Growth law of step bunches induced by the Ehrlich–Schwoebel effect in growth

Surface Science 493 (2001) 494±498

www.elsevier.com/locate/susc

Growth law of step bunches induced by the Ehrlich±Schwoebel e€ect in growth Masahide Sato a,b,*, Makio Uwaha a a b

Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan Computer Center, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan Received 21 September 2000; accepted for publication 5 February 2001

Abstract We study growth law of step bunches formed in growth due to the Ehrlich±Schwoebel e€ect by using a onedimensional step ¯ow model. We neglect evaporation of adatoms and assume that the repulsive interaction potential between steps is given by Al m , where A is the strength of the repulsion and l is the step distance. When the adatoms attach to the step more easily from the upper terrace than from the lower terrace, an equidistant train of steps is unstable to bunching instability in growth. The terrace width between bunches increases as the bunches grow via collision and coalescence. Time dependence of the terrace width is given by L  tb with b  1=2. The value of b is independent of the power of the step interaction potential m, which a€ects L dependence of the step distance in a bunch. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Step formation and bunching; Stepped single crystal surfaces; Surface di€usion; Surface structure, morphology, roughness, and topography; Vicinal single crystal surfaces

1. Introduction In vicinal faces of a crystal, asymmetry of the surface di€usion ®eld causes bunching instabilities. In the observed step bunching in sublimated Si(1 1 1) vicinal faces [1±4] heated with direct electric current, the asymmetry is induced by the drift of adatoms. If di€usion of adatoms over a step bypassing solidi®cation at the step is negligible (i.e. steps are impermeable), the step bunching occurs with the step-down drift [5±9]. * Corresponding author. Address: Department of Computational Science, Kanazawa University, Kakuma-machi, Kanazawa, 920-1192, Japan. Tel.: +81-76-264-5476; fax: +8176-234-4133. E-mail address: [email protected] (M. Sato).

When the bunching occurs with fast drift, the bunch grows via collision and coalescence of bunches. Growth law of the bunch size has been studied with a one-dimensional step ¯ow model. If evaporation of adatoms and impingement of atoms are negligible, the average terrace width L, which is roughly proportional to the average number of steps in a bunch N, increases with time as L  tb with b ˆ 1=2 [10,11]. The exponent is independent of the step interaction potential. The form of the step interaction potential a€ects the time dependence of the average step distance in a bunch lb . When the step interaction potential is given by A=lm , where l is the step distance, lb decreases with time as lb  t 1=…m‡1† [10±12]. Then lb decreases with increasing L as lb  L c with c ˆ 2=…m ‡ 1†. When evaporation of adatoms is

0039-6028/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 1 ) 0 1 2 5 8 - 4

M. Sato, M. Uwaha / Surface Science 493 (2001) 494±498

present, the value of b changes with the drift velocity. With fast drift, b  1=2, the same as the value without evaporation of adatoms, and it decreases gradually with decreasing the drift velocity [13]. The asymmetry of the surface di€usion ®eld is also produced by the asymmetry of step kinetics (the Ehrlich±Schwoebel (ES) e€ect [14,15]). In sublimation, bunching instability occurs when detachment of atoms to the lower terrace is easier than to the upper terrace. When evaporation of adatoms exceeds a critical strength, the bunching instability occurs [16±18]. Near the threshold of the instability, a vicinal face is unstable for long wavelength ¯uctuations. Modulation of the step density obeys the Benney equation [19] and formation of an equidistant train of bunches is expected from its solution. Far from the threshold, large bunches are produced in a late stage of the instability. In growth, bunching instability is observed in many materials such as SiC [20] and GaAs [21]. In these systems, the ES e€ect is one of the candidates for the cause of instability. In this paper we study the growth law of step bunches induced by the ES e€ect in growth. We use a one dimensional step ¯ow model [16,17]. By linear analysis, we ®nd the condition to induce the bunching instability in growth. By numerical integration of the step velocity, we study time evolution of step bunches and ®nd a power law growth. We determine the dependence of lb on the step interaction potential and the growth exponent b.

2. Model When we neglect evaporation of adatoms, the di€usion equation for the adatom density c…r; t† is given by 2

oc oc ˆ Ds 2 ‡ F ; ot oy

…2:1†

where Ds is the di€usion constant and F the impingement rate of atoms from the vapor. We assume that solidi®cation of adatoms occurs only at a step edge. When the direct adatom di€usion onto

495

the neighboring terraces is neglected, boundary conditions for Eq. (2.1) are given by [22] oc …2:2† Ds ˆ K …cjyn cneq †; oy yn where cneq is the equilibrium adatom density for the nth step, K are the kinetic coecients, yn is the position of the nth down step and the subscripts ‡ and indicate the lower and upper side terraces. The di€erent values of the kinetic coecients K represent the ES e€ect [14,15], which induces step instabilities. When the steps interact with the potential A=lm , the equilibrium adatom density is given by    mXA 1 1 cneq ˆ c0eq 1 ‡ ; …2:3† kB T lm‡1 lm‡1 n n 1 where c0eq is the equilibrium adatom density for an isolated step, ln ˆ jyn‡1 yn j and X is the atomic area. By solving Eq. (2.1) with Eq. (2.2), the step velocity Vn is given by Vn ˆ XK‡ …cjyn‡

cneq † ‡ XK …cjyn

cneq †:

…2:4†

In a quasi-static approximation (oc=ot ˆ 0), the step velocity is given by   XDs F 2 F l ‡d ln Vn ˆ Ds d‡ ‡ d ‡ ln 2Ds n XDs ‡ …cn‡1 cneq † d‡ ‡ d ‡ ln eq   XDs F 2 F ‡ l ‡ d ‡ ln 1 Ds d‡ ‡ d ‡ ln 1 2Ds n 1 XDs n 1 ‡ …cn ceq †; d‡ ‡ d ‡ ln 1 eq …2:5† where d ˆ Ds =K . When the steps are equidistant, ln ˆ l, the step velocity is given by V0 ˆ XFl. 3. Linear analysis We give a small perturbation to the equidistant step train and study the linear stability. When the

496

M. Sato, M. Uwaha / Surface Science 493 (2001) 494±498

position of the nth step is expressed as yn ˆ nl ‡ V0 t ‡ dyk eiknl‡xk t , the ampli®cation rate xk is given by xk ˆ iF X sin kl

F X …d 2

d‡2 †

…1 cos kl† 2 …d‡ ‡ d ‡ l† 2c0eq m…m ‡ 1†A …1 cos kl†2 : …3:1† …d‡ ‡ d ‡ l†kB Tlm‡1

The real part of Eq. (3.1) represents growth rate of the perturbation. For long wavelength ¯uctuation, kl  1, the real part of Eq. (3.1) is given by Re xk ˆ a2 k 2 ‡ a4 k 4 ;

…3:2†

where F X…d 2

a2 ˆ

a4 ˆ

d‡2 †l2

2…d‡ ‡ d ‡ l† F X…d 2

d‡2 †l4

4!…d‡ ‡ d ‡ l†

2

2

;

…3:3†

c0eq m…m ‡ 1†A 2…d‡ ‡ d ‡ l†kB Tlm

2

: …3:4†

The stability of the vicinal face is determined by a2 . If d < d‡ , i.e. adatoms attach to steps from the upper terrace easier than from the lower terrace, a2 > 0 and the vicinal face is unstable. When a2 is positive, the ®rst term in Eq. (3.4) is negative. Since the second term, which comes from the repulsive interaction, is always negative, a4 < 0 and the vicinal face is stable for short wavelength ¯uctuations. The wavelength the most unstable  pof mode is given by kmax ˆ 2p 2ja4 j=a2 . The imaginary part of Eq. (3.1) represents the drift of ¯uctuation. For long wavelength ¯uctuation, the imaginary part is given by Im xk  F Xl3 k 3 =3! Since the velocity of the ¯uctuation is given by V ˆ V0

Im xk XF 2 3 k l;  3! k

…3:5†

the ¯uctuation of the step density moves slowly in the same direction as the steps near the threshold of the instability.

4. Numerical simulation We carry out numerical simulation to follow time evolution of the step position once the instability occurs. Fig. 1 shows a snapshot of step bunches obtained by Monte Carlo simulation, where we forbid solidi®cation of adatoms from the lower terrace in order to emphasize the ES e€ect. The evaporation of adatoms is neglected and the steps advances with impingement of adatoms. The algorithm is similar to that in Ref. [23]. The step wandering occurs when the steps move in the direction of the steeper surface di€usion ®eld as shown in the linear analysis [24]. The wandering instability does not occur and the bunches are straight in growth. Then, it is justi®ed to use the one-dimensional model and calculate Eq. (2.5) numerically to study growth law of step bunches. Fig. 2 represents time evolution of the step position. Number of steps is 64 and parameters are Fl2 =c0eq Ds ˆ 0:25, d‡ =l ˆ 200, d =l ˆ 0:02, mXA= kB Tlm‡1 ˆ 0:4 with m ˆ 2, which correspond to the large impingement and the strong ES e€ect. The wavelength of the most unstable mode is short because of the large impingement rate. Then, step

128

96

y

64

32

0 0

32

64

96

128

x Fig. 1. A snapshot of step bunches in a one-sided model obtained by Monte Carlo simulation.

M. Sato, M. Uwaha / Surface Science 493 (2001) 494±498

497

100 80

t

L 40

20

1/2

t 0 0

8

16

y

24

32

Fig. 2. Time evolution of the step position obtained in the onedimensional step ¯ow model.

pairs appear in the initial stage and move fast. Pairs of steps collide with each other and form the step bunches. By collision and coalescence of the bunches, large bunches separated by large terraces are produced in a late stage. There are isolated steps in the large terraces and collision with the large bunches is repeated. The time evolution is rather similar to that in the bunching due to the fast drift in sublimation [13] except that both the steps and the bunches advance here. Fig. 3 represents time dependence of the averaged terrace width between the large bunches for various m. For each m the terrace width is averaged over 40 runs. When the bunch size grows by collision and coalescence, the terrace width L grows as L  tb with b  1=2. The value of b is independent of the power m of the step interaction potential. The value of m a€ects average step distance in a bunch lb (Fig. 4). The step distance lb decreases with power of the terrace width as lb  L c . The value of c changes with m and is given by c  2=…m ‡ 1†. Not only the feature of time evolution of step position but also the value of b and c are similar to that in the bunching induced by the fast drift.

10 10

1

2

10

t Fig. 3. Time dependence of the terrace width: ( ) m ˆ 2, mA=kB T ˆ 0:1; () m ˆ 4, mA=kB T ˆ 5  10 2 ; ( ) m ˆ 6, mA=kB T ˆ 2  10 3 . The other parameters are the same as in Fig. 2.

4 2

L

1 0.8 lb 0.6 0.4

L

0.2

L

0.1 10

–2/7

–2/5

–2/3

20

40

60 80100

L Fig. 4. Time dependence of the step distance in a bunch: ( ) m ˆ 2, mA=kB T ˆ 0:1; () m ˆ 4, mA=kB T ˆ 5  10 2 ; ( ) m ˆ 6, mA=kB T ˆ 2  10 3 . The other parameters are the same as in Fig. 2.

498

M. Sato, M. Uwaha / Surface Science 493 (2001) 494±498

5. Summary

References

In this paper we studied the step bunching ingrowth induced by the ES e€ect. Without evaporation of adatoms the step bunching occurs when attachment of adatoms is easier from the upper terrace than from the lower terrace. Though the direction of the motion of steps is di€erent, the growth law is similar to that of bunching induced by the fast drift. The terrace width increases as L  tb with b  1=2, which is independent of the power of the step interaction potential. The power of the step interaction potential m a€ects the dependence of lb on L. The step distance in the bunch decreases with time as lb  L 2=…m‡1† . Isolated steps exist on large terraces and collision with large bunches repeats. In several systems [20,21], where the ES e€ect is one of candidates for the cause of the instability, neither the growth law of step bunches nor the dependence of lb on L has been obtained so far. Quantitative study of the bunching in growth is desired.

[1] A.V. Latyshev, A.L. Aseev, A.B. Krasilnikov, S.I. Stenin, Surf. Sci. 213 (1989) 157. [2] Y. Homma, R. McClelland, H. Hibino, Jpn. J. Appl. Phys. 29 (1990) L2254. [3] E.D. Williams, E. Fu, Y.-N. Yang, D. Kandel, D.J. Weeks, Surf. Sci. 336 (1995) L746. [4] Y.-N. Yang, E.S. Fu, E.D. Williams, Surf. Sci. 356 (1996) 101. [5] S. Stoyanov, Jpn. J. Appl. Phys. 30 (1991) 1. [6] A. Natori, Jpn. J. Appl. Phys. 33 (1994) 3538. [7] B. Houchmandzadeh, C. Misbah, A. Pimpinelli, J. Phys. I France 4 (1994) 1843. [8] C. Misbah, O. Pierre-Louis, Phys. Rev. E 53 (1996) R4319. [9] M. Sato, M. Uwaha, J. Phys. Soc. Jpn. 65 (1996) 1515. [10] D.-J. Liu, D.J. Weeks, Phys. Rev. B 57 (1998) 14891. [11] M. Sato, M. Uwaha, J. Phys. Soc. Jpn. 47 (1998) 3675. [12] S. Stoyanov, V. Tonchev, Phys. Rev. B 58 (1998) 1590. [13] M. Sato, M. Uwaha, Surf. Sci. 442 (1999) 318. [14] G. Ehrlich, F.G. Hudda, J. Chem. Phys. 44 (1966) 1039. [15] R.L. Schwoebel, E.J. Shipsey, J. Appl. Phys. 37 (1966) 3682. [16] M. Uwaha, J. Cryst. Growth 128 (1993) 92. [17] M. Sato, M. Uwaha, Phys. Rev. B 51 (1995) 11172. [18] M. Sato, M. Uwaha, Europhys. Lett. 38 (1995) 639. [19] D.J. Benney, J. Math. Phys. 45 (1966) 150. [20] T. Kimoto, A. Itoh, H. Matsunami, Appl. Phys. Lett. 66 (1995) 3645. [21] H.-W. Ren, X.-W. Shen, T. Nishinaga, J. Cryst. Growth 166 (1996) 217. [22] G.S. Bales, A. Zangwill, Phys. Rev. B 41 (1900) 7408. [23] M. Uwaha, Y. Saito, Phys. Rev. Lett. 68 (1992) 224. [24] O. Pierre-Louis, C. Misbah, Y. Saito, J. Krug, P. Politi, Phys. Rev. Lett. 80 (1998) 4233.

Acknowledgements This work is performed as a part of the program ``Research for the Future'' of the Japanese Society for the Promotion of Science.