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Surface Structures of Growth Models with Constraints From the Surface-Height Distribution

Surface Structures of Growth Models with Constraints From the Surface-Height Distribution. 허희범 , 윤수연 , 김 엽 경희대학교. 1. 1. Motivation. In equilibrium state, Normal restricted solid-on-solid model : Edward-Wilkinson universality class Two-particle correlated surface growth

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Surface Structures of Growth Models with Constraints From the Surface-Height Distribution

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  1. Surface Structures of Growth Models with Constraints From the Surface-Height Distribution 허희범, 윤수연, 김 엽 경희대학교

  2. 1 1. Motivation • In equilibrium state, • Normal restricted solid-on-solid model • : Edward-Wilkinson universality class • Two-particle correlated surface growth • - Yup Kim, T. S. Kim and H. Park, PRE 66, 046123 (2002) • Dimer-type surface growth • J. D. Noh, H. Park, D. Kim and M. den Nijs, PRE 64, 046131 (2001) • - J. D. Noh, H. Park and M. den Nijs, PRL 84, 3891 (2000) • Self-flattening surface growth • -Yup Kim, S. Y. Yoon and H. Park, PRE(RC) 66, 040602(R) (2002)

  3. 2 Steady state or Saturation regime , Partition function, nh: the number of columns with height h 1. Normal RSOS (z =1) Normal Random Walk (1D) (EW) 2. Two-particle correlated (dimer-type) growth (z = -1) nh=even number, Even-Visiting Random Walk (1D)

  4. 3 Phase diagram (1D) ? ? ? z z= 1 z= 0 z=-1 Normal Random Walk Even-Visiting Random Walk Self-attracting Random Walk 3. Self-flattening surface growth (z = 0) Self-attracting random walk (1D)

  5. 4 2. Generalized Model 크기가 L인 1차원 substrate의 한 column 를 임의로 선택한다. 확률p (1-p)로 deposition (evaporation)을 결정한다. Height distribution 중, 주어진 높이 h에 있는 column의 개수 nh를 계산한다. Weight를 ( nh : 높이 h를 갖는 column의 개수 ) 로 정의하고, 만일 에서 deposition (evaporation)이 일어났다고 가정했을 때, 새로운 configuration에 대하여 weight 을 구한다. 이때, 이 과정을 허용할 확률을 다음과 같이 정의한다.

  6. 5 (여기서는d-dimensional hypercubic lattice에서의 nearest-neighbor bond vectors 중, 한 site를 말한다.) p =1/2 L = 10 z = 0.5 hmax hmin w w´ n´+2 = 2 n´+1 = 2 n´0 = 2 n´-1 = 2 n´-2 = 2 n+2 = 1 n+1 = 3 n 0 = 2 n-1 = 2 n-2 = 2 만약 확률 P가 P1 일 경우, deposition (evaporation)의 과정을 무조건 허용한다. 반대로 P<1 이면, 임의의 random number R을 발생시켜 P  R일 경우에만 이 과정을 허용한다. (Metropolis algorithm) 모든 과정은 restricted solid-on-solid constraint를 만족하여야 한다. P  R

  7. 6 3. Simulation Results  Equilibrium model (1D, p=1/2)

  8. 7 z 1.5 1.1 1 0.9 0.5 0 -0.5 -1  (L)  0.33 0.33 1/2 0.33  0.33  0.33  0.34  0.33   0.22 0.22 1/4 0.22  0.22  0.22  0.22  0.19

  9. 8

  10. 8-1 Phase diagram in equilibrium (z =1/2) z = 1 -1/2 1/2 3/2 z = 0 z =-1 z = 0.9 z = 1.1 2-particle corr. growth Self-flattening surface growth Normal RSOS

  11. 9 • Growing (eroding) phase (1D, p=1(0) ) z  0 : Normal RSOS model (Kardar-Parisi-Zhang universality class)

  12. 10 z  0 Normal RSOS Model (KPZ)

  13. 11 z  0 z=-0.5 p=1 L=128

  14. 11-1 z  0 morphology z = 0.5 , L=256

  15. 11-2 z < 0 morphology z = - 0.5 , L=256

  16. 12 4. Conclusion  Equilibrium model (1D, p=1/2) z 3/2 0.9 1.1 1 1/2 0 -1 -1/2 2-particle corr. growth (EVRW) Normal RSOS (Normal RW) Self-flattening surface growth (SATW) • Growing (eroding) phase (1D, p= 1(0) ) 1. z  0 : Normal RSOS model (KPZ universality class) 2. z  0 : Groove phase ( = 1) Phase transition at z=0 (?)

  17. Scaling Collapse to in 1D equilibrium state. ( = 1/3 , z = 1.5) 12-1

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