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Growth and dissolution kinetics of step structure
Journal of Crystal Growth 198/199 (1999) 32—37
Growth and dissolution kinetics of step structure Lev N. Balykov *, Masao Kitamura, Igor L. Maksimov, Kazumi Nishioka Course in Materials Science and Engineering, Graduate School of the University of Tokushima, 2-1 Minamijosanjima, Tokushima 770, Japan Department of Geology and Mineralogy, Graduate School of Science, Kyoto University, Sakyo, Kyoto 606-01, Japan Faculty of Applied Physics and Microelectronics, Nizhny Novgorod University, 23 Gagarin Avenue, Nizhny Novgorod 603600, Russia Department of Optical Science and Technology, The University of Tokushima, 2-1 Minamijosanjima, Tokushima 770, Japan
Abstract The asymmetry between growth and dissolution is examined by studying the nonequilibrium structure of a [1 0 0] step on a (0 0 1) surface of the Kossel crystal by a model in which densities of adatoms and vacancies on the step are treated as independent variables and the motion of kinks along the step is taken into account. Calculation of the steady-state density of kinks and growth and dissolution rates shows strong asymmetry of growth and dissolution processes even for the case of low temperatures and small supersaturations. 1999 Elsevier Science B.V. All rights reserved. PACS: 81.10.A; 61.50.A; 02.30 Keywords: Crystal surface; Step structure; Growth; Dissolution; Asymmetry
1. Introduction The growth rate of a crystal depends on the ability of the fluid phase to provide material to the crystal surface and the ability of the crystal to adsorb this material. When a molecule reaches the crystal surface from the fluid phase, it must find a position where it is strongly bound, such as a kink site on a step, or it is likely to return to the fluid. The chance of reaching a kink site depends on the * Corresponding author. Fax: #81 886 56 9435; e-mail: [email protected]. After submission of the manuscript, Professor Nishioka passed away on 20 May 1998.
average number of these sites on the surface, and therefore the growth rate depends on the precise structure of steps on the surface during growth. Same arguments apply to the case of dissolution as well. Study of the step structure evolution during growth has deserved much attention since the breakthrough paper by Burton et al. [1]. In their paper authors introduced the classification of the sites and segments on the step — kink, vacancy and adatom — and succeeded to find equilibrium distribution of those elementary units. This approach has been presented in a more conventional form by Ohara and Reid [2]. With regard to the nonequilibrium step structure, the concept of one-dimensional
0022-0248/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 9 8 ) 0 1 0 5 9 - 8
L.N. Balykov et al. / Journal of Crystal Growth 198/199 (1999) 32–37
nucleation of atomic rows on a step was introduced [3—5], and in the recent work van der Eerden [6] introduced an equation for the kink population dynamics along a step. The BCF model predicts perfect symmetry from small supersaturations and undersaturations. The classical computer simulation by Gilmer and Bennema [7] agrees with the BCF model, but some asymmetry was observed at large undersaturations. Growth and dissolution have been also studied experimentally [8—10]. To establish the relevance in making any comparison between experimental results and theoretical models requires a very detailed analysis of the experiments. This is not, however, the purpose of this paper and, therefore, left for future investigation. In the present work growth and dissolution are studied by a model that describes the step structure evolution in terms of interconnected nonlinear evolution equations for the densities of kinks, vacancies and adatoms which are treated as independent variables. Growth and dissolution rates calculated from the model show a strong asymmetry even for small deviations from equilibrium.
2. Nonequilibrium step structure We consider a [1 0 0] step on a (0 0 1) surface of the Kossel crystal under the nearest neighbor approximation using a model developed in our previous paper [11]. The step structure is described in terms of densities per site of four elementary units: shift C , being a monatomic shift by $[0 1 0]; “no shift” n ; adatom n and vacancy n , each being a combination of two shifts at two successive positions along a step (see Fig. 1). It should be pointed out that we shall distinguish between a shift on a step and a kink that appears to be a shift but is at least two atomic distances away from another shift. Taking this as a step model, we see that there are some restrictions to its morphology and structural evolution: (i) adsorption and desorption processes which would locally form a “double shift” (i.e., two neighboring segments along [1 0 0] would be shifted by $2[0 1 0] with respect to each other) are forbidden; (ii) overhangs are forbidden.
33
Fig. 1. Schematic picture of a [1 0 0] step with different kinds of shifts: kinks, adatoms, vacancies. Bold lines indicate higher atomic layer.
There are several processes that change the structure of the step: adsorption to and desorption from a straight step segment which is at least three atomic distances long creates an adatom or a vacancy, respectively, and annihilates two “no shifts”; adsorption at a vacancy and desorption of an adatom creates two “no shifts”; adsorption at a step position neighboring an adatom annihilates this adatom and creates two kinks; adsorption into two kinks facing each other eventually annihilates these two kinks and creates a vacancy. Combining these processes and employing a conventional “random-rain” adsorption model, which gives the same adsorption frequency at all (allowed) step positions, we are able to write governing equations for the step structure evolution: dn "!2l(e I\( I2#e\( I2)n#2le\( I2n dt #2le I\( I2n ,
(1)
dn "le\( I2n!le I\( I2n #Cn le I\( I2 dt !2n n le\( I2,
(2)
34
L.N. Balykov et al. / Journal of Crystal Growth 198/199 (1999) 32–37
dn "le I\( I2n!n le\( I2#Cn le\( I2 dt !2n n le I\( I2, (3) C #n "1, (4) where *k represents *k"k¹ ln(1#p), p denotes supersaturation on the terrace near the step, l the jump frequency of an atom to a kink site from the neighboring terrace site, 2 the nearest neighbor bonding energy, k the Boltzmann constant and ¹ temperature. The number of kinks is then given by the expression C "C !2(n #n ), which implies that we do not intend to apply our model to the case of very high temperatures when the density of shifts is close to unity. Under the steady state condition that dn /dt"dC /dt"dn /dt"dn /dt"0, differential equations (1)—(4) are reduced to algebraic ones:
1 ! a# n#bn #an "0, b
(5)
1 n!an #aCn !2n n "0, b
(6)
an!bn #Cn !2an n "0, (7) C #n "1, (8) where a"e I I2, b"e( I2. The solution of the system (5)—(8) gives the steady state value for the density of shifts C as a function of supersaturation at a given temperature. The steady state value for the density of kinks C (see thin lines in Fig. 1) can then be calculated from C "C !2(n #n ). (9) As seen from Fig. 2, kink density monotonically increases with supersaturation and decreases with undersaturation. For small deviations from equilibrium in the case of low temperatures (b 1) an approximate formula for the steady state density of kinks may be obtained: (10) C "C (1#p, where C the equilibrium density of kinks is given by equation 2e\( I2 C " . (1#2e\( I2)
(11)
Fig. 2. Thin lines: numerically calculated steady-state density of kinks as a function of a supersaturation at different temperatures. Bold lines: steady-state kink density calculated using the approximate formula (10). Numerically calculated and approximation lines for b"10 000 and b"100 000 overlap.
L.N. Balykov et al. / Journal of Crystal Growth 198/199 (1999) 32–37
Eq. (10) shows very good agreement with numerically calculated dependence of the kink density C on the supersaturation p (see bold lines in Fig. 2).
3. Growth and dissolution According to modern theories of crystallization, growth occurs only at kink sites on a step. Therefore, the most important parameter that controls the growth or dissolution rate is the density of kinks on a step, and there are only two processes that contribute to the growth or dissolution of a crystal: adsorption to a kink site of an adatom from the neighboring position on a terrace and desorption from a kink site to the neighboring position on a terrace. Processes involving edge diffusion of atoms along a step are excluded from our consideration. The net flux J to the step due to direct exchanges between kinks and terrace is given by J"le\( I2pC .
(12)
We shall consider growth and dissolution in terms of net fluxes J and J. In order to compare growth and dissolution, we need to define the undersaturation p that corresponds to the supersaturation p. The condition for the correspondence between growth and dissolution processes is given by (*k) "!(*k) ,
(13)
which gives the following relation between p and p: !p p" . 1#p
(14)
Let us now introduce a parameter, which we shall use to characterize asymmetry between growth and dissolution, an absolute value of the ratio of corresponding growth and dissolution rates: J pC . r, " !J !pC
(15)
35
For small deviations from equilibrium the relation (14) may be approximated by p+!p. Then the parameter r is determined only by the ratio of kink densities C and C. The classical BCF model pre dicts perfect symmetry (r"1) for small deviations from equilibrium, because it uses an assumption that a step keeps precisely its equilibrium structure during growth or dissolution, i.e., C "C"C . Our model is not based on this assumption. Applying the approximate formula (10) for the kink density for the case of small supersaturations and low temperatures, we obtain the following expression for r: pC (1#p r" "(1#p)*1, !pC (1#p
(16)
which predicts that the growth rate at small supersaturations and low temperatures is higher than the corresponding dissolution rate (see bold line in Fig. 3). For example, the growth at 20% supersaturation (p"0.2) is about 40% faster than the corresponding dissolution. It is not easy, however, to understand what causes this asymmetry. The approximate formula (16) can also be derived from the one-dimensional nucleation theory [3] and van der Eerden’s model for the step dynamics. For the case of larger supersaturations and higher temperatures, when the approximate formula (10) for the kink density is not applicable, we have to use the general expression (15) for r, where kink densities C and C are obtained from the solution of systems (5)—(9). Thin lines in Fig. 2 show that the growth rate is always higher than the corresponding dissolution rate. Let us give a brief comment on the second, third and higher neighbor interactions and why they are not included in our consideration. Consideration of the fourth and higher neighbors would bring about a correlation of atomic positions along the step, the account of which would essentially complicate the model. However, as these bonds are in general very weak, we shall not make a large error by neglecting these interactions altogether. If we assume the monolayer growth of a crystal, all atoms on the [1 0 0] surface will have four third-neighbor bonds, irrespective of their position — on a terrace, on a step or in a kink. Therefore, the
36
L.N. Balykov et al. / Journal of Crystal Growth 198/199 (1999) 32–37
Fig. 3. Thin lines: numerically calculated ratio of growth and dissolution rates as a function of a supersaturation at different temperatures. Bold line: ratio of growth and dissolution rates calculated using the approximate formula (16).
processes changing the step structure do not change number third-neighbor bonds, and consequently the account of third neighbors will not result in any changes in the step structure. It is easy to notice that in all above described processes changing the step structure two secondneighbor bonds are broken or recovered. Hence, at a spelling of the kinetic equations all terms will have a common multiplier — an exponent, containing 4 /k¹. It means, that the account of the sec ond neighbors will result in a change of the equations of the step evolution, that is the relaxation time will be different, but will not render any influence on the steady-state step structure. Since the interaction between second and higher neighbors is assumed to be negligible and our analysis is based on the consideration of the Kossel crystal, the present model includes an essential simplification in comparison with real step structures of crystals. Also in real cases of multiple-step struc-
tures the distance between the steps has a significant influence on the density of kinks along the steps and the general dynamics of growth. Consequently, should the present model be applied for direct quantitative comparisons with experimental data, it must be first modified to include the second and higher neighbor interactions and adjusted to correspond to the respective experimental conditions. However, even in its present form our model gives a new basis for understanding nonequilibrium step structures.
4. Summary The asymmetry between growth and dissolution is examined by studying the nonequilibrium structure of a [1 0 0] step on a (0 0 1) surface of the Kossel crystal by a model in which densities of adatoms and vacancies on the step are treated as
L.N. Balykov et al. / Journal of Crystal Growth 198/199 (1999) 32–37
independent variables and the motion of kinks along the step is taken into account. Calculation of the steady-state density of kinks and growth and dissolution rates shows strong asymmetry of growth and dissolution processes even for the case of low temperatures and small supersaturations.
Acknowledgements L.N.B. wishes to thank the Japanese Ministry of Education for granting him the Monbusho scholarship. I.L.M. appreciates the support in accordance with the JSPS visiting researcher program. This work was supported by JSPS Research for the Future Program in the Area of Atomic Scale Surface and Interface Dynamics under the project of “Dynamic behavior of growth surface and interface, and atomic scale simulation”, Grant-in-Aids Scientific Research under No. 0887056, and partially by the Satellite Venture Business Laboratory of Tokushima University.
37
References [1] W.K. Burton, N. Cabrera, F.C. Frank, Phil. Trans. Roy. Soc. A 73 (1951) 119. [2] M. Ohara, R.C. Reid, Modeling Crystal Growth Rates from Solution, Prentice-Hall, Englewood Cliffs, NJ, 1973, pp. 229—245. [3] V.V. Voronkov, Sov. Phys. Crystallogr. 15 (1970) 13. [4] F.C. Frank, J. Crystal Growth 22 (1974) 233. [5] J. Zhang, G.H. Nancollas, J. Crystal Growth 106 (1990) 181. [6] J.P. van der Eerden, in: D.T.J. Hurle (Ed.), Handbook of Crystal Growth, vol. 1a, North-Holland, Amsterdam, 1973, p. 406. [7] G.H. Gilmer, P. Bennema, J. Appl. Phys. 43 (1972) 1348. [8] W.B. Hillig, Acta Metall. 14 (1966) 1968. [9] J. Garside, J.W. Mullin, Trans. Inst. Chem. Engrs. London 46 (1968) T11. [10] K. Tsukamoto, Morphology and Growth Unit of Crystals, Terra, Tokyo, 1989, pp. 451—478. [11] L.N. Balykov, M. Kitamura, I.L. Maksimov, K. Nishioka, Phil. Mag. Lett. 78 (1998) 411—418.
Growth and dissolution kinetics of step structure Lev N. Balykov *, Masao Kitamura, Igor L. Maksimov, Kazumi Nishioka Course in Materials Science and Engineering, Graduate School of the University of Tokushima, 2-1 Minamijosanjima, Tokushima 770, Japan Department of Geology and Mineralogy, Graduate School of Science, Kyoto University, Sakyo, Kyoto 606-01, Japan Faculty of Applied Physics and Microelectronics, Nizhny Novgorod University, 23 Gagarin Avenue, Nizhny Novgorod 603600, Russia Department of Optical Science and Technology, The University of Tokushima, 2-1 Minamijosanjima, Tokushima 770, Japan
Abstract The asymmetry between growth and dissolution is examined by studying the nonequilibrium structure of a [1 0 0] step on a (0 0 1) surface of the Kossel crystal by a model in which densities of adatoms and vacancies on the step are treated as independent variables and the motion of kinks along the step is taken into account. Calculation of the steady-state density of kinks and growth and dissolution rates shows strong asymmetry of growth and dissolution processes even for the case of low temperatures and small supersaturations. 1999 Elsevier Science B.V. All rights reserved. PACS: 81.10.A; 61.50.A; 02.30 Keywords: Crystal surface; Step structure; Growth; Dissolution; Asymmetry
1. Introduction The growth rate of a crystal depends on the ability of the fluid phase to provide material to the crystal surface and the ability of the crystal to adsorb this material. When a molecule reaches the crystal surface from the fluid phase, it must find a position where it is strongly bound, such as a kink site on a step, or it is likely to return to the fluid. The chance of reaching a kink site depends on the * Corresponding author. Fax: #81 886 56 9435; e-mail: [email protected]. After submission of the manuscript, Professor Nishioka passed away on 20 May 1998.
average number of these sites on the surface, and therefore the growth rate depends on the precise structure of steps on the surface during growth. Same arguments apply to the case of dissolution as well. Study of the step structure evolution during growth has deserved much attention since the breakthrough paper by Burton et al. [1]. In their paper authors introduced the classification of the sites and segments on the step — kink, vacancy and adatom — and succeeded to find equilibrium distribution of those elementary units. This approach has been presented in a more conventional form by Ohara and Reid [2]. With regard to the nonequilibrium step structure, the concept of one-dimensional
0022-0248/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 9 8 ) 0 1 0 5 9 - 8
L.N. Balykov et al. / Journal of Crystal Growth 198/199 (1999) 32–37
nucleation of atomic rows on a step was introduced [3—5], and in the recent work van der Eerden [6] introduced an equation for the kink population dynamics along a step. The BCF model predicts perfect symmetry from small supersaturations and undersaturations. The classical computer simulation by Gilmer and Bennema [7] agrees with the BCF model, but some asymmetry was observed at large undersaturations. Growth and dissolution have been also studied experimentally [8—10]. To establish the relevance in making any comparison between experimental results and theoretical models requires a very detailed analysis of the experiments. This is not, however, the purpose of this paper and, therefore, left for future investigation. In the present work growth and dissolution are studied by a model that describes the step structure evolution in terms of interconnected nonlinear evolution equations for the densities of kinks, vacancies and adatoms which are treated as independent variables. Growth and dissolution rates calculated from the model show a strong asymmetry even for small deviations from equilibrium.
2. Nonequilibrium step structure We consider a [1 0 0] step on a (0 0 1) surface of the Kossel crystal under the nearest neighbor approximation using a model developed in our previous paper [11]. The step structure is described in terms of densities per site of four elementary units: shift C , being a monatomic shift by $[0 1 0]; “no shift” n ; adatom n and vacancy n , each being a combination of two shifts at two successive positions along a step (see Fig. 1). It should be pointed out that we shall distinguish between a shift on a step and a kink that appears to be a shift but is at least two atomic distances away from another shift. Taking this as a step model, we see that there are some restrictions to its morphology and structural evolution: (i) adsorption and desorption processes which would locally form a “double shift” (i.e., two neighboring segments along [1 0 0] would be shifted by $2[0 1 0] with respect to each other) are forbidden; (ii) overhangs are forbidden.
33
Fig. 1. Schematic picture of a [1 0 0] step with different kinds of shifts: kinks, adatoms, vacancies. Bold lines indicate higher atomic layer.
There are several processes that change the structure of the step: adsorption to and desorption from a straight step segment which is at least three atomic distances long creates an adatom or a vacancy, respectively, and annihilates two “no shifts”; adsorption at a vacancy and desorption of an adatom creates two “no shifts”; adsorption at a step position neighboring an adatom annihilates this adatom and creates two kinks; adsorption into two kinks facing each other eventually annihilates these two kinks and creates a vacancy. Combining these processes and employing a conventional “random-rain” adsorption model, which gives the same adsorption frequency at all (allowed) step positions, we are able to write governing equations for the step structure evolution: dn "!2l(e I\( I2#e\( I2)n#2le\( I2n dt #2le I\( I2n ,
(1)
dn "le\( I2n!le I\( I2n #Cn le I\( I2 dt !2n n le\( I2,
(2)
34
L.N. Balykov et al. / Journal of Crystal Growth 198/199 (1999) 32–37
dn "le I\( I2n!n le\( I2#Cn le\( I2 dt !2n n le I\( I2, (3) C #n "1, (4) where *k represents *k"k¹ ln(1#p), p denotes supersaturation on the terrace near the step, l the jump frequency of an atom to a kink site from the neighboring terrace site, 2 the nearest neighbor bonding energy, k the Boltzmann constant and ¹ temperature. The number of kinks is then given by the expression C "C !2(n #n ), which implies that we do not intend to apply our model to the case of very high temperatures when the density of shifts is close to unity. Under the steady state condition that dn /dt"dC /dt"dn /dt"dn /dt"0, differential equations (1)—(4) are reduced to algebraic ones:
1 ! a# n#bn #an "0, b
(5)
1 n!an #aCn !2n n "0, b
(6)
an!bn #Cn !2an n "0, (7) C #n "1, (8) where a"e I I2, b"e( I2. The solution of the system (5)—(8) gives the steady state value for the density of shifts C as a function of supersaturation at a given temperature. The steady state value for the density of kinks C (see thin lines in Fig. 1) can then be calculated from C "C !2(n #n ). (9) As seen from Fig. 2, kink density monotonically increases with supersaturation and decreases with undersaturation. For small deviations from equilibrium in the case of low temperatures (b 1) an approximate formula for the steady state density of kinks may be obtained: (10) C "C (1#p, where C the equilibrium density of kinks is given by equation 2e\( I2 C " . (1#2e\( I2)
(11)
Fig. 2. Thin lines: numerically calculated steady-state density of kinks as a function of a supersaturation at different temperatures. Bold lines: steady-state kink density calculated using the approximate formula (10). Numerically calculated and approximation lines for b"10 000 and b"100 000 overlap.
L.N. Balykov et al. / Journal of Crystal Growth 198/199 (1999) 32–37
Eq. (10) shows very good agreement with numerically calculated dependence of the kink density C on the supersaturation p (see bold lines in Fig. 2).
3. Growth and dissolution According to modern theories of crystallization, growth occurs only at kink sites on a step. Therefore, the most important parameter that controls the growth or dissolution rate is the density of kinks on a step, and there are only two processes that contribute to the growth or dissolution of a crystal: adsorption to a kink site of an adatom from the neighboring position on a terrace and desorption from a kink site to the neighboring position on a terrace. Processes involving edge diffusion of atoms along a step are excluded from our consideration. The net flux J to the step due to direct exchanges between kinks and terrace is given by J"le\( I2pC .
(12)
We shall consider growth and dissolution in terms of net fluxes J and J. In order to compare growth and dissolution, we need to define the undersaturation p that corresponds to the supersaturation p. The condition for the correspondence between growth and dissolution processes is given by (*k) "!(*k) ,
(13)
which gives the following relation between p and p: !p p" . 1#p
(14)
Let us now introduce a parameter, which we shall use to characterize asymmetry between growth and dissolution, an absolute value of the ratio of corresponding growth and dissolution rates: J pC . r, " !J !pC
(15)
35
For small deviations from equilibrium the relation (14) may be approximated by p+!p. Then the parameter r is determined only by the ratio of kink densities C and C. The classical BCF model pre dicts perfect symmetry (r"1) for small deviations from equilibrium, because it uses an assumption that a step keeps precisely its equilibrium structure during growth or dissolution, i.e., C "C"C . Our model is not based on this assumption. Applying the approximate formula (10) for the kink density for the case of small supersaturations and low temperatures, we obtain the following expression for r: pC (1#p r" "(1#p)*1, !pC (1#p
(16)
which predicts that the growth rate at small supersaturations and low temperatures is higher than the corresponding dissolution rate (see bold line in Fig. 3). For example, the growth at 20% supersaturation (p"0.2) is about 40% faster than the corresponding dissolution. It is not easy, however, to understand what causes this asymmetry. The approximate formula (16) can also be derived from the one-dimensional nucleation theory [3] and van der Eerden’s model for the step dynamics. For the case of larger supersaturations and higher temperatures, when the approximate formula (10) for the kink density is not applicable, we have to use the general expression (15) for r, where kink densities C and C are obtained from the solution of systems (5)—(9). Thin lines in Fig. 2 show that the growth rate is always higher than the corresponding dissolution rate. Let us give a brief comment on the second, third and higher neighbor interactions and why they are not included in our consideration. Consideration of the fourth and higher neighbors would bring about a correlation of atomic positions along the step, the account of which would essentially complicate the model. However, as these bonds are in general very weak, we shall not make a large error by neglecting these interactions altogether. If we assume the monolayer growth of a crystal, all atoms on the [1 0 0] surface will have four third-neighbor bonds, irrespective of their position — on a terrace, on a step or in a kink. Therefore, the
36
L.N. Balykov et al. / Journal of Crystal Growth 198/199 (1999) 32–37
Fig. 3. Thin lines: numerically calculated ratio of growth and dissolution rates as a function of a supersaturation at different temperatures. Bold line: ratio of growth and dissolution rates calculated using the approximate formula (16).
processes changing the step structure do not change number third-neighbor bonds, and consequently the account of third neighbors will not result in any changes in the step structure. It is easy to notice that in all above described processes changing the step structure two secondneighbor bonds are broken or recovered. Hence, at a spelling of the kinetic equations all terms will have a common multiplier — an exponent, containing 4 /k¹. It means, that the account of the sec ond neighbors will result in a change of the equations of the step evolution, that is the relaxation time will be different, but will not render any influence on the steady-state step structure. Since the interaction between second and higher neighbors is assumed to be negligible and our analysis is based on the consideration of the Kossel crystal, the present model includes an essential simplification in comparison with real step structures of crystals. Also in real cases of multiple-step struc-
tures the distance between the steps has a significant influence on the density of kinks along the steps and the general dynamics of growth. Consequently, should the present model be applied for direct quantitative comparisons with experimental data, it must be first modified to include the second and higher neighbor interactions and adjusted to correspond to the respective experimental conditions. However, even in its present form our model gives a new basis for understanding nonequilibrium step structures.
4. Summary The asymmetry between growth and dissolution is examined by studying the nonequilibrium structure of a [1 0 0] step on a (0 0 1) surface of the Kossel crystal by a model in which densities of adatoms and vacancies on the step are treated as
L.N. Balykov et al. / Journal of Crystal Growth 198/199 (1999) 32–37
independent variables and the motion of kinks along the step is taken into account. Calculation of the steady-state density of kinks and growth and dissolution rates shows strong asymmetry of growth and dissolution processes even for the case of low temperatures and small supersaturations.
Acknowledgements L.N.B. wishes to thank the Japanese Ministry of Education for granting him the Monbusho scholarship. I.L.M. appreciates the support in accordance with the JSPS visiting researcher program. This work was supported by JSPS Research for the Future Program in the Area of Atomic Scale Surface and Interface Dynamics under the project of “Dynamic behavior of growth surface and interface, and atomic scale simulation”, Grant-in-Aids Scientific Research under No. 0887056, and partially by the Satellite Venture Business Laboratory of Tokushima University.
37
References [1] W.K. Burton, N. Cabrera, F.C. Frank, Phil. Trans. Roy. Soc. A 73 (1951) 119. [2] M. Ohara, R.C. Reid, Modeling Crystal Growth Rates from Solution, Prentice-Hall, Englewood Cliffs, NJ, 1973, pp. 229—245. [3] V.V. Voronkov, Sov. Phys. Crystallogr. 15 (1970) 13. [4] F.C. Frank, J. Crystal Growth 22 (1974) 233. [5] J. Zhang, G.H. Nancollas, J. Crystal Growth 106 (1990) 181. [6] J.P. van der Eerden, in: D.T.J. Hurle (Ed.), Handbook of Crystal Growth, vol. 1a, North-Holland, Amsterdam, 1973, p. 406. [7] G.H. Gilmer, P. Bennema, J. Appl. Phys. 43 (1972) 1348. [8] W.B. Hillig, Acta Metall. 14 (1966) 1968. [9] J. Garside, J.W. Mullin, Trans. Inst. Chem. Engrs. London 46 (1968) T11. [10] K. Tsukamoto, Morphology and Growth Unit of Crystals, Terra, Tokyo, 1989, pp. 451—478. [11] L.N. Balykov, M. Kitamura, I.L. Maksimov, K. Nishioka, Phil. Mag. Lett. 78 (1998) 411—418.