Numerical Study of Impurity-Induced Growth Hysteresis on a

Subscriber access provided by ORTA DOGU TEKNIK UNIVERSITESI KUTUPHANESI

Article

Numerical study of impurity-induced growth hysteresis on a growing crystal surface Hitoshi Miura Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.5b01683 • Publication Date (Web): 20 Jan 2016 Downloaded from http://pubs.acs.org on January 28, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Crystal Growth & Design is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Crystal Growth & Design

Numerical study of impurity-induced growth hysteresis on a growing crystal surface Hitoshi Miura∗ Graduate School of Natural Sciences, Nagoya City University, Yamanohata 1, Mizuho-cho, Mizuho-ku, Nagoya, Aichi 467-8501, Japan E-mail: [email protected] Phone: +81 (52)872 5822. Fax: none

Abstract Growth hysteresis is one of the remarkable phenomena of crystal growth induced by impurities; namely, the growth rates of crystal take different values between when supersaturation is decreased and is increased. The appearance of hysteresis has been explained on the basis of the mean field theory, in which the physical quantities such as step velocity, step interval, and density of impurities adsorbed on the crystal surface are averaged over space and time. Here, we propose a new method for the numerical simulation of step dynamics with impurity adsorption–desorption kinetics on a growing crystal surface. We numerically reproduce the growth hysteresis in a more realistic situation, in which the physical quantities fluctuate. The calculated hysteresis agrees with the prediction of the mean field theory. We have also found that the supersaturation should be changed within a period that is similar to the impurity adsorption time scale or slightly longer in order to observe obvious growth hysteresis. Measurement of growth hysteresis may be useful in obtaining data of impurity adsorption–desorption kinetics.

1 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Introduction Crystal growth is affected by impurities present in the crystallization system even in amounts that do not infl

the bulk properties.1 One of the remarkable impurity-induced phenom-

ena is growth hysteresis, which means that the crystal growth velocity when the driving force (e.g., supersaturation) is increased is different from that when it is decreased. As listed in Table 1, there is some experimental evidence of growth hysteresis in various systems.2–6 Theoretically, it has been considered that growth hysteresis is caused by a slow impurity adsorption from the mother phase onto the crystal surface.5,7–11 However, previous theories were too simplified to simulate an actual system of crystal growth. We describe the essence of the previous theories of growth hysteresis as follows. Let us consider two physical quantities: the number of impurities adsorbed on the crystal surface per unit area, Ni, and the step velocity on the surface polluted by impurities, Vi. The impurities are transported from the mother phase to be adsorbed on the surface, which increases Ni. Impurities on the surface can be desorbed thermally back to the mother phase, which decreases Ni. In addition, it is considered that Ni drops to zero by a step passage because the adsorbed impurities will be incorporated into the crystal or removed to the mother phase; then it begins to increase by the impurity adsorption on the renewed surface until the adsorption rate reaches a balance with the desorption rate. However, if the impurity adsorption is slow, the following step passage will occur at the same area before the impurity adsorption– desorption equilibrium is established. In other words, frequent step passages result in a less-impurity surface (impurity sweeping). This suggests that Ni is a monotonically decreasing function of Vi. On the other hand, Vi is also a monotonically decreasing function of Ni because impurities typically inhibit the step advancement. The previous theories formulated these functions, Ni(Vi) and Vi(Ni), and obtained a steady growth solution that satisfied both functions simultaneously. 5,9–11 These theories considered quantities such as Ni and Vi that are averaged over time and space; therefore, we term them mean fi theories hereafter. However, these physical quantities should take different values at different positions on the 2 ACS Paragon Plus Environment

Page 2 of 23

Page 3 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


crystal surface, and fluctuate with time. There is no theoretical verification of the occurrence of growth hysteresis in such fluctuating fields. Recently, we developed numerical schemes for simulating the step dynamics that includes the impurity effect.12,13 The phase-field (PF) method proposed by Miura and Kobayashi 12 enables us to accurately calculate the velocities of straight and curved steps and the normal growth rate of the crystal surface due to spiral steps. Miura 13 introduced the impurity effect into the former PF method and reproduced the inhibition of step advancement by immobile impurities adsorbed on the crystal surface as suggested by some theories.14,15 If we further introduce the impurity adsorption–desorption kinetics into these numerical schemes, we can verify the mean field theories of growth hysteresis in more realistic situations. The purpose of this paper is to confirm whether or not growth hysteresis occurs as suggested by the mean field theories even in fluctuating fields. To achieve this objective, we introduce a numerical scheme of impurity adsorption–desorption kinetics based on a Monte Carlo (MC) method. By using the PF method for the step dynamics in combination with the MC method for the impurity adsorption–desorption kinetics, we found that the growth hysteresis appears even in realistic situations. The supersaturation at which the growth hysteresis appears in the numerical results agrees with the prediction by the mean field theory of Miura and Tsukamoto. 11 Table 1: Combination of crystals and impurities with which the growth hysteresis was observed in experiments. Refs.: [1] Dugua and Simon,2 [2] Land et al.,3 [3] Guzman et al.,4 [4] Friddle et al.,5 [5] Vorontsov et al..6 Crystal sodium perborate KDP (KH2PO4) potassium sulfate (K2SO4) COM (CaC2O4·H2O) water ice

Impurity sodium salt of butyl ester sulfate of oleic acid (SEO) Fe3+ chromium(III) aspartic-rich peptides carboxylated ε-poly-L-lysine (COOH-PLL)


Ref. [1] [2] [3] [4] [5]


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 23

Model Impurity adsorption–desorption kinetics We consider the time variation of Ni due to the adsorption of impurities onto the crystal surface and due to the desorption from there. In the mean field theories,5,9–11 the governing equation of this time variation is given by dNi = k (N — N )c − k N , a site i i d i dt

(1)

where ka is the absorption coefficient; kd, the desorption coefficient; Nsite, the density of absorption sites for impurities; and ci, the impurity density in the mother phase. The impurity flux from the mother phase to the crystal surface, Fi, is given by

Fi = kaNsiteci.

(2)

i In the equilibrium state ( dN = 0), we obtain the equilibrium density of adsorbed impurities, dt

Ne, as follows (Langmuir isotherm): Ne =

ka Nsite ci k c +k . a i

(3)

d

When ka, kd, Nsite, and ci are constant, eq 1 has an analytic solution Ni(t) = Ne − (Ne − Ni(0))e−t/t0 , where t0 is the typical time scale for Ni to approach Ne and is given by t0 = (kaci + kd)−1.


(4)

(5)

Page 5 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Monte Carlo approach Fig 1 shows the concept of impurity adsorption and desorption based on the MC method. The crystal surface is discretized by square computational cells with an equal width, ∆x. Each cell is labeled by i and j in the x- and y-directions, respectively. We consider that each cell provides a site for impurity adsorption; therefore, we obtain Nsite = 1/(∆x)2.

(6)

The existence of impurity at each cell is denoted by a logical variable, wi,j : wi,j = T if an impurity exists at the cell (i, j), and wi,j = F if not. The probability, Pa, that impurity adsorption to the cell occurs during a time step, ∆t, is given by

Pa =

Fi(∆x)2 ∆t when wi,j = F,

(7)

when wi,j = T.

0

The probability, Pd, that impurity desorption from the cell (i, j) occurs during ∆t is given by Pd =

kd ∆t when wi,j = T, 0

(8)

when wi,j = F.

At the impurity adsorption–desorption equilibrium, the fraction of the cells where wi,j = T, χ, is given by χ = Ne/Nsite.

(9)

Substitution of eqs 2, 3, and 6–8 into eq 9 yields 1−χ Pd = χ

P. a


(10)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 23

In addition, we obtain Ne = Fit0 from eqs 2, 3, and 5, and χ = Ne(∆x)2 from eqs 6 and 9. Therefore, t0 =

Ne Fi

=

χ Fi(∆x)2

.

(11)

By using eqs 7, 10, and 11, we can determine parameters Pa, Pd, and t0 from given parameters Fi and χ. We adopt the probabilities Pa and Pd for the MC calculation of impurity adsorption–desorption kinetics. The impurity distribution is updated at every time step using the MC method as described below. We generate random numbers for all cells, ri,j , which distribute uniformly from 0 to 1 using xorshift random number generators.16 ri,j is generated using xor128 from the seed set for each cell, si,j (m) (m = 1, 2, 3, 4), which is four 32-bit integers and correspond to x, y, z, and w in the original paper,16 respectively. In this study, we set ′ si,j (m) = r m+4(i−1)+nx(j−1) , where rk′ is a random number sequence generated by xor32 with

a seed of r0′ = 1, and nx is the number of cells in the x-direction. When wi,j = T, we switch the value to “F” if ri,j < Pd; namely, an impurity is desorbed. When wi,j = F, we switch the value to “T” if ri,j < Pa; namely, an impurity is adsorbed.

Normalization The PF method for the step dynamics used physical quantities that are normalized by units of length and time, ϵ′ and τ ′, respectively, which are given by12

′

ϵ =

π2κsmδ 4kBT

π2 δ , and τ = 4βst Ωce ′

(12)

where κ is the step edge free energy; sm, the area of a growth unit; δ, the step width; kB, the Boltzmann constant; T , the absolute temperature; βst, the step kinetic coefficient; Ω, the specific volume of the growth units; and ce, the solubility. From these units, the quantities ˜site = Nsite ϵ′2 , ∆x ˜i = Fi ϵ′2 τ ′ , ∆t˜ = ∆t/τ ′ , k ˜d = kd τ ′ , are normalized as follows: N ˜ = ∆x/ϵ′ , F N˜e = Neϵ′2 , and t˜0 = t0 /τ ′ . We can replace the physical quantities in the section “Monte 6 ACS Paragon Plus Environment

Page 7 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 1: Concept of impurity absorption and desorption. The crystal surface is discretized by square computational cells with an equal width, ∆x. Each cell provides a site for impurity adsorption. A logical variable, wi,j , denotes the existence of impurity at the cell (i, j): wi,j = T if an impurity exists, and wi,j = F if not.

Carlo approach” with these dimension-less quantities without any modification in the forms of these formulae.

Step dynamics We adopt the PF method to solve the step dynamics on a crystal surface polluted by impurities. In the PF method, the structure on the crystal surface is described by a continuum PF variable, ϕ(r, t), where r is the position vector on the surface and t is time, which represents the two-dimensional height profile. Note that ϕ is uniform on a terrace and changes smoothly at steps. The step has a finite thickness, δ. See our previous papers12,13 for a detailed description. The step dynamics is described by the time evolution of ϕ(r, t). The non-dimensional



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 23

governing equation discretized on a uniform Cartesian (square) grid is given by13 [ ϕn+1 i,j

n

− ϕi,j

n

n

n

n

n

= ν (ϕi+1,j + ϕi−1,j + ϕi,j+1 + ϕi,j−1 − 4ϕi,j ) + )] ( h σ s ˜ i,j i m n , +(1 + cos(πϕi,j )) Zi σ(∆x˜)2 −

R2 π

n

sin(πϕi,j ) (13)

where ϕni,j is the value of ϕ on the cell (i, j) at the nth time step; ν = ∆t˜/(∆x˜)2 , the Courant number; ∆t˜, the time step; ∆x˜, the cell width; R, the ratio of the cell width to the step width; and σ, the supersaturation. The term of hi,j σis˜m/Zi is non-zero only at cells occupied by impurities, where Zi is the number of cells that are occupied by an impurity. We use Zi = 7 × 7 to accurately reproduce the pinning effect on the step dynamics.13 The state of the impurity occupation at each cell is denoted by an integer variable hi,j : hi,j = 1 for occupied and hi,j = 0 for unoccupied. Fig 2 shows the relationship between wi,j and hi,j . The product of parameters σis˜m determines the step-stopping ability of an impurity.

Figure 2: Description of impurities adsorbed on the crystal surface. The red squares (wi,j = T) denote the cells on which an impurity exists. The size of each impurity is 7 × 7 cells. The gray squares (hi,j = 1) denote the cells occupied by impurities. Although overlapping of impurities may occur with a small probability, for simplicity we do not consider any scheme to avoid it.

The non-dimensional forms of the advancing velocity of a straight step, V˜∞, and the


Page 9 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


critical radius of a two-dimensional nucleus, ρ˜c, are respectively given by 12 õ ˜ = π 2 ∆x V ∞

and ρ ˜ = c

4 R . 2 π (∆x˜)σ

(14)

4R

Impurity removal by step passages To simulate impurity sweeping, we need to remove the adsorbed impurities at every step passage. Fig 3 is a schematic of the step passage at a position where an impurity is adsorbed on the terrace at ϕ = −1. The existence of the impurity is expressed by wi,j = T, which does not specify the height. This means that the impurity will stay on the upper terrace at ϕ = +1 after the step passage if it is not removed. To avoid this problem, the software automatically sets wi,j = F when ϕi,j exceeds 0.99 to remove the impurity, for example. The numerical procedure is as follows. We define an integer variable, Li,j , that gives the level of terrace. Li,j is given at the beginning of the calculation as a minimum integer that satisfies Li,j ≥ (ϕi,j − 0.99)/2. When the condition ϕi,j > 2Li,j + 0.99 is satisfied by the step passage, we add unity to Li,j and set wi,j = F.

Calculation Fig 4 shows a configuration of the calculation. The rectangular computational domain has an ˜x × L ˜ y and is divided by 512 × 256 square cells. We set a straight step parallel to the area of L y-axis. The left and right sides of the step are the upper (ϕi,j = +1) and lower (ϕi,j = −1) terraces, respectively. We impose a periodic boundary condition in the y-direction. In the x-

direction, we consider a periodic vicinal surface boundary condition (ϕ0,j = ϕ512,j +

2); namely, the step proceeds past the right boundary, followed by the appearance of a new step on the upper terrace from the left boundary. Because of the periodic vicinal surface boundary condition, the step interval λ˜ is equal to L˜ x = 512∆x˜. The step velocity V˜i is ˜ using the relationship of V ˜i = R ˜ /p, where p is the calculated from the normal growth rate R ˜i without slope of the crystal surface. Since p is constant during calculation, we can obtain V



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 23

Figure 3: Schematic of step passage at the position where an impurity is adsorbed on the terrace at ϕ = −1. The existence of the impurity is expressed by wi,j = T, which does not specify the height. This means that the impurity will stay on the upper terrace at ϕ = +1 after the step passage. To simulate the impurity sweeping, we remove the impurity manually when ϕi,j exceeds 0.99. An integer variable, Li,j , gives the level of terrace (see section “Impurity removal by step passages”).


Page 11 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


determining the step position. The relationship can be rewritten as12 ˜i = λ˜ d⟨ϕ⟩ , V 2 dt˜

(15)

where ⟨ϕ⟩ is the phase value averaged over the computational domain. The input parameters that we used in this study are as follows: R = 0.4, ν = 0.2, ∆x˜ = 0.2, and σis˜m = 4.20. 13 The parameters for the impurity adsorption–desorption kinetics that we explore in this paper are listed in Table 2. Parameter #2 is the default, parameter #1 indicates more rapid impurity kinetics because t0 is shorter, and parameter #3 indicates slower impurity kinetics because t0 is longer.

Figure 4: Schematic configuration of the calculation. The rectangular computational domain ˜ x × L˜ y and is divided by 512 × 256 square cells. A periodic boundary has an area of L condition is imposed in the y-direction. We also impose a periodic boundary condition in the x-direction so that the step proceeds past the right boundary, followed by the appearance of a new step on the upper terrace from the left boundary (periodic vicinal surface boundary condition). We start the calculation with initially impurity-free crystal surface.

The numerical procedure is as follows: 1. Set initial values of ϕi,j (see fig 4)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 23

2. Set initial values of Li,j (see section “Impurity removal by step passages”) 3. Update the impurity distribution wi,j using MC method (see section “Monte Carlo approach”) 4. Set hi,j using the updated impurity distribution (see section “Step dynamics”) 5. Update the phase field ϕi,j using PF method (see section “Step dynamics”) 6. Update Li,j and wi,j (see section “Impurity removal by step passages”) 7. Back to step 3 ˆ and Vˆ0 are parameters Table 2: Parameters for impurity adsorption–desorption kinetics. κ that characterize the mean field theory of growth hysteresis (see Supporting Information). χ and F˜i are given by eq S.13 (see Supporting Information). Pa , Pd, and t˜0 are given by eqs 7, 10, and 11, respectively. ˆ0 No. κ ˆ V #1 0.1 50 #2 0.1 100 #3 0.1 200

χ 1.522 × 10−4 1.522 × 10−4

˜i F 9.168 × 10−7 4.584 × 10−7

Pa 2.934 × 10−10 1.467 × 10−10

Pd 1.927 × 10−6 9.637 × 10−7

t˜0 4.15 × 103 8.30 × 103

1.522 × 10−4

2.292 × 10−7

7.335 × 10−11

4.818 × 10−7

1.66 × 104

Results and discussion Impurity sweeping We investigate the number of adsorbed impurities in the computational domain, ni, with and without step passage. The impurity density is given by N˜i = ni/(L˜ x L˜ y ). We adopt parameter #2 in Table 2. In the adsorption–desorption equilibrium, we expect that ni = 512 × 256χ ≃ 19.9. Fig 5a shows the time variations of ni from calculations with step (solid curve) and without step (thin dashed curve) at a fixed supersaturation of σ = 0.12. In the case without step, ni increases with time and reaches the adsorption–desorption equilibrium at t˜ ≃ 1×104. 12 ACS Paragon Plus Environment

Page 13 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The numerical result agrees with the solution of the mean field theory (thick dashed curve). On the other hand, it is found that the periodic step passages significantly reduce ni; the time-averaged value of ni is ≃ 20 for without step and ≃ 2 for with step. The reduction of adsorbed impurities is due to the impurity sweeping. The period of step passages, which ˜ ≃ 7 × 102 if corresponds to the exposure time of the terrace, can be estimated by ∼ V˜ ∞ /λ we can ignore the retardation of the step advancement, which is much shorter than the time scale of the impurity adsorption, ˜t0 ≃ 8 × 103. This suggests that the crystal surface is swept repeatedly before the impurity adsorption–desorption equilibrium is achieved.

Figure 5: (a) Time variation of the total number of impurities adsorbed on the surface, ni. The solid and thin dashed curves are the numerical results with and without step, respectively. The thick dashed curve is the analytic solution of the mean field theory. Bottom panels show snapshots from calculations with (b) and without step (c). Red points are adsorbed impurities. Parameter #2 is used. The supersaturation is fixed at σ = 0.12.

The decrease of supersaturation increases the exposure time, so it results in the adsorption of more impurities. The impurities will finally stop the step advancement completely, and



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 23

then the impurity adsorption–desorption equilibrium will be achieved. The step dynamics accompanied with the decrease of supersaturation is described in the next section in detail.

Catastrophe We investigate the changes in the step velocity accompanied by the changes in supersaturation. We reduce σ from σ = 0.12 to 0.02 at a constant rate during the4 first 4 × 104 period, then increase it to σ = 0.12 at the same rate during the next 4 × 10 period; namely, the period of one cycle is t˜cycle = 8 × 104. We repeat the down-and-up cycle of σ ten times. Fig 6 shows the time variations of V˜i (solid) and ni (dashed) during the first two cycles. It is found that V˜i repeatedly varies between two end states, namely, impurity-free growth (V˜i ≃ V˜ ∞) and no growth (V˜i ≃ 0). The changes in V˜i occur suddenly despite σ changing linearly with time. This figure also shows the periodic changes in ni between two end states, namely, impurity-free surface (ni ≃ 0) and impurity adsorption–desorption equilibrium (ni ≃ ne). After V˜i drops to zero, ni increases gradually up to ≃ ne. After V˜i recovers to ≃ V˜ , ni drops to zero rapidly. ∞ The behavior can be interpreted by “catastrophe”; it is caused by the positive feedback between the decrease in V˜i and the increase in ni, and vice versa. 11,17 When V˜i decreases with a decrease in σ, the impurity sweeping becomes ineffective, so ni increases because of the impurity adsorption. The increase in ni retards the step advancement, so V˜i is forced to decrease further; then the feedback loop continues until the step advancement is stopped completely. On the other hand, a step that has been stopped completely by impurities can begin to advance again because of the increase in σ when the critical radius becomes smaller than approximately half of the interval of impurities that pin the step. The moving step sweeps the impurities, so ni decreases rapidly, and then the impurity-free growth is recovered.


Page 15 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 6: Variations of step velocity, V˜i (solid), and the number of adsorbed impurities, ni (dashed), during down-and-up cycles of supersaturation σ. V˜i and ni are normalized by the velocity of a straight step, V˜∞ , and the number of impurities at adsorption–desorption equilibrium, ne, respectively. σ is decreased in the gray regions at a constant rate and is increased in others at the same rate (see top label).



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 23

Growth hysteresis We verify the appearance of growth hysteresis in the numerical results. Fig 7 shows the traces of V˜i as a function of supersaturation from the calculation shown in fig 6. The red and blue curves denote the traces when σ is decreased and increased, respectively. The thick dashed curve is the steady growth solution derived by the mean field theory of growth hysteresis (Miura and Tsukamoto, 11 modified, see Supporting Information). Fig 7a shows the result during the first cycle. When σ is decreased, V˜i decreases gradually along the steady growth solution zero suddenly at σrapidly ≃ 0.05.up When σ is increased, V˜i remains to zero untiland σ ≃drops 0.08, to and then increases to the steady growth solution. close These two traces are clearly different from each other. The hysteresis is also identified during the second cycle (Fig 7b). In fig 7c, we display all traces during ten cycles. The traces when σ is decreased scatter because the distribution of impurities on the crystal surface is changed in every cycle. The supersaturation at which the catastrophic drop of V˜i occurs ranges from ≃ 0.04 to ≃ 0.06. The scattering of the traces is also confirmed when σ is increased. The supersaturation at which the catastrophic increase of V˜i occurs ranges from ≃ 0.05 to ≃ 0.09, which is systematically larger than the former. The traces of V˜i from the calculations agree with the mean field theory. Fig 7d shows the averaged trace of the step velocity during the ten cycles. When σ is decreased, the average of traces decreases along the theoretical curve and drops to zero at around the transition point (see Supporting Information). On the other hand, when σ is increased, the catastrophic change in V˜i occurs at supersaturations that are smaller than the theoretical prediction (σ = 0.1). The reason why the recovery of V˜i occurs at smaller supersaturations may be the randomness of the impurity distribution. The mean field theory implicitly assumes the impurity distribution of a regular square array. It is known that a random distribution of impurities exhibits a weaker step-stopping ability than a regular distribution.13,18 In addition, the desorption of impurities that pin the step advancement (stoppers) may account for the difference between the calculation and the mean field theory. The mean field theory does not


Page 17 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 7: Traces of the step velocity, V˜i, from calculations as a function of supersaturation, σ, during the first (a), second (b), all of ten down-and-up cycles (c), and the average of the ten cycles (d). The red and blue curves denote the traces when σ is decreased and is increased, respectively. The thick dashed curve is the steady growth solution derived by the mean field theory (Miura and Tsukamoto, 11 modified, see Supporting Information). Parameter #2 is used.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 23

consider the desorption; however, in the calculation it is assumed that the desorption of the stoppers can occur as well as other impurities adsorbed on terraces. If one of the stoppers is desorbed from the pinned step, the step can temporarily advance until it is stopped again by other stoppers even if the supersaturation is too low to restart the step advancement.

Slower or faster impurity adsorption We investigate the growth hysteresis in the case of slower and faster impurity adsorption. Figs 8a and 8b show the growth hysteresis from a calculation using parameter #1, which indicates faster impurity adsorption than parameter #2. The fast impurity adsorption–desorption kinetics shifts supersaturation at which the catastrophic drop of V˜i occurs rightward, as suggested by the thick dashed curve. Figs 8c and 8d show the results using parameter #3, which indicates slower impurity adsorption. The slower kinetics shifts the supersaturation leftward. In both cases, the numerical results agree with the mean field theory.

Figure 8: The same as fig 7c except for the parameters used in calculations: (a, b) parameter #1 and (c, d) parameter #3.


Page 19 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Effect of down-and-up period of supersaturation We change the period of the down-and-up cycle of supersaturation, ˜tcycle , to investigate the effect on the growth hysteresis. Fig 9 shows the numerical results with ˜tcycle = 8 × 102 (a), 8 × 103 (b), 8 × 104 (c), and 8 × 105 (d), using parameter #2. In fig 9a, the down-and-up cycle is much faster than the impurity adsorption, so ni cannot sufficiently increase at low supersaturation phases. As a result, V˜i does not drop to zero and no clear hysteresis is confirmed. Fig 9b shows the case where t˜cycle is comparable to t˜0; however, traces of V˜i at the low supersaturation phases do not converge around zero and have large scatter. This suggests that the duration at low supersaturation phases is still too short to establish the impurity adsorption–desorption equilibrium. These results suggest that we should change the supersaturation more slowly than the impurity adsorption in order to observe the obvious growth hysteresis like fig 9c. However, as shown in fig 9d, the growth hysteresis becomes narrower when the down-and-up cycle is much slower than the impurity adsorption. This can be interpreted as follows. The catastrophic change of V˜i occurs accidentally, accompanied by random impurity adsorption and desorption; therefore, the exact timing is unpredictable. Compared with the faster cycle, there are many more chances during the slower down-and-up cycle for the occurrence of the catastrophic jump while the supersaturation is changed by the same amount. As a result, when σ is decreased, the catastrophic drop of V˜i occurs at a higher supersaturation in the slower cycle, and when σ is increased, the catastrophic increase occurs at a lower supersaturation. Therefore, the hysteresis becomes narrower. From these results, we conclude that the period of the down-and-up cycle of supersaturation should be the same order of magnitude as the time scale of the impurity adsorption or slightly longer in order to observe the clear growth hysteresis.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 23

Figure 9: The same as fig 7c except for the period of down-and-up cycle of supersaturation, t˜cycle : 8 × 102 (a), 8 × 103 (b), 8 × 104 (c), and 8 × 105 (d). Panel (c) is the same as fig 7c.

Concluding remarks We have successfully reproduced impurity-induced growth hysteresis on step dynamics using a newly developed numerical scheme that is composed of two methods: a phase-field method for the step dynamics with the impurity-induced step pinning and a Monte Carlo method for the impurity adsorption–desorption kinetics. The calculated traces of the step velocity accompanied by down-and-up cycles of supersaturation agreed with the prediction of the mean field theory of growth hysteresis (Miura and Tsukamoto, 11 modified, see Supporting Information). We have found that the period of the down-and-up cycle should be the same order of magnitude as the time scale of the impurity adsorption or slightly longer in order to observe the clear growth hysteresis if the impurity adsorption and desorption occur at the step front in the same way as on the terrace. The next target of this study is more global calculations. In this paper, we have considered a local area on the crystal surface within which the step interval can be regarded as being 20 ACS Paragon Plus Environment

Page 21 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


constant. However, it is known that impurities cause fluctuation of the step interval, namely, impurity-induced step bunching.19 We have not confirmed whether growth hysteresis appears even if the step bunching is taken into consideration. For the confirmation, we are planning to perform numerical simulations with a wider computational area in which more than one step is included in a forthcoming study.

Supporting Information Available Additional text (description of the mean field theory of growth hysteresis) and two videos. This material is available free of charge via the Internet at http://pubs.acs.org/.

References (1) Chernov, A. A.; Givargizov, E. I.; Bagdasarov, K. S.; Kuznetsov, V. A.;

Demi-

anets, L. N.; Lobachev, A. N. Modern Crystallography III: Crystal Growth; SpringerVerlag: Berlin, 1984; p 159. (2) Dugua, J.; Simon, B. J. Cryst. Growth 1978, 44, 280–286. (3) Land, T. A.; Martin, T. L.; Potapenko, S.; Tayhas Polmore, G.; De Yoreo, J. J. Nature 1999, 399, 442–445. (4) Guzman, L. A.; Kubota, N.; Yokota, M.; Sato, A.; Ando, K. Cryst. Growth Des. 2001, 1, 225–229. (5) Friddle, R. W.; Weaver, M. L.; Qiu, S. R.; Wierzbicki, A.; Casey, W. H.; De Yoreo, J. J. Proc. Natl. Acad. Sci. U. S. A. 2010, 107, 11–15. (6) Vorontsov, D. A.; Sazaki, G.; Hyon, S.-H.; Matsumura, K.; Furukawa, Y. J. Phys. Chem. B 2014, 118, 10240–10249.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 23

(7) Punin, Y. O.; Artamonova, O. I. Kristallographiya 1989, 34, 1262–1266. (8) Derksen, A. J.; van Enckevort, W. J. P.; Couto, M. S. J. Phys. D: Appl. Phys. 1994, 27, 2580–2591. (9) Kubota, N. Cryst. Res. Technol. 2001, 36, 749–769. (10) Kubota, N.; Yokota, M.; Doki, N.; Guzman, L. A.; Sasaki, S. Cryst. Growth Des. 2003, 3, 397–402. (11) Miura, H.; Tsukamoto, K. Cryst. Growth Des. 2013, 13, 3588–3595. (12) Miura, H.; Kobayashi, R. Cryst. Growth Des. 2015, 15, 2165–2175. (13) Miura, H. Cryst. Growth Des. 2015, 15, 4142–4148. (14) Cabrera, N.; Vermilyea, D. A. Growth and Perfection of Crystals; Wiley: New York, 1958; pp 393–410. (15) Potapenko, S. Y. J. Cryst. Growth 1993, 133, 147–154. (16) Marsaglia,

G.

J.

Stat.

Software

2003,

8,

1–9.

URL

http://www.jstatsoft.org/article/view/v008i14. (17) Weaver, M. L.; Qiu, S. R.; Friddle, R. W.; Casey, W. H.; De Yoreo, J. J. Cryst. Growth Des. 2010, 10, 2954–2959. (18) van Enckevort, W. J. P.; van den Berg, A. C. J. F. J. Cryst. Growth 1998, 183, 441–455. (19) Kandel, D.; Weeks, J. D. Phys. Rev. B: Condens. Matter Mater. Phys. 1994, 49, 5554– 5564.


Page 23 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


For Table of Contents Use Only Title: Numerical study of impurity-induced growth hysteresis on a growing crystal surface Authors: Hitoshi Miura

Brief Synopsis: We numerically reproduce impurity-induced growth hysteresis on a growing crystal surface. The step velocities take different values between when supersaturation is decreased (red curve) and is increased (blue curve). The calculated hysteresis agrees with the prediction of previous mean field theory of growth hysteresis (dashed curve).


Numerical Study of Impurity-Induced Growth Hysteresis on a

Recommend Documents