have been defined in a recent study of the low-coverage tracer diffusivity for complex energy landscapes with any number of distinct hop rates. 32 It is remarkable that Equation 17, which is valid out of equilibrium and contains all correlations between the hopping particles at any coverage, has the same form as Equation (29) in Ref. 32 for non-interacting particles (θ ≈ 0) in equilibrium, where fT = 1. We also point out that, for the reverse process, i.e. anisotropic etching, we previously explained the origin of the apparent activation energy of the etch rate based on a similar formulation. 40 Although future image processing techniques might enable analyzing sequences of brieflyspaced, detailed microscopy images, eventually providing a reasonable description of the motion of all adparticles, we argue that a sequence of just two such images should contain essential information. For this purpose, let us consider the Master equation solved in this study (i = 0, 1, ..., S): dθi = Fi θiφ + Σj6=i θj mji νji − Gi θi − θi Σj6=i mij νij , dt (18) nφ

where θiφ = Lx Li y is the density of empty sites of type i. Equation 18 means that the density of adparticles of type i increases due to direct adsorption from the gas phase (Fi θiφ ) and/or due to hops by particles of type j 6= i ending at sites of type i (Σj6=i θj mji νji ), while it decreases due to direct desorption to the gas phase (-Gi θi ) and/or due to hops by atoms of type i ending at sites of type j 6= i (-θi Σj6=i mij νij ). Now, we


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note that during typical monolayer growth (at constant deposition and negligible desorption), the morphology of the system (e.g. the perimeter shapes) at any given coverage directly reflect a particular density of empty and occupied sites (θiφ and θi ), as well as a particular set of values for the local multiplicities mij (i.e. the number of sites of type j surrounding the particles of type i). This opens the possibility for determining the hop rates (νij ) by directly extracting the values of θi , θiφ and mij from two images separated by small increments in θi and solving: θi Σj6=i mij νij − Σj6=i θj mji νji = F θiφ −

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(the time-average), and (ii) traditional studies have mainly focused on the density of islands, Nisl , and/or variables related to it, such as the 0 ≈ Nθisl , the rate average island size s = θ−θ Nisl

(1−θ) ≈ FN ), of growth of the average size ( ds dt isl etc... These variables are defined only between island nucleation and island coalescence, thus reducing the coverage window where the studies can be performed. On the contrary, the total hop rate–and the tracer diffusivity–are properly defined for any coverage value, well before island nucleation and well after island coalescence. Thus, we conclude that the tracer diffusivity is the natural transport coefficient that describes monolayer growth at constant deposition flux. Finally, we stress the idea that the actual shapes of the clusters and islands at any given coverage for both OSS and CVD processes directly reflect particular values for the density of empty and occupied sites, as well as for the numbers of sites of type j surrounding the particles of type i. This should help extracting the hop rates directly from microscopy images in the future. Supporting Information. A PDF file is provided containing: the details of the KMC simulations, Table S1 (atomistic activation energies used in the simulations), Figure S1 (equivalent to Figure 1, now for the square lattice), Figure S2 (coverage-dependence of the correlation factor for the triangular and square lattices), Figure S3 (complementary to Figure 2, now for additional coverage values), and Figure S4 (complementary to Figure 4, now for nine temperature subregions).

dθi . dt (19)

Thus, by directly constraining the search space, these equations should be useful to improve the efficiency and accuracy of evolutionary procedures aimed at determining the hop rates. 17 In this manner, extracting the reactivity directly from microscopy images might be possible in the future for both OSS and CVD processes. This might be useful as an automated alternative to existing approaches for extracting elementary hop rates. 18,21,29,36

Conclusions In summary, we show that the tracer diffusivity, traditionally defined as DT = ghR2 i/t = gl2 fT hNh i/t, is exactly described by the less familiar expression DT = gl2 fT hRh i, where hRh i = hNh i/t is the time average of the total hop rate. Based on this, we are able to describe accurately the apparent activation energy of DT in terms of the microscopic activation barriers (Equation 13). As a result, we show that the semi-empirical expression DT = l2 fT 2d Γ, based on loosely defining Γ as an average single-particle hop rate in previous studies, becomes an exact formulation once Γ is understood as the time average of the total hop rate per particle, Γ = hRh i/Na . Presumably, our formulation has eluded previous studies because (i) the tracer diffusivity is proportional to a slightly hidden version of the total hop rate

Acknowledgement We acknowledge support by the Ramón y Cajal Fellowship Program by the Spanish Ministry of Science and Innovation, and the 2015/01 contract by the DIPC. The KMC calculations were performed on the ATLAS supercomputer in the DIPC.

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Microscopic Origin of the Apparent Activation Energy in Diffusion

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