Please use this identifier to cite or link to this item: http://hdl.handle.net/11455/10713
標題: 以蒙地卡羅法模擬原子在奈米薄膜中經由空孔之擴散
Monte Carlo simulation of atom diffusion via vacancies in nanofilms
作者: 揭由志
Chieh, Yu-Chih
關鍵字: Monte Carlo simulation
蒙地卡羅模擬
nanofilms
correlation factor
diffusion
vacancies
奈米薄膜
相關因子
擴散
空孔
出版社: 材料科學與工程學系所
引用: [1].M. E. Glicksman, Diffusion in Solids: Field Theory, Solid-State Principles, and Applications, New York, Wiley & Son, (2000) 191-208, 235-243. [2].R. Brown, "A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies," Phil. Mag. 4, 161 (1828). [3].S. A. Akbar, “A Generalized View of the Correlation Factor in Solid-State Diffusion,” J. Appl. Phys., 75, 2851 (1994). [4].J. Philibert, S. J. Rothman, Atom movements: diffusion and mass transport in solids, Les Ulis, France, Editions de Physique, (1991) 61, 71-83 [5].Paul Shewmon, Diffusion in Solids, 2nd Ed., Warrendale, Pa., Minerals, Metals & Materials Society, (1989) 67. [6].A.D. Le Claire, Physical Chemistry edited by H. Eyring, New York, Academic Press, (1970) 282. [7].J. Bardeen and C. Herring, Imperfections in Nearly Perfect Crystals, edited by W. Shockley, New York, Wiley & Son, (1952) 281. [8].K. Compaan and Y. Haven, “Correlation factors for diffusion in solids,” Trans. Faraday Society. 54, 1498 (1958). [9].G. E. Murch, Diffusion in Crystalline Solids, edited by G. E. Murch and A. S. Nowick, Orlando, Academic Press, (1984) 399. [10].A.D. Le Claire and A.B. Lidiard, “Correlation effects in diffusion in crystals,” Phil. Mag. 1, 518 (1956). [11].J.R. Manning, “Correlation Effects in Impurity Diffusion,” Phys. Rev. 116, 819 (1959). [12].J.R. Manning, “Correlation Factors for Impurity Diffusion─fcc Lattice,” Phys. Rev. 128, 2169 (1962). [13].J. R. Manning, “Correlation Factors for Diffusion in Nondilute Alloys,” Phys. Rev. B 4, 1111 (1971). [14].H. J. de Bruin, G. E. Murch, H. Bakker, and L. P. Van der Mey, “Diffusion correlation in random alloys,” Thin Solid Films 25, 47 (1975). [15].H. J. de Bruin, H. Bakker, and L. P. Van der Mey, “Diffusion Correlation in Random Primitive Cubic Alloys,” Phys. Status Solidi B 82, 581 (1977). [16].H. Bakker, N. A. Stolwijk, L. P. Van der Mey, and T. J. Zuurendonk, “Microstructural dependence of vacancy diffusion in ordered alloys,” Nucl. Metall. 20, 96 (1976). [17].N A. Stolwijk, H. Bakker, and M. van Gend, “Monte Carlo computations on long- and short-range order in ordering binary alloys with vacancies,” J. Phys. C: Solid State Phys. 13, 5207 (1980). [18].G.E. Murch and S.J. Rothman, “Diffusion, correlation, and percolation in a random alloy,” Phil. Mag. A 43, 229 (1981). [19].P. Franke and R. Dieckmann, “Correlation factors for diffusion in binary random alloys with fcc structure,” J. Appl. Phys. 70, 787 (1991). [20].Y. Mishin and D. Farkas, “Monte Carlo simulation of correlation effects in a random bcc alloy”, Phil. Mag. A 75, 201 (1997). [21].G. E. Murch, “Computer simulation of tracer diffusion in a concentrated alloy with long-range order,” Phil. Mag. A 46, 565 (1982). [22].G. E. Murch, “Computer simulation of tracer diffusion in a concentrated alloy with short-range order”, Phil. Mag. A 45, 941 (1982). [23].R. Allnatt and E. L. Allnatt, “Comparison of computer simulated and theoretical tracer diffusion coefficients for a strictly regular solution model of a concentrated alloy,” Phil. Mag. A 66, 165 (1992). [24].F.-H. Lu, P. Franke, C. S. Nichols, and R. Dieckmann, “Monte Carlo Simulation of Cation Transport via Vacancies in Spinel Solid Solutions: One Type of Cation Exchange Prevails,” J. Appl. Phys. 75 (2), 819 (1993). [25].F.-H. Lu, P. Franke, C. S. Nichols, and R. Dieckmann, “Monte Carlo Simulation of Cation Diffusion via Vacancies in Quasibinary Spinel Solid Solutions Involving Multiple Types of Cation-Vacancy Exchanges,” J. Appl. Phys. 76 (2), 848 (1994). [26].F.-H. Lu, “Monte Carlo Simulation on the Cation Diffusion via Vacancies in Simple Spinels,” Comp. Mater. Sci 14, 48 (1999). [27].A. Einstein, “On the movement of small particles suspended in stationary liquids required by the molecular-kinetic theory of heat,” Annalen der Physik, 17, 549 (1905) [28].R. E. Howard and A.B. Lidiard, “Matter transport in solids,” Rep. Prog. Phys. 27, 161 (1964). [29].R. E. Howard, “Random-Walk Method for Calculating Correlation Factors: Tracer Diffusion by Divacancy and Impurity-Vacancy Pairs in Cubic Crystals,” Phys. Rev. 144, 650 (1966). [30].J.G. Mullen, “Effect of Bardeen-Herring Correlation on Vacancy Diffusion in Anisotropic Crystals,” Phys. Rev. 124 (6), 1723 (1961) [31].Helmut Mehrer, Diffusion In Solids: Fundamentals, Methods, Materials, Diffusion-controlled Processes, 1st Ed., New York, Springer, (2007) 64, 70. [32].A.G. Mikhin and N. de Diego, “Surface vacancy diffusion on Cu (111): a computer simulation study,” Surf. Sci. 418, 166 (1998). [33].P. Stoltze, “Simulation of surface defects,” J. Phys.-Condes. Matter 6, 9495 (1994). [34].P. Ballo, N. Kioussis, and G. Lu, “Interaction of vacancies with a grain boundary in aluminum,” Phys. Rev. B 64, 024104 (2001). [35].N. Metropolis, “The beginning of the Monte Carlo method,” Los Alamos Science, 15, 122 (1987) [36].N. Metropolis, and S. J. Ulam, “The Monte Carlo method,” J. Amer. Stat. Assoc. 44, 335 (1949). [37].D.W. Hubbard, How to Measure Anything: Finding the Value of Intangibles in Business, Hoboken, NJ, Wiley & Sons, (2007) 46. [38].D.E. Knuth, The Art of Computer Programming, Volume 2 / Seminumerical Algorithms, Reading MA, Addison-Wesley, (1969) 78. [39].F. James, “A review of pseudorandom number generators,” Comput. Phys. Commun. 60, 329 (1990). [40].D. H. Lehmer, Large-Scale Digital Calculating Machinery, Cambridge, Harvard University Press, (1951) 141. [41].M. Matsumoto and T. Nishimura, “Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator,” TOMACS 8, 3 (1998) [42].W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in FORTRAN and Numerical Recipes in C, 2nd. Ed., England, Cambridge University Press, (1992). [43].E. Kreyszig, Advanced Engineering Mathematics, New York, Wiley and Sons, (1972). [44].G. Lu, N. Kioussis, V. V. Bulatov, and E. Kaxiras, “Generalized stacking fault energy surface and dislocation properties of aluminum,” Phys. Rev. B 62, 3099 (2000). [45].Y. Shimomura, K. Sugio, and M. Kiritani, “Vacancy generation in deformed thin metal,” Comp. Mater. Sci. 14, 97 (1999). [46].M.I. Pascuet, R.C. Pasianot, and A.M. Monti, “Computer simulation of surface-point defects interaction in hcp metals,” J. Mol. Cata. A: Chemical 167, 165 (2001). [47].J. Kiefer, “Conditional Confidence Statements and Confidence Estimators,” J. Am. Stat. Assoc. 72, 789 (1977). [48].G..K. Robinson, “Some counterexamples to the theory of confidence intervals,” Biometrika 62, 155 (1975). [49].R.A. Fisher, “Applications of Student''s distribution,” Metron, 5, 90 (1925). [50].S.W. Kelly and C.A. Sholl, “Theory and Monte Carlo calculation of the tracer correlation factor,” J. Phys. C 20, 5293 (1987). [51].D.A. Porter, K.E. Easterling, Phase transformations in metals and alloys, 2nd Ed., London, Chapman & Hall, (1992) 43. [52].S.I. Masharov, “Equilibrium vacancy concentration in disordered binary alloys,” Russ. Phys. J. 15, 694 (1972). [53].R Balzer and H Sigvaldason, “Equilibrium vacancy concentration measurements on zinc single crystals,” J. Phys. F: Met. Phys. 9, 171 (1979). [54].B.V. Guérard, H. Peisl, R. Zitzmann, “Equilibrium vacancy concentration measurements on aluminum,” J. Appl. Phys. 3, 37 (1974).
摘要: 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.
URI: http://hdl.handle.net/11455/10713
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