铌酸锂晶体中的分形几何观察_英文_

铌酸锂晶体中的分形几何观察_英文_ 第 33 卷 第 4 期人工晶体学报 Vol . 33 4 No . JOURNAL OF SYNTHETIC CRYSTALS August ,2004 2004 年 8 月 Observation of Fractal Geometry in L ithium Niobate Crystal 1 2 2 2 HUAN G Hui, XU J ing2jun, KON G Yong2f a, ZHAN G Guo2quan, 2 2 2 2S HU Yong2chun, SUN J un, XU Xiao2xuan, ZHAN G Guang2yin (11 Kunming Institute of Physics , Kunming 650223 , China ; )21 Photonics Center , College of Physics Science , Nankai University , Tianjin 300071 , China ( )Received 12 April 2004 Abstract :We investigated both a true grown interface of lithium niobate crystal and a etch pit pattern on axial c surface of lithium () niobate crystal . We observed the fractal pattern that demonstrates all of the geometric characters of Sierpinski gasket triangle. The fractal dimension of grown interface and etching pit pattern in lithium niobate crystal have been calculated to be ln3/ ln2? 1158. It is suggested that an etching potential plays a key role in the formation of fractal geometry in the etch pit pattern. Key words :lithium niobate ; Sierpinski triangle gasket ; fractal geometry CLC number :O781 Document code :A () Article ID :10002985X20040420647204 铌酸锂晶体中的分形几何观察 1 2 2 2 2 2 2 2黄 晖,许京军,孔勇发,张国权,舒永春,孙 军,徐晓轩,张光寅 () 11 昆明物理研究所 ,昆明 650223 ; 21 南开大学物理科学学院光子学中心 ,天津 300071 摘要 :研究了铌酸锂 晶 体 沿 c 轴 的 真 实 生 长 界 面 及 腐 蚀 坑 模 式 , 在 两 种 情 况 中 均 观 察 到 了 明 显 的 塞 尔 宾 斯 基 ( ) Sierpinski三角垫分形几何特征 ,两种情况的分形维度通过计算得出均为 ln3/ ln2?1158 。 ( ) 关键词 :铌酸锂晶体 ;塞尔宾斯基 Sierpinski三角垫 ;分形几何 1 Introduction 1 ,2, Due to an attractive combination of piezoelectric , electro2optical , linear and nonlinear optical properties () lithium niobate LiNbOis one of the most interesting inorganic materials for a wide range of applications. Optical 3 devices based on LiNbOsuch as holographic memories , optical demultiplexers , photorefractive devices , waveguide 3 structure , electro2optic modulators , solid2state lasers , frequency doublers and mixers and parametric oscillators have been fabricated. Both the physical properties of LiNbOand the performance of the devices based on LiNbOare significantly 3 3 affected by the crystal structural quality. It is obvious that analysis of the crystal structural quality of the LiNbOcrystal 3 will take an important role to improve the crystal growth technology and the crystal structural quality. In this paper , we investigated a true grown interface of LiNbOcrystal and an etch pit pattern on axial c surface of 3 LiNbOcrystal wafer . In both the true grown interface and the etch pit pattern on axial c surface , we found there were 3 () fractal geometry structure that have all of the geometric characters of Sierpinski gasket triangle. The fractal dimension of the grown interface and the etching pit pattern of LiNbOcrystal were calculated to be ln3/ ln2?1158 . 3 Received date : 2004204212 () Foundation item :Project supported by the National High Technology Development Program of China No . 2001AA313020and partially by the key Natural ()Science Foundation of Yunnan Province No . 2003 E0012Z () Biogra phy :HUANG Hui 19762,Male ,from Yunnan province ,Sci . Doctor . E2mail : hhuang @nankai . edu. cn of the grown interface and the etching pit pattern of LiNbOcrystal were calculated to be ln3/ ln2?1158 . 3 2 Growth of crystals and observation of fractal geometry in LiNbO 3 () A congruent LiNbOcrystal Li/ Nb = 48145/ 51155 with a 76mm diameter was grown by the conventional 3 3 () Czochralski technique with the pulling direction along the c2axis. The starting materials were LiCO4N purityand 2 3 () NbO4N purity. They were thoroughly mixed at room temperature , calcined at 800 ? for 5h and then sintered at25 1100 ?for 8h. The pulling rate was kept at 3mm/ h that was controlled by monitoring the weight loss of melt using an electronic balance . During the growth of the shoulder part , the crystal rotation rate was reduced gradually from 25 to 17 r/ min in order to maintain a slightly convex melt2crystal interface . This procedure would result in an increase of the ) ((crystal diameter . The temperature difference in the melt2crystal interface 20 ?and the temperature gradient 10 ?/ cm ) above the melt surface and 15 ?/ cm in the melt volume near the surfacewere adjusted by the power both the after2 furnace and the heater beneath the bottom of the crucible . In the end process of the crystal growth , the pulling rate was out of control and became faster than 3mm/ h and the sintered LiNbOraw materials were depleted. The crystal bottom 3 ( ) ( ) surface was observed to be abnormal , as shown in Fig. 1 aand Fig. 1 b. It is seen that the interface is of fractal ( )geometry. The geometric characters of the interface coincided with the Sierpinski gasket triangle and the fractal dimension was calculated to be ln3/ ln2?1158 . Fig. 1 Fractal interfaces of LiNbOcrystal 3 () ( )Fig. 2 Etching fractal pattern in LiNbOaand a single etching fractal pit in LiNbOb 3 3 3 Observation of fractal geometric structure in the etch pit pattern The structure quality of crystals such as dislocation density could be given by means of chemical etch. The etching procedure was performed on a normal growth LiNbOcrystal wafer . The wafer was cut from a normal congruent crystal 3 HUANG Hui et al :Observation of Fractal Geometry in Lithium Niobate Crystal 第 4 期649 with the parallel faces perpendicular to the c2axis. The c2faces of the wafer was polished and then etched by etchant of ) ( ) ( ) ( HF and HNOHF/ HNO= 1/ 2. Fig. 2 aand bshow the etching pits. It is seen that there are obvious fractal 3 3 geometric structure in the etching pits. The fractal geometric structure was found to also coincide with the Sierpinski () gasket triangle. The fractal dimension was calculated to be ln3/ ln2?1158 . Mineralogists believed that asymmetry of electrochemistry on the crystal surface were the main reasons that lead to ( ) the phenomena of etch pits. There were always some defects dislocations , subgrain boundaries , impurityon the surface of crystals and these defects served as the etching cores. When the defects met active anion etchant , the etching electric potential etched the defects and the etching pits pattern was formed. The progress of etching process was in a closed 4 system. In the closed etching pit system , as the concentration of metallic cation became large , the anions moved into etching pit more and more in order to keep the balance of charge . This would result in the centralization of anions in the pit . On the other hand , the metallic compound hydrolyzed induced the stronger acidity in the etching pit . And atoms in the pit solved more and more into the solution. This would make etching pit become larger and larger . The tip of etching pit on the surface of crystal formed ,then the process of etching carried through according to the symmetry of crystal and formed etching pits as shown in Fig. 2 . Now there was a complicated etching potential of defects that controlled the process of etching and resulted in difference shape of pits. In pure lithium niobate of stoichiometric composition , the ideal cation stacking sequence along the polar c axis of the crystal can be described as —Li —Nb —?—Li —Nb —?— 5 ,6 () () , where ?represents a structural vacancy an empty oxygen octahedronas shown in Fig. 3 a. The Li / Nb is 5 + 2 - + less than 1 in congruent LiNbOcrystal , because Nbions combine much tighter with Othan that of Li . The crystal 3 structure situation changes in the congruent LiNbOcrystal as compared to the stacking sequence of stoichiometric crystal 3 ( ) as shown in Fig. 3 b. () ( ( ) Fig. 3 aStereoscopic view to be viewed with crossed Fig. 3 bChemical bonds in lithium niobate . Sketched ) eyesof the ideal crytal stacking sequence of lithium niobate c2 is one oxygen plane perpendicular to the crystallographic ( along the crystallographic c2axis light gray : oxygen , dark ( axis and the adjacent Li and Nb sites light gray : oxygen , )gray : niobium , black : lithium ) dark gray : niobium , black : lithium. Li sites below the oxygen plane are hidden by the corresponding Nb ions above the plane . M denotes a Nb ion occupying a Li site , V: a Li lithium vacancy ) () ( As we have known there are a lot of lithium vacancies Vand Nb ions occupying Li sites Nb. The Vand Li Li Li Nblocate in the center of triangle oxygen plane and the Vand Nbwill be etched firstly and result in effective curved2 Li Li Li ρface tiny etching pit with a curvature radii of . As the solving of the atoms around , the etching pit became larger and larger. The progress of the etching pit is determined by the etching potential . The averaged intension P of the etching 4 potential obeys the following expression: σ2V ( ) ()P = Pexp 1 0 ρk T ρ ρ σ where Pis the average intension of the etching potential when = 0 or ? ?, is the surface energy , V is the 0 volume of atom , k is Boltzmann’s constant and T is the etching temperature . () ρσEquation 1indicates that the etching pit becomes smaller with the increase of if parameters , V , T are fixed. So there is obviously fractal self2similar limit in the etching pit . In fact , the depth of surface layer is only of 324 atom thickness. After the Vand Nbare etched out , the three oxygen atoms in triangle plane will be etched secondly. The Li Li 7 ( ) progress of etching coincides with the controlling process of the Sierpinski gasket triangle. The fractal set can be expressed as : 3 ( )()F = ?S F2 j j = 1 ( ) Equation 2is composed of three similar2mapping sets having 1/ 2 condensation ratio . The Hausdorff’s fractal dimension and the Box’s fractal dimension of self2similar set deduced from the fractal theory is expressed as : m s ( )3 ρ C= 1 j j = 1 s Where Cis the condensation ratio of self2similar set , and the subscript j denotes the three condensation ratio of j Sierpinski gasket that satisfies : 3 1 s ( ) ()ρ = 14 j j = 1 2 () () () The solution of equation set Eq. 2, 3and 4are : ()5 S = dimF = dimF = ln3/ ln2?1158 HB It is obvious that the fractal dimension of etching pit pattern in LiNbOc2axis crystal is fractional to be ln3/ ln2? 3 1158 . 4 Conclusions () We observe fractal geometry with geometric characters of Sierpinski gasket trianglein both a true grown interface of LiNbOcrystal and the etch pit pattern on axial c surface of LiNbOcrystal . These phenomena are also observed and 3 3 studied in some other kinds of semiconductor crystals. However , to the best of our knowledge , such phenomena have not been observed and related investigations have not been carried out in LiNbOup to now. On the other hand , these 3 physical processes are very important to improve the quality of LiNbOcrystal and to understand the defect structure of 3 LiNbOcrystal that demonstrates many crystal growth dynamic processes. As a result , these phenomena give us a new 3 important and effective way to study the structural quality of LiNbOcrystal . 3 References 2Holland ,1978 : 481 . Rauber A. Current Topics in Materials Science M. E. Kaldised. Amsterdam :North 1 2 Kratzig E ,Schirmer O F. Photorefractive Material and Their ApplicationsM. Gunter P. , Huignard J P eds. Heidelberg , Berlin : Springer , 1988 :131 . 2doped LiNbOCrystalsJ . J . Cryst . Growth , 1994 ,140 : 45250 . 3 Kong Y F , Wen J K , et al . Dislocations and Subgrain Boundaries in Highly Magnesium 3 2Holland , 1980 : Chap . 2 . 4 Nabarro F R N. Dislocation in SolidsM. Amsterdam : North Acta . Cryst . , 1986 , B42 : 61268 . 5 Abrahams S C , Marsh P. Defect Structure Dependence on Composition in Lithium Niobate J . 2damage2resistant Dopants on the Nonlinear Optical Properties of LiNbOJ . Applied Physics B , 2001 , 72 : 6412645 . 6 Xue D ,et al . Influence of Optical 3 7 Mandebrot B B. The Fractal Geometry of Nature M. Freeman W Hed. New York , 1982 :373 .