- Email: [email protected]
Non-equilibrium shape of two-dimensional islands: kinetic modeling of morphological changes
Journal of Crystal Growth 237–239 (2002) 39–42
Non-equilibrium shape of two-dimensional islands: kinetic modeling of morphological changes Lev N. Balykova,*, Masao Kitamurab, Igor L. Maksimovc a Strukturforschung SF7, Hahn-Meitner–Institut Berlin GmbH, Glienicker Strasse 100, D-14109 Berlin, Germany Department of Geology and Mineralogy, Graduate School of Science, Kyoto University Sakyo, Kyoto606-8502, Japan c Theoretical Physics Department, Nizhny Novgorod University, 23 Gagarin Avenue, Nizhny Novgorod603000, Russian Federation b
Abstract It is a common observation that many crystals undergo a change in habit from the equilibrium polyhedral shape to another, kinetically favorable one during their growth. By analyzing the structures and rates of advance of [1 0 0] and [1 1 0] steps on the (0 0 1) surface of the Kossel crystal, and taking into account additional channel of incorporation of growth units into kink sites via the edge of the step, we demonstrate that the rate of advance of [1 0 0] step may become larger than one of the [1 1 0] step due to the effect of the activation energies for the exchange between terrace and steps. The change in habit of two-dimensional island from the shape bounded by [1 0 0] steps at equilibrium to one bounded by [1 1 0] steps at higher supersaturations is thus explained. r 2002 Elsevier Science B.V. All rights reserved. PACS: 81.10.Aj; 68.45.Da; 68.45.v Keywords: A1. Crystal morphology; A1. Growth models; A1. Roughening; A1. Smoothening; A1. Surface structure
1. Introduction The shape of a crystal or a two-dimensional island in thermodynamic equilibrium can be determined by minimizing the edge energy [1]. This shape is usually bounded by flat facets normal to low-index directions connected to each other by rounded regions [2]. Out of equilibrium, the shape of a crystal or a two-dimensional island is determined by the kinetics of the system. In this case, crystal faces may become roughened during growth at high supersaturations below their roughening temperature. This is the so-called *Corresponding author. Tel.: +49-30-8062-2720; fax: +4930-8062-2293. E-mail address: [email protected] (L.N. Balykov).
kinetic roughening caused by the driving force for crystallization [3–5]. It is also a common observation that many crystals undergo a change in habit from the equilibrium polyhedral shape to another, kinetically favorable one during their growth [6–10]. There have been several attempts to explain the habit change of crystals during growth based on a selective impurity effect [6,11] or changes in nature of growth surfaces such as surface roughening [8,10], surface melting [12], change in size of growth units [13], changes due to dipolar solvents [14] and two different states of adsorbed molecules [15]. These suggestions, however, all require special circumstances and are insufficient to generally clarify the controlling physical processes responsible for the habit change. Recently, a new growth mechanism was
0022-0248/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 1 8 4 6 - 2
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L.N. Balykov et al. / Journal of Crystal Growth 237–239 (2002) 39–42
proposed [16] and quantitatively described [17], in which growth units can be incorporated into kinks both directly from solution and via a neighboring site on the step by series adsorption, resulting in kinetic smoothening of crystal faces that are rough in equilibrium. In this paper, we apply this approach to give a description of the mechanism of habit change of crystals during growth. To illustrate the concept that is essentially due to a series process, we consider a two-dimensional island on a (0 0 1) surface of a Kossel crystal.
(a)
(b)
(c)
2. Modeling of growth steps kinetics We shall only consider [1 0 0] and [1 1 0] faces of the island, since other faces due to obvious geometrical reasons always tend to disappear from the shape of the island. We consider [1 0 0] and [1 1 0] steps separately and then apply the results to a large enough two-dimensional island, thus neglecting the corner effects of the rounded regions connecting [1 0 0] and [1 1 0] facets together. In describing [1 0 0] step, we apply our treatment previously developed in Ref. [18] extending it to take into account effect of activation energies for different elementary processes occurring on the step, and allowing exchange of growth units between the step and a terrace by both direct and series processes. Accounting for a series process means that after formation of an adatom by a direct process it can then move along the step. In considering [1 1 0] step, we follow our treatment developed in Ref. [17]. Elementary processes that change the step structures are shown in Fig. 1. The activation barriers for adatom formation, vacancy formation, motion of adatom along the edge of the step, and adsorption into a kink are DGaþ ; DGvþ ; DGsþ ; and DGkþ ; respectively. It has already been shown [18–20] that when series processes are very slow, DGsþ -N; density of kinks along [1 0 0]pstep ffiffiffiffiffiffiffiffiffiffiffiincreases with supersaturation as G1 0 0 E 1 þ sGe1 0 0 ; where Ge1 0 0 is the equilibrium density of kinks and s is the supersaturation on the terrace near the step. This is true for vapor growth, where we may neglect the differences in activation barriers for different
Fig. 1. Elementary processes occurring at [1 0 0] and [1 1 0] steps during growth. Motion of atoms along steps is neglected, when activation barrier for series process is very high. Desorption processes are neglected at higher supersaturations.
processes. In case of solution growth, activation energies may differ appreciably [21]. In what follows we consider solution growth and use the following relationship between activation energies: 1=2 DGaþ ¼ 12 DGkþ ¼ 13 DGvþ ; or da ¼ dk ¼ d1=3 v d; þ where dj ¼ exp ðDGj =kB TÞ; j ¼ a; v; s; k; kB the Boltzmann constant and T the temperature. It can be shown for that case that [1 0 0] step will roughen as G1 0 0 Eð1 þ sÞGe1 0 0 : Under the same conditions, number of kinks along [1 1 0] step will decrease as pffiffiffiffiffiffiffiffiffiffiffi G1 1 0 EGe1 1 0 = 1 þ s; where Ge1 1 0 is the equilibrium density of kinks along [1 1 0] step [17]. Thus, in the absence of series processes both [1 0 0] and [1 1 0] steps will roughen with increase of supersaturation, resulting in a rounded shape of an island at higher supersaturations [22,23].
L.N. Balykov et al. / Journal of Crystal Growth 237–239 (2002) 39–42
[100] step
[110] step
0.8
Growth velocity
Kink density Γ
1.0
41
0.6 0.4 0.2
[110] step
[100] step 0.0 0
(a)
5
10
15
20
Activation barrier lnδ
0
(b)
5
10
15
20
Activation barrier lnδ
Fig. 2. Kink densities (a) and growth velocities (b) of [1 0 0] and [1 1 0] steps at higher supersaturations in case of both direct and series processes being important. Crossover of growth velocities indicates the change in habit of two-dimensional island.
When series processes are appreciably fast, for example, ds ¼ da ; the situation will be different. [1 0 0] step will keep its equilibrium structure with increase of supersaturation: G1 0 0 EGe1 0 0 [22]. Number of kinks along [1 1 0]pstep, ffiffiffiffiffiffiffiffiffiffiffion the other hand, will increase as G1 1 0 E 1 þ sGe1 1 0 ; resulting in kinetic smoothening [17]. This means that the motion of adatoms along the step, i.e., edge diffusion, has a great stabilizing effect on structures of both [1 0 0] and [1 0 0] steps: while for [1 0 0] step roughening is suppressed up to high supersaturations, for [1 1 0] step it is even reversed to smoothening. This allows us to consider structures of [1 0 0] and [1 1 0] steps at higher supersaturations. In the high supersaturation limit, we conventionally assume that the step structure is determined mainly by the incoming flux from the terrace to the step. Fig. 2a shows kink densities along [1 0 0] and [1 1 0] steps calculated under such approximation. [1 0 0] step is well defined up to higher activation barriers. Formula for the density of kinks along [1 1 0] steppcan ffiffiffiffiffi be obtained from Ref. [17] as G1 1 0 E1 1= 2d; suggesting that the step becomes covered mainly by kinks at high supersaturations.
3. Non-equilibrium shape of two-dimensional island The rate of advance of [1 0 0] step can be given by a general expression V1 0 0 ¼ aðJ1dir0 0 þ J1ser0 0 Þ; where a is the lattice constant, J1dir0 0 and J1ser0 0 are
the net fluxes from the terrace into kink sites on the step by direct and series process, respectively. Similarly, a general expression for the rate of advance of [1 1 0] step can be given as V1 1 0 ¼ pffiffiffi 1= 2aðJ1dir1 0 þ J1ser1 0 Þ: Standard interpretation of crystal growth theory suggests that [1 1 0] steps should grow faster than [1 0 0] steps, although asymptotically tending to the same growth rate as [1 0 0] steps with increasing supersaturation [22,23]. Such relationships cannot by themselves explain the habit change from a polygon bounded by [1 0 0] steps to one bounded by [1 1 0] steps, because the crystal shape is determined by those faces with the lowest growth rate. In the case of series process being considerably fast, however, [1 0 0] and [1 1 0] steps exhibit a behavior different from the conventional picture. Therefore, in the case of series process being fast, we may expect the crossover in the rates of advance of [1 0 0] and [1 1 0] steps at high supersaturations. To confirm that, we calculate the rates of advance of [1 0 0] and [1 1 0] steps. As seen in Fig. 2b, at higher supersaturations, if the activation barriers are sufficiently high, [1 0 0] step will grow faster than [1 1 0] step, which will result in a change in habit: equilibrium square-like shape of two-dimensional island will change into kinetically favorable diamond-like shape. Thus, the possibility of a change in habit of growth shape of two-dimensional island, as an effect of high activation barrier for incorporation of growth units onto steps bounding the island, is confirmed on the basis of considerations of elementary kinetic processes
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L.N. Balykov et al. / Journal of Crystal Growth 237–239 (2002) 39–42
occurring during crystal growth. Detailed mathematical treatment of the problem will be reported shortly in a separate publication.
4. Summary We have studied the structures and rates of advance of [1 0 0] and [1 1 0] steps that bound a two-dimensional island on the (0 0 1) surface of the Kossel crystal. By taking into account additional channel of incorporation of growth units into kink sites via the edge of the step, we have demonstrated that the rate of advance of [1 0 0] step may become larger than one of the [1 1 0] step due to the effect of activation energies for exchange of growth units between terrace and steps. This explains the change in habit of two-dimensional island from the shape bounded by [1 0 0] steps at equilibrium to one bounded by [1 1 0] steps at higher supersaturations, without requiring any special circumstances. This approach, with minor modifications, can be applied to a two-dimensional crystal, and can be extended to treat a threedimensional system. The performed analysis, although conducted on the far from realistic Kossel crystal model, gives a new insight into understanding of crystal growth mechanisms.
Acknowledgements Authors would like to thank Prof. B. Mutaftschiev and Dr. A. Mori for useful discussions. This work was supported by the Foundation for Promotion of Material Science and Technology of Japan and by a Grant-in-Aids for Scientific Research from the Ministry of Education, Science, Sport and Culture in Japan.
References [1] G. Wulff, Z. Kristallogr. 34 (1901) 449.
[2] I. Markov, Crystal Growth for Beginners, World Scientific, Singapore, 1995. [3] B. Mutaftschiev, in: D.T.J. Hurle (Ed.), Handbook of Crystal Growth, Vol. 1a, North-Holland, Amsterdam, 1993. [4] K. Izumi, Gan Ping, M. Hashimoto, A. Toda, H. Miyaji, Y. Miyamoto, Y. Nakagawa, in: T. Nishinaga, K. Nishioka, J. Harada, A. Sasaki, H. Takei (Eds.), Advances in the Understanding of Crystal Growth Mechanisms, North-Holland, Amsterdam, 1997. [5] H.E. Buckley, Crystal Growth, Wiley, New York, 1951. [6] R. Kern, Bull. Soc. Franc. Mineral Crist. 76 (1953) 325, 391. [7] I. Sunagawa, in: I. Sunagawa (Ed.), Morphology of Crystals, Terra, Tokyo, 1987, p. 509. [8] L.I. Lian, K. Tsukamoto, I. Sunagawa, J. Crystal Growth 99 (1990) 150. [9] P. Bennema, in: D.T.J. Hurle (Ed.), Handbook of Crystal Growth, Vol. 1a, North-Holland, Amsterdam, 1993, p. 477. [10] F.C. Frank, in: J. Burls (Ed.), Proceedings of the International Diamond Conference, Oxford, 1966, Industrial Diamond Information Bureau, London, 1967, p. 119. [11] T. Kuroda, R.J. Lacmann, J. Crystal Growth 56 (1982) 189. [12] Y. Aoki, Mem. Fac. Sci. Kyushu Univ. Ser. D 24 (1979) 75. [13] T. Miyata, M. Kitamura, I. Sunagawa, in: I. Sunagawa (Ed.), Morphology and Growth Unit of Crystals, Terra, Tokyo, 1989, p. 287. [14] M. Bienfait, R. Boistelle, R. Kern, Adsorption et Croissance Cristalline, CNRS, Paris, 1965, p. 515. [15] X.Y. Liu, E.S. Boek, W.J. Briels, P. Bennema, Nature 374 (1995) 342. [16] M. Kitamura, K. Nishioka, Proc. Jpn. Acad. Ser. B 74 (1998) 100. [17] L.N. Balykov, M. Kitamura, I.L. Maksimov, K. Nishioka, Europhys. Lett. 52 (6) (2000) 692. [18] L.N. Balykov, M. Kitamura, I.L. Maksimov, K. Nishioka, Philos. Mag. Lett. 78 (5) (1998) 411. [19] V.V. Voronkov, Sov. Phys. Crystallogr. 15 (1970) 13. [20] J.P. van der Eerden, in: D.T.J. Hurle (Ed.), Handbook of Crystal Growth, Vol. 1a, North-Holland, Amsterdam, 1993, p. 406. [21] M. Kitamura, K. Nishioka, Proc. Jpn. Acad. Ser. B 76 (2000) 108. [22] W.K. Burton, N. Cabrera, F.C. Frank, Philos. Trans. Roy. Soc. A 243 (1951) 299. [23] A.A. Chernov, Modern Crystallography III Crystal Growth, Springer, Berlin, 1984.
Non-equilibrium shape of two-dimensional islands: kinetic modeling of morphological changes Lev N. Balykova,*, Masao Kitamurab, Igor L. Maksimovc a Strukturforschung SF7, Hahn-Meitner–Institut Berlin GmbH, Glienicker Strasse 100, D-14109 Berlin, Germany Department of Geology and Mineralogy, Graduate School of Science, Kyoto University Sakyo, Kyoto606-8502, Japan c Theoretical Physics Department, Nizhny Novgorod University, 23 Gagarin Avenue, Nizhny Novgorod603000, Russian Federation b
Abstract It is a common observation that many crystals undergo a change in habit from the equilibrium polyhedral shape to another, kinetically favorable one during their growth. By analyzing the structures and rates of advance of [1 0 0] and [1 1 0] steps on the (0 0 1) surface of the Kossel crystal, and taking into account additional channel of incorporation of growth units into kink sites via the edge of the step, we demonstrate that the rate of advance of [1 0 0] step may become larger than one of the [1 1 0] step due to the effect of the activation energies for the exchange between terrace and steps. The change in habit of two-dimensional island from the shape bounded by [1 0 0] steps at equilibrium to one bounded by [1 1 0] steps at higher supersaturations is thus explained. r 2002 Elsevier Science B.V. All rights reserved. PACS: 81.10.Aj; 68.45.Da; 68.45.v Keywords: A1. Crystal morphology; A1. Growth models; A1. Roughening; A1. Smoothening; A1. Surface structure
1. Introduction The shape of a crystal or a two-dimensional island in thermodynamic equilibrium can be determined by minimizing the edge energy [1]. This shape is usually bounded by flat facets normal to low-index directions connected to each other by rounded regions [2]. Out of equilibrium, the shape of a crystal or a two-dimensional island is determined by the kinetics of the system. In this case, crystal faces may become roughened during growth at high supersaturations below their roughening temperature. This is the so-called *Corresponding author. Tel.: +49-30-8062-2720; fax: +4930-8062-2293. E-mail address: [email protected] (L.N. Balykov).
kinetic roughening caused by the driving force for crystallization [3–5]. It is also a common observation that many crystals undergo a change in habit from the equilibrium polyhedral shape to another, kinetically favorable one during their growth [6–10]. There have been several attempts to explain the habit change of crystals during growth based on a selective impurity effect [6,11] or changes in nature of growth surfaces such as surface roughening [8,10], surface melting [12], change in size of growth units [13], changes due to dipolar solvents [14] and two different states of adsorbed molecules [15]. These suggestions, however, all require special circumstances and are insufficient to generally clarify the controlling physical processes responsible for the habit change. Recently, a new growth mechanism was
0022-0248/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 1 8 4 6 - 2
40
L.N. Balykov et al. / Journal of Crystal Growth 237–239 (2002) 39–42
proposed [16] and quantitatively described [17], in which growth units can be incorporated into kinks both directly from solution and via a neighboring site on the step by series adsorption, resulting in kinetic smoothening of crystal faces that are rough in equilibrium. In this paper, we apply this approach to give a description of the mechanism of habit change of crystals during growth. To illustrate the concept that is essentially due to a series process, we consider a two-dimensional island on a (0 0 1) surface of a Kossel crystal.
(a)
(b)
(c)
2. Modeling of growth steps kinetics We shall only consider [1 0 0] and [1 1 0] faces of the island, since other faces due to obvious geometrical reasons always tend to disappear from the shape of the island. We consider [1 0 0] and [1 1 0] steps separately and then apply the results to a large enough two-dimensional island, thus neglecting the corner effects of the rounded regions connecting [1 0 0] and [1 1 0] facets together. In describing [1 0 0] step, we apply our treatment previously developed in Ref. [18] extending it to take into account effect of activation energies for different elementary processes occurring on the step, and allowing exchange of growth units between the step and a terrace by both direct and series processes. Accounting for a series process means that after formation of an adatom by a direct process it can then move along the step. In considering [1 1 0] step, we follow our treatment developed in Ref. [17]. Elementary processes that change the step structures are shown in Fig. 1. The activation barriers for adatom formation, vacancy formation, motion of adatom along the edge of the step, and adsorption into a kink are DGaþ ; DGvþ ; DGsþ ; and DGkþ ; respectively. It has already been shown [18–20] that when series processes are very slow, DGsþ -N; density of kinks along [1 0 0]pstep ffiffiffiffiffiffiffiffiffiffiffiincreases with supersaturation as G1 0 0 E 1 þ sGe1 0 0 ; where Ge1 0 0 is the equilibrium density of kinks and s is the supersaturation on the terrace near the step. This is true for vapor growth, where we may neglect the differences in activation barriers for different
Fig. 1. Elementary processes occurring at [1 0 0] and [1 1 0] steps during growth. Motion of atoms along steps is neglected, when activation barrier for series process is very high. Desorption processes are neglected at higher supersaturations.
processes. In case of solution growth, activation energies may differ appreciably [21]. In what follows we consider solution growth and use the following relationship between activation energies: 1=2 DGaþ ¼ 12 DGkþ ¼ 13 DGvþ ; or da ¼ dk ¼ d1=3 v d; þ where dj ¼ exp ðDGj =kB TÞ; j ¼ a; v; s; k; kB the Boltzmann constant and T the temperature. It can be shown for that case that [1 0 0] step will roughen as G1 0 0 Eð1 þ sÞGe1 0 0 : Under the same conditions, number of kinks along [1 1 0] step will decrease as pffiffiffiffiffiffiffiffiffiffiffi G1 1 0 EGe1 1 0 = 1 þ s; where Ge1 1 0 is the equilibrium density of kinks along [1 1 0] step [17]. Thus, in the absence of series processes both [1 0 0] and [1 1 0] steps will roughen with increase of supersaturation, resulting in a rounded shape of an island at higher supersaturations [22,23].
L.N. Balykov et al. / Journal of Crystal Growth 237–239 (2002) 39–42
[100] step
[110] step
0.8
Growth velocity
Kink density Γ
1.0
41
0.6 0.4 0.2
[110] step
[100] step 0.0 0
(a)
5
10
15
20
Activation barrier lnδ
0
(b)
5
10
15
20
Activation barrier lnδ
Fig. 2. Kink densities (a) and growth velocities (b) of [1 0 0] and [1 1 0] steps at higher supersaturations in case of both direct and series processes being important. Crossover of growth velocities indicates the change in habit of two-dimensional island.
When series processes are appreciably fast, for example, ds ¼ da ; the situation will be different. [1 0 0] step will keep its equilibrium structure with increase of supersaturation: G1 0 0 EGe1 0 0 [22]. Number of kinks along [1 1 0]pstep, ffiffiffiffiffiffiffiffiffiffiffion the other hand, will increase as G1 1 0 E 1 þ sGe1 1 0 ; resulting in kinetic smoothening [17]. This means that the motion of adatoms along the step, i.e., edge diffusion, has a great stabilizing effect on structures of both [1 0 0] and [1 0 0] steps: while for [1 0 0] step roughening is suppressed up to high supersaturations, for [1 1 0] step it is even reversed to smoothening. This allows us to consider structures of [1 0 0] and [1 1 0] steps at higher supersaturations. In the high supersaturation limit, we conventionally assume that the step structure is determined mainly by the incoming flux from the terrace to the step. Fig. 2a shows kink densities along [1 0 0] and [1 1 0] steps calculated under such approximation. [1 0 0] step is well defined up to higher activation barriers. Formula for the density of kinks along [1 1 0] steppcan ffiffiffiffiffi be obtained from Ref. [17] as G1 1 0 E1 1= 2d; suggesting that the step becomes covered mainly by kinks at high supersaturations.
3. Non-equilibrium shape of two-dimensional island The rate of advance of [1 0 0] step can be given by a general expression V1 0 0 ¼ aðJ1dir0 0 þ J1ser0 0 Þ; where a is the lattice constant, J1dir0 0 and J1ser0 0 are
the net fluxes from the terrace into kink sites on the step by direct and series process, respectively. Similarly, a general expression for the rate of advance of [1 1 0] step can be given as V1 1 0 ¼ pffiffiffi 1= 2aðJ1dir1 0 þ J1ser1 0 Þ: Standard interpretation of crystal growth theory suggests that [1 1 0] steps should grow faster than [1 0 0] steps, although asymptotically tending to the same growth rate as [1 0 0] steps with increasing supersaturation [22,23]. Such relationships cannot by themselves explain the habit change from a polygon bounded by [1 0 0] steps to one bounded by [1 1 0] steps, because the crystal shape is determined by those faces with the lowest growth rate. In the case of series process being considerably fast, however, [1 0 0] and [1 1 0] steps exhibit a behavior different from the conventional picture. Therefore, in the case of series process being fast, we may expect the crossover in the rates of advance of [1 0 0] and [1 1 0] steps at high supersaturations. To confirm that, we calculate the rates of advance of [1 0 0] and [1 1 0] steps. As seen in Fig. 2b, at higher supersaturations, if the activation barriers are sufficiently high, [1 0 0] step will grow faster than [1 1 0] step, which will result in a change in habit: equilibrium square-like shape of two-dimensional island will change into kinetically favorable diamond-like shape. Thus, the possibility of a change in habit of growth shape of two-dimensional island, as an effect of high activation barrier for incorporation of growth units onto steps bounding the island, is confirmed on the basis of considerations of elementary kinetic processes
42
L.N. Balykov et al. / Journal of Crystal Growth 237–239 (2002) 39–42
occurring during crystal growth. Detailed mathematical treatment of the problem will be reported shortly in a separate publication.
4. Summary We have studied the structures and rates of advance of [1 0 0] and [1 1 0] steps that bound a two-dimensional island on the (0 0 1) surface of the Kossel crystal. By taking into account additional channel of incorporation of growth units into kink sites via the edge of the step, we have demonstrated that the rate of advance of [1 0 0] step may become larger than one of the [1 1 0] step due to the effect of activation energies for exchange of growth units between terrace and steps. This explains the change in habit of two-dimensional island from the shape bounded by [1 0 0] steps at equilibrium to one bounded by [1 1 0] steps at higher supersaturations, without requiring any special circumstances. This approach, with minor modifications, can be applied to a two-dimensional crystal, and can be extended to treat a threedimensional system. The performed analysis, although conducted on the far from realistic Kossel crystal model, gives a new insight into understanding of crystal growth mechanisms.
Acknowledgements Authors would like to thank Prof. B. Mutaftschiev and Dr. A. Mori for useful discussions. This work was supported by the Foundation for Promotion of Material Science and Technology of Japan and by a Grant-in-Aids for Scientific Research from the Ministry of Education, Science, Sport and Culture in Japan.
References [1] G. Wulff, Z. Kristallogr. 34 (1901) 449.
[2] I. Markov, Crystal Growth for Beginners, World Scientific, Singapore, 1995. [3] B. Mutaftschiev, in: D.T.J. Hurle (Ed.), Handbook of Crystal Growth, Vol. 1a, North-Holland, Amsterdam, 1993. [4] K. Izumi, Gan Ping, M. Hashimoto, A. Toda, H. Miyaji, Y. Miyamoto, Y. Nakagawa, in: T. Nishinaga, K. Nishioka, J. Harada, A. Sasaki, H. Takei (Eds.), Advances in the Understanding of Crystal Growth Mechanisms, North-Holland, Amsterdam, 1997. [5] H.E. Buckley, Crystal Growth, Wiley, New York, 1951. [6] R. Kern, Bull. Soc. Franc. Mineral Crist. 76 (1953) 325, 391. [7] I. Sunagawa, in: I. Sunagawa (Ed.), Morphology of Crystals, Terra, Tokyo, 1987, p. 509. [8] L.I. Lian, K. Tsukamoto, I. Sunagawa, J. Crystal Growth 99 (1990) 150. [9] P. Bennema, in: D.T.J. Hurle (Ed.), Handbook of Crystal Growth, Vol. 1a, North-Holland, Amsterdam, 1993, p. 477. [10] F.C. Frank, in: J. Burls (Ed.), Proceedings of the International Diamond Conference, Oxford, 1966, Industrial Diamond Information Bureau, London, 1967, p. 119. [11] T. Kuroda, R.J. Lacmann, J. Crystal Growth 56 (1982) 189. [12] Y. Aoki, Mem. Fac. Sci. Kyushu Univ. Ser. D 24 (1979) 75. [13] T. Miyata, M. Kitamura, I. Sunagawa, in: I. Sunagawa (Ed.), Morphology and Growth Unit of Crystals, Terra, Tokyo, 1989, p. 287. [14] M. Bienfait, R. Boistelle, R. Kern, Adsorption et Croissance Cristalline, CNRS, Paris, 1965, p. 515. [15] X.Y. Liu, E.S. Boek, W.J. Briels, P. Bennema, Nature 374 (1995) 342. [16] M. Kitamura, K. Nishioka, Proc. Jpn. Acad. Ser. B 74 (1998) 100. [17] L.N. Balykov, M. Kitamura, I.L. Maksimov, K. Nishioka, Europhys. Lett. 52 (6) (2000) 692. [18] L.N. Balykov, M. Kitamura, I.L. Maksimov, K. Nishioka, Philos. Mag. Lett. 78 (5) (1998) 411. [19] V.V. Voronkov, Sov. Phys. Crystallogr. 15 (1970) 13. [20] J.P. van der Eerden, in: D.T.J. Hurle (Ed.), Handbook of Crystal Growth, Vol. 1a, North-Holland, Amsterdam, 1993, p. 406. [21] M. Kitamura, K. Nishioka, Proc. Jpn. Acad. Ser. B 76 (2000) 108. [22] W.K. Burton, N. Cabrera, F.C. Frank, Philos. Trans. Roy. Soc. A 243 (1951) 299. [23] A.A. Chernov, Modern Crystallography III Crystal Growth, Springer, Berlin, 1984.