USOO5774696A United States Patent (19) 11 Patent Number: 5,774,696 Akiyama (45) Date of Patent: Jun. 30, 1998 54) TRIANGLE AND TETRAHEDRON MESH Shigyo et al., “TRIMEDES: A triangular Mesh Device GENERATION METHOD Simulator Linked w/ Topography/Process Simulation”, Trans of IEICE, vol. E71 No. 10, Oct. 88, pp. 992–999. 75 Inventor: Yutaka Akiyama, Tokyo, Japan Machek et al., “A Novel Finite-Element approach to Device Modeling", IEEE Trans on Elec. Dev., vol. ED-30 73 Assignee: NEC Corporation, Tokyo, Japan No. 9, Sep. 1983, pp. 1083–1092. Kójima et al., “Dual Mesh Approach for Semiconductor 21 Appl. No.: 654,190 Device Simulator", IEEE Trans on Magnetics, vol. 25 No. 4, 22 Filed: May 28, 1996 Jul. 1989, pp. 2953-2955. Chou et al., “Tangential Vector Finite Elements for Semi 30 Foreign Application Priority Data conductor Device Simulation', IEEE Trans on CAD, vol. 10 May 29, 1995 JP Japan .................................... 7-13O372 No. 9, Sep. 1991, pp. 1193–1200. (51) Int. Cl. ................................................ G06F 17/10 (List continued on next page.) 52 U.S. Cl. ........................... 395/500; 395/123; 364/578 58 Field of Search ..................................... 364/488, 489, Primary Examiner Kevin J. Teska 364/490, 491, 578; 395/500, 920, 123 Assistant Examiner Tyrone V. Walker Attorney, Agent, or Firm-Ostrolenk, Faber, Gerb & Soffen, 56) References Cited LLP U.S. PATENT DOCUMENTS 57 ABSTRACT 4,912,664 3/1990 Weiss et al. ............................ 395/141 A method for eliminating interSections between a Substance 4,933,889 6/1990 Meshkat et al. ........................ 364/578 boundary and triangles (or tetrahedra) of a triangle mesh (or 4.941,114 7/1990 Shigyo et al. .. ... 395/123 tetrahedron mesh) which satisfies a condition of Delaunay 5,214,752 5/1993 Meshkat et al. ... 395/123 partition and is used for a finite difference method. First, 5,315,537 5/1994 Blacker ....... ... 364/578 triangles interSecting with the Substance boundary are 5,442,569 8/1995 Osano ..................................... 364/578 5,553,009 9/1996 Meshkat et al. ........................ 364/578 Searched out. One of the vertices of any of the triangles is 5,553,206 9/1996 Meshkat ...... ... 395/123 Selected as a moving node P and the moving node is 5,579,249 11/1996 Edwards .................................. 364/578 projected to the Substance boundary to obtain a projected 5,617,322 4/1997 Yokota .................................... 364/578 point P. Processing object triangles which commonly have the moving node and peripheral triangles which are posi FOREIGN PATENT DOCUMENTS tioned around the processing object triangles are listed. 1-106266 4/1989 Japan. Then, checking to detect whether or not the projected point 4-268674 9/1992 Japan. is included in a circumscribed circle about any of the 4-309183 10/1992 Japan. peripheral triangles is performed. When the projected point 7-219977 8/1995 Japan. is included in a circumscribed circle, a node is added at the 7-319947 12/1995 Japan. projected point and triangles are produced using the node. But when the projected point is included in none of circum OTHER PUBLICATIONS Scribed circles, all of the processing object triangles are C.S. Rafferty, et al., “Iterative Methods in Semiconductor deleted, the moving node is shifted to the projected point, Device Simulation', IEEE Transactions on Electron and triangles in a region from which the processing object Devices, vol. ED-32, No. 10, Oct. 1985, pp. 2018-2027. triangles have been removed are reconstructed by using a M.S. Mock, et al., Tetrahadral Elements and the Scharfet maximum included angle method. ter-Gummel Method, Proceeding of the NASECODE IV, 1985, pp. 36–47. 7 Claims, 8 Drawing Sheets O star ) SEARCH FOR TRAnsle INTERSECTING WITH -- SUBSTANCE BOUNARY Project weRTExp of NTER SECTING TRIANSLE ON SUB STANCE BOUNDARY PolNT P') -- LSTRANGESPROCESSING OBJECT TRANSES).NODE P HAVING - LisT TRANGEs creRipheral TRIANGLES: ARQND --> PROCESSING 08JEC TRIANGLES FINRY cussibois-YESINGLJCE IN CR ANY PERIPHERAL - TREANGLE? s DELETE PRocessing l---56 OBJECT TRIANGLES MOVE NODE P to P -- 57 53 Reconstruct Transes As NCOE AT PRKJECTED USING MAXIMUM LINCLUDEC): POIN P' AN GEMERATE ANGE METHOD ANGLES 59 - END y 5,774,696 Page 2 OTHER PUBLICATIONS Tanimoto et al., “Discretization Error in MOSFET Device Yuan et al., “A Mesh Generator for Tetrahedral Elements Simulation', IEEE Trans on CAD, vol. 11 No. 7, Jul. 1992, Using Delaone Triangulation', IEEE Trans on Magnetics, pp. 921–925. vol. 29 No. 2, Mar. 1993, pp. 1906–1909. Kumashiro et al., “A Triangular Mesh Generation Method Cianpolini et al., “Efficient 3-D Simulation of Complex Suitable for the Analysis of Complex MOS Device Struc- Structures", IEEE Trans on CAD, vol. 10 No. 9, Sep.1991, tures', NUPAD V, pp. 167–170, 1994. pp. 1141-1149. U.S. Patent Jun. 30, 1998 Sheet 1 of 8 5,774,696 SiO2 X2e2S23S3is sasa saatavass sease335SYSSESSX was are as asayev WMANZAAYAASZVNSVVVSKYYAZX2 ZYANES (YYYA SAAAASKYYAA2&2VXXSSRNVSNs SYYYYYYYYY2 SYYYYXXYYYYCC1VVVVVMAAAZ) CVVVVVVVMAYYAS 2VVVVVVVVVNKVVVVVVVAAA NAAAAAAZVS/NAZ 7VVVVVAZR2NN/M/N FG 1 A F.G. 1 B (PRIOR ART) (PRIOR ART) XE3353. SAASE&SKSAAAAEGSANY SYYYYS(XYY&ESNDS S25 53. $4MyAS355; 27&X2V7S7R7RNZNZNSKXXXXXW Y %%0000%XXSSS KKK (AAAAAAZ KXXXXXXXXXX,AAAAAAA2SN/VN F.G. 1 C (PRIOR ART) U.S. Patent Jun. 30, 1998 Sheet 2 of 8 5,774,696 FIG. 2 (PRIOR ART) (PRIOR ART) U.S. Patent Jun. 30, 1998 Sheet 3 of 8 5,774,696 FG. 4A (PRIOR ART) U.S. Patent Jun. 30, 1998 Sheet 4 of 8 5,774,696 START SEARCH FOR TRIANGLE 91 (TRTRAHEDRON) INTERSECTING WITH SUBSTANCE BOUNDARY PROJECT VERTEX P OF INTER 92 SECTING TRIANGLE (TRTRAHEDRON) ON SUBSTANCE BOUNDARY 93 ADD NODE AT PROJECTED PONT P' RE-CONSTRUCT TRIANGLES 94 (RTRAHEDRA) USING METHOD OF MOCK F.G. 5 (PRIOR ART) F.G. 6B (PRIOR ART) U.S. Patent Jun. 30, 1998 Sheet 5 of 8 5,774,696 START SEARCH FOR TRIANGLE INTERSECTING WITH 51 SUBSTANCE BOUNDARY PROJECT VERTEX P OF INTER SECTING TRIANGLE ON SUB- 52 STANCE BOUNDARY (POINT P') LIST TRANGLES (PROCESSING OBJECT TRIANGLES) HAVING 53 NODE P LIST TRIANGLES (PERIPHERAL TRIANGLES) AROUND 54 PROCESSING OBJECT TRIANGLES PONT P' INCLUDED N CR CUMSCRIBEO CRCLE ABOUT YES ANY PERIPHERAL TRIANGLE 2 NO DELETE PROCESSING OBJECT TRIANGLES MOVE NODE P TO P' RE-CONSTRUCT TRIANGLES ADD NODE AT PROJECTED USNG MAXIMUM INCLUDED POINT P' AND GENERATE ANGLE METHOD TRIANGLES END FG. 7 59 U.S. Patent Jun. 30, 1998 Sheet 6 of 8 5,774, 696 7%XX B VNKOY A.FG. 8D FIG. 8BM2, A7 P' /NN/ (3 4&AAAA "YNX P' CY V FG. 8E U.S. Patent Jun. 30, 1998 Sheet 8 of 8 5,774,696 START SEARCH FOR TRTRAHEDRON INTERSECTING WITH 61 SUBSANCE BOUNDARY PROJECT VERTEX P OF INTER SECTING TETRAHEDRON ON SUB- 62 STANCE BOUNDARY (POINT P') LIST TERAHEDRA (PROCESSING OBJECT TETRA- 63 HEDRA) HAVING NODE P LIST TETRAHEDRA (PERIPHERAL TETRAHEDRA) AROUND PRO- 64 CESSING OBJECT TETRAHEDRA POINT P' RE-CONSTRUCT TETRAHEDRA ADD NODE AT PROJECTED USING MENMUM CIRCUM POINT P' AND GENERATE SCRIBED SPHERE METHOD TETRAHEDRA END FG 1 O 69 5,774,696 1 2 TRIANGLE AND TETRAHEDRON MESH 2 is connected to a plurality of grid points positioned GENERATION METHOD there around by branches (sides of triangles) indicated by Solid lines in FIG. 2, and the current J is defined on each BACKGROUND OF THE INVENTION: branch. Further, the current J between grid points is inte grated with respect to a croSS Section (indicated by a broken 1. Field of the Invention: line) of a current path covered by each branch. The cross This invention relates to a method of generating a triangle Section of the current path is represented by line Segments mesh or a tetrahedron mesh in a simulation which is based each interconnecting the circumcenters (represented by on a finite difference method or a like method. Small blank triangles) of individual two triangles positioned 2. Description of the Prior Art: 1O on the opposite sides of the current path as a common edge. A device Simulator for a Semiconductor device calculates Accordingly, in order to perform a device Simulation physical quantities in the inside of a Semiconductor device correctly, it is an essential condition that the circumcenters using a computer to calculate electric characteristics Such as of adjacent triangles do not interSect with each other. This is a terminal current and a threshold Voltage of a transistor. because, if the circumcenters of adjacent triangles interSect 15 with each other, then the croSS Section of the current path When it is attempted to optimize transistors in a Semicon with respect to which the current is integrated becomes ductor device represented by an LSI (large Scale integration) negative. Where the condition that the circumcenters of So that the Semiconductor device may exhibit its highest adjacent triangles do not interSect with each other is not electric characteristics, use of the device Simulator can satisfied, as seen in FIG. 3 (cited from the paper of C. S. Significantly reduce both the cost and the term comparing Rafferty et al. mentioned hereinabove), a result of the with an actual production of a prototype of an LSI. Further, analysis yields a physically impossible Voltage Spike at Since the device Simulator calculates physical quantities in the inside of a Semiconductor transistor, it can be investi which the quasifermi potential is 50 V. FIG. 3 is a view gated in what manner electrons or holes behave in the inside showing a result of a simulation by which a distribution of of the Semiconductor. Accordingly, the device Simulator can the quasi-Fermi potential of a device is detected. In order to 25 Satisfy the condition that the circumcenters of adjacent be used to make clear, for example, a cause of an impact triangles do not interSect with each other, the Delaunay ionization phenomenon which becomes an issue with regard partition wherein a circumscribed circle about a triangle to a fine MOSFET (metal-oxide-semiconductor field effect does not have a vertex of another triangle in the inside transistor).