Lateral and Longitudinal Damage Spreading in Surface Growth Models C.K. Lee, Yup Kim

v Abstract Kyung-Hee Univ. DSRG The lateral and vertical damage spreadings [1] of an initial small perturbation in various growth models are defined and studied. Vertical damage spreading d^, which is defined as the height difference between the height of a damage site and the average height of the interface, is considered as a function of the lateral damage distance d|| and time t, as d^ = d^(d||,t). In normal roughening models such as Family model and restricted solid-on-solid model d^ satisfies the scaling relation d^= d||a f (d || / t 1/z) rather well, where a is the rougheness exponent and z is the dynamic exponent of the kinetic surface roughening phenomena. Averaged damage spreadings, D^ and D||, are also considered as the function of the lateral substrate width L and t. DSRG 1

v Introduction I Scaling relations [5,6] Kyung-Hee Univ. DSRG Surface width Height-Height Correlation Function ln (W / La) ln (t / Lz) scaling L1 L2 L3 tb ln W ln t DSRG 2

I Damage Spreading [1] Kyung-Hee Univ. DSRG 1. Consider two systems A and B 2. Start from two different initial conditions, which are the same except one point at r0 3. Evolve under the same growth rules and under the same sequence of random numbers 4. The surface configurations of them evolve differently due to the different initial conditions 5. A damage site is defined as the point where the surface heights hA(r,t) and hB(r,t) are not the same t = 0 r0 A B d|| average t ¹ 0 d^ r0 A B Damage Site DSRG 3

Kyung-Hee Univ. v Results for RSOS [7] b = 0.34 a = 0.52 DSRG 4

Kyung-Hee Univ. 1/z = 0.66 기울기 = 1.0 DSRG 5

Kyung-Hee Univ. DSRG 6

Kyung-Hee Univ. DSRG a = 0.49 Damaged Cluster Distribution r0=64 L=128 7

I Scaling Ansatz in Damage Spreading Kyung-Hee Univ. I Super Roughening [8-10] 1) a >1 , W sat /L diverges in the thermodynamic limit 2) Height-Height Correlation Function in Super Roughening [7] I Scaling Ansatz in Damage Spreading I DT1[11] DSRG 8

Kyung-Hee Univ. DSRG v Results for DT1 b = 0.36 a = 1.3 9

Kyung-Hee Univ. 1/z  = 0.57 기울기 = 1.0 DSRG 10

Kyung-Hee Univ. k = 0.27 DSRG 11

Kyung-Hee Univ. Damaged Cluster Distribution r0=16 L=32 DSRG 12

v Summary and Discussion Kyung-Hee Univ. v Summary and Discussion 1) Damaged cluster distribution을 보면, general한 경우에는 damage들이 뭉쳐 다니는 반면, super roughening한 경우에는 damage들이 결합과 해체를 반복한다는 것을 알 수 있다. 2) D^ vs t 로부터 b 를, D^ vs L 로부터 a 를 구할수 있다. 3) D|| vs L 의 exponent가 1인 것으로 부터 t>>L인 경우 damage가 끝까지 퍼짐을 알 수 있다. 4) D|| vs t 로부터 general case에서는 1/z를 얻을 수 있었지만, super roughening한 경우에는 1/z=(2+k )/z 와 같은 anomalous exponent가 나온다. 5) 이러한 이유는 probability distribution이 와 같은 scaling property를 가지기 때문이다. DSRG 13

v Reference Kyung-Hee Univ. DSRG [1] J. M. Kim, Youngki Lee and In-mook Kim, Phys. Rev. E 54, 4603 (1997) [2] Haye Hinrichsen, Eytan Domany and Dietrich Stauffer, Cond-mat/9802115 [3] N.Jan and L. de Arcangelis, Ann. Rev. Comp. Phys. 1.1 (ed. D. Stauffer, World Scientific, Singapore 1994). [4] A. Coniglio, L. de Arcangelis, H. J. Gerrmann and N. Jan, Europhys. Lett. 8, 315(1989) [5] F.Famil and T. vicsek, J.Phys. A 18,L75(1985) [6] J. Krug and H. Spohn in Solids Far From Equilibrium : Growth, Morphology and Defects, edited by C. Godreche (Cambridge University Press, New York, 1991) [7] J. M. Kim and J. M. Kosterlitz. Phys. Rev. Lett. 62. 2289 [8] J. G. Amar, P.-M. Lam, and F. Family, Phys. Rev. E, 47, 3242 (1993) [9] M.Schroeder, M. Siegert, D. E. Wolf, J. D. Shore, and M. Plischke, Europhys. Lett., 24, 563(1993) [10] S. Das Sarma, S. V. Ghaisas, and J. M. Kim, 1994 Phys. Rev. E, 49, 122. [11] S. Das Sarma, and P. Tamborenea, Phys. Rev. Lett., 66, 325(1991) DSRG 14