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Simulation of the kinetics of surface roughening
surface science ELSEVIER
Surface Science 388 (1997) L 1126-L 1129
Surface Science Letters
Simulation of the kinetics of surface roughening V.P. Z h d a n o v a,b,, a Department of Applied Physies, Chalmers University of Technology, S-412 96 G6teborg, Sweden b Boreskov Institute of Catalysis, Russian Academy of Sciences, Novosibirsk 630090, Russia Received 3 February 1997; accepted for publication 23 June 1997
Abstract
Analysis using the Monte Carlo (MC) technique for the restricted-solid-on-solid model, with surface diffusion occurring via jumps to nearest-neighbour and next-nearest-neighbour sites, shows that the kinetics of surface roughening with conserved density are very slow not only just above the roughening temperature TR, but also at rather high temperatures. In particular, the deviations of the column heights from the average value are usually _<2 for T= 1.4TR at t = l06 MC steps per site. For these times, corresponding to the onset of roughening, the time dependence of the mean-square height difference is demonstrated to be close to logarithmic, w2odn t. © 1997 Elsevier Science B.V.
Keywords. Computer simulations; Surface roughening; Surface thermodynamics (including phase transitions)
Roughening of perfect single-crystal surfaces is a very soft (infinite-order) phase transition. A rigorous proof of its existence was given by van Beijeren [1] by mapping the body-centred solidon-solid (BCSOS) model on the symmetric sixvertex model. Since then, this phenomenon has been simulated by employing various models and techniques (see the reviews in Refs. [2,3]), and now its basic features at equilibrium are well understood, at least in the framework of the lattice approximation (roughening of real surfaces is often complicated by enhanced anharmonic vibrations of substrate atoms in the top layer (see the discussion in Ref. [4]). In particular, surface roughening is known to be characterized by the behaviour of the correlation function relating the mean-square height difference between two points separated by * Fax: (+7) 3832 355756 e-mail: [email protected]
or
(+7)
3832
357687;
0039-6028/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PH S0039-6028 ( 9 7 ) 0 0 5 1 7 - 7
distance r:
G(r)= ([h(r) - h(0)] 2).
(1)
Below the roughening temperature, G(r) reaches a finite asymptotic value at r-~oo. While at T> TR, this correlation function diverges at r-* oo as
G(r)=A(T)+B(T) In r.
(2)
Experimental studies based on this equation indicate that surface roughening usually occurs at temperatures just below the melting temperature (e.g. for fcc metals the ratio TR/TMis about 0.7-0.8 for the (110) face [5,6] (the (111) and (100) faces do not seem to roughen up at all up to TM). The kinetics of surface roughening have been studied theoretically by Chui and Weeks [7] and Kotrla and Levi [8]. The former phenomenological analysis [7] (see also a review in Ref. [9]) was based on the renormalization-group method. The latter Monte Carlo (MC) results [ 8] were obtained
V.P. Zhdanov / Smface Science 388 (1997) L1126- L1129
for the BCSOS model by employing the Glauber dynamics. The main goal of both groups [7, 8] was to explore crystal growth complicated by a roughening phase transition. For this reason, the dynamics employed in Refs. [7, 8] corresponded to evaporation and condensation (no surface diffusion), i.e. the density was not conserved. In reality, surface diffusion is usually very rapid at T > TR. Thus, it is of interest to simulate the diffusionrelated kinetics of surface roughening with conserved density. In this Letter, we present the first results of such simulations. Our analysis is based on the restricted-solid-onsolid model (RSOSM) introduced by Rommelse and den Nijs [10]. This model is more realistic compared with the conventional solid-on-solid model and, simultaneously, very convenient for MC simulations. In the framework of the RSOSM model, the crystal is represented as a two-dimensional array of columns of varying integer heights hi. For the nearest-neighbour columns, the heights are allowed to differ by at most one, hi-hi=O, _+ 1. The substrate energy is accordingly given by
HM = (K/2) ~
6(Ih,-hjl- l)
(3)
i,j
where K > 0 is the interaction constant (referring to the nearest-neighbour attractive interaction between metal atoms, eMM<0, one has K = leMMI/2), (ij) denotes nearest-neighbour sites on a square lattice, and f(x) = 1 when x = 0 and vanishes otherwise (the factor ½ is introduced in order to avoid double counting). In our MC simulations, we assume that initially (at t = 0 ) the surface is perfect. The algorithm for describing the system at t > 0 consists of attempts of diffusion jumps. A metal atom on the surface is chosen at random. One of the nearest-neighbour or next-nearest-neighbour columns is chosen at random. If a jump to this column violates the RSOSM constraints, the trial ends. Otherwise, the jump is realized with the probability given by the standard Metropolis rule (W= 1 for AE<0, and W=exp(-AE/T) for AE>O, where AE is the energy difference between the initial and final states). Employing the algorithm above, we have calcu-
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0.8 07 ~0.6 0.5 0.4 0.3
2
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10 r
15
20
Fig. 1. Correlation function G(r) for roughening of the 200×200 lattice at T=I.44TR and different times (up to t = 100 MCSs).
lated the kinetics of surface roughening of the (200 × 200) lattice (with periodic boundary conditions) for times up to 10 6 MC steps (MCSs: one MCS corresponds by definition to L x L attempts at diffusion, where L is the lattice size). To get TR (this was our first goal ), we have used the procedure originally proposed by Gilmer and coworkers [11 ]. It is based on calculation of G(r) ( Eq. ( 1 )) at different temperatures and approximation of G(r) by Eq. (2). At T= TR, the value of B in Eq.(2) is known [3,6] to be universal B(TR)=2/rc 2. The latter makes it possible to find TR. To obtain TR, we need G(r) at least at 5 < r < 15. For these distances, the time dependence of G(r) was found to be relatively strong at t < 105 MCSs. For t=105 the results are, however, almost the same as for 10 6 MCSs (provided that T is equal or slightly higher than TR), i.e. the system is close to equilibrium (at r < 1 5 ) in the latter case. For this reason, the data obtained at 10 6 MCSs can be used to calculate TR. On the basis of our simulations, we have found that TR=(0.90_+ 0.05)IeMMI (kB = 1 ). This value is in good agreement with TR=0.SIeMMI obtained by den Nijs and coworkers [10] on the basis of the transfer-matrix technique. Our main goal was to clarify features of the kinetics of surface roughening. The roughening process was found to be very slow not only just above TR but also at rather high temperatures. As
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,#:,t;,t #+%W-~$~++~+,~tl 8~1~.+o+ ++++÷+,.,.+++..++'~+~l~tttt r+3+%-~-+~4+o ++~p÷~Ul~.+++++~i ,~+,o4+o+.~+.++++++++.+÷0+++++.~ • . +o@4 ++4÷ +o4 +O@C)++ +;.+.+.+.++~ +~++~.+tt++.%+$t~.+|++, ,,,++,+4 .... ++ .... oo..~++o+o~o4~, • ~ O0 ~÷++++4~I++O0÷O0++++00÷÷+++++++O÷ +O÷÷3©,+,,+G++÷ ÷+++ 4' • O÷ ÷ ÷oo÷ 0+÷÷÷@+~+÷÷@÷÷÷÷ ~t~-0.+o.°'3="~o++4~++o++41++÷ ' HIO++400+++o++++++ooo+÷÷+~÷++c +÷c n ..~.l-l-l-o.i-++ + +.i-+ o o ÷ + ÷ o c,++ + L)++( k~++ll÷+÷oo+÷:>++o+oo++++÷eo4o++÷÷.j ~+++coo,u,u++ +~,~o.+ +÷+~ + +++:)++0 o+ ++++++÷++++++ {~t~ .~ ~t ++-+ _i~'~%+o+.~++++ti++"+'+'+'+'+'+'+'~+ . . . . . 4 ~ $ I I I I . ÷ + , , + + 4 + +++ + . , + .o++~.:.+'÷ +oo '3÷+++@~÷+÷oco,~Goo÷÷c++++÷o ' o o oo÷÷÷÷÷++o÷o ÷÷ + + + + + ÷ o 0"-:~,7.,200o o + o o~ @ l ÷
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c ++o
.
+ , - .c~.+~,~,,+
t c
+,:,G++ ~++++++++++
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+ .+.+++'~ . + t + t otS++~+o+++i+i~+..¢~.~+S T o o o o <, ;l+;0 c c c o oo,+" c ÷ . . . . . . . . . ...........
l +' '[~
+ l + + ~j++l + O C 0 CS:: < { () 0.3 O O 0 0,+ 0 + { O O C. : S : : ; ; } + + ~
Fig. 2. Top view of a 40 x 60 fragment of the 200 x 200 lattice at T = 1.44TR and t = 10 (a), 102 (b), 103 (C), 10 4 (d), 105 (e), and 106 MESs (f). Initially (at t =0), the surface is flat (h =0). Columns with h = 0 are shown by the plus signs. Filled circles and diamonds indicate columns with h = - 1 and - 2 . Open circles and diamonds correspond to h = 1 and 2 respectively.
V.P. Zhdanov / Sur/ace Science 388 (1997) L l I 26-L1129
( B C S O S model with non-conserved density) indicate that w 2 ~ : l n L at t ~ . For L = 1 2 8 and T = 1.4TR, this asymptotic regime, with w 2 ~ 2 , is reached already at t = 3 × 103 MCSs. Finally, it is reasonable to note that the logarithmic law (Eq. (5)) was originally predicted by Chui and Weeks [7] when analysing crystal growth on the basis o f a phenomenological equation with non-conserved density. Our simulations seem to indicate that this law holds for conserved density as well.
0.7
ii J
0.6
I
i
fl 0.5
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i
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t
0.2 1 I ,i i
0.1
O0
a 1
n 2
,
i 3
o 4
,
i 5
,
i 6
LogJt)
Fig. 3. Mean-square height difference as a function of logl0(t) (t is in MCSs) for T= 1.44TR.
Acknowledgements The a u t h o r thanks B. K a s e m o for useful discussions. This work was supported by T F R (contract N o 281-95-782).
an example, we present (Figs. 1-3) the results for T=I.3IeMMI (i.e. at T = 1.44TR). F r o m the time dependence of G(r) (Fig. 1 ), one can conclude that at t = 10 2, l 0 4 and 10 6 M C S s , the correlations in the arrangement o f atoms take place respectively at r < 4 , 16 and 20. Typical arrangements o f particles on times up to t = 106 MCSs are shown in Fig. 2. Surface roughening is seen to be rather weak. Even for t = 10 6 M C S s , the deviations o f h from the average value ( ( h ) = 0 ) are usually _<2. In fact, this corresponds to the onset of roughening. The asymptotic time dependence o f the meansquare height difference w 2 = (h 2) _ (/7) 2
(4)
seems (Fig. 3) to be logarithmic w Z = C + D ln t.
(5)
This dependence is expected to hold if the typical length characterizing correlations in the arrangement o f particles is lower than L (in our case, L is much larger than the correlation length). For t = 106 MCSs, the simulations yield w2=0.64 . If the density is not conserved, the roughening occurs much faster. In particular, the M C simulations [8]
References [1] H. van Beijeren, Phys. Rev. Lett. 38 (1977) 993. [2] M. den Nijs, in: D.A. King, D.P. Woodruff (Eds.), Phase Transitions and Adsorbate Restructuring at Metal Surfaces, Vol. 7, The Chemical Physics of Solid Surfaces, Elsevier, Amsterdam, 1994, p. 115. [3] A.C. Levi, in: D.A. King, D.P. Woodruff (Eds.), Phase Transitions and Adsorbate Restructuring at Metal Surfaces, Vol. 7, The Chemical Physics of Solid Surfaces, Elsevier, Amsterdam, 1994, p. 341. [4] G. Bracco, L. Bruschi, L. Pedemonte, R. Tatarek, Surf. Sci. 352 (1996) 964. [5] K. Kern, in: D.A. King, D.P. Woodruff (Eds.), Phase Transitions and Adsorbate Restructuring at Metal Surfaces, Vol. 7, The Chemical Physics of Solid Surfaces, Elsevier, Amsterdam, 1994, p. 292. [6] J. Lapujoulade, Surf. Sci. Rep. 20 (1994) 191. [7] S.T. Chui, J.D. Weeks, Phys. Rev. Lett. 40 (1978) 733. [8] M. Kotrla, A.C. Levi, J. Phys. A 25 (1992) 3121. [9] A.-E. Barabasi, H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, 1995, Chapter 18. [10] K. Rommelse, M. den Nijs, Phys. Rev. Lett. 59 (1987) 2578. M. den Nijs, Phys. Rev. Lett. 64 (1990) 435. [11] W.J. Shugard, J.D. Weeks, G.H. Gilmer, Phys. Rev. Lett. 41 (1978) 1399.
Surface Science 388 (1997) L 1126-L 1129
Surface Science Letters
Simulation of the kinetics of surface roughening V.P. Z h d a n o v a,b,, a Department of Applied Physies, Chalmers University of Technology, S-412 96 G6teborg, Sweden b Boreskov Institute of Catalysis, Russian Academy of Sciences, Novosibirsk 630090, Russia Received 3 February 1997; accepted for publication 23 June 1997
Abstract
Analysis using the Monte Carlo (MC) technique for the restricted-solid-on-solid model, with surface diffusion occurring via jumps to nearest-neighbour and next-nearest-neighbour sites, shows that the kinetics of surface roughening with conserved density are very slow not only just above the roughening temperature TR, but also at rather high temperatures. In particular, the deviations of the column heights from the average value are usually _<2 for T= 1.4TR at t = l06 MC steps per site. For these times, corresponding to the onset of roughening, the time dependence of the mean-square height difference is demonstrated to be close to logarithmic, w2odn t. © 1997 Elsevier Science B.V.
Keywords. Computer simulations; Surface roughening; Surface thermodynamics (including phase transitions)
Roughening of perfect single-crystal surfaces is a very soft (infinite-order) phase transition. A rigorous proof of its existence was given by van Beijeren [1] by mapping the body-centred solidon-solid (BCSOS) model on the symmetric sixvertex model. Since then, this phenomenon has been simulated by employing various models and techniques (see the reviews in Refs. [2,3]), and now its basic features at equilibrium are well understood, at least in the framework of the lattice approximation (roughening of real surfaces is often complicated by enhanced anharmonic vibrations of substrate atoms in the top layer (see the discussion in Ref. [4]). In particular, surface roughening is known to be characterized by the behaviour of the correlation function relating the mean-square height difference between two points separated by * Fax: (+7) 3832 355756 e-mail: [email protected]
or
(+7)
3832
357687;
0039-6028/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PH S0039-6028 ( 9 7 ) 0 0 5 1 7 - 7
distance r:
G(r)= ([h(r) - h(0)] 2).
(1)
Below the roughening temperature, G(r) reaches a finite asymptotic value at r-~oo. While at T> TR, this correlation function diverges at r-* oo as
G(r)=A(T)+B(T) In r.
(2)
Experimental studies based on this equation indicate that surface roughening usually occurs at temperatures just below the melting temperature (e.g. for fcc metals the ratio TR/TMis about 0.7-0.8 for the (110) face [5,6] (the (111) and (100) faces do not seem to roughen up at all up to TM). The kinetics of surface roughening have been studied theoretically by Chui and Weeks [7] and Kotrla and Levi [8]. The former phenomenological analysis [7] (see also a review in Ref. [9]) was based on the renormalization-group method. The latter Monte Carlo (MC) results [ 8] were obtained
V.P. Zhdanov / Smface Science 388 (1997) L1126- L1129
for the BCSOS model by employing the Glauber dynamics. The main goal of both groups [7, 8] was to explore crystal growth complicated by a roughening phase transition. For this reason, the dynamics employed in Refs. [7, 8] corresponded to evaporation and condensation (no surface diffusion), i.e. the density was not conserved. In reality, surface diffusion is usually very rapid at T > TR. Thus, it is of interest to simulate the diffusionrelated kinetics of surface roughening with conserved density. In this Letter, we present the first results of such simulations. Our analysis is based on the restricted-solid-onsolid model (RSOSM) introduced by Rommelse and den Nijs [10]. This model is more realistic compared with the conventional solid-on-solid model and, simultaneously, very convenient for MC simulations. In the framework of the RSOSM model, the crystal is represented as a two-dimensional array of columns of varying integer heights hi. For the nearest-neighbour columns, the heights are allowed to differ by at most one, hi-hi=O, _+ 1. The substrate energy is accordingly given by
HM = (K/2) ~
6(Ih,-hjl- l)
(3)
i,j
where K > 0 is the interaction constant (referring to the nearest-neighbour attractive interaction between metal atoms, eMM<0, one has K = leMMI/2), (ij) denotes nearest-neighbour sites on a square lattice, and f(x) = 1 when x = 0 and vanishes otherwise (the factor ½ is introduced in order to avoid double counting). In our MC simulations, we assume that initially (at t = 0 ) the surface is perfect. The algorithm for describing the system at t > 0 consists of attempts of diffusion jumps. A metal atom on the surface is chosen at random. One of the nearest-neighbour or next-nearest-neighbour columns is chosen at random. If a jump to this column violates the RSOSM constraints, the trial ends. Otherwise, the jump is realized with the probability given by the standard Metropolis rule (W= 1 for AE<0, and W=exp(-AE/T) for AE>O, where AE is the energy difference between the initial and final states). Employing the algorithm above, we have calcu-
1.3 12
I t=10°MCS i
1.1 1.0
4
0.9 3
0.8 07 ~0.6 0.5 0.4 0.3
2
//
1
0.2 0.1 0.0
n=0 r
5
10 r
15
20
Fig. 1. Correlation function G(r) for roughening of the 200×200 lattice at T=I.44TR and different times (up to t = 100 MCSs).
lated the kinetics of surface roughening of the (200 × 200) lattice (with periodic boundary conditions) for times up to 10 6 MC steps (MCSs: one MCS corresponds by definition to L x L attempts at diffusion, where L is the lattice size). To get TR (this was our first goal ), we have used the procedure originally proposed by Gilmer and coworkers [11 ]. It is based on calculation of G(r) ( Eq. ( 1 )) at different temperatures and approximation of G(r) by Eq. (2). At T= TR, the value of B in Eq.(2) is known [3,6] to be universal B(TR)=2/rc 2. The latter makes it possible to find TR. To obtain TR, we need G(r) at least at 5 < r < 15. For these distances, the time dependence of G(r) was found to be relatively strong at t < 105 MCSs. For t=105 the results are, however, almost the same as for 10 6 MCSs (provided that T is equal or slightly higher than TR), i.e. the system is close to equilibrium (at r < 1 5 ) in the latter case. For this reason, the data obtained at 10 6 MCSs can be used to calculate TR. On the basis of our simulations, we have found that TR=(0.90_+ 0.05)IeMMI (kB = 1 ). This value is in good agreement with TR=0.SIeMMI obtained by den Nijs and coworkers [10] on the basis of the transfer-matrix technique. Our main goal was to clarify features of the kinetics of surface roughening. The roughening process was found to be very slow not only just above TR but also at rather high temperatures. As
K P. Zhdanov / Sur[ace Science 388 (1997) L1126-L1129 (a) t=lO MCS
(b) t=lOaMCS
+++++++I:I++++~++~++O++++++~+++++++
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~OO++~++~o++++++++I+o+~,++++++@~+++o+~+~4+++++~+++~II8~++ ~;~++~+++++++0o+++ .... ...... '.+99 44+++ ++++e++++++e++ ++ +0++o++ooc+ +4++++++~ ~'i. . . . . . . . + + 0 + 0 " + + l + + ~ . . . . . . . • . . . . . . . . + ~ : , ~ + + $ ~ . . . . . . + ...... ~O++++I+++
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Fig. 2. Top view of a 40 x 60 fragment of the 200 x 200 lattice at T = 1.44TR and t = 10 (a), 102 (b), 103 (C), 10 4 (d), 105 (e), and 106 MESs (f). Initially (at t =0), the surface is flat (h =0). Columns with h = 0 are shown by the plus signs. Filled circles and diamonds indicate columns with h = - 1 and - 2 . Open circles and diamonds correspond to h = 1 and 2 respectively.
V.P. Zhdanov / Sur/ace Science 388 (1997) L l I 26-L1129
( B C S O S model with non-conserved density) indicate that w 2 ~ : l n L at t ~ . For L = 1 2 8 and T = 1.4TR, this asymptotic regime, with w 2 ~ 2 , is reached already at t = 3 × 103 MCSs. Finally, it is reasonable to note that the logarithmic law (Eq. (5)) was originally predicted by Chui and Weeks [7] when analysing crystal growth on the basis o f a phenomenological equation with non-conserved density. Our simulations seem to indicate that this law holds for conserved density as well.
0.7
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Fig. 3. Mean-square height difference as a function of logl0(t) (t is in MCSs) for T= 1.44TR.
Acknowledgements The a u t h o r thanks B. K a s e m o for useful discussions. This work was supported by T F R (contract N o 281-95-782).
an example, we present (Figs. 1-3) the results for T=I.3IeMMI (i.e. at T = 1.44TR). F r o m the time dependence of G(r) (Fig. 1 ), one can conclude that at t = 10 2, l 0 4 and 10 6 M C S s , the correlations in the arrangement o f atoms take place respectively at r < 4 , 16 and 20. Typical arrangements o f particles on times up to t = 106 MCSs are shown in Fig. 2. Surface roughening is seen to be rather weak. Even for t = 10 6 M C S s , the deviations o f h from the average value ( ( h ) = 0 ) are usually _<2. In fact, this corresponds to the onset of roughening. The asymptotic time dependence o f the meansquare height difference w 2 = (h 2) _ (/7) 2
(4)
seems (Fig. 3) to be logarithmic w Z = C + D ln t.
(5)
This dependence is expected to hold if the typical length characterizing correlations in the arrangement o f particles is lower than L (in our case, L is much larger than the correlation length). For t = 106 MCSs, the simulations yield w2=0.64 . If the density is not conserved, the roughening occurs much faster. In particular, the M C simulations [8]
References [1] H. van Beijeren, Phys. Rev. Lett. 38 (1977) 993. [2] M. den Nijs, in: D.A. King, D.P. Woodruff (Eds.), Phase Transitions and Adsorbate Restructuring at Metal Surfaces, Vol. 7, The Chemical Physics of Solid Surfaces, Elsevier, Amsterdam, 1994, p. 115. [3] A.C. Levi, in: D.A. King, D.P. Woodruff (Eds.), Phase Transitions and Adsorbate Restructuring at Metal Surfaces, Vol. 7, The Chemical Physics of Solid Surfaces, Elsevier, Amsterdam, 1994, p. 341. [4] G. Bracco, L. Bruschi, L. Pedemonte, R. Tatarek, Surf. Sci. 352 (1996) 964. [5] K. Kern, in: D.A. King, D.P. Woodruff (Eds.), Phase Transitions and Adsorbate Restructuring at Metal Surfaces, Vol. 7, The Chemical Physics of Solid Surfaces, Elsevier, Amsterdam, 1994, p. 292. [6] J. Lapujoulade, Surf. Sci. Rep. 20 (1994) 191. [7] S.T. Chui, J.D. Weeks, Phys. Rev. Lett. 40 (1978) 733. [8] M. Kotrla, A.C. Levi, J. Phys. A 25 (1992) 3121. [9] A.-E. Barabasi, H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, 1995, Chapter 18. [10] K. Rommelse, M. den Nijs, Phys. Rev. Lett. 59 (1987) 2578. M. den Nijs, Phys. Rev. Lett. 64 (1990) 435. [11] W.J. Shugard, J.D. Weeks, G.H. Gilmer, Phys. Rev. Lett. 41 (1978) 1399.