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Effect of the roughening transition on the vicinal surface in the step droplet zone
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/crysgro
Effect of the roughening transition on the vicinal surface in the step droplet zone 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 BS T RAC T
Communicated by S. Uda
For vicinal surfaces around the (001) surface inclined towards the 〈111〉 direction, the influence of roughening transitions on the surface tension and on step droplets is studied numerically. The surface tension is calculated using a restricted solid-on-solid model with a point-contact type step–step attraction (p-RSOS model) on a square lattice. To ensure the reliability of the calculations, the density matrix renormalization group method is used. The growth rate of the vicinal surface near equilibrium is also calculated by the Monte Carlo method. It is found that the roughening transition changes the morphology around the (001) surface, and the roughening transition affects the size of locally merged steps (step droplets).
Keywords: Computer simulation Surface structure Crystal morphology Surface processes Growth models
1. Introduction In 1963, Cabrera and Coleman [1] showed that if the surface free energy has a “type II” anomaly, the equilibrium shape of a crystal droplet (equilibrium crystal shape, ECS) has a first-order shape transition1; moreover, the instability with respect to macrostep formation may result from the anomalous surface free energy at equilibrium. There has long been a need for a means of calculating anisotropic surface tension using a microscopic model in order to better understand the morphology and rate of growth of such crystals. However, due to the low dimensionality of the surface and of steps, neither mean field calculations nor quasi-chemical approximations are reliable for calculating surface thermodynamic quantities [3]. In our previous work [4–7], we calculated the slope dependence of the surface tension using a restricted solid-on-solid (RSOS) model with a point-contact type step–step attraction (p-RSOS model). To obtain reliable results, we used the density matrix renormalization group (DMRG) method [8,9], which is known to be effective for onedimensional quantum spin systems. From the results on the slope dependence of the surface tension, we found that the surface tension is discontinuous at low temperatures for a vicinal surface inclined towards the 〈111〉 direction. The discontinuity of the surface tension leads to the formation of faceted macrosteps on a vicinal surface, which inhibits crystal growth because of the significant reduction in the number of kinks, thus degrading the crystal quality [10]. By varying the strength of the step–step attraction, a faceting
diagram was constructed [7] based on a singularity in the surface tension (Fig. 1). The surface tension is discontinuous around the (111) surface for T < Tf ,1; the surface tension is discontinuous around the (001) surface for T < Tf ,2 . The step-faceting zone represents the temperature range T < Tf ,2 , the step droplet zone represents the range Tf ,2 ≤ T < Tf ,1, and the Gruber–Mullins–Pokrovsky–Talapov (GMPT) [11] zone represents the range Tf ,1 ≤ T . In the present work, we consider how the roughening transition on the (001) surface affects the morphology of the crystal and the locally merged steps. In the step droplet zone, the surface tension is continuous near the (001) surface, whereas it is discontinuous around the (111) surface. This discontinuity reflects the formation of locally merged steps. We will consider two cases in detail: a typical example of step droplet zone I (ϵ int /ϵ = −1.5 and kB T /ϵ = 1.0 ), and a typical example of step droplet zone II (ϵ int /ϵ = −1.5 and kB T /ϵ = 1.4 ). 2. p-RSOS model The microscopic model considered here is the restricted solid-onsolid (RSOS) model with the point-contact-type step–step attraction (p-RSOS model). Here, “restricted” means that the height difference between any two neighboring lattice sites is restricted to values of { ± 1, 0}. Point-contact-type step–step attraction refers to the attraction caused by the energy gain for a bonding state formed by overlapping orbitals at the meeting point of neighboring steps. In the p-RSOS model, the Hamiltonian for the (001) surface can be written as follows:
E-mail address: [email protected]. In general, an ECS is similar to the Andreev (surface) free energy [2] (see the caption of Fig. 2 for more details). When the surface slope jumps at the facet edge on the ECS, this is referred to as a first-order shape transition after the fashion of a phase transition. 1
http://dx.doi.org/10.1016/j.jcrysgro.2016.10.014 Received 30 July 2016; Received in revised form 1 October 2016; Accepted 11 October 2016 Available online xxxx 0022-0248/ © 2016 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by/4.0/).
Please cite this article as: Akutsu, N., Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j.jcrysgro.2016.10.014
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
N. Akutsu
Fig. 1. Faceting diagram of the p-RSOS model for a vicinal surface obtained by the DMRG calculations. Squares: calculated points T f ,1. For T < T f ,1, the surface tension is discontinuous around the (111) surface. Triangles: calculated points T f ,2 . For T < T f ,2 , the surface tension is discontinuous around the (001) surface. Open circles: calculated roughening transition temperatures of the (001) surface. Solid line: zone boundary calculated using the two-dimensional Ising model. The definition of a QI Bose solid, QI Bose liquid, and QI Bose gas are given in a previous publication, from which this figure is taken [7].
Fig. 2. Profile of the Andreev free energy, which is also the ECS profile. (a) k B T /ϵ = 1.0 . Pale lines near Q and Q′ indicate metastable surfaces. (b) k B T /ϵ = 1.4 . ∼ Z = f (ηx , ηy )/k B T = λz (x, y)/k B T , R = ± X2 + Y 2 with X = Y, (X , Y ) = (ηx , ηy )/k B T = −λ (x, y)/k B T , where z (x, y) represents the ECS, and λ represents the Lagrange multiplier related to the crystal volume. We assume ϵsurf is equal to ϵ. ϵ int /ϵ = −1.5. The points P and P′ indicate the (001) facet edge, and the points Q and Q′ indicate the (111) facet edge.
/ p−RSOS = 5 ϵsurf +
∼ of h (m, n ). The Andreev free energy f (η ) [2] is the thermodynamic (grand) potential calculated from the partition function A , as follows [12]:
∑ ϵ[|h (n + 1, m) − h (n, m)| + |h (n, m + 1) − h (n, n, m
m )|] +
∑ ϵint [δ (|h (n + 1, m + 1) − h (n, m)|, 2) + δ (|h (n + 1, n, m
1 ∼ ∼ f (η ) = f (ηx , ηy ) = − lim kB T ln A , 5 →∞ 5
(1)
m − 1) − h (n , m )|, 2)],
where 5 is the total number of lattice points, ϵsurf is the surface energy per unit cell on the planar (001) surface, ϵ is the microscopic step energy, δ (a, b ) is the Kronecker delta, and ϵ int is the microscopic step– step interaction energy. The summation with respect to (n,m) is performed over all sites on the square lattice. The RSOS condition is implicitly required. When ϵ int is negative, the step–step interaction becomes attractive (sticky steps). In order to evaluate a vicinal surface, we add the terms for the Andreev field [2]: η = (ηx , ηy ) in order to introduce a step in the surface; note that this is an external field, similar to the chemical potential in a gas–liquid system. The adjusted Hamiltonian for the vicinal surface is [12]:
/ vicinal = / p−RSOS − ηx
where 5 is the number of lattice points on the square lattice. As mentioned in the introduction, in low-dimensional cases, it is necessary to have more precision than that provided by a mean field calculation of the partition function. Fortunately, since the Hamiltonian is quite simple, we can map the p-RSOS model to a 19vertex model [5] and then use the transfer matrix method to calculate it. We note that the transfer matrix version of the DMRG method, which is known as the product wave-function renormalization group (PWFRG) method [9], provides a way to efficiently calculate the largest eigenvalue of the transfer matrix. Examples of the Andreev free energy calculated by the DMRG method are shown in Fig. 2 as a function of (ηx / kB T , ηy / kB T ) for ηx = ηy .
∑ [h (n + 1, m) − h (n, m)] − ηy ∑ [h (n, n, m
m + 1) − h (n , m )].
(3)
3. Shape transitions around the (001) and (111) surfaces: DMRG calculations
n, m
(2)
From statistical mechanics, the (grand) partition function A is calculated as follows: A = ∑{h (m, n)} e−β / vicinal where β = 1/ kB T . The summation with respect to {h (m, n )} is taken over all possible values
One of the effects of the roughening transition on the (001) surface TR(001) is a change in the crystal habit. In Fig. 2, we show the calculated ECS around the (001) facet for T < TR(001) (step droplet zone I) and 2
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
N. Akutsu
∼ Fig. 3. Surface free energy f ( p) obtained from f (η ) , where p = ± px2 + py2 and px = py on the surface. (a) k B T /ϵ = 1.0 (T < TR(001) ). (b) k B T /ϵ = 1.4 (TR(001) < T ). Pale lines in (a) indicate f ( p) for the metastable surfaces. The end points of the pale lines show the spinodal points approximately. We assume that ϵsurf is equal to ϵ, and ϵ int /ϵ = −1.5.
Fig. 4. Monte Carlo simulation of a vicinal surface inclined towards the 〈111� direction. (a) k B T /ϵ = 1.0 . (b) k B T /ϵ = 1.4 . The number of steps equals 90. Size: 60 2 × 60 2 . Snapshots at 2 × 108 Monte Carlo steps (MCS) per site. ϵ int /ϵ = −1.5. The surface height is represented by brightness with 10 gradations, with brighter regions being higher. The darkest areas next to the brightest ones represent terraces that are actually higher by a value of unity, because of the finite gradation. Around the center of (a), wide (001) terraces and steps can be seen.
TR(001) < T (step droplet zone II). In Fig. 2(a), there is a (001) facet and its edge is indicated by points P and P′, whereas in Fig. 2(b), there is no (001) facet. The faceting transition with respect to the (001) surface occurs at TR(001) [7], and this behavior is universal [13–15]. At the points Q and Q′ in Figs. 2(a) and (b), the surface slope jumps, and the facet edge of the (111) surface shows a first-order shape
transition, which is an identifying characteristic of the step droplet zone. In the step-faceting zone, the (111) facets directly contact the (001) facet. That is, a first-order shape transition also occurs at the points P and P′. ∼ Using the inverse Legendre transform with respect to f (η ), ∼ f ( p) = f (η ) + η·p, we obtain the surface free energy per unit area,
3
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Fig. 5. (a) k B T /ϵ = 1.0 . (b) k B T /ϵ = 1.4 . Solid lines and filled squares: polar graph of the surface tension, γsurf ( px , py )/ϵ with px=py. A rotational symmetry of order two is assumed. Dotted lines: the ECS. We assume ϵsurf is equal to ϵ. ϵ int /ϵ = −1.5.
Fig. 6. (a) Example of a negative step. (b) Side view of faceted steps. (c) View of (b) rotated by 54.74°. The side surfaces of the steps in (b) become terraces. This figure is taken from Ref.
Fig. 7. Slope dependence of the average size of locally merged steps as calculated by the Monte Carlo method. (a) Around the (001) surface. (b) Around the (111) surface. Time at averaging = 2 × 108 MCS/site. Size: 160 2 × 160 2 . Open circles: k B T /ϵ = 1.4 and ϵ int = −1.5. Open squares: k B T /ϵ = 1.0 and ϵ int = −1.5. Filled triangles: k B T /ϵ = 1.4 and ϵ int = 0 (original RSOS model). Filled reversed triangles: k B T /ϵ = 1.0 and ϵ int = 0 (original RSOS model).
follows:
f ( p). In Fig. 3, we show the surface free energy. There is a cusp singularity at p=0 on the (001) surface when T < TR(001) , as shown in Fig. 3(a). Hence, a flat (001) “singular surface” appears on the ECS. In Fig. 3(b), when TR(001) < T , the curve is parabolic around p=0. The surface on the ECS in the direction of 〈001〉 is spherical. There is no flat (001) surface. A characteristic of the surface free energy in the step droplet zone is a discontinuity near a (111) surface. By using the inverse Legendre transform, the external thermodynamic parameter is changed from η (the chemical potential of a step) to p (the number of steps). Near the (111) surface, the structure of a regular train of steps is thermodynamically unstable [5–7]. In that location, the vicinal surface has two surfaces, as shown at point Q in Fig. 2. In this manner, the steps selfassemble to form a macrostep on a vicinal surface at equilibrium. The results of a Monte Carlo simulation of macrostep formation are shown in Fig. 4. The surface tension is calculated from the surface free energy, as
γsurf ( px , py ) = f ( px , py )/ 1 + px2 + py2 ;
(4)
the polar graph for which is shown in Fig. 5, and rotational symmetry of order two is assumed. The ECS calculated as the Andreev free energy (shown by dotted lines) agrees with the ECS obtained by the Wulff construction [16]. The surface energy for the (111) surface is small because ϵ int /ϵ = −1.5. The energy gain due to the point contact type step–step attraction reduces the energy of the (111) surface. 4. Step droplets In order to better understand the steps near the (111) surface, we investigate the microscopic structure of the vicinal surface. To describe the steps from the (111) surface, we introduce negative steps (Fig. 6) 4
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Fig. 8. Time evolution of the mean height of the surface, as calculated by the Monte Carlo method. (a) k B T /ϵ = 1.0 . (b) k B T /ϵ = 1.4 . Black lines: ϵ int /ϵ = −1.5. Gray (blue) lines: ϵ int = 0 . Δμ /ϵ = 0.0005. Size: 160 2 × 160 2 . Nstep = 160 .
[7]. In the step-faceting zone, the “terrace” and the “side surface” of a macrostep are interchangeable. In the step droplet zone, we consider the size of the locally merged steps (negative steps with a high step density). Near p=0, we consider the average size of the locally merged steps, 〈n〉 = Nstep /〈nstep 〉, where Nstep is the total number of elementary steps, and nstep is the number of merged steps. Similarly, we consider the average size of the locally merged negative steps near p = 2 , notated as 〈nneg 〉. The relationship between 〈n〉 and 〈nneg 〉 is 〈n neg 〉 = 〈n〉( 2 − p )/ p [7]. In Fig. 7(a), we show the slope dependence of 〈n〉 near the (001) surface for a vicinal surface with a low step density, as calculated by the Monte Carlo method; in Fig. 7(b), we show the slope dependence of 〈n neg 〉 near the (111) surface for a vicinal surface with a high step density. The roughening transition on the (001) surface affects the locally merged steps on a vicinal surface with a low step density. As can be seen in Fig. 7(a), 〈n〉 − 1 for T < TR(001) depends linearly on p. In contrast, 〈n〉 − 1 for TR(001) < T depends on p2, which is similar to the behavior of ϵ int = 0 (original RSOS model). When 〈n〉 − 1 depends linearly on p, a locally merged step may break the universal GMPT behavior and cause the p2 term to become f ( p) [5]. That is, when there is low step density, the locally self-assembled steps and the negative steps in step droplet zone I contribute to the thermodynamic properties. In step droplet zone II, however, the locally self-assembled steps are negligible. Note that relatively wide (001) terraces, (111) terraces, steps, and negative steps are clearly visible in Fig. 4(a) (T < TR(001) ), whereas relatively wide (001) terraces cannot be seen in Fig. 4(b) (TR(001) < T ). For a vicinal surface with a high step density, TR(001) does not cause an essential change in the slope dependence of 〈n neg 〉, although the value of 〈n neg 〉 decreases as the temperature increases. As can be seen in Fig. 7(b), 〈n neg 〉 for the p-RSOS model is large because a (111) macrostep is formed. The slope dependence of 〈n neg 〉 is clearly different from that of ϵ int = 0 . In Figs. 4(a) and (b), (111) terraces and negative steps are clearly visible.
merged negative steps. At kB T /ϵ = 1.4 (TR(001) < T ), the growth rate for the p-RSOS model is slightly larger than it is at kB T /ϵ = 1.0 . The locally merged steps and the locally merged negative steps are partially separated, and this increases the kink density for the vicinal surface. In contrast, the growth rate for the original RSOS model is smaller than that for kB T /ϵ = 1.0 . The roughness of the vicinal surface reduces the number of kinks that are able to contribute to crystal growth; this is due to the geometrical restriction, Δh = {0, ± 1}. 6. Discussion Returning to Eq. (2), we can see that, at the level of a step, the Andreev field corresponds to the chemical potential, and the Andreev free energy, defined in Eq. (3), corresponds to the grand potential per unit area [7]. Hence, the value of the Andreev free energy can be interpreted as the negative pressure −P ( p) on a vicinal surface with a slope p. This pressure P ( p) is a thermodynamic force that enlarges the area of a surface with a slope p. As a result of this force, on a vicinal surface with high step density (Fig. 4), the pressure on the (111) surface balances the pressure on the vicinal surface, as shown at points Q and Q′ in Fig. 2. 7. Conclusion At the roughening transition temperature for the (001) surface, TR(001), the crystal habit changes. For T < TR(001) , a (001) facet exists on the ECS; whereas for TR(001) < T , there is no (001) facet on the ECS. On a vicinal surface, a (001) terrace and steps are visible when T < TR(001) , whereas they are not visible above this temperature. When TR(001) < T , the locally merged steps dissociate at a low step density. For a vicinal surface in the step droplet zone, step flow growth can be observed. However, when T < TR(001) , the growth rate is significantly smaller than that for the original RSOS model (ϵ int = 0 ). Acknowledgements
5. Growth rate
This work was supported by Japan Society for Promotion of Science (JSPS) KAKENHI Grant Number JP25400413.
In order to study the growth rate for a vicinal surface near equilibrium, we used the Monte Carlo method to calculate the time evolution of the mean surface height h (t ). The results are shown in Fig. 8. At kB T /ϵ = 1.0 (T < TR(001) ), the vicinal surface grows steadily, similar to what is seen in step flow growth. However, the growth rate for the p-RSOS model in Fig. 8(a) is only 47% that for the original RSOS model. This is because the kink density for the vicinal surface decreases due to the formation of locally merged steps and locally
References [1] N. Cabrera, R.V. Coleman, The Art and Science of Growing Crystals, J.J. Gilman (Ed.), John Wiley & Sons, New York, London, 1963, p. 3; N. Cabrera, Surf. Sci. 2, 1964, 320 [2] A.F. Andreev, Zh. Eksp. Teor. Fiz. 80, 1981, 2042. Sov. Phys. JETP 53, 1981, 1063; N. Akutsu and T. Yamamoto, Rough-Smooth Transition of Step and Surface, ed. T. Nishinaga, Handbook of Crystal Growth, vol. I, Elsevier, Amsterdam, 2015, p. 265.
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[11] E.E. Gruber, W.W. Mullins, J. Phys. Chem. Solids 28 (1967) 6549; V.L. Pokrovsky, A.L. Talapov, Phys. Rev. Lett. 42 (1979) 65. [12] N. Akutsu, Y. Akutsu, Phys. Rev. B 57, 1998, R4233; Prog. Theor. Phys. 116, 2006, 983 [13] C. Jayaprakash, W.F. Saam, S. Teitel, Phys. Rev. Lett. 50 (1983) 2017. [14] H. van Beijeren, I. Nolden, Structure and Dynamics of Surfaces II, ed. W. Schommers and P. von Blanckenhagen, Springer-Verlag, Berlin Heidelberg, 1987, p. 259; P. Görnert and F. Voigt, Current Topics in Materials Science, E. Kaldis (Ed. ), vol. 11, North-Holland, Amsterdam, 1984. Ch. 1, 1. [15] Yashuhiro Akutsu, J. Phys. Soc. Jpn. 58 (1989) 2219. [16] M. von Laue, Z. Kristallogr. 105, 1944, 124; C. Herring, Phys. Rev. 82 1951 87; J.K. MacKenzie, A.J.W. Moore, J.F. Nicholas, J. Chem. Phys. Solids 23 1962 185; S. Toschev, Crystal Growth, An Introduction, P. Hartman (Ed.), North-Holland, 1973, p. 328
[3] N.D. Mermin, H. Wagner, Phys. Rev. Lett. 17, 1966, 1133; erratum: Phys. Rev. Lett. 17, 1966, 1307 [4] Noriko Akutsu, Appl. Surf. Sci. 256, 2009, 1205, J. Cryst. Growth, 318, 2011, 10 [5] Noriko Akutsu, J. Phys.: Condens. Matter 23 (2011) 485004. [6] Noriko Akutsu, Phys. Rev. E 86, 2012, 061604, J. Cryst. Growth, 401, 2014, 72 [7] N. Akutsu, AIP Adv. 6 (2016) 035301. [8] S.R. White, Phys. Rev. Lett. 69 (1992) 2863; T. Nishino, J. Phys. Soc. Jpn. 64 (1995) 3598; U. Schollwöck, Rev. Mod. Phys. 77 (2005) 259. [9] T. Nishino, K. Okunishi, J. Phys. Soc. Jpn. 64 (1995) 4084; Y. Hieida, K. Okunishi, Y. Akutsu, Phys. Lett. A233 (1997) 464; Y. Hieida, K. Okunishi, Y. Akutsu, New J. Phys. 1 (7) (1999) 1. [10] Takeshi Mitani, Naoyoshi Komatsu, Tetsuo Takahashi, Tomohisa Kato, Shunta Harada, Toru Ujihara, Yuji Matsumoto, Kazuhisa Kurashige, Hajime Okumura, J. Cryst. Growth 423 (2015) 45.
6
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/crysgro
Effect of the roughening transition on the vicinal surface in the step droplet zone 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 BS T RAC T
Communicated by S. Uda
For vicinal surfaces around the (001) surface inclined towards the 〈111〉 direction, the influence of roughening transitions on the surface tension and on step droplets is studied numerically. The surface tension is calculated using a restricted solid-on-solid model with a point-contact type step–step attraction (p-RSOS model) on a square lattice. To ensure the reliability of the calculations, the density matrix renormalization group method is used. The growth rate of the vicinal surface near equilibrium is also calculated by the Monte Carlo method. It is found that the roughening transition changes the morphology around the (001) surface, and the roughening transition affects the size of locally merged steps (step droplets).
Keywords: Computer simulation Surface structure Crystal morphology Surface processes Growth models
1. Introduction In 1963, Cabrera and Coleman [1] showed that if the surface free energy has a “type II” anomaly, the equilibrium shape of a crystal droplet (equilibrium crystal shape, ECS) has a first-order shape transition1; moreover, the instability with respect to macrostep formation may result from the anomalous surface free energy at equilibrium. There has long been a need for a means of calculating anisotropic surface tension using a microscopic model in order to better understand the morphology and rate of growth of such crystals. However, due to the low dimensionality of the surface and of steps, neither mean field calculations nor quasi-chemical approximations are reliable for calculating surface thermodynamic quantities [3]. In our previous work [4–7], we calculated the slope dependence of the surface tension using a restricted solid-on-solid (RSOS) model with a point-contact type step–step attraction (p-RSOS model). To obtain reliable results, we used the density matrix renormalization group (DMRG) method [8,9], which is known to be effective for onedimensional quantum spin systems. From the results on the slope dependence of the surface tension, we found that the surface tension is discontinuous at low temperatures for a vicinal surface inclined towards the 〈111〉 direction. The discontinuity of the surface tension leads to the formation of faceted macrosteps on a vicinal surface, which inhibits crystal growth because of the significant reduction in the number of kinks, thus degrading the crystal quality [10]. By varying the strength of the step–step attraction, a faceting
diagram was constructed [7] based on a singularity in the surface tension (Fig. 1). The surface tension is discontinuous around the (111) surface for T < Tf ,1; the surface tension is discontinuous around the (001) surface for T < Tf ,2 . The step-faceting zone represents the temperature range T < Tf ,2 , the step droplet zone represents the range Tf ,2 ≤ T < Tf ,1, and the Gruber–Mullins–Pokrovsky–Talapov (GMPT) [11] zone represents the range Tf ,1 ≤ T . In the present work, we consider how the roughening transition on the (001) surface affects the morphology of the crystal and the locally merged steps. In the step droplet zone, the surface tension is continuous near the (001) surface, whereas it is discontinuous around the (111) surface. This discontinuity reflects the formation of locally merged steps. We will consider two cases in detail: a typical example of step droplet zone I (ϵ int /ϵ = −1.5 and kB T /ϵ = 1.0 ), and a typical example of step droplet zone II (ϵ int /ϵ = −1.5 and kB T /ϵ = 1.4 ). 2. p-RSOS model The microscopic model considered here is the restricted solid-onsolid (RSOS) model with the point-contact-type step–step attraction (p-RSOS model). Here, “restricted” means that the height difference between any two neighboring lattice sites is restricted to values of { ± 1, 0}. Point-contact-type step–step attraction refers to the attraction caused by the energy gain for a bonding state formed by overlapping orbitals at the meeting point of neighboring steps. In the p-RSOS model, the Hamiltonian for the (001) surface can be written as follows:
E-mail address: [email protected]. In general, an ECS is similar to the Andreev (surface) free energy [2] (see the caption of Fig. 2 for more details). When the surface slope jumps at the facet edge on the ECS, this is referred to as a first-order shape transition after the fashion of a phase transition. 1
http://dx.doi.org/10.1016/j.jcrysgro.2016.10.014 Received 30 July 2016; Received in revised form 1 October 2016; Accepted 11 October 2016 Available online xxxx 0022-0248/ © 2016 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by/4.0/).
Please cite this article as: Akutsu, N., Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j.jcrysgro.2016.10.014
Journal of Crystal Growth xx (xxxx) xxxx–xxxx
N. Akutsu
Fig. 1. Faceting diagram of the p-RSOS model for a vicinal surface obtained by the DMRG calculations. Squares: calculated points T f ,1. For T < T f ,1, the surface tension is discontinuous around the (111) surface. Triangles: calculated points T f ,2 . For T < T f ,2 , the surface tension is discontinuous around the (001) surface. Open circles: calculated roughening transition temperatures of the (001) surface. Solid line: zone boundary calculated using the two-dimensional Ising model. The definition of a QI Bose solid, QI Bose liquid, and QI Bose gas are given in a previous publication, from which this figure is taken [7].
Fig. 2. Profile of the Andreev free energy, which is also the ECS profile. (a) k B T /ϵ = 1.0 . Pale lines near Q and Q′ indicate metastable surfaces. (b) k B T /ϵ = 1.4 . ∼ Z = f (ηx , ηy )/k B T = λz (x, y)/k B T , R = ± X2 + Y 2 with X = Y, (X , Y ) = (ηx , ηy )/k B T = −λ (x, y)/k B T , where z (x, y) represents the ECS, and λ represents the Lagrange multiplier related to the crystal volume. We assume ϵsurf is equal to ϵ. ϵ int /ϵ = −1.5. The points P and P′ indicate the (001) facet edge, and the points Q and Q′ indicate the (111) facet edge.
/ p−RSOS = 5 ϵsurf +
∼ of h (m, n ). The Andreev free energy f (η ) [2] is the thermodynamic (grand) potential calculated from the partition function A , as follows [12]:
∑ ϵ[|h (n + 1, m) − h (n, m)| + |h (n, m + 1) − h (n, n, m
m )|] +
∑ ϵint [δ (|h (n + 1, m + 1) − h (n, m)|, 2) + δ (|h (n + 1, n, m
1 ∼ ∼ f (η ) = f (ηx , ηy ) = − lim kB T ln A , 5 →∞ 5
(1)
m − 1) − h (n , m )|, 2)],
where 5 is the total number of lattice points, ϵsurf is the surface energy per unit cell on the planar (001) surface, ϵ is the microscopic step energy, δ (a, b ) is the Kronecker delta, and ϵ int is the microscopic step– step interaction energy. The summation with respect to (n,m) is performed over all sites on the square lattice. The RSOS condition is implicitly required. When ϵ int is negative, the step–step interaction becomes attractive (sticky steps). In order to evaluate a vicinal surface, we add the terms for the Andreev field [2]: η = (ηx , ηy ) in order to introduce a step in the surface; note that this is an external field, similar to the chemical potential in a gas–liquid system. The adjusted Hamiltonian for the vicinal surface is [12]:
/ vicinal = / p−RSOS − ηx
where 5 is the number of lattice points on the square lattice. As mentioned in the introduction, in low-dimensional cases, it is necessary to have more precision than that provided by a mean field calculation of the partition function. Fortunately, since the Hamiltonian is quite simple, we can map the p-RSOS model to a 19vertex model [5] and then use the transfer matrix method to calculate it. We note that the transfer matrix version of the DMRG method, which is known as the product wave-function renormalization group (PWFRG) method [9], provides a way to efficiently calculate the largest eigenvalue of the transfer matrix. Examples of the Andreev free energy calculated by the DMRG method are shown in Fig. 2 as a function of (ηx / kB T , ηy / kB T ) for ηx = ηy .
∑ [h (n + 1, m) − h (n, m)] − ηy ∑ [h (n, n, m
m + 1) − h (n , m )].
(3)
3. Shape transitions around the (001) and (111) surfaces: DMRG calculations
n, m
(2)
From statistical mechanics, the (grand) partition function A is calculated as follows: A = ∑{h (m, n)} e−β / vicinal where β = 1/ kB T . The summation with respect to {h (m, n )} is taken over all possible values
One of the effects of the roughening transition on the (001) surface TR(001) is a change in the crystal habit. In Fig. 2, we show the calculated ECS around the (001) facet for T < TR(001) (step droplet zone I) and 2
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∼ Fig. 3. Surface free energy f ( p) obtained from f (η ) , where p = ± px2 + py2 and px = py on the surface. (a) k B T /ϵ = 1.0 (T < TR(001) ). (b) k B T /ϵ = 1.4 (TR(001) < T ). Pale lines in (a) indicate f ( p) for the metastable surfaces. The end points of the pale lines show the spinodal points approximately. We assume that ϵsurf is equal to ϵ, and ϵ int /ϵ = −1.5.
Fig. 4. Monte Carlo simulation of a vicinal surface inclined towards the 〈111� direction. (a) k B T /ϵ = 1.0 . (b) k B T /ϵ = 1.4 . The number of steps equals 90. Size: 60 2 × 60 2 . Snapshots at 2 × 108 Monte Carlo steps (MCS) per site. ϵ int /ϵ = −1.5. The surface height is represented by brightness with 10 gradations, with brighter regions being higher. The darkest areas next to the brightest ones represent terraces that are actually higher by a value of unity, because of the finite gradation. Around the center of (a), wide (001) terraces and steps can be seen.
TR(001) < T (step droplet zone II). In Fig. 2(a), there is a (001) facet and its edge is indicated by points P and P′, whereas in Fig. 2(b), there is no (001) facet. The faceting transition with respect to the (001) surface occurs at TR(001) [7], and this behavior is universal [13–15]. At the points Q and Q′ in Figs. 2(a) and (b), the surface slope jumps, and the facet edge of the (111) surface shows a first-order shape
transition, which is an identifying characteristic of the step droplet zone. In the step-faceting zone, the (111) facets directly contact the (001) facet. That is, a first-order shape transition also occurs at the points P and P′. ∼ Using the inverse Legendre transform with respect to f (η ), ∼ f ( p) = f (η ) + η·p, we obtain the surface free energy per unit area,
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Fig. 5. (a) k B T /ϵ = 1.0 . (b) k B T /ϵ = 1.4 . Solid lines and filled squares: polar graph of the surface tension, γsurf ( px , py )/ϵ with px=py. A rotational symmetry of order two is assumed. Dotted lines: the ECS. We assume ϵsurf is equal to ϵ. ϵ int /ϵ = −1.5.
Fig. 6. (a) Example of a negative step. (b) Side view of faceted steps. (c) View of (b) rotated by 54.74°. The side surfaces of the steps in (b) become terraces. This figure is taken from Ref.
Fig. 7. Slope dependence of the average size of locally merged steps as calculated by the Monte Carlo method. (a) Around the (001) surface. (b) Around the (111) surface. Time at averaging = 2 × 108 MCS/site. Size: 160 2 × 160 2 . Open circles: k B T /ϵ = 1.4 and ϵ int = −1.5. Open squares: k B T /ϵ = 1.0 and ϵ int = −1.5. Filled triangles: k B T /ϵ = 1.4 and ϵ int = 0 (original RSOS model). Filled reversed triangles: k B T /ϵ = 1.0 and ϵ int = 0 (original RSOS model).
follows:
f ( p). In Fig. 3, we show the surface free energy. There is a cusp singularity at p=0 on the (001) surface when T < TR(001) , as shown in Fig. 3(a). Hence, a flat (001) “singular surface” appears on the ECS. In Fig. 3(b), when TR(001) < T , the curve is parabolic around p=0. The surface on the ECS in the direction of 〈001〉 is spherical. There is no flat (001) surface. A characteristic of the surface free energy in the step droplet zone is a discontinuity near a (111) surface. By using the inverse Legendre transform, the external thermodynamic parameter is changed from η (the chemical potential of a step) to p (the number of steps). Near the (111) surface, the structure of a regular train of steps is thermodynamically unstable [5–7]. In that location, the vicinal surface has two surfaces, as shown at point Q in Fig. 2. In this manner, the steps selfassemble to form a macrostep on a vicinal surface at equilibrium. The results of a Monte Carlo simulation of macrostep formation are shown in Fig. 4. The surface tension is calculated from the surface free energy, as
γsurf ( px , py ) = f ( px , py )/ 1 + px2 + py2 ;
(4)
the polar graph for which is shown in Fig. 5, and rotational symmetry of order two is assumed. The ECS calculated as the Andreev free energy (shown by dotted lines) agrees with the ECS obtained by the Wulff construction [16]. The surface energy for the (111) surface is small because ϵ int /ϵ = −1.5. The energy gain due to the point contact type step–step attraction reduces the energy of the (111) surface. 4. Step droplets In order to better understand the steps near the (111) surface, we investigate the microscopic structure of the vicinal surface. To describe the steps from the (111) surface, we introduce negative steps (Fig. 6) 4
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Fig. 8. Time evolution of the mean height of the surface, as calculated by the Monte Carlo method. (a) k B T /ϵ = 1.0 . (b) k B T /ϵ = 1.4 . Black lines: ϵ int /ϵ = −1.5. Gray (blue) lines: ϵ int = 0 . Δμ /ϵ = 0.0005. Size: 160 2 × 160 2 . Nstep = 160 .
[7]. In the step-faceting zone, the “terrace” and the “side surface” of a macrostep are interchangeable. In the step droplet zone, we consider the size of the locally merged steps (negative steps with a high step density). Near p=0, we consider the average size of the locally merged steps, 〈n〉 = Nstep /〈nstep 〉, where Nstep is the total number of elementary steps, and nstep is the number of merged steps. Similarly, we consider the average size of the locally merged negative steps near p = 2 , notated as 〈nneg 〉. The relationship between 〈n〉 and 〈nneg 〉 is 〈n neg 〉 = 〈n〉( 2 − p )/ p [7]. In Fig. 7(a), we show the slope dependence of 〈n〉 near the (001) surface for a vicinal surface with a low step density, as calculated by the Monte Carlo method; in Fig. 7(b), we show the slope dependence of 〈n neg 〉 near the (111) surface for a vicinal surface with a high step density. The roughening transition on the (001) surface affects the locally merged steps on a vicinal surface with a low step density. As can be seen in Fig. 7(a), 〈n〉 − 1 for T < TR(001) depends linearly on p. In contrast, 〈n〉 − 1 for TR(001) < T depends on p2, which is similar to the behavior of ϵ int = 0 (original RSOS model). When 〈n〉 − 1 depends linearly on p, a locally merged step may break the universal GMPT behavior and cause the p2 term to become f ( p) [5]. That is, when there is low step density, the locally self-assembled steps and the negative steps in step droplet zone I contribute to the thermodynamic properties. In step droplet zone II, however, the locally self-assembled steps are negligible. Note that relatively wide (001) terraces, (111) terraces, steps, and negative steps are clearly visible in Fig. 4(a) (T < TR(001) ), whereas relatively wide (001) terraces cannot be seen in Fig. 4(b) (TR(001) < T ). For a vicinal surface with a high step density, TR(001) does not cause an essential change in the slope dependence of 〈n neg 〉, although the value of 〈n neg 〉 decreases as the temperature increases. As can be seen in Fig. 7(b), 〈n neg 〉 for the p-RSOS model is large because a (111) macrostep is formed. The slope dependence of 〈n neg 〉 is clearly different from that of ϵ int = 0 . In Figs. 4(a) and (b), (111) terraces and negative steps are clearly visible.
merged negative steps. At kB T /ϵ = 1.4 (TR(001) < T ), the growth rate for the p-RSOS model is slightly larger than it is at kB T /ϵ = 1.0 . The locally merged steps and the locally merged negative steps are partially separated, and this increases the kink density for the vicinal surface. In contrast, the growth rate for the original RSOS model is smaller than that for kB T /ϵ = 1.0 . The roughness of the vicinal surface reduces the number of kinks that are able to contribute to crystal growth; this is due to the geometrical restriction, Δh = {0, ± 1}. 6. Discussion Returning to Eq. (2), we can see that, at the level of a step, the Andreev field corresponds to the chemical potential, and the Andreev free energy, defined in Eq. (3), corresponds to the grand potential per unit area [7]. Hence, the value of the Andreev free energy can be interpreted as the negative pressure −P ( p) on a vicinal surface with a slope p. This pressure P ( p) is a thermodynamic force that enlarges the area of a surface with a slope p. As a result of this force, on a vicinal surface with high step density (Fig. 4), the pressure on the (111) surface balances the pressure on the vicinal surface, as shown at points Q and Q′ in Fig. 2. 7. Conclusion At the roughening transition temperature for the (001) surface, TR(001), the crystal habit changes. For T < TR(001) , a (001) facet exists on the ECS; whereas for TR(001) < T , there is no (001) facet on the ECS. On a vicinal surface, a (001) terrace and steps are visible when T < TR(001) , whereas they are not visible above this temperature. When TR(001) < T , the locally merged steps dissociate at a low step density. For a vicinal surface in the step droplet zone, step flow growth can be observed. However, when T < TR(001) , the growth rate is significantly smaller than that for the original RSOS model (ϵ int = 0 ). Acknowledgements
5. Growth rate
This work was supported by Japan Society for Promotion of Science (JSPS) KAKENHI Grant Number JP25400413.
In order to study the growth rate for a vicinal surface near equilibrium, we used the Monte Carlo method to calculate the time evolution of the mean surface height h (t ). The results are shown in Fig. 8. At kB T /ϵ = 1.0 (T < TR(001) ), the vicinal surface grows steadily, similar to what is seen in step flow growth. However, the growth rate for the p-RSOS model in Fig. 8(a) is only 47% that for the original RSOS model. This is because the kink density for the vicinal surface decreases due to the formation of locally merged steps and locally
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