Natural scaling of size distributions in homogeneous and heterogeneous rate equations with size-linear capture rates , V. G. Dubrovskii and Yu. S. Berdnikov
Citation: The Journal of Chemical Physics 142, 124110 (2015); doi: 10.1063/1.4916323 View online: http://dx.doi.org/10.1063/1.4916323 View Table of Contents: http://aip.scitation.org/toc/jcp/142/12 Published by the American Institute of Physics
THE JOURNAL OF CHEMICAL PHYSICS 142, 124110 (2015)
Natural scaling of size distributions in homogeneous and heterogeneous rate equations with size-linear capture rates V. G. Dubrovskii1,2,3,a) and Yu. S. Berdnikov1 1
St. Petersburg Academic University, Khlopina 8/3, 194021 St. Petersburg, Russia Ioffe Physical Technical Institute of the Russian Academy of Sciences, Politekhnicheskaya 26, 194021 St. Petersburg, Russia 3 ITMO University, Kronverkskiy pr. 49, 197101 St. Petersburg, Russia 2
(Received 9 December 2014; accepted 14 March 2015; published online 31 March 2015) We obtain exact solutions of the rate equations for homogeneous and heterogeneous irreversible growth models with linear size dependences of the capture rates. In the limit of high ratios of diffusion constant over deposition rate, both solutions yield simple analytical scaling functions with the correct normalizations. These are given by the cumulative distribution function and the probability density function of the gamma-distribution in homogeneous and heterogeneous cases, respectively. Our size distributions depend on the value of the capture rate a in the reaction of joining two mobile monomers A1 (A1 + A1 → A2) or the monomer attachment to the reactive defect B (A1 + B → AB). In homogeneous cases, the size distribution is monotonically decreasing regardless of a. In heterogeneous growth, the distribution is monotonically decreasing when a ≤ 1 and monomodal when a > 1. The obtained solutions describe fairly well the experimental data on the length distributions of Al, Ga, In, and Mn adatom chains on Si(100)-2 × 1 surfaces. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4916323]
I. INTRODUCTION
Nucleation and growth of two-dimensional (2D) surface islands has been of interest for a long time.1–4 Many important aspects of the island growth kinetics on the pre-coalescence stage of epitaxial process can be well understood within the rate equation (RE) approach5 for the island size distribution (SD) n s . One of the most important features of irreversible growth is the scaling property that suggests a particular behavior of the SD as a function of the scaled size s/ ⟨s⟩ (with ⟨s⟩ as the mean size) and the coverage Θ in the limiting regime.6–14 This scaling is usually formulated as follows: in the limit of large ratios of the adatom diffusion coefficient (D) over the deposition rate (F), Λ = D/F → ∞, the SD n s (⟨s⟩ , Θ) is expected to scale as n s (⟨s⟩ , Θ) = [Θ/⟨s⟩2] f (s/ ⟨s⟩), with f (x) independent of Λ. Particular forms of the SDs and the scaling properties depend on the capture rates σ s which describe the strength of islands of a given size to capture adatoms from the feeding zones around the islands. It was soon realized that the standard mean-field methods for determining the capture rates often yield non-analytic scaling with singular functions f (x). In particular, non-analytic scaling behaviors have been confirmed within the RE approach for mean-field capture rates, σ s = const by Bartelt and Evans,7 σ s ∝ s p with 0 ≤ p < 1/2 by Vvedensky,11 and σ s ∝ s p with 0 ≤ p < 1 in our recent work.14 Mean-field treatments15–17 assume that the local environment of an island does not depend on its neighbors. In this case, the capture rate can be obtained as a solution to the diffusion equation for an isolated island. This implies that the characteristic diffusion length of adatoms a)[email protected]
0021-9606/2015/142(12)/124110/9/$30.00
is much smaller than the distance between the islands, the usual assumption in nucleation theory with decay of islands.18–21 However, this approach may fail in irreversible growth at very high diffusivities, where islands always compete for diffusion flux. Kinetic Monte Carlo simulations13,22 and theoretical calculations of the capture zones based on the Voronoi tessellation22–24 show that in many cases, σ s is a linear function of s at large enough s. Larger islands have larger capture zones due to the wider empty regions surrounding such islands,22 and the capture rate is roughly proportional to the island area. Furthermore, the comprehensive analysis of Korner et al.13 reveals that the capture rates are almost independent of the coverage for compact islands. According to the current view,13 self-consistent REs with the correct capture rates are capable of describing the scaling properties of the SDs. Of course, this requires small enough Θ to avoid the coalescence effects. When the RE approach applies, simple analytical SDs and the corresponding scaling functions can be obtained in some cases;7,11,14,16 however, the exactly solvable models are rare. Consequently, in this work, we present a fully analytical description of the SDs in homogeneous and heterogeneous irreversible growth which becomes possible for a model system with size-linear and coverage-independent capture rates for all s. We show that the corresponding REs can be integrated explicitly and their shapes are strongly affected by one and single control parameter—the size-independent constant a in the linear interpolation of σ s . Both homogeneous and heterogeneous REs naturally yield analytic scaling behavior of the SDs in the limit Λ → ∞. The forms of the scaled SDs are very different. We obtain monotonically decreasing shapes in homogeneous case, while heterogeneous SDs feature either
142, 124110-1
© 2015 AIP Publishing LLC
124110-2
V. G. Dubrovskii and Yu. S. Berdnikov
J. Chem. Phys. 142, 124110 (2015)
monotonically decreasing or monomodal behaviors depending on the value of a. Finally, we discuss how our analytical solutions might be useful for interpreting experimental data and modeling of the SDs of linear chains of metal adatoms (Mn, Ga, In, and Al) on Si(100)-2 × 1.25–33,36 In particular, we consider the interplay between monotonically decreasing and monomodal shapes of the SDs depending on the homogeneous or heterogeneous character of nucleation.
II. RATE EQUATIONS
In homogeneous irreversible growth, we study the set of reactions, As + A1 → As+1, s = 1, 2, 3, . . ., which yield the formation of immobile islands consisting of s monomers (“size”) As by attaching mobile adatoms (or monomers) A1. The corresponding REs have the form5,7,14 dn s = Dn1(σ s−1n s−1 − σ s n s ), s ≥ 2. (1) dt In heterogeneous growth, islands are formed on the immobile nucleation centers B (reactive point defects, impurities, etc.) via the scheme As B + A1 → As+1 B, s = 0, 1, 2, 3, . . .. The set of heterogeneous REs writes down as33 dn0 = −Dn Aσ0n0, dt (2) dn s = Dn A(σ s−1n s−1 − σ s n s ), s ≥ 1, dt where n0 and n A denote the concentration of free nucleation centers and free monomers, respectively. Direct capture of monomers by deposition on top of (or adjacent to) an island of size s is disregarded as unimportant for the scaling behavior.13 The surface coverage by homogeneous and heterogeneous islands (including monomers) is defined as Θ = n1 + sn s , Θ = n A + sn s , (3) s ≥2
s ≥1
respectively, and obeys the equation dΘ = F(1 − Θ). (4) dt The mean value of any s-dependent value bs over the ensemble of homogeneous islands is obtained as bs n s s ≥2 ⟨b⟩ ≡ , N= ns . (5) N s ≥2
III. ASYMPTOTIC ANALYSIS IN HOMOGENEOUS CASE
Generally, the capture rates that enter the REs, given by Eq. (1) or (2), depend on the island size s, ratio Λ ≡ D/F, and coverage Θ. Precise analytical description of σ s (Λ, Θ) is beyond reach to this end, unless the capture rates are determined within the mean field scheme.15–21 Nevertheless, one can use Eqs. (1) and (3)–(5) for σ s to derive dn1 (7) + 2Dσ1n12 + Dn1 N ⟨σ⟩ = F(1 − Θ). dt We now assume that n1 tends to zero for all but short times, in which case Eqs. (7) and (4) yield dn1 N ⟨σ⟩ n1 +Λ = 1. (8) dΘ 1−Θ At large Λ, the concentration of free monomers rapidly reaches the stationary state with n1
1−Θ . ΛN ⟨σ⟩
(9)
As discussed in the Introduction, kinetic Monte Carlo simulations13,22 and geometrical models for capture zones22–24 show that σ s is proportional to s at large s and often independent of Θ. Therefore, below we assume that Θ . (10) N The coverage-independent coefficient α in the linear expression ⟨σ⟩ = α ⟨s⟩ can be always put to one by the appropriate renormalization of time in Eq. (1). If the slope of the linear dependence of σ s on s depends on Θ, the situation becomes more complex and will be studied elsewhere. Inserting Eq. (10) into Eq. (8) and solving it at large Λ yields ⟨σ⟩ ⟨s⟩ =
( ) 2 1 − exp − ΛΘ 1 − Θ . (11) 1−Θ ΛΘ √ The last expression holds for Θ ≥ Θ0 ∼ 1/ Λ and shows that the asymptotic stage is established already when √ the coverage reaches a very small value of the order of 1/ Λ.13 Figure 1 demonstrates how the asymptotes match the solutions at smaller Θ as Λ increases. n1 =
1−Θ ΛΘ
Obviously, N is the total concentration of all immobile islands at a given time. In the heterogeneous case, we write instead bs n s s ≥0 tot ⟨b⟩ ≡ , n + N = n , N = ns . 0 B ntot B s ≥1 (6) Here, = const denotes the total concentration of all immobile islands including free nucleation centers, which equals the total concentration of nucleation centers at any time. Therefore, heterogeneous SD can be normalized to one by dividing it to ntot B : s ≥0 n s /ntot = 1, while the normalization of homogeneous SD is time-dependent. ntot B
FIG. 1. Monomer concentration versus coverage at different Λ (solid lines); the dotted lines show the asymptotes n 1 = (1 − Θ)/(ΛΘ).
124110-3
V. G. Dubrovskii and Yu. S. Berdnikov
J. Chem. Phys. 142, 124110 (2015)
As in Ref. 14, we now consider the ansatz that linearizes Eq. (1) to dn s = σ s−1n s−1 − σ s n s , dz The z variable is given by
s ≥ 2.
(12)
dz F(1 − Θ) = Dn1 . (13) dt Θ Here, the first equation is exact and the approximate expression holds when the monomer concentration is obtained from Eq. (11) on the asymptotic stage. Using Eq. (4) and putting √ z to zero at Θ0 ∼ 1/ Λ, we find that z depends logarithmically on the coverage whenever the mean capture rate equals the mean size, as given by our Eq. (10): z ln(Θ/Θ0). As shown in Appendix A, the choice of Θ0 that ensures the correct normalization of homogeneous SD is given by Θ0 = [a(a + 1)/2Λ]1/2, where a ≡ σ1 is the capture rate of the dimerization reaction A1 + A1 → A2. This yields 1/2 a(a + 1) ez . (14) Θ 2Λ Using Eq. (14) in asymptotic Eq. (11) results in 1/2 1/2 1 2 2 −z e − e−z , n1 Λa(a + 1) Λ Λa(a + 1) (15) where the 1/Λ term can be safely neglected in the scaling limit Λ → ∞. From Eq. (15), the ratio of the diffusion term over the direct impingement term scales as Λn1 ∼ Λ1/2 and tends to infinity at Λ → ∞. This allows us to neglect direct impingement when considering the scaling behavior in our model. The surface density of islands obeys the equation dN/dt = aDn12 or, in view of Eq. (13), dN/dz = an1. Using Eq. (15) and integrating, we find 1/2 1/2 2a 2a −z N (1 − e ) . (16) Λ(a + 1) Λ(a + 1) Therefore, the island density saturates at large enough z. Finally, from Eqs. (14) and (16), the asymptotic mean size ⟨s⟩ Θ/N increases exponentially with z, ⟨s⟩
(a + 1) z e . a
discussed in Refs. 2, 13, and 22, heterogeneous growth model considered in Ref. 33, and with our major assumption ⟨σ⟩ ⟨s⟩ introduced in Sec. III. Of course, these capture rates can hardly be rigorously justified in concrete systems for any size s and should be considered as approximate models yielding exact solutions which are rarely achieved in more complex systems. For small sizes, Eqs. (18) and (19) can be treated as the first terms in the Taylor expansion of arbitrary σ s in s near s = 1 or 0. With arbitrary diffusivities of monomers, size-linear capture rates σ s would apply only for some one-dimensional model systems such as shown in Fig. 2: vertical nanowires fed from a vapor phase through their sidewalls,19 linear peptide chains growing from solutions with a continuous influx of “monomers” due to chemical reactions and surface adatom rows in situations where adatoms are captured on the sides of rows and then diffuse to the ends. If the adatom capture just occurs at the ends of surface rows, the capture rates become size-independent corresponding to the formal limit a → ∞ in Eqs. (18) and (19). This case yields monomodal distributions with non-analytic scaling.7,14 In systems with high diffusivities (this property is essential for considering the scaling limit D/F → ∞), sizelinear capture rates might be due to a competition between the islands for accommodating the diffusion fluxes of monomers as discussed above. Here, we write the capture rates as functions of s as in Refs. 5, 11, 14, 18–21, 34, and 35 rather than in the form σ s / ⟨σ⟩ = f (s/ ⟨s⟩) as suggested by Bales and Chrzan.15 However, the case σ s s at large s and therefore ⟨σ⟩ ⟨s⟩ is the simple particular case of the Bales-Chrzan representation. In our model, the a constant gives the probability of nucleation of the very first particle A2 or A1 B, while σ s s at s >> 1 describes the aggregation of large clusters. Small a values relate to situations where aggregation is preferred to nucleation, while at large a, nucleation is preferred to aggregation. As shown in Appendix B, the sets of discrete REs given by Eq. (12), for s ≥ 2 in the homogeneous case and s ≥ 1 in the heterogeneous case and with our size-linear capture rates, have the following exact solutions:
(17)
IV. EXACT SOLUTIONS FOR SIZE-LINEAR CAPTURE RATES
Let us now consider a simplified case where the capture rates are linear in s not only for large but also for all sizes, and independent of Θ. In the homogeneous case, we write σ s = a + s − 1,
s ≥ 1,
(18)
thus, the dimerization rate σ1 = a, with arbitrary positive a. In the heterogeneous case, we use instead σ s = a + s,
s ≥ 0,
(19)
with the rate of monomer attachment to the nucleation center σ0 = a. Such models correlate with the simulation results
FIG. 2. Illustration of growth systems with linear size dependence of the capture rates σ s = a + s: catalyst-assisted vertical nanowires which collect growth species from its sidewalls (a), linear chains of molecules growing from water solutions (b), and one-dimensional surface rows in situations where adatoms are captured on the sides of chains and then diffuse to the ends (c).
124110-4
V. G. Dubrovskii and Yu. S. Berdnikov
n s+2(z) =
J. Chem. Phys. 142, 124110 (2015)
Γ(a + s + 1) Γ(a)s! z d yn1(z − y)e−(a+1)y (1 − e−y )s , ×
Λ. Obviously, our φa (x) satisfies the well-known properties required from the scaling functions2,8,13 s ≥ 0,
0
(20) n s+1(z) =
Γ(a + s + 1) Γ(a)s! z × d yn0(z − y)e−(a+1)y (1 − e−y )s ,
s ≥ 0.
0
(21) Here, Γ(a + s + 1) denotes gamma-function and s! = Γ(s + 1) is the factorial. Clearly, the properties of these SDs are determined by the behaviors of the monomer concentration n1(z) in the homogeneous case and the concentration of free nucleation centers n0(z) in the heterogeneous case. The latter is easily calculated exactly from Eq. (2) for n0—indeed, in view of dz = Dn A dt, we have dn0/dz = −an0 and therefore, −az n0(z) = ntot . Be
(22)
Using this result in Eq. (21), we obtain the exact solution in the form of the Polya distribution33 n s (⟨s⟩) = ntot B
Γ(a + s) (⟨s⟩ /a)s , Γ(a)s! (1 + ⟨s⟩ /a)a+s
s ≥ 0.
(23)
∞
∞
⟨s⟩ = a(e − 1) z
(24)
is the mean size of the SD. Homogeneous growth is more complex for the analysis because no exact solution is known for n1(z) and we have to resort to some approximations. However, asymptotic Eq. (15) leads to a simple continuum approximation of the homogeneous SD, as will be discussed in Sec. V.
V. CONTINUUM SIZE DISTRIBUTIONS AND SCALING
In the heterogeneous case, using Γ(a+s)/s! s a−1 at a/s 1. At a = 1, the discrete Polya SD is reduced to geometrical distribution which is monotonically decreasing and has the continuum form given by Eq. (25) with φ1(x) = exp(−x). At a → ∞, corresponding to size-independent capture rates, the Polya SD is reduced to Poissonian, s ntot (s − ⟨s⟩)2 B −⟨s⟩ ⟨s⟩ → exp − , e n s (⟨s⟩) = ntot B s! 2 ⟨s⟩ 2π ⟨s⟩ (28) and yields the Gaussian continuum SD which does not feature analytic scaling in terms of the s/ ⟨s⟩ variable.7,14,33 Explicit representation of the homogeneous SD given by Eq. (20) with the monomer concentration defined by Eq. (15) is derived in Appendix C and has the form n s+1(⟨s⟩ , Θ)
Here,
dx xϕa (x) = 1
dxϕa (x) =
Θ (a + 1) I1−(a+1)/2⟨s⟩(s, a). ⟨s⟩2 2a
(29)
Here, I y (s, a) is the regularized incomplete beta function in standard notations.36 When ⟨s⟩ → ∞ and s → ∞, this SD automatically acquires the scaling form n(s, ⟨s⟩ , Θ)
Θ ⟨s⟩2
f a (x) ,
(30)
with x = s/ ⟨s⟩. As shown in Appendix C, the scaling function f a (x, Λ) can be well approximated by ( ) a + 1 (a + 1)x f a (x) = γ ,a , (31) 2a 2
(25)
where x = s/ ⟨s⟩, with the scaling function a a a−1 −a x x e . (26) Γ(a) Thus, the scaling SD is given by the probability density function of the gamma-distribution. The first equality in Eq. (25) does not require any assumption on the value of Λ. Thus, the scaling form of the Polya SD is the intrinsic property of the heterogeneous growth model with size-linear capture rates.33 However, we need ⟨s⟩ Θ/ntot B to obtain the traditional scaling behavior6 given by the second equality in Eq. (25), which requires n A → 0 according to Eqs. (3) and (6). This holds for all but very short times, only at large ϕa (x) =
FIG. 3. Polya SDs (solid lines) and their continuum approximations (dashed lines) at a = 2 and different ⟨s⟩. The discrete Polya SD is defined only for integer s, so the solid lines are guide for the eye only.
124110-5
V. G. Dubrovskii and Yu. S. Berdnikov
J. Chem. Phys. 142, 124110 (2015)
FIG. 4. Continuum SDs in the homogeneous model, obtained from Eq. (29) at ⟨s⟩ = 3, 5, and 10 for a = 0.5 and 1 and ⟨s⟩ = 3, 5, 10, 15, and 30 for a = 3 and re-plotted in scaling variables, compared to the scaling SDs (bold lines) given by Eqs. (31) and (32).
where γ( y, a) =
1 Γ(a)
∞ dtt a−1e−t
(32)
y
is the regularized incomplete gamma function. Thus, the homogeneous scaling function is given by the cumulative distribution function of the gamma-distribution. Very importantly, the scaling function f a (x) obeys both normalization conditions given by Eq. (27) for any a. Figure 4 demonstrates how the non-scaled SD given by Eq. (29) approaches the scaling shapes at different a. Opposite to the heterogeneous case, this scaling function is monotonically decreasing regardless of the value of a, as shown in Fig. 5. The observed difference in the shapes of the scaling functions can be well understood from elementary considerations. Indeed, in the heterogeneous case, the dimers A1 B can be formed only on the free nucleation centers B whose concentration is fixed and thus the value of n0 rapidly tends to zero. If a is large enough (greater than one), the n0 decreases faster than the concentrations of larger clusters. This creates a peak in the SD which subsequently propagates toward larger s. In the homogeneous case, the dimers A2 are built from two monomers whose number is continuously refilled from
FIG. 5. Scaling functions in homogeneous (solid lines) and heterogeneous (dashed lines) growth at different a. The scaling function exp(−x) is the same in both cases for a = 1.
vapor, resulting in a purely decreasing SD. It is interesting to note that the two models yield exactly identical exponential scaling function when a = 1, with the capture rates being strictly proportional to the island size in both cases. VI. LINEAR CHAINS OF METAL ADATOMS ON SILICON
Previous works25–28 revealed that room temperature deposition of group III (Ga, In, and Al) and some transition metals (Mn) produces stable linear chains of surface adatoms which lie orthogonal to the Si-dimer rows on 2 × 1 reconstructed Si(100) surface. Scanning tunneling microscopy imaging allowed for precise determination of the SDs of these chains over sizes (i.e., lengths) at low coverages (typically less than 0.13). In certain cases, for example, for linear chains of Al25 and Ga26 adatoms, the scaled SDs were found monotonically decreasing, although a “weak” peak for small sizes was not excluded. In contrast, the time-dependent SDs of Mn adatom chains were decreasing only at the beginning of growth, while for longer times, the SD shapes transformed to the pronounced monomodal.28 The role of heterogeneous nucleation at C-type defects versus homogeneous nucleation of metal surface chains on Si
FIG. 6. SDs of Ga adatom chains26 at different coverages (symbols), and their best fits (lines) by the homogeneous beta (a) and heterogeneous Polya (b) distributions with the experimental mean sizes and different a.
124110-6
V. G. Dubrovskii and Yu. S. Berdnikov
FIG. 7. Size histograms f s (s) of Mn adatom chains on Si(100)-2 × 1 after 2 (a), 7 (b), and 12 (c) depositions from Ref. 28, fitted by the Polya SD given by Eq. (23) at a = 3 and different ⟨s⟩ = 2.50 at 2 s, 4.96 at 5 s, and 6.44 at 12 s..
has been widely discussed. In particular, Albao et al.29 argued that the growth of Ga chains on Si(100) is not dominated by heterogeneous nucleation, contrasting the proposal of Kocán et al.30 and the experimental findings of Javorský et al.27 in the case of In chains which were shown to terminate at C defects. Probability of heterogeneous nucleation in a given system should depend on the concentration of the
J. Chem. Phys. 142, 124110 (2015)
reactive defects. As demonstrated theoretically in Ref. 37, concentration of the defects larger than 0.0025 per site may change the island SDs from monomodal to monotonically decreasing in the case of the irreversible growth model. Fragmentation of islands has also been found important for the SD shapes.31,32,37 For example, theoretical results of Ref. 31 suggest that the scaling functions are monotonously decreasing in quasi-equilibrium case with fragmentation, while in the case of irreversible growth, they exhibit a monomodal character. As mentioned already, our irreversible growth model with size-linear and coverage-independent capture rates is expected to apply to the case of compact islands,13 including linear adatom chains. The obtained analytical SDs are monotonically decreasing in homogeneous case and are capable of describing both monotonically decreasing and monomodal shapes in heterogeneous case depending on whether the a value is smaller or greater than one. Therefore, within the model, monomodal shapes should correspond to homogeneous nucleation, while purely decreasing SDs can be described by both homogeneous and heterogeneous solutions. In particular, Fig. 6 shows experimental SDs of Ga adatom chains26 which could be reasonably fitted by both homogeneous beta and heterogeneous Polya distributions with very similar a values. Figure 7 demonstrates a good fit of the time-dependent SDs of Mn chains28 obtained with the Polya distribution at a = 3 in the heterogeneous case (the normalization is adjusted to the experimental size histograms). Of course, the monomodal SDs at longer growth times cannot be described within the homogeneous model.
FIG. 8. Scaled SDs of Al25 (a), Ga26 (b), In27 (c), and Mn28 (d) adatom chains versus scaled size, fitted with homogeneous and heterogeneous scaling functions.
124110-7
V. G. Dubrovskii and Yu. S. Berdnikov
Figure 8 shows the measured SDs of Al, Ga, In, and Mn adatom chains,25–28 plotted in scaling variables and fitted by the continuum homogeneous and heterogeneous scaling functions given by Eqs. (31) and (26), respectively. It is seen that homogeneous scaling function reproduces the experimental data slightly better for Al and Ga. The In chains are described much better by heterogeneous growth model, while for Mn, the only reasonable fit is obtained within heterogeneous model. These results correlate with Figs. 6 and 7 and the conclusions drawn in the original works, i.e., more probable homogeneous mechanism in the case of Ga26 and heterogeneous nucleation on C-defects for In27 and Mn28 adatom chains.
J. Chem. Phys. 142, 124110 (2015)
APPENDIX A: CHOICE OF Θ0
For small enough Θ and in the scaling limit Λ → ∞, Eq. (11) of the main text is reduced to 1 . (A1) ΛΘ Integrating Eq. (13) in view of dΘ = F(1 − Θ)dt yields z ln(Θ/Θ0), or n1
Θ Θ0 e z .
The value of Θ0 should be determined self-consistently to ensure the correct normalization of the mean size. Using Eq. (A2) in Eq. (A1), we have
VII. CONCLUSION
To sum up, we have obtained exact analytical solutions for SDs in homogeneous and heterogeneous irreversible growth for the model system described by the REs with sizelinear capture rates for all s. These solutions are given by the beta and Polya distributions, respectively. Both solutions naturally yield the scaling form of the continuum SDs in the limit of D/F → ∞, with the correct normalization of the island density and mean size. The homogeneous SD is monotonically decreasing for all a, while the heterogeneous one is decreasing at a ≤ 1 and monomodal at a > 1. Both scaling solutions are reduced to f (x) = exp(−x) when a = 1. The obtained SD shapes describe quite well the experimental data on Al, Ga, In, and Mn adatom chains on 2 × 1 reconstructed Si(100) surfaces and support the conclusion on the homogeneous character of nucleation for Al and Ga and heterogeneous nucleation for In and Mn. Our exact solutions of the REs and the corresponding scaling functions depend critically on the assumption of size-linear σ s . Indeed, the analytic scaling behavior of the SDs is not expected to hold when σ s ∝ s p with p < 1,11,14 so the case of σ s ∝ s at large s seems to be the best possibility for scaling within the RE approach. We have used the coverage independence of the capture rates, the assumption which looks relevant for compact but neither for point nor fractal island.13 The coverage-dependent and sizelinear capture rates will be studied elsewhere. The obtained SDs are highly sensitive to the value of the constant a. This effect has been disregarded in some continuous models.8,11,14 Such a property can be understood through the “drift” form of the boundary condition in the continuous RE at zero size.14 We also point out that the kinetic Monte Carlo simulations for compact islands13 yield monomodal SDs in most cases, which contradicts our conclusion for homogeneous growth. This discrepancy needs to be investigated in more detail.
(A2)
n1
e−z . ΛΘ0
(A3)
We now note that Eq. (3) reduces the generating function (see Appendix B for the detail) for the discrete SD given by Eq. (20) to n(x, z) =
z
s ≥0
dy
n s+2(z)x = a s
ae−z ΛΘ0
0
z dy 0
n1(z − y)e−(a+1)y [1 − x(1 − e−y )]a+1
e−a y . [1 − x(1 − e−y )]a+1
(A4)
Obviously, from the definition of the generating function, it follows that n(1, z) N at n1 → 0. Using Eq. (A4), we obtain N
a . ΛΘ0
(A5)
From Eqs. (A2), (A5), and ⟨s⟩ Θ/N, the asymptotic mean size should equal ΛΘ20 z e . (A6) a On the other hand, the mean size can be obtained as ⟨s⟩
N ⟨s⟩
∂n(x, z) . ∂ x x=1
(A7)
Using Eqs. (A4) and (A5), we find (a + 1) z e . 2 Comparing this to Eq. (A6) yields 1/2 a(a + 1) Θ0 = , 2Λ ⟨s⟩
(A8)
(A9)
the equation used in the main text. APPENDIX B: SOLUTION OF RATE EQUATIONS FOR SIZE-LINEAR CAPTURE RATES
ACKNOWLEDGMENTS
Yu.S.B. thanks the FP7 UC project “Nanoembrace” (Grant Agreement No. 316751). V.G.D. gratefully acknowledges financial support received from the Russian Science Foundation under the Grant No. 14-22-00018.
Let σ s be given by Eq. (18) of the main text. Inserting these σ s into the linearized REs of homogeneous growth given by Eq. (12), we have dn s = (a + s − 2)n s−1 − (a + s − 1)n s , dz
s ≥ 2. (B1)
124110-8
V. G. Dubrovskii and Yu. S. Berdnikov
J. Chem. Phys. 142, 124110 (2015)
We now introduce the generating function for concentrations by the definition n(x, z) = n s+2(z)x s . (B2) s ≥0
Using Eq. (B1), after simple calculation, we obtain the closed differential equation in partial derivatives for the generating function, ∂n(x, z) ∂n(x, z) − (x − 1)x = (x − 1)(a + 1)n(x, z) + an1(z). ∂z ∂x (B3)
APPENDIX C: CONTINUUM SIZE DISTRIBUTION IN HOMOGENEOUS CASE
Starting from the exact solution given by Eq. (20), we first use Eq. (15) and then make the substitution of variables e−y = t under the integral. Together with the known formula for the beta function,34 B(s, a) =
1 I y (s, a) = B(s, a) (B4)
yielding that no islands are present on the surface at zero moment of time. The equivalent system of ordinary differential equations for Eq. (B3) writes down as dx dn dz =− = . 1 (x − 1)x (x − 1)(a + 1)n + an1
(B5)
From the first equation, we get the first integral of the form ( x ) Ψ1 = z − ln . (B6) 1−x From the second equation, with the initial condition given by Eq. (B4) and using Eq. (B6), we obtain n(x, z) = a
dy 0
n1(z − y)e−(a+1)y [1 − x(1 − e−y )]a+1
.
Γ(a + s + 1) 1 = ηs a+1 Γ(a + 1)s! (1 − η) s ≥0
(B7)
(B8)
for η = x(1 − e−y ) in Eq. (B7). The result has the form Γ(a + s + 1) xs Γ(a)s! s ≥0 z d yn1(z − y)e−(a+1)y (1 − e−y )s .
×
(B9)
0
Comparing this to Eq. (B2) yields the final result given by Eq. (20) of the main text. The described procedure is exactly identical for heterogeneous growth except for the definition of the z variable and the generating function which should be changed to dz = Dn A, dt n(x, z) = n s+1(z)x s . s ≥0
dtt s−1(1 − t)a−1,
1 B(s, a) =
(B10) (B11)
(C2)
dtt s−1(1 − t)a−1 0
as the incomplete and complete beta functions, respectively. To obtain the scaling form of Eq. (29), we note that whenever ⟨s⟩ → ∞, s → ∞, and a is finite, the known asymptote of incomplete beta function34 yields ( ) (a + 1) s I1−(a+1)/2⟨s⟩(s, a) γ ,a . (C3) 2 ⟨s⟩ This leads to the limiting curve given by Eqs. (30) and (31) of the main text.
1H.
Brune, Surf. Sci. Rep. 31, 121 (1998). W. Evans, P. A. Thiel, and M. C. Bartelt, Surf. Sci. Rep. 61, 1 (2006). 3W. Dieterich, M. Einax, and P. Maass, Eur. Phys. J.: Spec. Top. 161, 151 (2008). 4M. Einax, W. Dieterich, and P. Maass, Rev. Mod. Phys. 85, 921 (2013). 5A. Venables, G. D. T. Spiller, and M. Hanbucken, Rep. Prog. Phys. 47, 399 (1984). 6T. Vicsek and F. Family, Phys. Rev. Lett. 52, 1669 (1984). 7M. C. Bartelt and J. W. Evans, Phys. Rev. B 46, 12675 (1992). 8M. C. Bartelt and J. W. Evans, Phys. Rev. B 54, R17359 (1996). 9J. W. Evans and M. C. Bartelt, Phys. Rev. B 63, 235408 (2001). 10J. G. Amar and F. Family, Phys. Rev. Lett. 74, 2066 (1995). 11D. D. Vvedensky, Phys. Rev. B 62, 15435 (2000). 12M. Korner, M. Einax, and P. Maass, Phys. Rev. B 82, 201401 (2010). 13M. Korner, M. Einax, and P. Maass, Phys. Rev. B 86, 085403 (2012). 14V. G. Dubrovskii and N. V. Sibirev, Phys. Rev. B 89, 054305 (2014). 15G. S. Bales and D. C. Chrzan, Phys. Rev. B 50, 6057 (1994). 16G. S. Bales and A. Zangwill, Phys. Rev. B 55, R1973 (1997). 17P. Jensen, H. Larralde, and A. Pimpinelli, Phys. Rev. B 55, 2556 (1997). 18S. A. Kukushkin and A. V. Osipov, Prog. Surf. Sci. 51, 1 (1996). 19V. G. Dubrovskii, Nucleation Theory and Growth of Nanostructures (Springer, Heidelberg, New York, Dordrecht, London, 2014). 20V. G. Dubrovskii, J. Chem. Phys. 131, 164514 (2009). 21V. G. Dubrovskii and M. V. Nazarenko, J. Chem. Phys. 132, 114507 (2010). 22M. C. Bartelt, C. R. Stoldt, C. J. Jenks, P. A. Tiel, and J. W. Evans, Phys. Rev. B 59, 3125 (1999). 23F. G. Gibou, C. Ratsch, M. F. Gyure, S. Chen, and R. E. Caflisch, Phys. Rev. B 63, 115401 (2001). 24F. G. Gibou, C. Ratsch, and R. E. Caflisch, Phys. Rev. B 67, 155403 (2003). 25T.-C. Shen, C. Wang, and J. R. Tucker, Phys. Rev. Lett. 78, 1271 (1997). 26M. A. Albao, M. M. R. Evans, J. Nogami, D. Zorn, M. S. Gordon, and J. W. Evans, Phys. Rev. B 72, 035426 (2005). 27J. Javorský, M. Setvín, I. Ošt’ádal, P. Sobotík, and M. Kotrla, Phys. Rev. B 79, 165424 (2009). 28H. Liu and P. Reinke, Surf. Sci. 602, 986 (2008). 2J.
The series expansion of this n(x, z) is performed by applying the formula
n(x, z) =
y 0
n(x, z = 0) = 0,
(C1)
and Eqs. (14), (16), and (17), this results in Eq. (29) of the main text, with
This should be solved with the initial condition,
z
Γ(s)Γ(a) , Γ(s + a)
124110-9
V. G. Dubrovskii and Yu. S. Berdnikov
29M. A. Albao, M. M. R. Evans, J. Nogami, D. Zorn, M. S. Gordon, and J. W.
Evans, Phys. Rev. B 74, 037402 (2006). 30P. Kocán, P. Sobotík, and I. Ošt’ádal, Phys. Rev. B 74, 037401 (2006). 31V. I. Tokar and H. Dreyssé, Phys. Rev. B 74, 115414 (2006). 32V. I. Tokar and H. Dreyssé, Phys. Rev. B 84, 085456 (2011). 33V. G. Dubrovskii and Yu. S. Berdnikov, Mater. Phys. Mech. 21, 207 (2014). 34J. R. King and J. A. D. Wattis, J. Phys. A: Math. Gen. 35, 1357 (2002).
J. Chem. Phys. 142, 124110 (2015) 35C.
Ratsch, A. Zangwill, P. Šmilauer, and D. D. Vvedensky, Phys. Rev. Lett. 72, 3194 (1994). 36Handbook of Mathematical Functions, Applied Mathematics Series Vol. 55, edited by M. Abramowitz and I. A. Stegun (National Bureau of Standards, 1964). 37P. Kocán, P. Sobotík, I. Ošt’ádal, M. Setvín, and S. Haviar, Phys. Rev. E 80, 061603 (2009).
Citation: The Journal of Chemical Physics 142, 124110 (2015); doi: 10.1063/1.4916323 View online: http://dx.doi.org/10.1063/1.4916323 View Table of Contents: http://aip.scitation.org/toc/jcp/142/12 Published by the American Institute of Physics
THE JOURNAL OF CHEMICAL PHYSICS 142, 124110 (2015)
Natural scaling of size distributions in homogeneous and heterogeneous rate equations with size-linear capture rates V. G. Dubrovskii1,2,3,a) and Yu. S. Berdnikov1 1
St. Petersburg Academic University, Khlopina 8/3, 194021 St. Petersburg, Russia Ioffe Physical Technical Institute of the Russian Academy of Sciences, Politekhnicheskaya 26, 194021 St. Petersburg, Russia 3 ITMO University, Kronverkskiy pr. 49, 197101 St. Petersburg, Russia 2
(Received 9 December 2014; accepted 14 March 2015; published online 31 March 2015) We obtain exact solutions of the rate equations for homogeneous and heterogeneous irreversible growth models with linear size dependences of the capture rates. In the limit of high ratios of diffusion constant over deposition rate, both solutions yield simple analytical scaling functions with the correct normalizations. These are given by the cumulative distribution function and the probability density function of the gamma-distribution in homogeneous and heterogeneous cases, respectively. Our size distributions depend on the value of the capture rate a in the reaction of joining two mobile monomers A1 (A1 + A1 → A2) or the monomer attachment to the reactive defect B (A1 + B → AB). In homogeneous cases, the size distribution is monotonically decreasing regardless of a. In heterogeneous growth, the distribution is monotonically decreasing when a ≤ 1 and monomodal when a > 1. The obtained solutions describe fairly well the experimental data on the length distributions of Al, Ga, In, and Mn adatom chains on Si(100)-2 × 1 surfaces. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4916323]
I. INTRODUCTION
Nucleation and growth of two-dimensional (2D) surface islands has been of interest for a long time.1–4 Many important aspects of the island growth kinetics on the pre-coalescence stage of epitaxial process can be well understood within the rate equation (RE) approach5 for the island size distribution (SD) n s . One of the most important features of irreversible growth is the scaling property that suggests a particular behavior of the SD as a function of the scaled size s/ ⟨s⟩ (with ⟨s⟩ as the mean size) and the coverage Θ in the limiting regime.6–14 This scaling is usually formulated as follows: in the limit of large ratios of the adatom diffusion coefficient (D) over the deposition rate (F), Λ = D/F → ∞, the SD n s (⟨s⟩ , Θ) is expected to scale as n s (⟨s⟩ , Θ) = [Θ/⟨s⟩2] f (s/ ⟨s⟩), with f (x) independent of Λ. Particular forms of the SDs and the scaling properties depend on the capture rates σ s which describe the strength of islands of a given size to capture adatoms from the feeding zones around the islands. It was soon realized that the standard mean-field methods for determining the capture rates often yield non-analytic scaling with singular functions f (x). In particular, non-analytic scaling behaviors have been confirmed within the RE approach for mean-field capture rates, σ s = const by Bartelt and Evans,7 σ s ∝ s p with 0 ≤ p < 1/2 by Vvedensky,11 and σ s ∝ s p with 0 ≤ p < 1 in our recent work.14 Mean-field treatments15–17 assume that the local environment of an island does not depend on its neighbors. In this case, the capture rate can be obtained as a solution to the diffusion equation for an isolated island. This implies that the characteristic diffusion length of adatoms a)[email protected]
0021-9606/2015/142(12)/124110/9/$30.00
is much smaller than the distance between the islands, the usual assumption in nucleation theory with decay of islands.18–21 However, this approach may fail in irreversible growth at very high diffusivities, where islands always compete for diffusion flux. Kinetic Monte Carlo simulations13,22 and theoretical calculations of the capture zones based on the Voronoi tessellation22–24 show that in many cases, σ s is a linear function of s at large enough s. Larger islands have larger capture zones due to the wider empty regions surrounding such islands,22 and the capture rate is roughly proportional to the island area. Furthermore, the comprehensive analysis of Korner et al.13 reveals that the capture rates are almost independent of the coverage for compact islands. According to the current view,13 self-consistent REs with the correct capture rates are capable of describing the scaling properties of the SDs. Of course, this requires small enough Θ to avoid the coalescence effects. When the RE approach applies, simple analytical SDs and the corresponding scaling functions can be obtained in some cases;7,11,14,16 however, the exactly solvable models are rare. Consequently, in this work, we present a fully analytical description of the SDs in homogeneous and heterogeneous irreversible growth which becomes possible for a model system with size-linear and coverage-independent capture rates for all s. We show that the corresponding REs can be integrated explicitly and their shapes are strongly affected by one and single control parameter—the size-independent constant a in the linear interpolation of σ s . Both homogeneous and heterogeneous REs naturally yield analytic scaling behavior of the SDs in the limit Λ → ∞. The forms of the scaled SDs are very different. We obtain monotonically decreasing shapes in homogeneous case, while heterogeneous SDs feature either
142, 124110-1
© 2015 AIP Publishing LLC
124110-2
V. G. Dubrovskii and Yu. S. Berdnikov
J. Chem. Phys. 142, 124110 (2015)
monotonically decreasing or monomodal behaviors depending on the value of a. Finally, we discuss how our analytical solutions might be useful for interpreting experimental data and modeling of the SDs of linear chains of metal adatoms (Mn, Ga, In, and Al) on Si(100)-2 × 1.25–33,36 In particular, we consider the interplay between monotonically decreasing and monomodal shapes of the SDs depending on the homogeneous or heterogeneous character of nucleation.
II. RATE EQUATIONS
In homogeneous irreversible growth, we study the set of reactions, As + A1 → As+1, s = 1, 2, 3, . . ., which yield the formation of immobile islands consisting of s monomers (“size”) As by attaching mobile adatoms (or monomers) A1. The corresponding REs have the form5,7,14 dn s = Dn1(σ s−1n s−1 − σ s n s ), s ≥ 2. (1) dt In heterogeneous growth, islands are formed on the immobile nucleation centers B (reactive point defects, impurities, etc.) via the scheme As B + A1 → As+1 B, s = 0, 1, 2, 3, . . .. The set of heterogeneous REs writes down as33 dn0 = −Dn Aσ0n0, dt (2) dn s = Dn A(σ s−1n s−1 − σ s n s ), s ≥ 1, dt where n0 and n A denote the concentration of free nucleation centers and free monomers, respectively. Direct capture of monomers by deposition on top of (or adjacent to) an island of size s is disregarded as unimportant for the scaling behavior.13 The surface coverage by homogeneous and heterogeneous islands (including monomers) is defined as Θ = n1 + sn s , Θ = n A + sn s , (3) s ≥2
s ≥1
respectively, and obeys the equation dΘ = F(1 − Θ). (4) dt The mean value of any s-dependent value bs over the ensemble of homogeneous islands is obtained as bs n s s ≥2 ⟨b⟩ ≡ , N= ns . (5) N s ≥2
III. ASYMPTOTIC ANALYSIS IN HOMOGENEOUS CASE
Generally, the capture rates that enter the REs, given by Eq. (1) or (2), depend on the island size s, ratio Λ ≡ D/F, and coverage Θ. Precise analytical description of σ s (Λ, Θ) is beyond reach to this end, unless the capture rates are determined within the mean field scheme.15–21 Nevertheless, one can use Eqs. (1) and (3)–(5) for σ s to derive dn1 (7) + 2Dσ1n12 + Dn1 N ⟨σ⟩ = F(1 − Θ). dt We now assume that n1 tends to zero for all but short times, in which case Eqs. (7) and (4) yield dn1 N ⟨σ⟩ n1 +Λ = 1. (8) dΘ 1−Θ At large Λ, the concentration of free monomers rapidly reaches the stationary state with n1
1−Θ . ΛN ⟨σ⟩
(9)
As discussed in the Introduction, kinetic Monte Carlo simulations13,22 and geometrical models for capture zones22–24 show that σ s is proportional to s at large s and often independent of Θ. Therefore, below we assume that Θ . (10) N The coverage-independent coefficient α in the linear expression ⟨σ⟩ = α ⟨s⟩ can be always put to one by the appropriate renormalization of time in Eq. (1). If the slope of the linear dependence of σ s on s depends on Θ, the situation becomes more complex and will be studied elsewhere. Inserting Eq. (10) into Eq. (8) and solving it at large Λ yields ⟨σ⟩ ⟨s⟩ =
( ) 2 1 − exp − ΛΘ 1 − Θ . (11) 1−Θ ΛΘ √ The last expression holds for Θ ≥ Θ0 ∼ 1/ Λ and shows that the asymptotic stage is established already when √ the coverage reaches a very small value of the order of 1/ Λ.13 Figure 1 demonstrates how the asymptotes match the solutions at smaller Θ as Λ increases. n1 =
1−Θ ΛΘ
Obviously, N is the total concentration of all immobile islands at a given time. In the heterogeneous case, we write instead bs n s s ≥0 tot ⟨b⟩ ≡ , n + N = n , N = ns . 0 B ntot B s ≥1 (6) Here, = const denotes the total concentration of all immobile islands including free nucleation centers, which equals the total concentration of nucleation centers at any time. Therefore, heterogeneous SD can be normalized to one by dividing it to ntot B : s ≥0 n s /ntot = 1, while the normalization of homogeneous SD is time-dependent. ntot B
FIG. 1. Monomer concentration versus coverage at different Λ (solid lines); the dotted lines show the asymptotes n 1 = (1 − Θ)/(ΛΘ).
124110-3
V. G. Dubrovskii and Yu. S. Berdnikov
J. Chem. Phys. 142, 124110 (2015)
As in Ref. 14, we now consider the ansatz that linearizes Eq. (1) to dn s = σ s−1n s−1 − σ s n s , dz The z variable is given by
s ≥ 2.
(12)
dz F(1 − Θ) = Dn1 . (13) dt Θ Here, the first equation is exact and the approximate expression holds when the monomer concentration is obtained from Eq. (11) on the asymptotic stage. Using Eq. (4) and putting √ z to zero at Θ0 ∼ 1/ Λ, we find that z depends logarithmically on the coverage whenever the mean capture rate equals the mean size, as given by our Eq. (10): z ln(Θ/Θ0). As shown in Appendix A, the choice of Θ0 that ensures the correct normalization of homogeneous SD is given by Θ0 = [a(a + 1)/2Λ]1/2, where a ≡ σ1 is the capture rate of the dimerization reaction A1 + A1 → A2. This yields 1/2 a(a + 1) ez . (14) Θ 2Λ Using Eq. (14) in asymptotic Eq. (11) results in 1/2 1/2 1 2 2 −z e − e−z , n1 Λa(a + 1) Λ Λa(a + 1) (15) where the 1/Λ term can be safely neglected in the scaling limit Λ → ∞. From Eq. (15), the ratio of the diffusion term over the direct impingement term scales as Λn1 ∼ Λ1/2 and tends to infinity at Λ → ∞. This allows us to neglect direct impingement when considering the scaling behavior in our model. The surface density of islands obeys the equation dN/dt = aDn12 or, in view of Eq. (13), dN/dz = an1. Using Eq. (15) and integrating, we find 1/2 1/2 2a 2a −z N (1 − e ) . (16) Λ(a + 1) Λ(a + 1) Therefore, the island density saturates at large enough z. Finally, from Eqs. (14) and (16), the asymptotic mean size ⟨s⟩ Θ/N increases exponentially with z, ⟨s⟩
(a + 1) z e . a
discussed in Refs. 2, 13, and 22, heterogeneous growth model considered in Ref. 33, and with our major assumption ⟨σ⟩ ⟨s⟩ introduced in Sec. III. Of course, these capture rates can hardly be rigorously justified in concrete systems for any size s and should be considered as approximate models yielding exact solutions which are rarely achieved in more complex systems. For small sizes, Eqs. (18) and (19) can be treated as the first terms in the Taylor expansion of arbitrary σ s in s near s = 1 or 0. With arbitrary diffusivities of monomers, size-linear capture rates σ s would apply only for some one-dimensional model systems such as shown in Fig. 2: vertical nanowires fed from a vapor phase through their sidewalls,19 linear peptide chains growing from solutions with a continuous influx of “monomers” due to chemical reactions and surface adatom rows in situations where adatoms are captured on the sides of rows and then diffuse to the ends. If the adatom capture just occurs at the ends of surface rows, the capture rates become size-independent corresponding to the formal limit a → ∞ in Eqs. (18) and (19). This case yields monomodal distributions with non-analytic scaling.7,14 In systems with high diffusivities (this property is essential for considering the scaling limit D/F → ∞), sizelinear capture rates might be due to a competition between the islands for accommodating the diffusion fluxes of monomers as discussed above. Here, we write the capture rates as functions of s as in Refs. 5, 11, 14, 18–21, 34, and 35 rather than in the form σ s / ⟨σ⟩ = f (s/ ⟨s⟩) as suggested by Bales and Chrzan.15 However, the case σ s s at large s and therefore ⟨σ⟩ ⟨s⟩ is the simple particular case of the Bales-Chrzan representation. In our model, the a constant gives the probability of nucleation of the very first particle A2 or A1 B, while σ s s at s >> 1 describes the aggregation of large clusters. Small a values relate to situations where aggregation is preferred to nucleation, while at large a, nucleation is preferred to aggregation. As shown in Appendix B, the sets of discrete REs given by Eq. (12), for s ≥ 2 in the homogeneous case and s ≥ 1 in the heterogeneous case and with our size-linear capture rates, have the following exact solutions:
(17)
IV. EXACT SOLUTIONS FOR SIZE-LINEAR CAPTURE RATES
Let us now consider a simplified case where the capture rates are linear in s not only for large but also for all sizes, and independent of Θ. In the homogeneous case, we write σ s = a + s − 1,
s ≥ 1,
(18)
thus, the dimerization rate σ1 = a, with arbitrary positive a. In the heterogeneous case, we use instead σ s = a + s,
s ≥ 0,
(19)
with the rate of monomer attachment to the nucleation center σ0 = a. Such models correlate with the simulation results
FIG. 2. Illustration of growth systems with linear size dependence of the capture rates σ s = a + s: catalyst-assisted vertical nanowires which collect growth species from its sidewalls (a), linear chains of molecules growing from water solutions (b), and one-dimensional surface rows in situations where adatoms are captured on the sides of chains and then diffuse to the ends (c).
124110-4
V. G. Dubrovskii and Yu. S. Berdnikov
n s+2(z) =
J. Chem. Phys. 142, 124110 (2015)
Γ(a + s + 1) Γ(a)s! z d yn1(z − y)e−(a+1)y (1 − e−y )s , ×
Λ. Obviously, our φa (x) satisfies the well-known properties required from the scaling functions2,8,13 s ≥ 0,
0
(20) n s+1(z) =
Γ(a + s + 1) Γ(a)s! z × d yn0(z − y)e−(a+1)y (1 − e−y )s ,
s ≥ 0.
0
(21) Here, Γ(a + s + 1) denotes gamma-function and s! = Γ(s + 1) is the factorial. Clearly, the properties of these SDs are determined by the behaviors of the monomer concentration n1(z) in the homogeneous case and the concentration of free nucleation centers n0(z) in the heterogeneous case. The latter is easily calculated exactly from Eq. (2) for n0—indeed, in view of dz = Dn A dt, we have dn0/dz = −an0 and therefore, −az n0(z) = ntot . Be
(22)
Using this result in Eq. (21), we obtain the exact solution in the form of the Polya distribution33 n s (⟨s⟩) = ntot B
Γ(a + s) (⟨s⟩ /a)s , Γ(a)s! (1 + ⟨s⟩ /a)a+s
s ≥ 0.
(23)
∞
∞
⟨s⟩ = a(e − 1) z
(24)
is the mean size of the SD. Homogeneous growth is more complex for the analysis because no exact solution is known for n1(z) and we have to resort to some approximations. However, asymptotic Eq. (15) leads to a simple continuum approximation of the homogeneous SD, as will be discussed in Sec. V.
V. CONTINUUM SIZE DISTRIBUTIONS AND SCALING
In the heterogeneous case, using Γ(a+s)/s! s a−1 at a/s 1. At a = 1, the discrete Polya SD is reduced to geometrical distribution which is monotonically decreasing and has the continuum form given by Eq. (25) with φ1(x) = exp(−x). At a → ∞, corresponding to size-independent capture rates, the Polya SD is reduced to Poissonian, s ntot (s − ⟨s⟩)2 B −⟨s⟩ ⟨s⟩ → exp − , e n s (⟨s⟩) = ntot B s! 2 ⟨s⟩ 2π ⟨s⟩ (28) and yields the Gaussian continuum SD which does not feature analytic scaling in terms of the s/ ⟨s⟩ variable.7,14,33 Explicit representation of the homogeneous SD given by Eq. (20) with the monomer concentration defined by Eq. (15) is derived in Appendix C and has the form n s+1(⟨s⟩ , Θ)
Here,
dx xϕa (x) = 1
dxϕa (x) =
Θ (a + 1) I1−(a+1)/2⟨s⟩(s, a). ⟨s⟩2 2a
(29)
Here, I y (s, a) is the regularized incomplete beta function in standard notations.36 When ⟨s⟩ → ∞ and s → ∞, this SD automatically acquires the scaling form n(s, ⟨s⟩ , Θ)
Θ ⟨s⟩2
f a (x) ,
(30)
with x = s/ ⟨s⟩. As shown in Appendix C, the scaling function f a (x, Λ) can be well approximated by ( ) a + 1 (a + 1)x f a (x) = γ ,a , (31) 2a 2
(25)
where x = s/ ⟨s⟩, with the scaling function a a a−1 −a x x e . (26) Γ(a) Thus, the scaling SD is given by the probability density function of the gamma-distribution. The first equality in Eq. (25) does not require any assumption on the value of Λ. Thus, the scaling form of the Polya SD is the intrinsic property of the heterogeneous growth model with size-linear capture rates.33 However, we need ⟨s⟩ Θ/ntot B to obtain the traditional scaling behavior6 given by the second equality in Eq. (25), which requires n A → 0 according to Eqs. (3) and (6). This holds for all but very short times, only at large ϕa (x) =
FIG. 3. Polya SDs (solid lines) and their continuum approximations (dashed lines) at a = 2 and different ⟨s⟩. The discrete Polya SD is defined only for integer s, so the solid lines are guide for the eye only.
124110-5
V. G. Dubrovskii and Yu. S. Berdnikov
J. Chem. Phys. 142, 124110 (2015)
FIG. 4. Continuum SDs in the homogeneous model, obtained from Eq. (29) at ⟨s⟩ = 3, 5, and 10 for a = 0.5 and 1 and ⟨s⟩ = 3, 5, 10, 15, and 30 for a = 3 and re-plotted in scaling variables, compared to the scaling SDs (bold lines) given by Eqs. (31) and (32).
where γ( y, a) =
1 Γ(a)
∞ dtt a−1e−t
(32)
y
is the regularized incomplete gamma function. Thus, the homogeneous scaling function is given by the cumulative distribution function of the gamma-distribution. Very importantly, the scaling function f a (x) obeys both normalization conditions given by Eq. (27) for any a. Figure 4 demonstrates how the non-scaled SD given by Eq. (29) approaches the scaling shapes at different a. Opposite to the heterogeneous case, this scaling function is monotonically decreasing regardless of the value of a, as shown in Fig. 5. The observed difference in the shapes of the scaling functions can be well understood from elementary considerations. Indeed, in the heterogeneous case, the dimers A1 B can be formed only on the free nucleation centers B whose concentration is fixed and thus the value of n0 rapidly tends to zero. If a is large enough (greater than one), the n0 decreases faster than the concentrations of larger clusters. This creates a peak in the SD which subsequently propagates toward larger s. In the homogeneous case, the dimers A2 are built from two monomers whose number is continuously refilled from
FIG. 5. Scaling functions in homogeneous (solid lines) and heterogeneous (dashed lines) growth at different a. The scaling function exp(−x) is the same in both cases for a = 1.
vapor, resulting in a purely decreasing SD. It is interesting to note that the two models yield exactly identical exponential scaling function when a = 1, with the capture rates being strictly proportional to the island size in both cases. VI. LINEAR CHAINS OF METAL ADATOMS ON SILICON
Previous works25–28 revealed that room temperature deposition of group III (Ga, In, and Al) and some transition metals (Mn) produces stable linear chains of surface adatoms which lie orthogonal to the Si-dimer rows on 2 × 1 reconstructed Si(100) surface. Scanning tunneling microscopy imaging allowed for precise determination of the SDs of these chains over sizes (i.e., lengths) at low coverages (typically less than 0.13). In certain cases, for example, for linear chains of Al25 and Ga26 adatoms, the scaled SDs were found monotonically decreasing, although a “weak” peak for small sizes was not excluded. In contrast, the time-dependent SDs of Mn adatom chains were decreasing only at the beginning of growth, while for longer times, the SD shapes transformed to the pronounced monomodal.28 The role of heterogeneous nucleation at C-type defects versus homogeneous nucleation of metal surface chains on Si
FIG. 6. SDs of Ga adatom chains26 at different coverages (symbols), and their best fits (lines) by the homogeneous beta (a) and heterogeneous Polya (b) distributions with the experimental mean sizes and different a.
124110-6
V. G. Dubrovskii and Yu. S. Berdnikov
FIG. 7. Size histograms f s (s) of Mn adatom chains on Si(100)-2 × 1 after 2 (a), 7 (b), and 12 (c) depositions from Ref. 28, fitted by the Polya SD given by Eq. (23) at a = 3 and different ⟨s⟩ = 2.50 at 2 s, 4.96 at 5 s, and 6.44 at 12 s..
has been widely discussed. In particular, Albao et al.29 argued that the growth of Ga chains on Si(100) is not dominated by heterogeneous nucleation, contrasting the proposal of Kocán et al.30 and the experimental findings of Javorský et al.27 in the case of In chains which were shown to terminate at C defects. Probability of heterogeneous nucleation in a given system should depend on the concentration of the
J. Chem. Phys. 142, 124110 (2015)
reactive defects. As demonstrated theoretically in Ref. 37, concentration of the defects larger than 0.0025 per site may change the island SDs from monomodal to monotonically decreasing in the case of the irreversible growth model. Fragmentation of islands has also been found important for the SD shapes.31,32,37 For example, theoretical results of Ref. 31 suggest that the scaling functions are monotonously decreasing in quasi-equilibrium case with fragmentation, while in the case of irreversible growth, they exhibit a monomodal character. As mentioned already, our irreversible growth model with size-linear and coverage-independent capture rates is expected to apply to the case of compact islands,13 including linear adatom chains. The obtained analytical SDs are monotonically decreasing in homogeneous case and are capable of describing both monotonically decreasing and monomodal shapes in heterogeneous case depending on whether the a value is smaller or greater than one. Therefore, within the model, monomodal shapes should correspond to homogeneous nucleation, while purely decreasing SDs can be described by both homogeneous and heterogeneous solutions. In particular, Fig. 6 shows experimental SDs of Ga adatom chains26 which could be reasonably fitted by both homogeneous beta and heterogeneous Polya distributions with very similar a values. Figure 7 demonstrates a good fit of the time-dependent SDs of Mn chains28 obtained with the Polya distribution at a = 3 in the heterogeneous case (the normalization is adjusted to the experimental size histograms). Of course, the monomodal SDs at longer growth times cannot be described within the homogeneous model.
FIG. 8. Scaled SDs of Al25 (a), Ga26 (b), In27 (c), and Mn28 (d) adatom chains versus scaled size, fitted with homogeneous and heterogeneous scaling functions.
124110-7
V. G. Dubrovskii and Yu. S. Berdnikov
Figure 8 shows the measured SDs of Al, Ga, In, and Mn adatom chains,25–28 plotted in scaling variables and fitted by the continuum homogeneous and heterogeneous scaling functions given by Eqs. (31) and (26), respectively. It is seen that homogeneous scaling function reproduces the experimental data slightly better for Al and Ga. The In chains are described much better by heterogeneous growth model, while for Mn, the only reasonable fit is obtained within heterogeneous model. These results correlate with Figs. 6 and 7 and the conclusions drawn in the original works, i.e., more probable homogeneous mechanism in the case of Ga26 and heterogeneous nucleation on C-defects for In27 and Mn28 adatom chains.
J. Chem. Phys. 142, 124110 (2015)
APPENDIX A: CHOICE OF Θ0
For small enough Θ and in the scaling limit Λ → ∞, Eq. (11) of the main text is reduced to 1 . (A1) ΛΘ Integrating Eq. (13) in view of dΘ = F(1 − Θ)dt yields z ln(Θ/Θ0), or n1
Θ Θ0 e z .
The value of Θ0 should be determined self-consistently to ensure the correct normalization of the mean size. Using Eq. (A2) in Eq. (A1), we have
VII. CONCLUSION
To sum up, we have obtained exact analytical solutions for SDs in homogeneous and heterogeneous irreversible growth for the model system described by the REs with sizelinear capture rates for all s. These solutions are given by the beta and Polya distributions, respectively. Both solutions naturally yield the scaling form of the continuum SDs in the limit of D/F → ∞, with the correct normalization of the island density and mean size. The homogeneous SD is monotonically decreasing for all a, while the heterogeneous one is decreasing at a ≤ 1 and monomodal at a > 1. Both scaling solutions are reduced to f (x) = exp(−x) when a = 1. The obtained SD shapes describe quite well the experimental data on Al, Ga, In, and Mn adatom chains on 2 × 1 reconstructed Si(100) surfaces and support the conclusion on the homogeneous character of nucleation for Al and Ga and heterogeneous nucleation for In and Mn. Our exact solutions of the REs and the corresponding scaling functions depend critically on the assumption of size-linear σ s . Indeed, the analytic scaling behavior of the SDs is not expected to hold when σ s ∝ s p with p < 1,11,14 so the case of σ s ∝ s at large s seems to be the best possibility for scaling within the RE approach. We have used the coverage independence of the capture rates, the assumption which looks relevant for compact but neither for point nor fractal island.13 The coverage-dependent and sizelinear capture rates will be studied elsewhere. The obtained SDs are highly sensitive to the value of the constant a. This effect has been disregarded in some continuous models.8,11,14 Such a property can be understood through the “drift” form of the boundary condition in the continuous RE at zero size.14 We also point out that the kinetic Monte Carlo simulations for compact islands13 yield monomodal SDs in most cases, which contradicts our conclusion for homogeneous growth. This discrepancy needs to be investigated in more detail.
(A2)
n1
e−z . ΛΘ0
(A3)
We now note that Eq. (3) reduces the generating function (see Appendix B for the detail) for the discrete SD given by Eq. (20) to n(x, z) =
z
s ≥0
dy
n s+2(z)x = a s
ae−z ΛΘ0
0
z dy 0
n1(z − y)e−(a+1)y [1 − x(1 − e−y )]a+1
e−a y . [1 − x(1 − e−y )]a+1
(A4)
Obviously, from the definition of the generating function, it follows that n(1, z) N at n1 → 0. Using Eq. (A4), we obtain N
a . ΛΘ0
(A5)
From Eqs. (A2), (A5), and ⟨s⟩ Θ/N, the asymptotic mean size should equal ΛΘ20 z e . (A6) a On the other hand, the mean size can be obtained as ⟨s⟩
N ⟨s⟩
∂n(x, z) . ∂ x x=1
(A7)
Using Eqs. (A4) and (A5), we find (a + 1) z e . 2 Comparing this to Eq. (A6) yields 1/2 a(a + 1) Θ0 = , 2Λ ⟨s⟩
(A8)
(A9)
the equation used in the main text. APPENDIX B: SOLUTION OF RATE EQUATIONS FOR SIZE-LINEAR CAPTURE RATES
ACKNOWLEDGMENTS
Yu.S.B. thanks the FP7 UC project “Nanoembrace” (Grant Agreement No. 316751). V.G.D. gratefully acknowledges financial support received from the Russian Science Foundation under the Grant No. 14-22-00018.
Let σ s be given by Eq. (18) of the main text. Inserting these σ s into the linearized REs of homogeneous growth given by Eq. (12), we have dn s = (a + s − 2)n s−1 − (a + s − 1)n s , dz
s ≥ 2. (B1)
124110-8
V. G. Dubrovskii and Yu. S. Berdnikov
J. Chem. Phys. 142, 124110 (2015)
We now introduce the generating function for concentrations by the definition n(x, z) = n s+2(z)x s . (B2) s ≥0
Using Eq. (B1), after simple calculation, we obtain the closed differential equation in partial derivatives for the generating function, ∂n(x, z) ∂n(x, z) − (x − 1)x = (x − 1)(a + 1)n(x, z) + an1(z). ∂z ∂x (B3)
APPENDIX C: CONTINUUM SIZE DISTRIBUTION IN HOMOGENEOUS CASE
Starting from the exact solution given by Eq. (20), we first use Eq. (15) and then make the substitution of variables e−y = t under the integral. Together with the known formula for the beta function,34 B(s, a) =
1 I y (s, a) = B(s, a) (B4)
yielding that no islands are present on the surface at zero moment of time. The equivalent system of ordinary differential equations for Eq. (B3) writes down as dx dn dz =− = . 1 (x − 1)x (x − 1)(a + 1)n + an1
(B5)
From the first equation, we get the first integral of the form ( x ) Ψ1 = z − ln . (B6) 1−x From the second equation, with the initial condition given by Eq. (B4) and using Eq. (B6), we obtain n(x, z) = a
dy 0
n1(z − y)e−(a+1)y [1 − x(1 − e−y )]a+1
.
Γ(a + s + 1) 1 = ηs a+1 Γ(a + 1)s! (1 − η) s ≥0
(B7)
(B8)
for η = x(1 − e−y ) in Eq. (B7). The result has the form Γ(a + s + 1) xs Γ(a)s! s ≥0 z d yn1(z − y)e−(a+1)y (1 − e−y )s .
×
(B9)
0
Comparing this to Eq. (B2) yields the final result given by Eq. (20) of the main text. The described procedure is exactly identical for heterogeneous growth except for the definition of the z variable and the generating function which should be changed to dz = Dn A, dt n(x, z) = n s+1(z)x s . s ≥0
dtt s−1(1 − t)a−1,
1 B(s, a) =
(B10) (B11)
(C2)
dtt s−1(1 − t)a−1 0
as the incomplete and complete beta functions, respectively. To obtain the scaling form of Eq. (29), we note that whenever ⟨s⟩ → ∞, s → ∞, and a is finite, the known asymptote of incomplete beta function34 yields ( ) (a + 1) s I1−(a+1)/2⟨s⟩(s, a) γ ,a . (C3) 2 ⟨s⟩ This leads to the limiting curve given by Eqs. (30) and (31) of the main text.
1H.
Brune, Surf. Sci. Rep. 31, 121 (1998). W. Evans, P. A. Thiel, and M. C. Bartelt, Surf. Sci. Rep. 61, 1 (2006). 3W. Dieterich, M. Einax, and P. Maass, Eur. Phys. J.: Spec. Top. 161, 151 (2008). 4M. Einax, W. Dieterich, and P. Maass, Rev. Mod. Phys. 85, 921 (2013). 5A. Venables, G. D. T. Spiller, and M. Hanbucken, Rep. Prog. Phys. 47, 399 (1984). 6T. Vicsek and F. Family, Phys. Rev. Lett. 52, 1669 (1984). 7M. C. Bartelt and J. W. Evans, Phys. Rev. B 46, 12675 (1992). 8M. C. Bartelt and J. W. Evans, Phys. Rev. B 54, R17359 (1996). 9J. W. Evans and M. C. Bartelt, Phys. Rev. B 63, 235408 (2001). 10J. G. Amar and F. Family, Phys. Rev. Lett. 74, 2066 (1995). 11D. D. Vvedensky, Phys. Rev. B 62, 15435 (2000). 12M. Korner, M. Einax, and P. Maass, Phys. Rev. B 82, 201401 (2010). 13M. Korner, M. Einax, and P. Maass, Phys. Rev. B 86, 085403 (2012). 14V. G. Dubrovskii and N. V. Sibirev, Phys. Rev. B 89, 054305 (2014). 15G. S. Bales and D. C. Chrzan, Phys. Rev. B 50, 6057 (1994). 16G. S. Bales and A. Zangwill, Phys. Rev. B 55, R1973 (1997). 17P. Jensen, H. Larralde, and A. Pimpinelli, Phys. Rev. B 55, 2556 (1997). 18S. A. Kukushkin and A. V. Osipov, Prog. Surf. Sci. 51, 1 (1996). 19V. G. Dubrovskii, Nucleation Theory and Growth of Nanostructures (Springer, Heidelberg, New York, Dordrecht, London, 2014). 20V. G. Dubrovskii, J. Chem. Phys. 131, 164514 (2009). 21V. G. Dubrovskii and M. V. Nazarenko, J. Chem. Phys. 132, 114507 (2010). 22M. C. Bartelt, C. R. Stoldt, C. J. Jenks, P. A. Tiel, and J. W. Evans, Phys. Rev. B 59, 3125 (1999). 23F. G. Gibou, C. Ratsch, M. F. Gyure, S. Chen, and R. E. Caflisch, Phys. Rev. B 63, 115401 (2001). 24F. G. Gibou, C. Ratsch, and R. E. Caflisch, Phys. Rev. B 67, 155403 (2003). 25T.-C. Shen, C. Wang, and J. R. Tucker, Phys. Rev. Lett. 78, 1271 (1997). 26M. A. Albao, M. M. R. Evans, J. Nogami, D. Zorn, M. S. Gordon, and J. W. Evans, Phys. Rev. B 72, 035426 (2005). 27J. Javorský, M. Setvín, I. Ošt’ádal, P. Sobotík, and M. Kotrla, Phys. Rev. B 79, 165424 (2009). 28H. Liu and P. Reinke, Surf. Sci. 602, 986 (2008). 2J.
The series expansion of this n(x, z) is performed by applying the formula
n(x, z) =
y 0
n(x, z = 0) = 0,
(C1)
and Eqs. (14), (16), and (17), this results in Eq. (29) of the main text, with
This should be solved with the initial condition,
z
Γ(s)Γ(a) , Γ(s + a)
124110-9
V. G. Dubrovskii and Yu. S. Berdnikov
29M. A. Albao, M. M. R. Evans, J. Nogami, D. Zorn, M. S. Gordon, and J. W.
Evans, Phys. Rev. B 74, 037402 (2006). 30P. Kocán, P. Sobotík, and I. Ošt’ádal, Phys. Rev. B 74, 037401 (2006). 31V. I. Tokar and H. Dreyssé, Phys. Rev. B 74, 115414 (2006). 32V. I. Tokar and H. Dreyssé, Phys. Rev. B 84, 085456 (2011). 33V. G. Dubrovskii and Yu. S. Berdnikov, Mater. Phys. Mech. 21, 207 (2014). 34J. R. King and J. A. D. Wattis, J. Phys. A: Math. Gen. 35, 1357 (2002).
J. Chem. Phys. 142, 124110 (2015) 35C.
Ratsch, A. Zangwill, P. Šmilauer, and D. D. Vvedensky, Phys. Rev. Lett. 72, 3194 (1994). 36Handbook of Mathematical Functions, Applied Mathematics Series Vol. 55, edited by M. Abramowitz and I. A. Stegun (National Bureau of Standards, 1964). 37P. Kocán, P. Sobotík, I. Ošt’ádal, M. Setvín, and S. Haviar, Phys. Rev. E 80, 061603 (2009).
