Monte Carlo simulation of atom diffusion via vacancies in nanofilms

本研究利用蒙地卡羅模擬法探討具有簡單立方(SC)、體心立方(BCC)、面心立方(FCC)、以及六方最密堆積(HCP)晶體結構薄膜中原子藉由空孔擴散之相關因子(fA)，先前從未有人以此主題進行討探。首先利用全週期性邊界條件計算不同型態晶格塊材中之fA值以檢測蒙地卡羅程式碼之正確性，所得到的fA值與文獻中之數值相當吻合。 本研究之目的在探討在薄膜系統中，特別是在原子及缺陷移動受限制之奈米薄膜區間內，原子經空孔擴散之相關因子。首先選用具有簡單立方(SC)晶格結構之薄膜作為模組系統進行在奈米薄膜中相關因子之探討。在模擬程序中薄膜之膜厚(MZ)是主要輸入變數，同時亦改變每個原子與空孔之平均交換次數(NJPA)以觀察fA值之收斂性。原子—空孔在薄膜表面以及內部交換之頻率(Γs/Γi)對相關因子之影響亦進行探討。計算過程中平行於薄膜表面之x與y方向採用週期性邊界條件，而膜厚之z方向則使用反射形式邊界條件以將原子—空孔之交換限制在薄膜之內。本研究亦對具有BCC、FCC、以及HCP晶體結構之薄膜之相關因子進行探討。初始階段將原子與空孔在表面及內部之交換頻率設定為相同 (Γs/Γi =1)。結果顯示在所有探討之晶格中，原子藉由空孔擴散之相關因子與膜厚以及交換頻率均有相當程度之關聯性。當薄膜厚度足夠大時，fA值接近其所對應3D塊材晶格之數值。當薄膜厚度非常小時，fA值如同預期的達到2D晶格之數值。若NJPA足夠大時，所得fA數值在奈米薄膜區間會收斂至所對應之3D塊材晶格數值之2/3。在具有SC及HCP晶格之薄膜中，fA值在奈米薄膜區間會產生最小值平線區，但此現象並未在具有BCC及FCC晶格之薄膜中發現。此最小值平線區是由於在奈米薄膜中原子擴散受限制所造成。研究發現最小fA值之發生與否，與晶格中平行表面之原子配位數之分率有關。當Γs/Γi比值足夠大時，無論 NJPA如何改變，所有在奈米薄膜區間之fA值均會收斂至對應之2D晶格數值。 除了fA之外，研究中亦針對具有SC及FCC晶格結構薄膜中之空孔擴散相關因子(fv)加以討論。所得之fv值與薄膜膜厚亦有相當明顯之關連性。在奈米薄膜區間fv對MZ之曲線同樣會出現最小值平線區。fA在x，y與z方向之分量fAx，fAy 與 fAz亦進行分析以解釋在薄膜中原子藉由空孔擴散之關聯效應。本研究將奈米薄膜區間中fA 與 fv對薄膜膜厚之關係以數學模式做進一步分析及探討。Monte Carlo (MC) simulation was used to investigate correlation factors of atoms (fA) diffusing via vacancies in the films possessing simple cubic (SC), body-centered cubic (BCC), face-centered cubic (FCC), and hexagonal close packed (HCP) lattices. This topic has never been investigated before. The MC codes were firstly verified by calculating of the correlation factors fA for various types of lattices as fully periodic boundary conditions were employed. The obtained fA-values were consistent with those available in the literatures. The aim of this work is to investigate the correlation factors of atom diffusion via vacancies in the film system, especially in the nanofilm regime, where the atom movement was constrained. A simple cubic lattice was selected as a model system for performing detail studies of the MC simulation. The film thickness (MZ) of the films was a prime input parameter in this study. Number of jumps per atom (NJPA) was varied to investigate the convergence of fA. Influences of jump frequency ratios of the atom-vacancy exchange on the film surface and in the interior of the films (Γs/Γi) on the correlation factors were also explored. Periodic boundary conditions were imposed in the directions (x and y) parallel the film surface while the reflective type boundary condition was applied in the film thickness (z-direction) to restrict atom-vacancy exchanges within the films. The correlation factors of atom diffusion of the films with BCC, FCC, and HCP lattices were also explored. Initially, the jump frequency ratio was set to be identical (Γs/Γi =1). The correlation factors show strong dependence of the film thickness, as well as jump frequency ratios in all investigated lattices. When the film thickness is sufficiently large, fA approaches reasonably to the values of the corresponding 3D bulk lattices. As the film thickness is considerably small, fA reaches to the values of the 2D lattices, as expected. The values of fA converge to 2/3 of 3D bulk lattices in the nanofilm regime at sufficiently large NJPA. Minimum plateau of the factors can be observed in the nanofilm regime for the SC and HCP lattices while cannot be found for BCC and FCC lattices. The minimum plateau is due to the confinement of atom diffusion in the nanofilms. The occurrence of the minimum is associated with the faction of the coordination number over the film surface. As Γs/Γi is sufficiently large all the values of fA converge to the limiting value of 2D lattices irrespective of NJPA in nanofilm regime,. In addition to fA, correlation factors of vacancy diffusion (fv) in such films with SC and FCC structures were also investigated. The factors fv also show strong dependence of the film thickness MZ. The fv vs. MZ curves also exhibit a minimum plateau in the nanofilm regime. The components of fAx, fAy, and fAz, the corresponding fA in x, y, and z directions, were further analyzed to interpret the correlation effect in the films. The relation of the correlation factors fA and fv with the film thickness MZ in the nanofilm regime were further explored and discussed.誌 謝 Abstract i 摘 要 iii Table of Contents v List of Tables vii List of Figures viii Chapter 1 Introduction 1 1.1 Correlation effect in diffusion 1 1.2 Correlation factor 2 1.3 Literature survey 3 1.4 Motivation 5 1.5 Thesis overview 5 Chapter 2 Theoretical Background 7 2.1 Random walk in diffusion 7 2.2 Correlated walk in diffusion 8 2.2.1 Definition of the correlation factor 8 2.2.2 Vacancy-assisted diffusion 9 2.3 Monte Carlo method and random number generator 15 2.3.1 Monte Carlo method 15 2.3.2 Random number and random number generator 16 2.3.3 Simulation of the correlation factors by Monte Carlo method 17 Chapter 3 Computational methodology 18 3.1 Examination of the random number generators 18 3.2. Monte Carlo simulation of the correlation factors for the bulk and film systems 21 3.3 Statistical test of the correlation factors 26 Chapter 4 Results and discussion 28 4.1 Verification of the MC codes 28 4.1.1 Examination of the uniformity of the random number generators 28 4.1.2 Correlation factors of atom diffusion fA via vacancies of 2D square and honeycomb lattices 28 4.1.3. Correlation factors of atom diffusion fA via vacancies of 3D SC, BCC, FCC, and HCP lattices 32 4.1.4. Correlation factors of atom diffusion fA via vacancies in the films with SC, BCC, FCC, and HCP lattices 35 4.2 Correlation factors of atom diffusion fA via vacancies in nanofilms 37 4.2.1 Films with simple cubic lattice 38 4.2.1.1 Correlation factors of atom diffusion fA via vacancies in the films 38 4.2.1.2 Correlation factors of vacancy diffusion fv in the films 53 4.2.1.3 Further analysis of obtained correlation factors 59 4.2.2 Films with other lattices 62 4.2.2.1 Correlation factors of atom diffusion fA via vacancies in a face-centered cubic lattice 62 4.2.2.2 Correlation factors of atom diffusion fA via vacancies in a body-centered cubic lattice 79 4.2.2.3 Correlation factors of atom diffusion fA via vacancies in a hexagonal close packed lattice 83 4.2.3 Summary 91 4.2.3.1 The convergence plateau of fA and fv in the nanofilm regime 91 4.2.3.2 The specific film thickness at which fA plateau occurs 93 Chapter 5 Conclusions 97 References 99 Curriculum Vitae 10