Pinning of steps near equilibrium without impurities, adsorbates, or dislocations

Journal of Crystal Growth ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Pinning of steps near equilibrium without impurities, adsorbates, or dislocations Noriko Akutsu n Faculty of Engineering, Osaka Electro-Communication University, Hatsu-cho, Neyagawa, Osaka 572-8530, Japan

art ic l e i nf o

Keywords: A1. Computer simulation A1. Crystal morphology A1. Growth models A1. Surface process A1. Surface structure

a b s t r a c t Discontinuous surface tension of a vicinal surface is calculated numerically based on a statistical mechanical method. The microscopic model is a restricted solid-on-solid (RSOS) model with a pointcontact type step–step attraction (p-RSOS model). The discontinuity in the surface tension leads to step faceting at low temperatures near equilibrium, where macrosteps with smooth side surfaces are formed, resulting in two-surface coexistence. Using the Monte Carlo method with the Metropolis algorithm, we show the formation of the merged steps (macrosteps) and the slow velocity of the merged steps. We demonstrate that steps on a vicinal surface are pinned without impurities, adsorbates, or dislocations. We also propose simple ideas to avoid non-uniform growing of a crystal caused by the discontinuous surface tension of the crystal. & 2014 Elsevier B.V. All rights reserved.

1. Introduction Surface tension is one of the most fundamental quantities for various surface phenomena near equilibrium. As for crystals, the surface tension becomes anisotropic due to the crystal lattice structure. The polar graph of the surface tension has been known as the Wulff figure [1–8]. The equilibrium crystal shape (ECS), which is the crystal droplet shape with the least surface free energy, is drawn from the Wulff figure by way of the Wulff construction. At temperature T¼ 0, the Wulff figure is calculated from the anisotropic surface energy, and the ECS drawn by way of the Wulff construction becomes polyhedron called the Wulff polyhedron. At higher temperature T 4 0, the effect of the surface entropy on the surface tension cannot be negligible, because the surface tension is connected to the surface free energy density. As a result, the ECS consists of several facets with low Miller indices and curved areas [1,7–11]. Near a facet, the facet is surrounded by vicinal surfaces which are slightly tilted surfaces. The vicinal surface is described by a train of steps with step density ρ. Due to the zigzag structure of a step caused by thermal fluctuations, steps are separated. This tendency to separate steps is called entropic repulsion and it leads to the Gruber–Mullins–Pokrovsky–Talapov (GMPT) universal

n

Tel.: þ 81 72 824 1140; fax: þ 81 72 825 4666. E-mail addresses: [email protected], [email protected]

behavior [12] or one-dimensional (1D) free fermion universal behavior [1,7–13]. It is interesting to consider the surface tension of the vicinal surface with step–step attraction. The aim of this paper is to give an example of a discontinuous surface tension calculated from a microscopic model with step–step attraction [14–16]; and to explain how the pinning of steps is caused by the discontinuity of the surface tension. It is also the aim of this paper to propose ideas to avoid non-uniform growing of a vicinal surface caused by the discontinuous surface tension.

2. Microscopic model 2.1. p-RSOS model The microscopic model we considered is the restricted solidon-solid (RSOS) model with point-contact type step–step attraction [14]. We call this model the p-RSOS model. The surface height hðn; mÞ is assigned to a site (n,m) on the lattice. The height difference between the neighboring lattice points is restricted to take the value of f 7 1; 0g. This is the meaning of “restricted” for the RSOS model. Fig. 1 shows the perspective view (a) and the top view (b) of a vicinal surface. The thick lines describe surface steps. The Hamiltonian of the surface is written as follows: HpRSOS ¼ ∑ ϵ½jhðn þ 1; mÞ hðn; mÞj þ jhðn; m þ 1Þ  hðn; mÞj n;m

http://dx.doi.org/10.1016/j.jcrysgro.2014.01.068 0022-0248 & 2014 Elsevier B.V. All rights reserved.

Please cite this article as: N. Akutsu, Journal of Crystal Growth (2014), http://dx.doi.org/10.1016/j.jcrysgro.2014.01.068i

N. Akutsu / Journal of Crystal Growth ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

a

b

[001] h

]

[100]

y

y

x

x

0 [01

[110]

Fig. 1. (a) Perspective view of a vicinal surface tilted towards the 〈100〉 direction on an RSOS model. Shaded areas show microscopic side surfaces of a step. (b) Top view of the vicinal surface tilted towards the 〈110〉 direction of the RSOS model on a squared lattice. Thick lines: surface steps. Filled square: the meeting point of the adjacent steps.

-[111]

[001]

[111]

γsurf /ε

θ 1.5 et ) fac (001

1 -1

0 p

1

Z Y X

Fig. 2. (a) Slope dependence of surface tension. Original RSOS model ðϵint ¼ 0Þ. kB T=ϵ ¼ 0:3. (b) Perspective view of an equilibrium crystal shape (ECS) around the (001) facet. ϵint ¼ 0. kB T=ϵ ¼ 0:3.

þ ∑ ϵint ½δðjhðn þ 1; m þ 1Þ  hðn; mÞj; 2Þ

function Z as follows [18]:

n;m

þ δðjhðn þ1; m  1Þ hðn; mÞj; 2Þ;

ð1Þ

where ϵ represents the microscopic step energy, δða; bÞ is Kronecker's delta, and ϵint represents the microscopic step–step interaction. The summation with respect to (n,m) is taken over all the sites on the square lattice. The RSOS condition is required implicitly. When ϵint is negative, the step–step interaction becomes attractive (sticky steps). We consider the origin of ϵint to be the orbital overlap of the dangling bonds. When the orbitals overlap in the same phase, they make a bonding state. We regard the energy gain by forming the bonding state as jϵint j. In order to treat the vicinal surface, we add the terms of the Andreev field [17] η ¼ ðηx ; ηy Þ, which is an external field (like pressure in the gas–liquid system), to tilt the surface. Then, the total Hamiltonian for the vicinal surface becomes as follows [18]: Hvicinal ¼ HpRSOS n;m

ð2Þ

n;m

2.2. Calculations based on statistical mechanics According to the formula of statistical mechanics, the partition function Z is calculated as follows: Z¼

∑ e  βHvicinal

ð4Þ

where N is the number of lattice points on the square lattice. Practically, the correct calculation of the partition function Eq. (3) is impossible in most of the cases. Fortunately, since the Hamiltonian is quite simple, we can apply the transfer matrix method to the calculation on the p-RSOS model by mapping it to the 19 vertex model [15]. Besides, the transfer matrix version of the density matrix renormalization group (DMRG) method [19], which is called the product wave-function renormalization group (PWFRG) method [20], is applicable for the efficient calculation of the largest eigenvalue of the transfer matrix.

3. Discontinuity of surface tension

 ηx ∑ ½hðn þ1; mÞ  hðn; mÞ  ηy ∑ ½hðn; m þ1Þ hðn; mÞ:

1 f~ ðηÞ ¼ f~ ðηx ; ηy Þ ¼  lim kB T ln Z; N -1N

ð3Þ

fhðm;nÞg

where β ¼ 1=kB T. The summation with respect to fhðm; nÞg is taken over all the possible values of hðm; nÞ. The Andreev free energy f~ ðηÞ [17] is the thermodynamic potential calculated from the partition

Instead of drawing a Wulff figure, we draw a figure like Fig. 2(a). The vertical axis represents the surface tension, and the horizontal axis represents the surface slope p ¼ tan θ ¼ ρd. θ represents the tilt angle of the vicinal surface (Fig. 2(b)). d represents the unit height of a single step. The surface slope is also described by the surface gradient ∇zðx; yÞ ¼ ð∂z=∂x; ∂z=∂yÞ ¼ ðpx ; py Þ as p ¼ jpj for

θ Z 0 and p ¼ jpj for θ o 0, where zðx; yÞ represents the ECS. The

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi normal vector n is described by p as n ¼ ð  px ;  py ; 1Þ= 1 þ p2x þ p2y

[21]. Before obtaining surface tension, we calculate the surface free energy per projected x–y area f ðpÞ [17]. We call f ðpÞ the vicinal surface free energy for short. f ðpÞ is obtained from the Andreev free energy f~ ðηÞ (Eq. (4)) by using the thermodynamic formula f ðpÞ ¼ f~ ðηÞ þ p  η [17]. The calculated vicinal surface free energy

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-[111]

[001]

-[111]

[111]

[001]

k T/ ε =0.35 1.5

1

Β

1.5

1

1

1

[111]

k T/ ε =0.37

γsurf /ε

γsurf /ε

γsurf /ε

[001]

Β

1.5

0 p

-[111]

[111]

k T/ ε =0.36

Β

-1

3

-1

0 p

1

-1

0 p

1

Fig. 3. The calculated surface tensions γ surf =ϵ [16]. Broken line: the values for the metastable surfaces.

becomes [15] f eff ðpÞ ¼ f ð0Þ þ γ ðϕÞjpj þ Aeff ðϕÞjpj2 =d þ Beff ðϕÞjpj3 =d þ Oðjpj4 Þ; 3

2

ð5Þ

where ϕ represents the tilt angle of the mean running direction of steps from the y-axis, γ ðϕÞ represents the step tension, Aeff ðϕÞ represents the step-coalescence factor and Beff ðϕÞ represents the step-interaction coefficient. The characteristic of Eq. (5) is the appearance of the quadratic term with respect to jpj. The stepcoalescence factor Aeff ðϕÞ depends on the mean number of steps in a locally merged step [15]. This term does not exist ðAeff ðϕÞ ¼ 0Þ in the free energy of the GMPT universal form. The surface tension is obtained from the vicinal surface free energy as follows [21]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ surf ðnÞ ¼ f eff ðpÞ= 1 þ jpj2 : ð6Þ The calculated surface tensions are shown in Fig. 3. As seen from the figures, the surface tension becomes discontinuous at low temperatures. There are two transition temperatures, kB T f ;1 =ϵ ¼ 0:3613 7 0:0005 and kB T f ;2 =ϵ ¼ 0:3585 7 0:0007 [15]. At temperatures lower than T f ;1 , the surface tension becomes discontinuous around the (111) facet. At temperatures lower than T f ;2 , the surface tension becomes discontinuous around the (001) facet. The sticky character of the steps ðϵint o 0Þ is the origin of the discontinuity of surface tension. The vicinal surface free energy f eff ðpÞ is described by the surface internal energy eeff ðpÞ and the surface entropy seff ðpÞ as f eff ðpÞ ¼ eeff ðpÞ  Tseff ðpÞ. At sufficiently low temperatures, eeff ðpÞ becomes dominant; and then all the steps condensate to form the (111) surface due to negative ϵint . Since the second derivative of the vicinal surface free energy with respect to p becomes negative, the homogeneous vicinal surface with slope p is thermodynamically unreachable. Hence, the surface tension for the homogeneous phase becomes discontinuous. The calculations based on statistical mechanics [14–16] give a microscopic foundation to Cabrera and Coleman's argument [22]. At sufficiently high temperatures, seff ðpÞ for the step wandering becomes dominant; and then the condensed steps melt into a “gas phase”.

4. Step faceting 4.1. Monte Carlo method Next, we demonstrate the connection between the discontinuous surface tension and the step faceting [22,23]. Since a Monte Carlo method is used for the demonstration, we explain the Monte Carlo method first.

As the simulation model, we consider the vicinal surface of the p-RSOS model on a square lattice, where the surface inclines to the 〈110〉 direction. We require a periodic boundary condition for the vertical direction in Fig. 4. As for the horizontal direction in Fig. 4, the surface height on the right side is connected to the left side by adding N step , where N step is the number of steps. To achieve the time change of the state near equilibrium, we adopt a simple Metropolis algorithm. We choose a site (n,m) randomly. We also choose the direction of the change of the height hðn; mÞ, “addition” ðΔN ¼ 1Þ or “subtraction” ðΔN ¼  1Þ, with the probability 0.5. Then, the height is updated by the Metropolis algorithm together with the RSOS condition. That is, if the height differences among nearest neighbor sites in the final state equal one of f1; 0;  1g, the event occurs in the following probability P : ( 1 ðΔEðn; mÞ r 0Þ; P¼ ð7Þ exp½  βΔEðn; mÞ ðΔEðn; mÞ 4 0Þ; where ΔEðn; mÞ ¼ Eðhðn; mÞ 71Þ Eðhðn; mÞÞ þ ΔNΔμ. The energy Eðhðn; mÞÞ is calculated by Eq. (1). Δμ ¼ μa  μc represents the difference of the bulk chemical potential between the ambient phase (μa) and the crystal (μc). Δμ becomes zero in the equilibrium. 4.2. Step faceting caused by discontinuous surface tension We show the vicinal surface free energy in Fig. 4 in addition to the surface tension, because the vicinal surface free energy is similar to the Helmholtz free energy in gas–liquid first-order transition. Let us consider a vicinal surface with the mean slope being 0.707 (the star in Fig. 4(a)). Since the homogeneous structure is thermodynamically unstable, the surface is realized as the mixture of two surfaces. Two-surface coexistence under equilibrium is demonstrated by a Monte Carlo simulation in Fig. 4(c) I. All steps stick together (Fig. 4(c) II). Hence, the side surface of a merged step forms a macroscopic (111) plane (Fig. 4(c) III). Under non-equilibrium but near equilibrium, several merged steps are formed (Fig. 4(d)). We can see the side surfaces of macrosteps of the (111) surfaces in Fig. 4(d) I and III. The (111) surfaces are clearly smooth. It should be noted that the slope fluctuations of the Gaussian type do not occur near equilibrium, because the surface tension is discontinuous. The lack of slope fluctuations stabilizes the faceted steps. In order to show the effect of smoothness, we calculated a time dependent mean surface height hðtÞ by using the Monte Carlo method (Fig. 5) [16]. The original RSOS model shows step flow growth (Fig. 5(a)). The surface height increases or decreases linearly as time passes. On the other hand, the surface of p-RSOS model hardly moves at this temperature. This is because there are few kinks on the side surface of macrosteps. Hence, the height of the vicinal surface does not grow or evaporate linearly. This non-linear height-change is demonstrated in

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4

-[111]

[001]

Equilibrium Configuration

[111]

Δμ/ε = 0.0005

[ f ( p)-f (0)] /ε

2 1

p =0.707 I -1

0

I ] 0] 10 [10 [0

0 1

p

II

II

γsurf /ε

1.5

(001)

p

III

(001)

) 11 (1

III

1

) 11 (1

0

) 11 (1

-1

(001)

) 11 (1

h(x) 1

(001)

(001)

x

(001)

Fig. 4. Descriptive figures for the step faceting. kB T=ϵ ¼ 0:35. ϵint ¼  0:5. The mean surface slope p ¼0.707 (the star in (a)). (a) The vicinal surface p free ffiffiffi energy. pffiffiffi (b) The surface tension. (c) The equilibrium configuration. I: a snapshot of the top view of a vicinal surface. 1  108 Monte Carlo steps per site (MCS). Size: 80 2  80 2. The number of steps is 80. The diagram depicts 80 steps. However, it is difficult to distinguish all 80 because there are only 10 color gradations. Therefore, they repeat in group of 10, making 8 macro steps immediately visible. II: one-dimensional representation of the step merging. III: the side view of the vicinal surface. (d) Non-equilibrium configuration. Similar figures as (c).

2250 1810 2000

h(t)

h(t)

1805 1750

1800

1500

1250

0

1

2

MCS/site

3

4 (x10 6 )

0.8

0.9

MCS/site

1 (x10 8 )

Fig. 5. Time dependence of the mean surface height (Monte Carlo calculations) [16]. kB T=ϵ ¼ 0:35. ϵint =ϵ ¼  0:5. (a) Solid lines: p-RSOS model. Two-dot chain lines: the original RSOS model ðϵint ¼ 0Þ. (b) Behavior of the p-RSOS model over a longer time period.

Fig. 5(b). Over a long time period, the vicinal surface grows (evaporates) intermittently. The 2D nucleus formed near the cross line of two surfaces is the trigger of the intermittent growth (evaporation). The summary of this section is given as follows: On the side surface of a merged step, the surface becomes smooth and the kink density decreases. Hence, the velocity of a merged step becomes about 1/50 of the velocity of a single step.

5. Pinning of steps 5.1. Pinning and depinning of steps Finally, we demonstrate the pinning of steps near equilibrium. Fig. 6 shows the snapshots of the Monte Carlo results. As seen from the figure, the meeting adjacent steps form merged steps. As shown in the previous section, the merged steps move with the velocity of about 1/50 of the velocity of a single step. Hence, on the surface, the steps are pinned by the locally formed merged steps.

When the driving force is large, the merged step dissociates. The 2D nucleation at the cross line of the two surfaces happens frequently (Fig. 7(b)). The side surface roughens kinetically. 5.2. Ideas to avoid non-uniform growing of crystal The self-pinned macrosteps may diminish the quality of a grown crystal. To avoid the pinning of steps near equilibrium due to the discontinuous surface tension, we present the following suggestions. (1) Raise the temperature for growth to increase the effect of the entropy of step wanderings. (2) Increase the driving force ðΔμÞ moderately so that frequent 2D nucleations (kinetic roughening) may occur.

6. Discussion At higher temperature T 4 T f ;1 , the steps dissociate and the quadratic term in f eff ðpÞ disappears ðAeff ¼ 0Þ [15]. In the middle temperature range T f ;2 o T o T f ;1 , a special phase appears as the

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Fig. 6. Pinning of steps demonstrated by pffiffiffi pffiffiffi the Monte Carlo simulation on the p-RSOS model [16]. The surface tilts in the direction of 〈110〉. Δμ=ϵ ¼ 0:1. kB T=ϵ ¼ 0:1. ϵint =ϵ ¼  0:5. System size: 240 2  240 2. N step ¼ 24. (a) The initial state. (b) 2000 MCS/site. (c) 4000 MCS/site.

Fig. 7. Dissociation of a merged step under p the driving force demonstrated by the Monte Carlo simulation [16]. The surface tilts in the direction of 〈110〉. Δμ=ϵ ¼ 0:35. ffiffiffi large p ffiffiffi kB T=ϵ ¼ 0:1. ϵint =ϵ ¼  0:5. System size: 240 2  240 2. N step ¼ 24. (a) The initial state. (b) 2000 MCS/site. (c) 8000 MCS/site.

mixture of the locally merged steps. We call the locally merged step the “step droplet”1 [24,15]. In the temperature range, though the surface tension around the (111) facet is discontinuous, the surface tension around the (001) facet becomes continuous. Slope fluctuations with infinitesimal energy cost occur frequently. Hence, the distribution function of the slope fluctuations becomes the Gaussian type. We discussed the phase in our previous paper [15,16]. In a real surface, various kinetic effects exist in addition to the step–step attraction [1,25]. Some of these kinetic effects cause step bunching. The step bunching, however, disappears under equilibrium. On the other hand, the point-contact-type step–step attraction causes merged steps in equilibrium [14]. Since the force-range of the p-RSOS model is “zero-range”, the discontinuity in surface tension occurs only for the vicinal surface tilted towards the 〈110〉 direction. For the vicinal surface tilted towards the 〈100〉 direction, the surface tension is almost the same as that of the original RSOS model [15]. Non-fermion like features in a vicinal surface with the multilayer steps (“touchy” steps) are reported by Sathiyanarayanan et al. [26]. The non-fermion like features are obtained in the p-RSOS model for T f ;2 o T. It is quite interesting, but the detailed discussion of the non-fermion like features is beyond the scope of this paper. We discuss the effect of the multilayer steps elsewhere [27]. When the step–step interaction is the short range, the surface tension can be discontinuous for the vicinal surface tilted towards 〈100〉. An example of this kind of behavior is found in a restricted

1 The step droplet is named after the 1D quantum n-mer with being locally bound.

solid-on-solid model coupled with the Ising system (RSOS-I model) [24,28,29], which is considered for the vicinal surface with adsorption. The model also has two transition temperatures corresponding to T f ;1 and T f ;2 . In the case of long range attractive interaction between steps 2 such as g 0 =l , where l is the distance between adjacent steps and g0 ( o 0 for the attractive case) is the coefficient, Jayaprakash et al. [30] and Lässig [31] treated the case on a terrace-step-kink model using the mean field calculation. They showed, referring to the exact result of Calogero–Sutherland [32], that the vicinal surface behaves as GMPT type at higher temperatures than a special temperature Tc; for T o T c , steps collapse into one; i.e. all steps stick together. They did not obtain the explicit form of the vicinal surface free energy for T o T c . Carlon and van Beijeren [33,34] presented two models and derived the surface free energy with a quadratic term using the mean field approximation. The first model they presented [33] is a staggered body-centered cubic solid-on-solid (BCSOS) model with long range step–step interaction. The quadratic term in the surface free energy suggests that there may exist a liquid like phase which corresponds to the phase for T f ;2 o T o T f ;1 in the p-RSOS model [15]. The latter model Carlon and van Beijeren presented [34] is also a staggered BCSOS model which describes surfaces of crystals with nearest neighbor attractions and next nearest neighbor repulsions. We studied an equilibrium crystal shape with a PWFRG calculation for a staggered BCSOS model [35]. As far as the results we had obtained [35], the new phase with the quadratic term in the surface free energy was not found. More detailed calculations may be required. It should be noted that g 0 ¼ 0 in both the p-RSOS model and the RSOS-I model [10,36]. In the continuous limit, the p-RSOS

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model and the RSOS-I model are mapped to the Calogero–Sutherland model with the kinetic term and the hard core interactionterm. In order to obtain a quadratic term and Aeff 4 0 in the vicinal surface free energy without long range step–step attraction, the non-uniformity due to the step droplets is crucial [15]. 7. Conclusion Surface tension becomes discontinuous at low temperatures, when there is the point-contact-type step–step attraction (sticky steps). The discontinuous surface tension leads the vicinal surface to become the mixture of two surfaces. On the side surface of a merged step, the surface becomes smooth and the kink density decreases. The velocity of a merged step can be less than 1/50 of the velocity of a single step near equilibrium. Therefore, without impurities, adsorbates, or dislocations, steps are pinned at the place where the steps locally merge.

[9] [10] [11] [12]

[13] [14] [15] [16] [17] [18] [19] [20]

[21] [22]

Acknowledgments

[23] [24] [25]

This work was supported by Japan society for promotion of science (JSPS) KAKENHI Grant number 25400413.

[26]

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Please cite this article as: N. Akutsu, Journal of Crystal Growth (2014), http://dx.doi.org/10.1016/j.jcrysgro.2014.01.068i