2007-05-28

Max Planck Institute of Microstructure Physics

Scientific Profile, History, People, Cooperation of Max Planck Institute of Microstructure Physics

Max Planck Institute of Microstructure Physics
Scientific Profile

Experimental and theoretical research carried out at the Max Planck Institute of Microstructure Physics is primarily focussed on solid state phenomena that are determined by small dimensions and surfaces and interfaces. The investigations concentrate on establishing relations between the magnetic, electronic, optical, and mechanical properties of solids and their microstructure. Thin films and surfaces are investigated as well as nanocrystalline materials, phase boundaries and defects in bulk crystals. The results of the research will provide the necessary information for creating new and improved functional or structural materials in application areas such as sensorics, opto- and microelectronics.

History

The Max Planck Institute of Microstructure Physics was founded in 1992 as the first institute of the Max-Planck-Gesellschaft in the eastern part of Germany based on the previous Academy of Science Institute of Solid Sate Physics and Electron Microscopy. The institute consists of two experimental departments (I and II) and the theory department. A new laboratory building for the experimental departments, including also workshop facilities and a hall for special experiments, were put into use in September 1997. Two other buildings were reconstructed and are available since the beginning of 1999. Guest houses of the institute was opened in 1995 and in 1999.


People

The staff of the institute including scientific, technical and administrative personnel, comprised 99 positions, partly occupied by non-tenured personnel (19 scientists and 4 technicians). In addition, 44 co-workers have been funded by outside sources (incl. 16 graduate students) and 26 graduate students and 38 postdocs by MPG fellowships. Furthermore, 99 scientists (61 person-years) from abroad worked at the institute (incl. 26 graduate students).

Cooperation

A joint German-French research association in the field of magnetic thin films called "Laboratoire Européen Associé" (LEA) and collaboration based on an official agreement in the area of wafer bonding technology between the Research Center of Advanced Science and Technology (RCAST) at the University of Tokyo and the Max Planck Institute in Halle are well established now. Furthermore, starting on April 4, 2005 the Max Planck Research School for Science of Nanostructures together with Martin Luther University Halle Wittenberg and the Institute for Mechanics of Materials, Halle was established.

For more information, please go to this website:
http://www.mpi-halle.mpg.de/mpi/mpi_f_abo.html

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A Maven in Electronic Thin Film Science--Professor King - Ning Tu


Professor; B.S., National Taiwan University; M.S., Brown University; Ph.D. in Applied Physics, Harvard University(1968) ; Senior Manager of Materials Science Department at IBM T.J. Watson Research Center; Science Research Council Senior Research Fellow and The Royal Society Guest Research Fellow at Cavendish Laboratory, UK; Fellow of American Physical Society; Fellow of the Metallurgical Society; Overseas Fellow of Churchill College; Application to Practice Award of the Metallurgical Society; Alexander von Humboldt Research Award for senior US scientists; President of the Materials Research Society in 1981, Member of Academia Sinica, Republic of China.

As of April 2007, the total number of citation N = 10186.
h-factor = 56.
a-factor = N/(h x h) = 3.25.


RESEARCH DESCRIPTION
Our research interest is in wafer-based and flux-driven materials science. Modern microelectronic, opto-electronic, bio-sensor, and MEMS devices are built on wafers, involving the growth or removal of mono-layers of atoms from the wafer surface or an interface. They are open systems, in which the initial wafer surface area is constant and the flux-driven processes can come from atoms, molecules, or energy beams. Specifically, we concentrate on interfacial reactions, including metal-Si reaction, Cu-Sn reaction in solder joints, nanoscale interdiffusion and reaction, polarity effect of electromigation on interfacial reaction, and kinetic theories of interfacial reaction.

Our major research areas are (1) Cu-Sn reactions in Pb-free solder metallurgy for electronic packaging technology, (2) Advanced materials reliability problems of microelectronic devices, especially the interaction among electromigration, chemical reaction, and mechanical stress in flip chip technology, and (3) Nanoscale interdiffusion and reactions. In addition, we also conduct exploratory research on (4) Interaction of implanted metallic atoms with dislocations and grain boundaries in Si, and (5) Kinetic theories of interdiffusion and reactions.

On Pb-free solder metallurgy, we study the applications of eutectic SnAg, SnAgCu, SnCu, SnZn as solder bumps to flip chip technology. The wetting reaction and solid state aging of these Pb-free alloys with thin film under-bump-metallization are of interest. Due to the large difference in thermal expansion coefficients between the Si chip and its packaging substrate, the solder joints are stressed. In turn, the stress affects chip-packaging interaction and the integrity of Cu/ultra low k multi-layered interconnect structure on the chip. The diameter of the solder balls is approaching 50 mm, so electromigration becomes a reliability issue. The advanced materials reliability problems due to a combined action from chemical, electrical, and mechanical forces in flip chip technology will be studied systematically. Electromigration induced microstructure evolution and grain rotation in Pb-free solder alloys requires investigation. A unique nature of most Pb-free solders is that they are very rich in Sn, hence the old topics of Sn whisker, Sn pest, and Sn cry are of interest again. We shall combine micro-diffraction in synchrotron radiation, focused ion beam imaging, and cross-sectional transmission electron microscopy to study these issues. In optical packaging, we interest in how to wet an optical fiber by molten solder and how to achieve high precision alignment by solder joints.

On interconnect technology, our research emphasizes the effect of current crowding on vacancy and solute diffusion in electromigration. The nature of the electromigration force along the direction of current density gradient, i.e., normal to the current flow, will be explored. Why actual failures tend to initiate in the low current density regions will be studied. The effect of current crowding on joule heating as well as on stress concentration will be analyzed. The nature of back stress induced by electromigration and whether or not there is back stress in Cu interconnect will be investigated.

On nanoscale interdiffusion and reactions, we study the reaction of ultra thin metal films on nano Si wires and vice versa, and the nanoscale explosion in multi-layered nano-thickness thin films. Hollow nanostructure formation based on the Kirkendall effect will be investigated.

On extended defects in Si, we study the nucleation, growth, and ripening of dislocation loops formed by ion implantation and post-implantation annealing. The interaction of these loops with metallic atoms such as Ni and Co will be investigated. We also investigate the nano-grid of screw dislocation network, or very small angle twist-type grain boundary in Si bicrystals formed by wafer bonding. Again we examine the interaction of implanted metallic atoms with these dislocation networks.

Kinetic theory of phase transformations in open systems under the constraint of a constant surface area applies to phase changes on a wafer or on a given area of surface or interface will be developed. For example, the constraint of constant area is fundamental to the ripening of hemispherical scallops during the reaction between molten solder and Cu. The existence of nano channels between the scallops will be studied. The constraint of constant area also applies to the linear rate of grain growth in thin film deposition.


2006-2007 Publications:

A. M. Gusak, G. V. Lutsenko, and K. N. Tu, “Ostwald ripening with non-equilibrium vacancies,” Acta Mat., 54, 785-791 (2006).



Lingyun Zhang, Shengquan Ou, Joanne Huang, K. N. Tu, Stephen Gee, and Luu Nguyen, “Effect of current crowding on void propagation at the interface between intermetallic compound and solder in flip chip solder joints, “ Appl. Phys. Lett., 88, 012106 (2006).



S. W. Liang, T. L. Shao, Chih Chen, Everett C. C. Yeh, and K. N. Tu, “Relieving the current crowding effect in flip-chip solder joints during current stressing, “ J. Mater. Res., 21, 137-146 (2006).



M. O. Alam, B. Y. Wu, Y. C. Chan, and K. N. Tu, "High electric current density induced interfacial reactions in the micro Ball Grid Array (µBGA) solder joint" Acta Mat., 54, 613-621 (2006).



J. W. Nah, Fei Ren, K. N. Tu, Sridharan Venk, and Gabe Camara, “Electromigration in Pb-free flip chip solder joints on flexible substrates,“ J. Appl. Phys., 99, 023520 (2006).



Young-Woo Okm Tae-Yeon Seong, Chel-Jong Choi, and K. N. Tu, “Field emission form Ni-disilicide nanorods formed by using implantation of Ni an Si couples with laser annealing,” Appl. Phys. Lett., 88, 043106 (2006).



Z. H. Gan, W. Shao, M. Y. Yan, A. V. Vairagar, T. Zaporozhets, M. A. Meyer, A. Krishnamoorthy, , K. N. Tu, A. Gusak, E. Zschech, and S. G. Mhailkar, “Understanding the impact of surface engineering, structure, and design on electromigration through Monte Carlo simulation and in-situ SEM studies,” in “Stress-induced phenomena in metallization,” AIP Proceedings of 8th Workshop on Stress-induced Phenomena in metallization, Dresden, Germany, vol. 817, p.34-42 (2006).



M. Y. Yan, K. N. Tu, A. V. Vairagar, M. A. Meyer, H. Geisler, A. Preusse, and E. Zschech, “Effect of overburden thicknss on the copper microstructure of dual-inlaid interconnect structures,” AIP Proceedings of 8th Workshop on Stress-induced Phenomena in metallization, Dresden, Germany, vol. 817, p. 211-216 (2006).



M. Y. Yan, K. N. Tu, A. V. Vairagar, S. G. Mhaisalkar, and Ahila Krishnamoorthy, “A direct measurement of electromigration induced drift velocity in Cu dual damascene interconnects,” Microelectronics Reliability, 46, 1392-1395 (2006).



K. N. Tu, M. Y. Yan, Fei Ren, Joannne Huang, Emily Ou, L. Y. Zhang, and J. W. Nah, “Electromigration in flip chip solder joints,” AIP Proceedings of 8th Workshop on Stress-induced Phenomena in metallization, Dresden, Germany, vol. 817, p. 327-338 (2006)



Jae-Woong Nah, Fei Ren, Kyung-Wook Paik, and K. N. Tu, “Effect of electromigration on mechanical shear behavior of flip chip solder joints,” J. Mater. Res., 21, 698-702 (2006).



Annie T. Huang, A. M. Gusak, K. N. Tu, and Yi-Shao Lai, “Thermomigration in SnPb composite flip chip solder joints,” Appl. Phys. Lett., 88, 141911 (2006).



S. W. Liang, Y. W. Chang, T. L. Shao, Chih Chen, and K. N. Tu, “Effect of three-dimensional current and temperature distribution on void formation and propagation in flip chip solder joints during electromigration,” Appl. Phys. Lett., 89, 022117 (2006).



Xi Zhang, K. N. Tu, Y. H. Xie, C. H. Tung, and S. Y. Xu, “Single-step fabrication of Ni films with arrayed macropores and nanostructured skeletons,” Adv. Mater., 18, 1905-1909 (2006).



Xi Zhang, K. N. Tu, Y. H. Xie, and C. H. Tung, “High aspect ration Ni structure fabricated by electrochemical replication of hydrofluoric acid etched Si,” Electrochemical and Solid State Letters, 9, C150-C152 (2006).



R. Agarwal, Shengquan E. Ou, and K. N. Tu, “Electromigration and critical product in eutectic SnPb solder lines at 100 C,” J. Appl. Phys., 100, 024909 (2006).



Annie T. Huang, K. N. Tu and Yi-Shao Lai, “Effect of the combination of electromigration and thermomigration on phase migration and partial melting in flip chip composite SnPb solder joints,” J. Appl. Phys., 100, 033512 (2006).



Fei Ren, Jae-Woong Nah, K. N. Tu, Bingshou Xiong, Luhua Xu, and John H. L. Pang, “Electromigration induced ductile-to-brittle transition in lead-free solder joints,” Appl. Phys. Lett., 89, 141914 (2006).



Luhua Xu, John H. L. Pang and K. N. Tu, “Effect of electromigration-induced back stress gradient on nano-indentation marker movement in SnAgCu solder joints,” Appl. Phys. Lett., 89, 221909 (2006).



Fan-Yi Ouyang, K. N. Tu, Yi-Shao Lai, and Andriy M. Gusak, “Effect of entropy production on microstructure change in eutectic SnPb flip chip solder joints by thermomigration.” Appl. Phys. Lett., 89, 221906 (2006).



Xi Zhang and K. N. Tu, “Preparation of hierarchically porous nickel from macroporous silicon,” J. of Am. Chem. Soc., Communication, 128 (47), 15306-15307, Nov. 2006.



Xi Zhang, F. Ren, M. S. Goorsky, and K. N. Tu, “Study of the initial stage of electroless nickel deposition on Si (100) substrates in aqueous alkaline solution”, Surface and Coatings Technology, 201 (6), 2724 -2732, Dec. 2006.



Jae-Woong Nah, J. O. Suh, K. N. Tu, Seung Wook Yoon, Vempati Srinivasa Rao, Vaidyanathan Kripesh, and Fay Hua, “Electromigration in flip chip solder joints having a thick Cu column bump and a shallow solder interconnect,” J. Appl. Phys., 100, 123513 (2006).



Zhenghao Gan, A. M. Gusak, W. Shao, Zhong Chen, S. G. Mhaisalkar, T. Zaporozhets, and K. N. Tu, “Analytical modeling of reservoir effect on electomigration in Cu interconnects,” J. Mater. Res., 22, 152-156 (2007).



Luhua Xu, Pradeep Dixit, Jianmin Miao, John H. L. Pang, Xi Zhang, and K. N. Tu, “Through-wafer electroplated copper interconnect with ultrafine grains and high density of nanotwins,” Appl. Phys. Lett., 90, 033111 (2007).



Chengkun Xu, Xi Zhang, K. N. Tu, and Y. H. Xie, “Nickel displacement deposition of porous silicon with ultrahigh aspect ratio,” J. of Electrochemical Society, 154(3), D170-D174 (2007).



K. N. Tu, Chin Chen, and Albert T. Wu, “Stress analysis of spontaneous Sn whisker growth,” J. Mater. Sci: Mater. Electron., 18, 269-281 (2007).



W. Shao, S. G. Mhaisalkar, T. Sritharan, A. V. Vairagar, H. J. Engelmann, O. Aubel, E. Zschech, A. M. Gusak, and K. N. Tu, „Direct evidence of Cu/cap/liner edge being the dominant electromigration path in dual damascene Cu interconnects,“ Appl. Phys. Lett., 90, 052106 (2007).



Xi Zhang, Zhong Chen, and K. N. Tu, "Immersion nickel deposition on blank silicon in aqueous solution containing ammonium fluoride", Thin Solid Films, 515, 4696-4701 (2007).



Jae-Woong Nah, Kai Chen, K. N. Tu, Bor-Rung Su, and Chih Chen, “Mechanism of electromigration-induced failure in flip chip solder joints with a 10 micron thick Cu under-bump-metallization”, J. Mater. Res., 22, 763-769 (2007).



Chengkun Xu, Mingheng Li, Xi Zhang, K. N. Tu, and Y. H. Xie, “Theoretical studies of displacement deposition of Ni into porous Si with ultrahigh aspect ratio”, Electrochimica Acta, 52, 3901-3909 (2007).



J. W. Jang, J. K. Lin, D. R. Frear, T. Y. Lee, and K. N. Tu, “Ripening-assisted void formation in the matrix of Pb-free solder joints during solid-state aging,” J. Mater. Res., 22, 826-830 (2007).







List of thesis students in the last five years
Ph.D. students:

Dr. Harqkyun Kim, 1996 Dissertation title: "Instability at wetting tip and wetting interface of Sn-based solders on Cu substrate." Now at BMR, Orange County, CA.

Dr. Jia-Sheng Huang, 1997 Dissertation title: "Polarity effect on failure of Ni and Ni2Si contacts on p+-Si and n+-Si under high current densities." Now at Agere, Alhambra, CA.

Dr. Patrick G. Kim, 1998 Dissertation title: "Wetting behaviors of Pb-based and Pb-free solders on Au, Pd, and Ni substrates." Now at Amkor Technology, Chandler, AZ.

Dr. Lowen Chow, 1999 Dissertation title: "Structure and mechanical propertes of low dielectric constant xerogel thin films" Now at Intel, Santa Clara, CA.

Dr. Chien-Neng Liao, 1999 Dissertation title: "Thermoelectric characterization of Si thin films in SOI wafers and thermal conductivity of low dielectric constant thin films." Now assistant professor at National Tsing Hua University, Hsinchu, Taiwan, ROC.

Dr. Chih Chen, 1999 Dissertation title: "Grain boundary structure in twist-type Si bicrystals made from SOI and dopant activation in SOI by high density currents." Now assistant professor at National Chiao Tung Univeristy, Hsinchu, Taiwan, ROC (starting 08/01/00).

Dr. Dawei Zheng, 1999 Dissertation title: “Measurement of local stress for microelectronics applications,” Now at Lightcrosss, Los Angeles, CA.

Dr. Chengyi Liu, 3/00, Ph.D. Dissertation on "Wetting behavior and electromigration of SnPb solders as a function of alloy composition." Now assistant professor at National Central University, Chungli, Taiwan, ROC.

Dr. Taek Yeong Lee, 2001 Dissertation title: "Electromigration and solid state aging of Pb-free flip chip solder joints and synchrotron radiation study of Sn whisker growth," Now at AT&T Bell Lab., Lucent Technologies, Murray Hill, NJ.

Dr. Peter Sangwoo Nam, 11/01, Dissertation title: "GaAs backside through chip via hole integration using inductively coupled plasma etching." Now at TRW, Redondo Beach, CA.

Dr. Woojin Choi, 2002 Dissertation title "Reliability of Pb-free solders in electronic packaging technology." Now at Intel, Chandra, AZ.

Dr. Hua Gan, 2004 Dissertation Title, “Polarity effect of electromigration on intermetallic compound formation in Pb-free solder V-groove samples.” Now at IBM T. J. Watson Research Center, Yorktown Heights, NY.

Albert T. C, Wu, 12/04, Ph.D. dissertation on "Electromigration and Microstructure Evolution in Anisotropic Conducting Tin Studied by Synchrotron X-ray Microdiffraction." Now at Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA.

Dr. Emily Shengquan Ou, 2005 Dissertation Tiltle, "The polarity effect of electromigration on intermetallic compound formation and back stress in V-groove solder lines." Now at Intel, Chandler, AZ.

M.Sc. students:

Wang Yang, 1995 Thesis title: "Ultra-fast soldering reaction of eutectic SnPb and eutectic SnBi on Pd surfaces." Now at Vitesse Semiconductor Corp., Camarillo, CA.

Ann A Liu, 1996 Thesis title: "Spalling of Cu-Sn compounds in the soldering reaction between eutectic SnPb and Au/Cu/Cr thin films." Now at TRW, Redondo Beach, CA.

Sharon S. Y. Huang, 1997 Thesis title: "Ultra-thin TaN as diffusion barrier for Cu interconnects." Now at Applied Materials, Santa Clara, CA.

Jessica P. Almaraz, 1998 Thesis title: "Morphology of the wetting reaction of Pb-free solder (eutectic SnAg and eutectic SnBi) on Ni substrates." Now at Northrop, Palmdale, CA.

Ben Zhengyi Jia, 1997 Thesis title: “Stress of Ni thin films on Si wafers,” Now a graduate student in Dept. EE, UCLA.

Yi-Pin Tsai, 1999 thesis title: “Microstructure and properties of low dielectric constant porous polymer PAE thin films,” Now at Intel, Los Angeles, CA.

Judy Pei-Yao Liu, 2000 Thesis title "Microstructure and property of Organically Modified Silicate Film Used for Interlayer Dielectric with Low dielectric Constant." Now at Applied Materials, Hsinchu, Taiwan, ROC.

Quyen Tang Huynh, 2000 Thesis title "Electromigration Study in PbSn Solder Lines." Now at Intel, Los Angeles, CA.

Gu Xu, 11/01, Thesis title "Effect of electromigration in V-shaped solder lines." Now at Dept. MSE, UCLA.

Cindy Wan-Ying Ma, 2002 Thesis title "The synthesis and characterization of porous low-k methylsilsesquioxane films for interlayer dielectric applications."

Jongsung Kim, 2002 Thesis title "Flow kinetics of molten Pb-free solders along V-groove etched on (001) Si surface." Now at UCLA.

Seung-Yub Lee, 2003 Thesis Title "Synthesis and Characterization of Organically Modified Silicates Thin Film for Low Dielectric Constant Materials." Now at Caltech.

Xi Zhang, 2004 Thesis Title, “Electroless Ni metallization of macro-porous silicon for the application to cross-talk isolation in mixed signal integrated circuits.” Now a Ph. D. candidate in Dept. of MSE, UCLA.

Minyu Yan, 2005 thesis on "The effect of immersion and evaporated Sn coating on the electromigration failure mechanism and lifetime of Cu damascene interconnects." Now a Ph. D. student in UCLA.

Mr. Rajat Agarwal, 2005 Thesis Title, "Electromigration in eutectic SnPb V-groove solder lines at 100 C." Now at Intel, Santa Clara, CA.

To see more detail information, please go to this webpage:
http://www.seas.ucla.edu/ms/faculty1/tu.html

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2007-05-19

越野之王--悍马(HUMMER)


一提起悍马(HUMMER),你立即会想到那一种具有越野之王称号的车辆,尤其是在伊拉克战争中及战争后,你总能从电视中看到美国大兵驾驶悍马军车巡逻于巴格达街头,或者被袭击后的现场。但在电视上看到的是军用悍马,你不可能买到,你出再高的价钱也只能买到民用版悍马。性能及外形最接近军用悍马的民用版悍马是H1型,这是一款真正的越野车,它在美国的基本售价为10.5万美元。如果你出不起这个价,可以买悍马H2型,这是一款现在正在中国流行的SUV,它的基本售价为48800美元(在亚运村市场曾标出150万元RMB,现在标价为118万RMB)。从今年夏天开始,悍马H2SUT也将开卖,它的价格至今还是个秘。SUT是什么?那是“运动型多功能卡车”的意思。在美国,卡车(Truck)一般是指我们中国人根本不正眼看的皮卡。在美国开皮卡就像是穿牛仔裤一样,不分贵贱和贫富,谁都可以开,而且从某种程度上讲,驾驶一辆皮卡比开轿车或SUV都更具“美国精神”。然而,悍马那高昂的价格把许多悍马迷拒之门外,它那宽大的车身也显得太霸道,它那“油老虎”式的大排量发动机也让不少人负担不起。



H3T——悍马小弟

哪里有需要,哪里就有产品。专为年轻人设计的小型悍马H3T概念车最近亮相,骤然间唤起了众多悍马车迷们的希望。H3T是一款单排皮卡车型,它虽是最小的悍马,长宽高仅为4443×1893×1795mm,但它继承了悍马H1的反叛造型特征,也拥有美国军车的DNA。H3T的两门两座式皮卡设计,是当今美国最为时尚的车身造型。

悍马H3T的外形就是皮卡,与它的三个哥哥H1、H2和H2 SUT都不同。它的顶篷可以电动打开,后窗也可以隐藏起来,从而形成一个完全开放的驾驶室。H3T的机盖仍像它的哥哥们一样是向前打开,而是是半自动的。它的通过性极强,配用34英寸的轮胎,离地间隙高达11.5英寸,从而使其接近角和离去角分别达到51度和50度。虽然你一眼就可看出H3T是悍马家族成员,但由于车身尺寸要小些,线条也稍柔顺些,因此比它的哥哥们看起来更亲切些,至少没有H1那么野蛮,也没有H2那样张扬。同时,较紧凑的车身尺寸也让H3T显得更具动感和灵敏。

H3T的轮胎和内饰设计是与世界运动服装名牌耐克合作的产物。耐克将它在运动鞋上的制作技术应用在
H3T轮胎上,使其不论在何种路面上都具有很强的抓地力;耐克在运动衣上的制作技术则运用在H3T的座
椅面料上,这种面料不仅轻质,而且不需机械手段即可达到温暖如春或清爽宜人的效果。


以往的悍马车都是在通用全尺寸卡车平台上开发的车型,而H3T则是在通用中型卡车平台上设计的车型。
它的四轮都靠近车身四角,采用四轮盘式刹车,前盘上设置了6活塞式卡钳,后盘则是4活塞式卡钳。



H3T配用通用的新型Vortec 3.5升直列五汽缸发动机,最大功率高达350马力,升功率达到100马力/升。

最大扭矩为474Nm。













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2007-05-14

A Brief Introduction of Fractal(分形几何学简介)


谁创立了分形几何学?
1973年,曼德勃罗(B.B.Mandelbrot)在法兰西学院讲课时,首次提出了分维和分形几何的设想。分形(Fractal)一词,是曼德勃罗创造出来的,其愿意具有不规则、支离破碎等意义,分形几何学是一门以非规则几何形态为研究对象的几何学。由于不规则现象在自然界是普遍存在的,因此分形几何又称为描述大自然的几何学。分形几何建立以后,很快就引起了许多学科的关注,这是由于它不仅在理论上,而且在实用上都具有重要价值。
分形几何与传统几何相比有什么特点:
⑴从整体上看,分形几何图形是处处不规则的。例如,海岸线和山川形状,从远距离观察,其形状是极不规则的。
⑵在不同尺度上,图形的规则性又是相同的。上述的海岸线和山川形状,从近距离观察,其局部形状又和整体形态相似,它们从整体到局部,都是自相似的。当然,也有一些分形几何图形,它们并不完全是自相似的。其中一些是用来描述一般随即现象的,还有一些是用来描述混沌和非线性系统的。


什么是分维?
在欧氏空间中,人们习惯把空间看成三维的,平面或球面看成二维,而把直线或曲线看成一维。也可以梢加推广,认为点是零维的,还可以引入高维空间,但通常人们习惯于整数的维数。分形理论把维数视为分数,这类维数是物理学家在研究混沌吸引子等理论时需要引入的重要概念。为了定量地描述客观事物的“非规则”程度,1919年,数学家从测度的角度引入了维数概念,将维数从整数扩大到分数,从而突破了一般拓扑集维数为整数的界限。

分维的概念我们可以从两方面建立起来:一方面,我们首先画一个线段、正方形和立方体,它们的边长都是1。将它们的边长二等分,此时,原图的线度缩小为原来的1/2,而将原图等分为若干个相似的图形。其线段、正方形、立方体分别被等分为2^1、2^2和2^3个相似的子图形,其中的指数1、2、3,正好等于与图形相应的经验维数。一般说来,如果某图形是由把原图缩小为1/a的相似的b个图形所组成,有:
a^D=b, D=logb/loga
的关系成立,则指数D称为相似性维数,D可以是整数,也可以是分数。另一方面,当我们画一根直线,如果我们用0维的点来量它,其结果为无穷大,因为直线中包含无穷多个点;如果我们用一块平面来量它,其结果是0,因为直线中不包含平面。那么,用怎样的尺度来量它才会得到有限值哪?看来只有用与其同维数的小线段来量它才会得到有限值,而这里直线的维数为1(大于0、小于2)。与此类似,如果我们画一个Koch曲线,其整体是一条无限长的线折叠而成,显然,用小直线段量,其结果是无穷大,而用平面量,其结果是0(此曲线中不包含平面),那么只有找一个与Koch曲线维数相同的尺子量它才会得到有限值,而这个维数显然大于1、小于2,那么只能是小数(即分数)了,所以存在分维。其实,Koch曲线的维数是1.2618……。

Fractal(分形)一词的由来
据曼德勃罗教授自己说,fractal一词是1975年夏天的一个寂静夜晚,他在冥思苦想之余偶翻他儿子的拉丁文字典时,突然想到的。此词源于拉丁文形容词fractus,对应的拉丁文动词是frangere(“破碎”、“产生无规碎片”)。此外与英文的fraction(“碎片”、“分数”)及fragment(“碎片”)具有相同的词根。在70年代中期以前,曼德勃罗一直使用英文fractional一词来表示他的分形思想。因此,取拉丁词之头,撷英文之尾的fractal,本意是不规则的、破碎的、分数的。曼德勃罗是想用此词来描述自然界中传统欧几里德几何学所不能描述的一大类复杂无规的几何对象。例如,弯弯曲曲的海岸线、起伏不平的山脉,粗糙不堪的断面,变幻无常的浮云,九曲回肠的河流,纵横交错的血管,令人眼花僚乱的满天繁星等。它们的特点是,极不规则或极不光滑。直观而粗略地说,这些对象都是分形。

分形的定义
曼德勃罗曾经为分形下过两个定义:
(1)满足下式条件
Dim(A)>dim(A)
的集合A,称为分形集。其中,Dim(A)为集合A的Hausdoff维数(或分维数),dim(A)为其拓扑维数。一般说来,Dim(A)不是整数,而是分数。

(2)部分与整体以某种形式相似的形,称为分形。
然而,经过理论和应用的检验,人们发现这两个定义很难包括分形如此丰富的内容。实际上,对于什么是分形,到目前为止还不能给出一个确切的定义,正如生物学中对“生命”也没有严格明确的定义一样,人们通常是列出生命体的一系列特性来加以说明。对分形的定义也可同样的处理。
(i)分形集都具有任意小尺度下的比例细节,或者说它具有精细的结构。
(ii)分形集不能用传统的几何语言来描述,它既不是满足某些条件的点的轨迹,也不是某些简单方程的解集。
(iii)分形集具有某种自相似形式,可能是近似的自相似或者统计的自相似。
(iv)一般,分形集的“分形维数”,严格大于它相应的拓扑维数。
(v)在大多数令人感兴趣的情形下,分形集由非常简单的方法定义,可能以变换的迭代产生。

转自:http://library.thinkquest.org/C006364/GB/fractal/intro.htm

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