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What causes step bunching––negative Ehrlich–Schwoebel barrier versus positive incorporation barrier
Surface Science 515 (2002) L459–L463 www.elsevier.com/locate/susc
Surface Science Letters
What causes step bunching––negative Ehrlich–Schwoebel barrier versus positive incorporation barrier M.H. Xie *, S.Y. Leung, S.Y. Tong
1
Department of Physics, HKU-CAS Joint Laboratory on New Materials, The University of Hong Kong, Pokfulam Road, Hong Kong, Hong Kong Received 28 March 2002; accepted for publication 14 June 2002
Abstract Step asymmetry induced step bunching of a vicinal surface during deposition is investigated analytically and by kinetic Monte Carlo simulation. Specifically, effect of a negative Ehrlich–Schwoebel (ES) barrier at a downward step versus a positive incorporation barrier at an upward step edge is compared. It is found that a positive incorporation barrier can in general result in step bunching, whereas a negative ES barrier usually does not. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Surface structure, morphology, roughness, and topography; Step formation and bunching; Monte Carlo simulations
Thin film growth by epitaxial methods such as molecular beam epitaxy (MBE) are known to be largely determined by kinetics of the surface, such as adatom surface diffusion and island nucleation etc. [1,2]. It is recognized that the presence of a step edge barrier [3] (i.e., the Ehrlich–Schwoebel (ES) barrier), where adatoms experience an additional energy barrier when diffuse down a step than when they hop on a flat terrace, can have important consequences in the evolution of surface
* Corresponding author. Tel.: +852-28597945; fax: +85225599152. E-mail address: [email protected] (M.H. Xie). 1 Department of Applied Physics and Materials Sciences, City University of Hong Kong, Kowloon Tang, Hong Kong.
morphology. For example, on a nominally flat surface, a positive ES barrier leads to mound formation [4,5], whereas on a vicinal surface, it may cause Bales–Zangwill instability [6]. If there exists a negative ES barrier, a step-bunched morphology arguably results for growth of a vicinal surface. Qualitatively, this is based on considerations of step growth (advance) rate. A step grows by capturing adatoms from its neighboring (upper and lower) terraces. Wider terraces mean more supply of adatoms to the step in the step flow growth mode and so the step advances faster. When there exists a step asymmetry due to, e.g., the existence of a negative ES barrier such that a step captures adatoms more efficiently from its upper terrace than from the lower terrace, the advance rate of the step becomes dominated by the
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upper terrace. If this terrace is wider to begin with, the growth rate of the step in front (i.e., the descending step) will be higher, leaving the terrace behind (i.e., its upper terrace) wider. Similarly, if the upper terrace is initially narrower than others, its descending step grows slowly, making the upper terrace even narrower. Continued growth thus ultimately leads to step bunching. Such an argument had been used to suggest a possible negative ES barrier on GaAs(1 1 0) surface, where macrostep formation was observed experimently [7]. It is noted that a similar upper-terrace dominated step asymmetry can also arise from the presence of a positive incorporation barrier––the additional energy barrier which impedes adatoms diffusing towards an ascending step (Fig. 1a). It is thus not obvious which of the above two cases (negative ES barrier versus positive incorporation barrier) is responsible for an experimently observed step bunching. It is the purpose of this paper to show that in general the step asymmetry induced step bunching is due to a positive incorporation barrier rather than negative ES barrier, although the two give rise to the same asymmetry in adatom capture rate from upper versus lower terraces.
The analysis is based on one-dimensional diffusion equation at steady state: D
d2 n þF ¼0 dx2
ð1Þ
where D ¼ m expðEd =kT Þ is the diffusion coefficient of adatoms, nðxÞ is adatom concentration at position x, and F is the deposition flux. m represents the attempt frequency for hopping, while Ed is the diffusion (hopping) energy barrier. k is the Boltzmann constant and T is the growth temperature. The lattice constant of the crystalline surface or the distance for a single hop is chosen as the length unit. The above equation is valid under the condition of complete condensation, i.e., when there is no desorption, which is a good approximation for most MBE growth conditions. The solution of Eq. (1) under the boundary condition at x ¼ W (representing the advance rate of a step) 2 D
dn ¼ b ðn n0 Þ dx
ð2Þ
reads [9] nðxÞ ¼
FW 2 1 þ 1 Fx2 ðn nþ Þx þ ðn þ n Þ þ 2 2W 2D 2D ð3Þ
where, n ¼
1 2FW 2 2W b 2þ H D D þ
2Wn0 þ ðb þ b þ bþ b Þ D
ð4Þ
represent adatom concentrations at ascending (þ) and descending () steps, with their kinetic coefficients bþ and b , respectively (see Fig. 1a). n0 is
Fig. 1. (a) Schematic diagram showing a terrace and steps and the associated energy diagram. It illustrates the negative ES barrier and positive incorporation energy barrier at descending and ascending steps, respectively. Note also the definition of parameters. (b) Schematic diagram of a vicinal surface, illustrating the growth of a step due to contributions from its neighboring terraces.
2
By writing such a boundary condition, it is implied that the steps are not permeable, i.e., the mass transport are local to a single terrace bounded by two steps. If mass transport are nonlocal, the concentration profile on a given terrace becomes affected by its neighboring terraces [8] and in general, it is more symmetrical than the non-local case.
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the equilibrium adatom concentration at the step. 2W is the terrace width, and 2W bþ 2W b H ¼ 1þ 1þ 1 ð5Þ D D The kinetic coefficient b of a step is related to kink density on step, 1=dk (dk represents average kink separation), as well as the energy barrier U for adatom incorporation [9]: b ¼
m exp U =kT dk
ð6Þ
where the barrier U is the sum of the surface diffusion barrier Ed and an extra term DE, i.e., U ¼ Ed þ DE, where DE is the extra energy required for adatoms diffusing toward a step than that when they hop on flat terraces (an example of the DE is the ES barrier described earlier). In general, DE can be both positive and negative. The superscripts ‘þ’ and ‘’ in Eq. (6) distinguish the cases between adatom capture by an ascending step and a descending step. By setting dn=dx ¼ 0, one easily finds the location xm at which the adatom concentration has the maximum: xm ¼
2W 2 bþ b D H
ð7Þ
from which, we find that if bþ ¼ b , xm ¼ 0, i.e., when ascending and descending steps are symmetrical in terms of their efficiencies in capturing adatoms from terraces, the concentration profile will be peaked at the center of the terrace. On the other hand, if bþ 6¼ b , distinction needs to be made: under the diffusion-limited regime, or specifically when 2W b =D 1, ðbþ b Þ H , we still have xm ! 0. This is understood by the fact that in this regime, growth is limited by the rate in which adatoms diffuse toward a step. So, even though the step kinetic coefficients are different between ascending and descending steps, both are sufficiently high compared to the diffusion rate, so both steps can be regarded as perfect traps. Then adatom concentration profile on terrace is not affected by the asymmetry in b and peaks at the terrace center. In the other extreme, i.e., when 2W b =D 1, the growth becomes attachmentrate limited (diffusion is rapid compared to the
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capturing rate of steps). Then from Eqs. (5) and (7) and ignoring high-order terms in H, i.e., H ð2W =DÞðbþ þ b Þ 1 b =bþ xm W ð8Þ 1 þ b =bþ Therefore, the position showing the maximum adatom concentration depends on the ratio of b =bþ . If b > bþ , i.e., when the descending step is more efficient in capturing adatoms, xm < 0, i.e., adatom concentration peaks nearer to the ascending step and few atoms are found close to the descending step. A high adatom concentration means a high nucleation rate, so when growth is carried out in the nucleation regime, more islands will be found near the ascending step. Finally, it can be expected that a cross-over from diffusionlimited to kinetic-limited regime will occur for 2W b =D 1, i.e., when diffusion rate and step attachment rate become comparable [10]. Let’s now determine the growth rate of a step bounded by an upper terrace with width 2Wu and a lower terrace having width 2Wl (A in Fig. 1b). The step advances by capturing atoms from these two terraces with respective kinetic coefficients b (from upper terrace) and bþ (from lower terrace). The overall growth rate of the step is then v ¼ vl þ vu
ð9Þ
where, according to Eq. (2), dn vlðuÞ ¼ b ðn n0 Þ ¼ D dx þ F b b 2 ¼ FWlðuÞ 2WlðuÞ D H
ð10Þ
Here, vlðuÞ represents the rate due to adatom capture from the lower(upper) terrace. The above formula can be simplified depending on the relative values of bþ and b . If bþ ¼ b or if bþ 6¼ b but 2W b =D 1 (diffusion-limited regime), we have vlðuÞ ffi FWlðuÞ
ð11Þ
If, on the other hand, 2W b =D 1 (attachmentrate limited), then 2b vlðuÞ ¼ F ð12Þ WlðuÞ bþ þ b
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This latter equation is usually written in the form of v ¼ k L [11], where L ¼ 2W is the terrace width, and k are characteristic constants (proportionality) reflecting the effectiveness of steps in capturing adatoms from neighboring terraces. Apparently by inspecting Eq. (12), one finds k ¼ F b =ðbþ þ b Þ. From the above equations, it is seen that only in attachment limited regime does the growth rate of a step become dependent on step kinetic coefficients b , and if they are different between ascending and descending steps or specifically, if b > bþ , step bunching can happen. This is because, according to Eq. (12), we have vu > vl given the same terrace width, Wl ¼ Wu . This means that the (total) growth rate of a step becomes dominated by adatom capture from its upper terrace (i.e., the contribution of upper terrace is higher than from the lower terrace). If, at a particular moment, Wu > Wl due to, e.g., growth fluctuation (noise), then the step in front of Wu (A in Fig. 1b) grows much faster than the front step of Wl (B in Fig. 1b). Consequently, terrace Wu is further widened but Wl is narrowed, leading to step bunching. This contrasts to growth in diffusion limited regime, where according to Eq. (11), vl is greater than vu only if Wl > Wu . But since the upper and lower terraces equally contribute to step growth rate, this makes the total rate v unchanged (refer to Eq. (9)). As pointed out earlier, both negative ES barrier and positive incorporation barrier can lead to the inequality b > bþ . However, in order for Eq. (12) to be applicable (the necessary condition for step bunching), the growth has to be in the attachmentlimited regime, i.e., when 2W b =D 1. While D is related to diffusion barrier, Ed , according to
expðEd =kT Þ, b depends on U ¼ Ed þ DE as well as kink density, 1=dk (see Eq. (6)). So, the condition of attachment-limited growth translates into U ¼ Ed þ DE > Ed þ kT lnð2W =dk Þ, or 2W DE > kT ln ð13Þ dk As long as the kink density, 1=dk , is not too small, or kink separation dk is not too large, this means DE > 0. The density of kinks on a step is dependent on the formation energy of kinks as well as on tem-
perature [9]. At thermal equilibrium, low kink density demands low temperature. On the other hand, at too low temperature, equilibrium consideration becomes inappropriate and kink generation is by kinetics. Low temperature leads to low edge diffusion of adatoms and hence more kinks at steps. Therefore, in general, the condition for dk < 2W is fulfilled and so the attachment-limited growth demands DE > 0. A negative ES barrier, where DE < 0, does not satisfy this condition and so, no step bunching will result. Consequently, one may conclude that the step-asymmetry induced step bunching is in general originated from the existence of a positive incorporation barrier at the ascending steps rather than the negative ES barrier at descending steps. The above consideration is tested by kinetic Monte Carlo (kMC) simulations in two-dimensions. 3 Fig. 2 show resulted surfaces following 10 (a, c) and 100 (b, d) monolayers (MLs) deposition on a starting vicinal surface with equally spaced 16 terraces on a 256 256 lattice. Periodic boundaries are adopted. In the simulation, no specific assumption is made about the property of the steps (e.g., its permeability and kink density, etc.). Instead, we follow a more general scheme where diffusion is treated by breaking chemical bonds between neighboring atoms [12]. The simulation parameters are as follows: flux F ¼ 0:2 MLs/s, surface temperature T ¼ 1000 K, diffusion barrier Ed0 ¼ Ed þ nEn , where Ed ¼ 1:05 eV representing the energy for diffusion for an isolated adatom (i.e., when there is no neighbor on the same level), while En ¼ 0:5 eV represents the binding energy per neighboring atom. n is the total number of such neighbors. Fig. 2(a) and (b) show the cases when there is a negative ES barrier of 0.3 eV while Fig. 2(c) and (d) show results when there is a positive incorporation barrier of 0.3 eV. It is clear that negative ES does not generate step bunching even after 100 MLs film growth whereas in the 3
Due to the relative straightness of the resulted surface steps (see Fig. 2), one-dimensional model can be approximately applied to the two-dimensional case. Here, however, we further demonstrate that the general conclusion made based on onedimensional model is also applicable to a more relevant twodimensional case.
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layer during growth. In such cases, an exchange action between atoms of the surfactant and the deposit at steps is needed for film growth [13,14], which costs extra energy than for adatoms merely diffusing on top of the surfactant layer. The incorporation barrier is in essence similar to the ES barrier, except that the latter refers to the case where adatoms incorporate at descending steps while the former refer to the situation where adatoms are captured by ascending steps.
Acknowledgements
Fig. 2. KMC simulated surfaces following 10 MLs (a and c) and 100 MLs (b and d) film deposition on a vicinal substrate surface with initially equally spaced terrace width. (a) and (b) show cases where there is a negative ES barrier of 0.3 eV, while (c) and (d) show results when there is a positive incorporation barrier of 0.3 eV (see text for other simulation parameters).
case of a positive incorporation barrier, 10 MLs deposition already leads to step bunching. These observations validate the analysis made above. It is noted that the energy parameters chosen above are meant to guarantee the condition for high kink density and thus DE > 0 for attachmentlimited regime. Indeed, kink separation dk ¼ 1 þ 0:5 expðEn =2kT Þ 10 at 1000 K [9], so Eq. (13) suggests DE > 0:1 eV. Our choice of DE ¼ 0:3 eV thus conforms to such a condition. Secondly, recall that the analysis based on Eq. (2) assumes that the steps are not permeable. To see this, we calculate the flux ratio between that from terrace u to l and that from a terrace to the step (refer to Fig. 1). Note that for an adatom to diffuse across a step, it needs to overcome at least an overall energy barrier of Ed þ En þ DE [9], whereas for the atom to diffuse towards a step, the barrier is maximumly Ed þ DE (Fig. 1a), so the ratio between the two fluxes is approximately expðEn =kT Þ, which gives about 103 under the conditions used for simulation. Finally, on a practical note, an incorporation barrier can be introduced by, e.g., a surfactant
We are grateful to Josef Myslivecek, Pavel milauer and Dimitri D. Vvedensky for providing S us the original kMC codes. The work is fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project no. HKU 7121/00P).
References [1] Z.Y. Zhang, M.G. Lagally, Science 276 (1997) 377. [2] J. Villain, A. Pimpinelli, D. Wolf, Comments Cond. Mat. Phys. 16 (1992) 1. [3] R.L. Schwoebel, E.J. Shipsey, J. Appl. Phys. 37 (1966) 3682; R.L. Schwoebel, J. Appl. Phys. 40 (1969) 614; G. Ehrlich, F.G. Hudda, J. Chem. Phys. 44 (1966) 1039. [4] M.D. Johnson, C. Orme, A.W. Hunt, D. Graff, J. Sudijono, L.M. Sander, B.G. Orr, Phys. Rev. Lett. 72 (1994) 116. [5] J.A. Stroscio, D.T. Pierce, M.D. Stiles, A. Zangwill, L.M. Sander, Phys. Rev. Lett. 23 (1995) 4246. [6] G.S. Bales, A. Zangwill, Phys. Rev. B 41 (1990) 5500. [7] M. Krishnamurthy, M. Wassermeier, D.R.M. Williams, P.M. Petroff, Appl. Phys. Lett. 62 (1993) 1922. [8] M.S. Altman, private communication. [9] I.V. Markov, Crystal Growth for Beginners, Fundamentals of Nucleation, Crystal Growth and Epitaxy, World Scientific, Singapore, 1995. [10] S.V. Khare, T.L. Einstein, Phys. Rev. B 57 (1998) 4782. [11] D. Kandel, J.D. Weeks, Phys. Rev. Lett. 74 (1995) 3632. [12] D.D. Vvedensky, S. Clarke, Surf. Sci. 225 (1990) 373. [13] D. Kandel, E. Kaxiras, Phys. Rev. Lett. 75 (1995) 2742. [14] B.G. Liu, J. Wu, E.G. Wang, Z. Zhang, Phys. Rev. Lett. 83 (1999) 1195.
Surface Science Letters
What causes step bunching––negative Ehrlich–Schwoebel barrier versus positive incorporation barrier M.H. Xie *, S.Y. Leung, S.Y. Tong
1
Department of Physics, HKU-CAS Joint Laboratory on New Materials, The University of Hong Kong, Pokfulam Road, Hong Kong, Hong Kong Received 28 March 2002; accepted for publication 14 June 2002
Abstract Step asymmetry induced step bunching of a vicinal surface during deposition is investigated analytically and by kinetic Monte Carlo simulation. Specifically, effect of a negative Ehrlich–Schwoebel (ES) barrier at a downward step versus a positive incorporation barrier at an upward step edge is compared. It is found that a positive incorporation barrier can in general result in step bunching, whereas a negative ES barrier usually does not. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Surface structure, morphology, roughness, and topography; Step formation and bunching; Monte Carlo simulations
Thin film growth by epitaxial methods such as molecular beam epitaxy (MBE) are known to be largely determined by kinetics of the surface, such as adatom surface diffusion and island nucleation etc. [1,2]. It is recognized that the presence of a step edge barrier [3] (i.e., the Ehrlich–Schwoebel (ES) barrier), where adatoms experience an additional energy barrier when diffuse down a step than when they hop on a flat terrace, can have important consequences in the evolution of surface
* Corresponding author. Tel.: +852-28597945; fax: +85225599152. E-mail address: [email protected] (M.H. Xie). 1 Department of Applied Physics and Materials Sciences, City University of Hong Kong, Kowloon Tang, Hong Kong.
morphology. For example, on a nominally flat surface, a positive ES barrier leads to mound formation [4,5], whereas on a vicinal surface, it may cause Bales–Zangwill instability [6]. If there exists a negative ES barrier, a step-bunched morphology arguably results for growth of a vicinal surface. Qualitatively, this is based on considerations of step growth (advance) rate. A step grows by capturing adatoms from its neighboring (upper and lower) terraces. Wider terraces mean more supply of adatoms to the step in the step flow growth mode and so the step advances faster. When there exists a step asymmetry due to, e.g., the existence of a negative ES barrier such that a step captures adatoms more efficiently from its upper terrace than from the lower terrace, the advance rate of the step becomes dominated by the
0039-6028/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 2 ) 0 1 9 7 6 - 3
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upper terrace. If this terrace is wider to begin with, the growth rate of the step in front (i.e., the descending step) will be higher, leaving the terrace behind (i.e., its upper terrace) wider. Similarly, if the upper terrace is initially narrower than others, its descending step grows slowly, making the upper terrace even narrower. Continued growth thus ultimately leads to step bunching. Such an argument had been used to suggest a possible negative ES barrier on GaAs(1 1 0) surface, where macrostep formation was observed experimently [7]. It is noted that a similar upper-terrace dominated step asymmetry can also arise from the presence of a positive incorporation barrier––the additional energy barrier which impedes adatoms diffusing towards an ascending step (Fig. 1a). It is thus not obvious which of the above two cases (negative ES barrier versus positive incorporation barrier) is responsible for an experimently observed step bunching. It is the purpose of this paper to show that in general the step asymmetry induced step bunching is due to a positive incorporation barrier rather than negative ES barrier, although the two give rise to the same asymmetry in adatom capture rate from upper versus lower terraces.
The analysis is based on one-dimensional diffusion equation at steady state: D
d2 n þF ¼0 dx2
ð1Þ
where D ¼ m expðEd =kT Þ is the diffusion coefficient of adatoms, nðxÞ is adatom concentration at position x, and F is the deposition flux. m represents the attempt frequency for hopping, while Ed is the diffusion (hopping) energy barrier. k is the Boltzmann constant and T is the growth temperature. The lattice constant of the crystalline surface or the distance for a single hop is chosen as the length unit. The above equation is valid under the condition of complete condensation, i.e., when there is no desorption, which is a good approximation for most MBE growth conditions. The solution of Eq. (1) under the boundary condition at x ¼ W (representing the advance rate of a step) 2 D
dn ¼ b ðn n0 Þ dx
ð2Þ
reads [9] nðxÞ ¼
FW 2 1 þ 1 Fx2 ðn nþ Þx þ ðn þ n Þ þ 2 2W 2D 2D ð3Þ
where, n ¼
1 2FW 2 2W b 2þ H D D þ
2Wn0 þ ðb þ b þ bþ b Þ D
ð4Þ
represent adatom concentrations at ascending (þ) and descending () steps, with their kinetic coefficients bþ and b , respectively (see Fig. 1a). n0 is
Fig. 1. (a) Schematic diagram showing a terrace and steps and the associated energy diagram. It illustrates the negative ES barrier and positive incorporation energy barrier at descending and ascending steps, respectively. Note also the definition of parameters. (b) Schematic diagram of a vicinal surface, illustrating the growth of a step due to contributions from its neighboring terraces.
2
By writing such a boundary condition, it is implied that the steps are not permeable, i.e., the mass transport are local to a single terrace bounded by two steps. If mass transport are nonlocal, the concentration profile on a given terrace becomes affected by its neighboring terraces [8] and in general, it is more symmetrical than the non-local case.
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the equilibrium adatom concentration at the step. 2W is the terrace width, and 2W bþ 2W b H ¼ 1þ 1þ 1 ð5Þ D D The kinetic coefficient b of a step is related to kink density on step, 1=dk (dk represents average kink separation), as well as the energy barrier U for adatom incorporation [9]: b ¼
m exp U =kT dk
ð6Þ
where the barrier U is the sum of the surface diffusion barrier Ed and an extra term DE, i.e., U ¼ Ed þ DE, where DE is the extra energy required for adatoms diffusing toward a step than that when they hop on flat terraces (an example of the DE is the ES barrier described earlier). In general, DE can be both positive and negative. The superscripts ‘þ’ and ‘’ in Eq. (6) distinguish the cases between adatom capture by an ascending step and a descending step. By setting dn=dx ¼ 0, one easily finds the location xm at which the adatom concentration has the maximum: xm ¼
2W 2 bþ b D H
ð7Þ
from which, we find that if bþ ¼ b , xm ¼ 0, i.e., when ascending and descending steps are symmetrical in terms of their efficiencies in capturing adatoms from terraces, the concentration profile will be peaked at the center of the terrace. On the other hand, if bþ 6¼ b , distinction needs to be made: under the diffusion-limited regime, or specifically when 2W b =D 1, ðbþ b Þ H , we still have xm ! 0. This is understood by the fact that in this regime, growth is limited by the rate in which adatoms diffuse toward a step. So, even though the step kinetic coefficients are different between ascending and descending steps, both are sufficiently high compared to the diffusion rate, so both steps can be regarded as perfect traps. Then adatom concentration profile on terrace is not affected by the asymmetry in b and peaks at the terrace center. In the other extreme, i.e., when 2W b =D 1, the growth becomes attachmentrate limited (diffusion is rapid compared to the
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capturing rate of steps). Then from Eqs. (5) and (7) and ignoring high-order terms in H, i.e., H ð2W =DÞðbþ þ b Þ 1 b =bþ xm W ð8Þ 1 þ b =bþ Therefore, the position showing the maximum adatom concentration depends on the ratio of b =bþ . If b > bþ , i.e., when the descending step is more efficient in capturing adatoms, xm < 0, i.e., adatom concentration peaks nearer to the ascending step and few atoms are found close to the descending step. A high adatom concentration means a high nucleation rate, so when growth is carried out in the nucleation regime, more islands will be found near the ascending step. Finally, it can be expected that a cross-over from diffusionlimited to kinetic-limited regime will occur for 2W b =D 1, i.e., when diffusion rate and step attachment rate become comparable [10]. Let’s now determine the growth rate of a step bounded by an upper terrace with width 2Wu and a lower terrace having width 2Wl (A in Fig. 1b). The step advances by capturing atoms from these two terraces with respective kinetic coefficients b (from upper terrace) and bþ (from lower terrace). The overall growth rate of the step is then v ¼ vl þ vu
ð9Þ
where, according to Eq. (2), dn vlðuÞ ¼ b ðn n0 Þ ¼ D dx þ F b b 2 ¼ FWlðuÞ 2WlðuÞ D H
ð10Þ
Here, vlðuÞ represents the rate due to adatom capture from the lower(upper) terrace. The above formula can be simplified depending on the relative values of bþ and b . If bþ ¼ b or if bþ 6¼ b but 2W b =D 1 (diffusion-limited regime), we have vlðuÞ ffi FWlðuÞ
ð11Þ
If, on the other hand, 2W b =D 1 (attachmentrate limited), then 2b vlðuÞ ¼ F ð12Þ WlðuÞ bþ þ b
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This latter equation is usually written in the form of v ¼ k L [11], where L ¼ 2W is the terrace width, and k are characteristic constants (proportionality) reflecting the effectiveness of steps in capturing adatoms from neighboring terraces. Apparently by inspecting Eq. (12), one finds k ¼ F b =ðbþ þ b Þ. From the above equations, it is seen that only in attachment limited regime does the growth rate of a step become dependent on step kinetic coefficients b , and if they are different between ascending and descending steps or specifically, if b > bþ , step bunching can happen. This is because, according to Eq. (12), we have vu > vl given the same terrace width, Wl ¼ Wu . This means that the (total) growth rate of a step becomes dominated by adatom capture from its upper terrace (i.e., the contribution of upper terrace is higher than from the lower terrace). If, at a particular moment, Wu > Wl due to, e.g., growth fluctuation (noise), then the step in front of Wu (A in Fig. 1b) grows much faster than the front step of Wl (B in Fig. 1b). Consequently, terrace Wu is further widened but Wl is narrowed, leading to step bunching. This contrasts to growth in diffusion limited regime, where according to Eq. (11), vl is greater than vu only if Wl > Wu . But since the upper and lower terraces equally contribute to step growth rate, this makes the total rate v unchanged (refer to Eq. (9)). As pointed out earlier, both negative ES barrier and positive incorporation barrier can lead to the inequality b > bþ . However, in order for Eq. (12) to be applicable (the necessary condition for step bunching), the growth has to be in the attachmentlimited regime, i.e., when 2W b =D 1. While D is related to diffusion barrier, Ed , according to
expðEd =kT Þ, b depends on U ¼ Ed þ DE as well as kink density, 1=dk (see Eq. (6)). So, the condition of attachment-limited growth translates into U ¼ Ed þ DE > Ed þ kT lnð2W =dk Þ, or 2W DE > kT ln ð13Þ dk As long as the kink density, 1=dk , is not too small, or kink separation dk is not too large, this means DE > 0. The density of kinks on a step is dependent on the formation energy of kinks as well as on tem-
perature [9]. At thermal equilibrium, low kink density demands low temperature. On the other hand, at too low temperature, equilibrium consideration becomes inappropriate and kink generation is by kinetics. Low temperature leads to low edge diffusion of adatoms and hence more kinks at steps. Therefore, in general, the condition for dk < 2W is fulfilled and so the attachment-limited growth demands DE > 0. A negative ES barrier, where DE < 0, does not satisfy this condition and so, no step bunching will result. Consequently, one may conclude that the step-asymmetry induced step bunching is in general originated from the existence of a positive incorporation barrier at the ascending steps rather than the negative ES barrier at descending steps. The above consideration is tested by kinetic Monte Carlo (kMC) simulations in two-dimensions. 3 Fig. 2 show resulted surfaces following 10 (a, c) and 100 (b, d) monolayers (MLs) deposition on a starting vicinal surface with equally spaced 16 terraces on a 256 256 lattice. Periodic boundaries are adopted. In the simulation, no specific assumption is made about the property of the steps (e.g., its permeability and kink density, etc.). Instead, we follow a more general scheme where diffusion is treated by breaking chemical bonds between neighboring atoms [12]. The simulation parameters are as follows: flux F ¼ 0:2 MLs/s, surface temperature T ¼ 1000 K, diffusion barrier Ed0 ¼ Ed þ nEn , where Ed ¼ 1:05 eV representing the energy for diffusion for an isolated adatom (i.e., when there is no neighbor on the same level), while En ¼ 0:5 eV represents the binding energy per neighboring atom. n is the total number of such neighbors. Fig. 2(a) and (b) show the cases when there is a negative ES barrier of 0.3 eV while Fig. 2(c) and (d) show results when there is a positive incorporation barrier of 0.3 eV. It is clear that negative ES does not generate step bunching even after 100 MLs film growth whereas in the 3
Due to the relative straightness of the resulted surface steps (see Fig. 2), one-dimensional model can be approximately applied to the two-dimensional case. Here, however, we further demonstrate that the general conclusion made based on onedimensional model is also applicable to a more relevant twodimensional case.
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layer during growth. In such cases, an exchange action between atoms of the surfactant and the deposit at steps is needed for film growth [13,14], which costs extra energy than for adatoms merely diffusing on top of the surfactant layer. The incorporation barrier is in essence similar to the ES barrier, except that the latter refers to the case where adatoms incorporate at descending steps while the former refer to the situation where adatoms are captured by ascending steps.
Acknowledgements
Fig. 2. KMC simulated surfaces following 10 MLs (a and c) and 100 MLs (b and d) film deposition on a vicinal substrate surface with initially equally spaced terrace width. (a) and (b) show cases where there is a negative ES barrier of 0.3 eV, while (c) and (d) show results when there is a positive incorporation barrier of 0.3 eV (see text for other simulation parameters).
case of a positive incorporation barrier, 10 MLs deposition already leads to step bunching. These observations validate the analysis made above. It is noted that the energy parameters chosen above are meant to guarantee the condition for high kink density and thus DE > 0 for attachmentlimited regime. Indeed, kink separation dk ¼ 1 þ 0:5 expðEn =2kT Þ 10 at 1000 K [9], so Eq. (13) suggests DE > 0:1 eV. Our choice of DE ¼ 0:3 eV thus conforms to such a condition. Secondly, recall that the analysis based on Eq. (2) assumes that the steps are not permeable. To see this, we calculate the flux ratio between that from terrace u to l and that from a terrace to the step (refer to Fig. 1). Note that for an adatom to diffuse across a step, it needs to overcome at least an overall energy barrier of Ed þ En þ DE [9], whereas for the atom to diffuse towards a step, the barrier is maximumly Ed þ DE (Fig. 1a), so the ratio between the two fluxes is approximately expðEn =kT Þ, which gives about 103 under the conditions used for simulation. Finally, on a practical note, an incorporation barrier can be introduced by, e.g., a surfactant
We are grateful to Josef Myslivecek, Pavel milauer and Dimitri D. Vvedensky for providing S us the original kMC codes. The work is fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project no. HKU 7121/00P).
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