Zipping process on the step bunching in the vicinal surface of the restricted solid-on-solid model with the step attraction of the point contact type

Journal of Crystal Growth 318 (2011) 10–13

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Zipping process on the step bunching in the vicinal surface of the restricted solid-on-solid model with the step attraction of the point contact type Noriko Akutsu  Faculty of Engineering, Osaka Electro-Communication University, Hatsu-cho, Neyagawa, Osaka 572-8530, Japan

a r t i c l e i n f o

a b s t r a c t

Available online 26 October 2010

Step dynamics on the vicinal surface at low temperature is studied by a restricted solid-on-solid model with the inter-ledge attraction of the point type (p-RSOS model). The p-RSOS model is known to show the thermal step bunching due to the singular property of the surface free energy, though the model contains no long range interaction such as elastic interactions among steps. We demonstrate the step-zipping process in the step bunching at low temperature by use of Monte Carlo simulations without special surface kinetics. & 2010 Elsevier B.V. All rights reserved.

Keywords: A1. Computer simulation A1. Surface structure A1. Phase equilibria A1. Step bunching A1. Vicinal surface A1. Surface free energy

1. Introduction The step bunching or the macro-step formation has long been studied, since the phenomena sometimes are considered to be the main cause of the introduction of the imperfection into the growing crystal. In addition, recently, the step bunching has lead attention from the view point of the development of the self-organization technique in the nano-scale. So far, it has been known that the step bunching is caused by the various mechanisms [1–18]. One is the Schwoebel effect [1], and others are the effect of impurities [2–5], the effect of the electromigration [6–8] on the vicinal surface, the effect of the phase transition on the surface reconstruction [9–12], the effect of strain [13], the long range inter-step attraction [14], and the short range adsorbates mediated inter-step attraction [15–17]. In our previous paper [18], we presented a restricted solid-on-solid (RSOS) model with the inter-ledge attraction of the point type (pRSOS model for short, Fig. 1) in order to study the step behavior on the vicinal surface with the local inter-step attraction. As the ‘‘local’’ interstep attraction, we consider the transient local bonds formed at the collision point of the adjacent steps, where the exchange of electrons between the steps occurs. The local bonds affect the anisotropic surface free energy, and lead the change of the equilibrium crystal shape (ECS). We obtained the first-order shape transition on the profile of the ECS from the detailed calculation of the surface free energy by use of the density matrix renormalization group (DMRG) method [19,20]. We also demonstrated the step bunching caused by the local inter-ledge attraction on the vicinal surface by use of the Monte Carlo (MC) method, though the model contains no long range inter-step interaction such as the elastic repulsion among steps.

While experimentally, the zipping process, which is one of the sticking processes of two adjacent steps like the slide fasteners on clothes, is often observed during the early stage of the step bunching [21]. The zipping process also can be explained by some kinetic models adopted to describe step bunching. However, it is unclear whether special kinetic processes are absolutely necessary or not. The aim of this paper is to demonstrate the zipping process in the early stage of the step bunching on the p-RSOS model by use of the Monte Carlo method with the simple Metropolis algorithm of the non-conserved system.

2. The p-RSOS model We consider the restricted solid-on-solid (RSOS) model on a two-dimensional (2D) square lattice. Let us assign the surface height h(n,m) to the site (n,m) in the lattice. The height difference between the neighboring lattice point is restricted to take the value of f 71,0g. The energy cost to make the height difference or the microscopic ledge energy is denoted by e. Here, we introduce the local inter-step attraction of the point type. On the vicinal surface, steps with zigzag structure run as shown in Fig. 1. When adjacent steps collide with each other at the point shown by the filled square in Fig. 1, the energy might be lowered due to the electron exchange between the adjacent steps. This allows us to assign eint o 0 at the collision site of the adjacent steps. This eint can be regarded as the energy of the inter-ledge interaction, and modifies the original RSOS model as follows [18]: X Hp-RSOS ¼ e½jhðn þ 1,mÞhðn,mÞj þ jhðn,mþ 1Þhðn,mÞj n,m

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E-mail address: [email protected] 0022-0248/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2010.10.088

X

eint ½dðjhðn þ 1,m þ 1Þhðn,mÞj,2Þ

n,m

þ dðjhðn þ1,m1Þhðn,mÞj,2Þ,

ð1Þ

N. Akutsu / Journal of Crystal Growth 318 (2011) 10–13

11

y ,m

(n ) x Fig. 1. Top view of the p-RSOS model. Thick gray lines: surface steps. Filled square: the collision point of the adjacent steps. x- and y-axes are set parallel to crystal axes (/1 0 0S and /0 1 0S), respectively. The lattice point of the square lattice locates at the center of the square.

where dða,bÞ is Kronecker’s delta. The summation with respect to (n,m) is taken all over the sites on the square lattice. The RSOS condition is required implicitly. Further, we add the terms of Andreev field [22] g ¼ ðZx , Zy Þ, which is the macroscopic field (like pressure in the gas–liquid system) to make the surface be tilted, to Eq. (1) in order to treat the vicinal surface. Then, the model Hamiltonian equation (1) for the vicinal surface becomes as follows: X X Hvicinal ¼ Hp-RSOS Zx ½hðn þ 1,mÞhðn,mÞZy ½hðn,m þ 1Þhðn,mÞ: n,m

3. Anisotropic surface free energies Before we demonstrate the results obtained by use of MC method, we show the surface free energies [18], the Andreev free energy and the vicinal surface free energy, calculated by the DMRG method [19,20,23]. The Andreev free energy f~ ðgÞ [22] is the thermodynamic potential calculated from the partition function Z of the p-RSOS model as follows [23]:

Z¼

X

ebHvicinal ,

ð3Þ ð4Þ

fhðm,nÞg

where N is the number of lattice points on the square lattice, and

b ¼ 1=kB T. We adopt the transfer matrix method to calculate Z to obtain f~ ðgÞ numerically. As an efficient diagonalization of the transfer matrix to obtain the largest eigenvalue of the matrix, we use the product wave-function renormalization group (PWFRG) method [20], which is a variant of the DMRG method [19]. The anisotropic surface free energy determines the equilibrium crystal shape (ECS), which is the crystal droplet shape with the least total surface free energy [22–25]. Let us describe the ECS as z(x,y) and the surface gradient p as p ¼ ðpx ,py Þ ¼ ð@z=@x,@z=@yÞ. From the thermodynamics of the ECS [22–25], we have the following equation: f~ ðZx , Zy Þ ¼ lzðlx,-lyÞ,

Zx ¼ lx, Zy ¼ ly,

Then, let us define the normalized ECS [23] as follows: ZðX,YÞ ¼ bf~ ðbZx , bZy Þ ¼ blzðblx,blyÞ, X ¼ bZx ,

Y ¼ bZy ,

b ¼ 1=kB T:

ð6Þ

n,m

ð2Þ

1 f~ ðgÞ ¼ f~ ðZx , Zy Þ ¼  lim kB TlnZ, N -1 N

Fig. 2. Perspective view of the equilibrium crystal shape (ECS) around (0 0 1) facet on the p-RSOS model. kB T=e ¼ 0:3. eint =e ¼ 0:5. The inset shows the ECS on the original RSOS model ðeint ¼ 0Þ at kB T=e ¼ 0:3.

In Fig. 2, we show the calculated normalized ECS around the (0 0 1) facet. The ECS of the p-RSOS model has large (1 1 1) facets. The local bonds at the collision point of adjacent steps change the ECS, though the bonds are transient. This means that the anisotropy of the surface tension is also changed by the local bonds. The vicinal surface free energy f(px,py), which is the surface free energy per projected area [25], is obtained from the Andreev free energy in the following equation: f ðpx ,py Þ ¼ f~ ðZx , Zy Þ þ Zx px þ Zy py :

ð7Þ

We show the Andreev free energy and the vicinal surface free energy calculated by the DMRG method in Fig. 3(a) and (b), respectively. Due to the first-order shape transition on the ECS, the surface slope of the surface between (0 0 1) facet (px ¼py ¼0) and (1 1 1) facet (px ¼py ¼1) jumps from (p1,p1) to (1,1), where p1 ¼0.349 is the value shown in Fig. 3(b) at kB T=e ¼ 0:36 and p1 ¼0 at kB T=e ¼ 0:35. We show the magnified figure near the facet edge at kB T=e ¼ 0:36 in the inset of Fig. 3(a) so that the curved region in the profile of the ECS at kB T=e ¼ 0:36 may be discernible. As temperature decreases, the first-order shape transition begins at kB Tf =e ¼ 0:36137 0:0005; then, the curved region decreases rapidly to vanish at kB Tf ,2 =e ¼ 0:35857 0:0007. As seen in Fig. 3(b), the homogeneous vicinal surface with slope p is thermodynamically unstable, if p has the value with p1 o p o1. In this case, the surface with the mean slope p is realized as the mixture of the surface with the slope p1 and the surface with the slope 1 ((1 1 1) surface). We call this kind of step bunching, which is caused by the singular property of the free energy, as the thermal step bunching [17].

4. Monte Carlo simulation ð5Þ

where l is the Lagrange multiplier relating to the volume of the droplet [24]. Therefore, the shape of f~ ðZx , Zy Þ as a function of Zx and Zx is similar to the ECS.

As the simulation model, we consider the vicinal surface of the p-RSOS model which slants to /1 1 0S direction. We show the top view of the initial pffiffiffi structure pffiffiffi of the surface in Fig. 4(a). The size of the system is 132 2  80 2. A given number of steps are put almost

12

N. Akutsu / Journal of Crystal Growth 318 (2011) 10–13

0.05 f (px ,py)/kBT

∼ f (X, Y )/kBT

X=Y

0 -0.05 0

-0.1

4 px = py

2

-0.002 -0.004

2.08

2.0

0

2.085

2.1 X

2.2

0

p1 0.5

1

p

Fig. 3. (a) The calculated Andreev free energy f~ ðX,YÞ divided by kBT (¼ the profile of the normalized ECS), where ðX,YÞ ¼ lbðZx , Zy Þ (Eq. (6)). eint =e ¼ 0:5. Solid lines: kB T=e ¼ 0:35 and 0.36 from the right to the left, respectively. Broken line: the values for the metastable surfaces. Inset: the magnified figure for the curve at kB T=e ¼ 0:36. (b) The calculated vicinal surface free energy f(px,py) divided by kBT. eint =e ¼ 0:5. Open square: the calculated values of f(0,0) and f(1,1) at kB T=e ¼ 0:35. Solid curve and the open circle: the calculated values of f(p,p) and f(1,1) at kB T=e ¼ 0:36. Broken curve: the values for the metastable surfaces.

in the same interval. We require 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 h(132,132) are connected to the left side as h(0,0)¼h(132,132)+Nstep, where Nstep is the number of steps. As the time development, we adopt a simple Metropolis Algorithm with no driving force for the crystal growth. We choose a site (n,m) randomly. We also choose the direction of the change of the height h(n,m), ‘‘addition’’ or ‘‘subtraction’’, 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 ones of {1,0, 1}, the event occurs in the following probability P: ( 1 ðDEðn,mÞ r 0Þ, P¼ ð8Þ exp½bDEðn,mÞ ðDEðn,mÞ 4 0Þ, where DEðn,mÞ ¼ Eðhðn,mÞ 7 1ÞEðhðn,mÞÞ. The energy E(h(n,m)) is calculated by Eq. (1).

5. Zipping process We demonstrate the zipping process in the early stage of the step bunching in Fig. 4. Initially, the steps are set apart (Fig. 4(a)) almost in the same interval. After starting the simulation, the steps travel and collide with other steps (Fig. 4(b)). At sufficiently low temperature, the neighboring steps stick each other at the collision point, and the united steps seldom detach. Then, the two steps zip up from the colliding point due to thermal fluctuations (Fig. 4(b)–(c)). In order to demonstrate the zipping process, we choose temperature so that the ‘‘unzipping process’’ seldom occurs; and we choose the step density ( ¼ the surface slope) so that the mean free length lfree with respect to the step-collision becomes to be less pffiffiffi than the vertical size of the system ð80 2Þ. Therefore, the multiple collision of steps occurs in the simulated area, and it causes several ‘‘bridge’’, which is the single step connecting one macrostep to another macro-step. Since the zipping up movements come to an end when the bridge with the least step free energy is formed between two macro-steps, the surface stands still after a time tq (Fig. 4(d)). Namely, for tq t t, the vicinal surface is quenched. If the step density becomes smaller than the one shown in Fig. 4, lfree becomes larger than the one shown in Fig. 4, and then lfree

Fig. 4. Snapshots of the MC simulation of the vicinal surface of the p-RSOS model (top view). The surface pffiffiffi is tilted pffiffiffi in the direction of (1 1 0). kB T=e ¼ 0:1. eint =e ¼ 0:5. System size: 132 2  80 2. Nstep ¼ 20. In the brightness, there are 10 gradation (the brighter the higher). Since the number of steps exceeds the number of gradations, the height higher by 1 than the height shown by the brightest color is shown by the darkest color. (a) The initial state. (b) 1000 MCS/site. (c) 2000 MCS/site. (d) 4  106 MCS=site.

exceeds the vertical size of the system (Nstep ¼16, Fig. 5(a)). The structure of the quenched surface contains less ‘‘bridge’’ than the one shown in Fig. 4. If the temperature is higher, the unzipping process occurs more frequently due to thermal fluctuations. The zipping process becomes difficult to see in the MC simulation. Steps in the vicinal surface, however, self-assemble as the thermal step bunching to form a macro-step (Fig. 5(b)) as the two-surface co-existence due to the vicinal surface free energy (Fig. 3(b)).

N. Akutsu / Journal of Crystal Growth 318 (2011) 10–13

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Acknowledgements This work was supported by Grants-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan (No. 15540323). The author would like to thank Prof. K. Sudoh for discussions on the inter-step attractions. References

Fig. 5. Snapshots of the MC simulation of the vicinal surface of the p-RSOS model (top view). The surface is tilted in the direction of (1 1 0). In the brightness, there are 10 gradation (the brighter the higher). Since the number of steps exceeds the number of gradations, the height higher by 1 than the height p shown by ffiffiffi pthe ffiffiffi brightest color is shown by the darkest color. eint =e ¼ 0:5. Size: 132 2  80 2. (a) Nstep ¼ 6 16. kB T=e ¼ 0:1. 4  10 MC steps per site. (b) kB T=e ¼ 0:3. Nstep ¼ 8. 6  106 MC steps per site.

6. Conclusion We can demonstrate the zipping process on the p-RSOS model without special surface kinetics. The essential quantity to cause the zipping process in the present model is the inter-step attraction. Though the attraction is local and transient, it causes the singular property in the surface free energy. The key parameters are the step density and temperature. Therefore, for various materials, the zipping process is expected to be observed on the vicinal surface at the sufficiently low temperature, when the p-RSOS model is applicable.

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