Internet and Parallel Computing in Semiconductor Device Simulation

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1 Iteret ad Parallel Comutig i Semicoductor Device Simulatio INN-LIANG LIU, YIMING LI 2, TIEN-SHENG CHAO 3, ad S. M. SZE 2 Deartmet of Alied Mathematics 2 Deartmet of Electroics Egieerig 3 Natioal Nao Device Laboratories Natioal Chiao Tug Uiversity 00 Ta Hsueh Rd., Hsichu TAIWAN Abstract: - A rototye of o-lie simulator is roosed for semicoductor device simulatio. The ricial cocet of the desig of the simulator is to rovide a Web-based, latform ideedet, ad distributed comutig eviromet for the user. The framework of the rototye cosists of a stadard Web browser iterface imlemeted via HTTP ad PHP, a Liux Web server, a MPI system of arallel rocessors, ad umerical ackages ecessary to erform a comlete device simulatio. The kerel of the simulator is MONOMOS so called that umerical solutios of the fudametal semicoductor euatios are obtaied by meas of the mootoe iterative method istead of Newto's method. Ideedet of the iitial guess for each I-V oit, the mootoe iterative method allows a simultaeous, ad hece arallel, simulatio of differet I-V curves with various biasig coditios. Simulatio results of sub-micro N-MOSFET devices will be give to demostrate the features of the rototye. Key Words: - Semicoductor Device Simulatio, Parallel Comutatio, MPI, Mootoe Iterative Method, Web-based Itroductio TCAD (techology of comuter-aided desig) is oe of the imortat techologies i semicoductor idustry for uderstadig ad develoig ew semicoductors. However, may TCAD tools ted to be used oly at laces where they were develoed because of the difficulties i ortig ad learig the evolvig software. We roose here a rototye of o-lie device simulator, which cosists of a stadard Web browser iterface imlemeted via HTTP ad PHP, a Liux Web server, a MPI system of arallel rocessors, ad umerical ackages ecessary to erform a comlete device simulatio. The kerel of the simulator is MONOMOS so called that umerical solutios of the fudametal semicoductor euatios are obtaied by meas of the mootoe iterative method istead of Newto's method []. The mai features of the simulator are latform ideedece, user-friedly iterface, distributed comutig system, ad arallel I-V curve simulatios. The first two features are direct coseueces of the curret state of the Web techologies while the third is due to Liux ad MPI. Ideedet of the iitial guess for each I-V oit, the mootoe iterative method allows a simultaeous, ad hece arallel, simulatio of differet I-V curves with various biasig coditios. This feature is very differet from that of the existig TCAD tools i which the solutio oits of I-V curves are comuted successively from lower biases to higher biases by meas of the cotiuatio method. I the ext sectio, we state the semicoductor device models that have bee imlemeted i the kerel solver. I Sec. 3, we briefly describe the umerical schemes ad solutio algorithm of MONOMOS. The system architecture of the rototye is give i Sec. 4. Simulatio results of some LDD N-MOSFET device models are reorted i Sec. 5. Cocludig remarks are give i the last sectio. Fially, we ote that the o-lie simulator ca be accessed at htt://moo.math.ctu.edu.tw. 2 Semicoductor Device Models The steady-state eergy trasort model [2] has bee alied to study hot electro effects i which the device scale is dow to 0.25µm regio. The followig is a commoly used model: φ = ( + D), () εs

2 = R(, ), (2) = R(, ), (3) ( ω ω S 0 = E ), τ ω ( T ) (4) ω ω0 S = E ( ). τ ω ( T ) (5) Here φ is the electrostatic otetial, ad are electro ad hole cocetratios, T ad T are electro ad hole temerature. E=- φ is the electric field, is the elemetary charge, ε s is the dielectric costat of semicoductor, D is the satially-deedet doig rofiles, ad R is the carrier recombiatio rate.,, S, ad S are carrier currets ad eergy flux desities defied as follows: = µ φ + D + µ k T, B (6) = µ φ D µ kb T, (7) S, = ω + kbt + Q (8) S, = ω + kbt + Q (9) where the µ, µ, D, D, ω, ω, ad Q, Q are carrier mobility fuctios, diffusio coefficiets, average eergies, ad heat flows, resectively. However, the most widely used device model i commercial software is still the drift-diffusio model: φ = ( + D), εs (0) = R(, ), () = R(, ), (2) where ad are simlified to = µ φ + D, (3) = µ φ D. (4) I device simulatio, most umerical methods have bee develoed for the aroximatio of the system ()-(5) with the rimal state variables φ,,, T ad T. O the other had, based o Boltzma statistics [3, 4], there are also some methods develoed for the uasi-fermi formulatio of the system (0)-(2) i terms of the uasi-fermi levels [3, 5, 6]. The formulatio results i self-adjoit oerators ad its oliear load fuctios are mootoe. These roerties are articularly favorable for the mootoe iterative method [7]. Note that these models are subject to some aroriate coditios o the boudary of a rectagular regio ABEF show i Fig. [8]. Homogeeous Neuma boudary coditio B C Robi boudary coditio D E C' D' Dirichlet boudary coditio + + Dirichlet boudary coditio B' A Source Gate Drai P Dirichlet boudary coditio Substrate Homogeeous Neuma boudary coditio Fig.. Domai of a 2-D N-MOSFET device. 3 Numerical Methods The classical Gummel s decoulig method for solvig the DD or ET models cosists of three mai rocedures: the box Scharfetter-Gummel discretizatio, three (DD) or five (ET) ier loos of Newto s iteratio for each ukow fuctio, ad a outer loo for all ukow fuctios. Our method relaces Newto s iteratio by the mootoe iteratio i which the corresodig discrete system is of the followig form [8]: ( m+ ) m) ( m) ( m) ( D + λ I ) Z = ( L + U ) F( Z ) + λz, (5) where Z is the ukow vector, F is the oliear vector form, ad D, L, U, ad I are diagoal, lower triagular, uer triagular, ad idetity matrices, resectively. Ad the mootoe iterative arameter λ is determied ode-by-ode deedig o the device structure, doig cocetratio, bias coditio, ad oliear roerty of each decouled euatio. Note that the system (5) is of acobi tye ad hece is highly arallel. It does ot reuire ay assemblig rocess for a global matrix. Coseuetly, the method is very cost effective i terms of both comutatioal time ad memory. Furthermore, the mootoe iterative method is a global method i the sese that it does ot ivolve ay acobia matrix, as that of Newto s method, that iheretly reuires a sufficietly accurate iitial guess to begi with. Moreover, it will ot roduce o-hysical egative values for the miority carrier cocetratio uder E' F

3 heavy recombiatio rocesses. These are the ovel features of the kerel solver MONOMOS. 4 The System Architecture The ricial cocet of the desig of the simulator is to rovide a Web-based, latform ideedet, ad distributed comutig eviromet for the user. As show i Fig. 2, the framework of the rototye cosists of a stadard Web browser iterface imlemeted via HTTP ad PHP, a Liux Web server, a MPI system of arallel rocessors, ad the simulatio kerel MONOMOS. Iteral Users PC/Mac/Uix HTTP HTML/PHP User s WWW I/O Iterface Web ad MPI Server MONOMOS MPI Cliets MONOMOS Exteral Users PC/Mac/Uix HTTP Processes Reuest Network Liks Data Trasmissio MPI Library Doig Profile (cm -3 ) oits reuested by the user. All iuired outut grahics will be dislayed back to the user s browser ad together with outut files will be stored i the user s workig directory that resides i the Web server ad ca be accessed for later use. The simulator is self-cotaied with all simulatio ackages such as mesh geeratio, umerical solvers, data maagemet, grahic tools etc. It does ot reuire ay maual iterrutios from the user or the server. 5 Simulatio Results ad Discussio We ow reset some tyical simulatio results of the simulator. The first examle is a 0.35µm LDD N-MOSFET device with the gate oxide thickess 70A ad with the doig rofile show i Fig. 3. Figs. 4 ad 5 illustrate the simulated electrostatic otetial ad electro desity at V DS = 2.0V ad V GS = 2.0V, resectively. Fig. 3. Surface lot of the doig cocetratio for 0.35µm LDD N-MOSFET device. Fig. 2. The system architecture of the simulator. The I/O iterface is imlemeted i HTML ad PHP ad is i either grahical, butto-selectio, or fill-i format. It cosists of a seuece of ste-by-ste ueries startig from doig rofile, device geometry, I-V format, iut biases, mobility model, mesh geeratio, to outut iformatio etc. All the iut data is the traslated to a text file by the PHP server. The text file is the fed ito the MPI MONOMOS server, which automatically distributes sub-tasks to its cliets that erform their idividual comutatios accordig to the sychroized MONOMOS rograms i a arallel fashio. The arallel algorithm is based o the umber of I-V Fig. 4. Surface lot of the electrostatic otetial for 0.35µm LDD N-MOSFET device with V DS = 2.0V ad V GS = 2.0V.

4 Electro Desity (cm -3 ) Fig. 5. Surface lot of the electro desity for 0.35µm LDD N-MOSFET device with V DS = 2.0V Fig. 6 is a comariso of the simulated ad the exerimetal drai curret versus drai voltage characteristics for W / L eff = 40/0.35 ad V BS = 0V. Note articularly that all of the simulated I-V oits i Fig. 6 were comuted ideedetly by our simulator, i.e. the iitial guesses reuired to start the comutatios of all I-V oits were all the same ad were chose to be the charge eutrality coditio. The followig is a summary of the features of our MPI imlemetatio for the tested model: 8 Liux-based 450-MHZ Petium CPU with 28 MB ersoal comuters, mesh odes o the solutio domai, ad the same stoig criteria for all rocessors. Partially illustrated i Fig. 6, there were totally 9 I-V curves with V GS = 0, , 4V. Each oe of which cosists of 2 I-V oits with V DS ragig from 0 to 4V at 0.2V sacig. Table is a summary of the executio times of a comlete simulatio of all 89 I-V oits versus differet umber of rocessors. A early otimal seedu i CPU time ca be observed i Fig. 7. The suerior scalability of the arallel rocessig is maily due to the ature of the mootoe iterative method. Table. Executio times of 9 I-V curves for 0.35µm LDD N-MOSFET device i a MPI eviromet. Number of Processors Ru Time (Sec) CPU Time (Sec) Measuremet Simulatio W / L eff = 40/0.35 V GS = 3.0 V Data Time (Sec) I DS (ma) V GS =.0 V V DS (V) V GS = 2.0 V Fig. 6. Simulated (dots) ad exerimetal (solid lies) drai curret versus drai voltage characteristics for 0.35µm LDD N-MOSFET device with W / L eff = 40/0.35 ad V BS = 0V. This is very differet from the curret ractice of device simulatio with which well-kow device simulators usually roceed the comutatios of I-V curves by usig the cotiuity techiue stratig from the revious I-V oit as a iitial fuctio to the ext I-V oit due to the local covergece roerty of Newto's iteratio. Such cotiuatio rocess from a low I-V oit to a desired high I-V oit is uite time-cosumig ad is very subtle with the choice of iitial guesses. The comutatios of the I-V oits as that of Fig. 6 ca be erformed i arallel based o the MPI system. Seedu Factor Ru Time CPU Time Number of Processors Fig. 7. Seedu factors of the executio time of 9 I-V curves for 0.35µm LDD N-MOSFET device versus the umber of rocessors. Next examle is to demostrate the simulator with a choice of ET model o a 0.25µm N-MOSFET for a study of the hot carrier effects. The device has ellitical 5E+20 cm -3 Gaussia doig rofiles i source ad drai regios, E+6 cm -3 i the -substrate regio, ad a shallow 5E+7 cm -3

5 imlatatio i chael surface. The juctio deth is 0.3µm ad the lateral diffusio uder gate is 0.09µm. The thickess of the gate oxide is 70A. A tyical simulatio result for electro temerature at V DS = 2.0V ad V GS = 2.0V is show i Fig. 8. Fig. 8. Surface lot of electro temerature rofile at V DS = 2.0V ad V GS = 2.0V. 6 Coclusios A rototye of o-lie semicoductor device simulator is reseted i this aer. The mai features of the simulator are latform-ideedece, WWW I/O iterface, PHP Web server, efficiet kerel MONOMOS, ad MPI distributed comutig eviromet. The stadard Web browser iterface ad Web server rovide the user a very friedly eviromet for various device simulatios ad I/O data maagemet. Ideedet of the iitial guess, MONOMOS allows arallel simulatios of differet I-V curves with various biasig coditios. Comutatioal results demostrate a excellet seedu with resect to the umber of rocessors i the MPI cofiguratio. Corresodig to drift-diffusio ad eergy balaced models, two bechmark N-MOSFET device examles were used to show the accuracy ad efficiecy of the simulator. distributio effects, IEEE Electro Device Letters, Vol. 3, No., 992, [3]. W. Slotboom, Comuter-aided two-dimesioal aalysis of biolar trasistor, IEEE Tras. Electro Devices, Vol. ED-20, 973, [4] S. M. Sze, Physics of Semicoductor Devices, 2d Ed., Wiley-Itersciece, New York, 98. [5] T. I. Seidma ad S. C. Choo, Iterative scheme for comuter simulatio of semicoductor devices, Solid-State Electroics, Vol. 5, 972, [6] R. Bak, D.. Rose, ad W. Fichter, Numerical methods for semicoductor simulatio, SIAM. Sci. Stat. Comut., Vol. 4, No. 3, 983, [7] C. V. Pao, Block mootoe iterative methods for umerical solutios of oliear ellitic euatios, Numer. Math., Vol. 72, 995, [8] Y. Li ad.-l. Liu, A mootoe iterative method for semicoductor device simulatio, 2000, rerit. Refereces: [] Y. Li, S. S. Chug, ad.-l. Liu, A ovel aroach for the two-dimesioal simulatio of submicro MOSFET's usig mootoe iterative method, Proc. It. Symosium o VLSI Techology, Systems, ad Alicatios, 999, [2] D. Che, E. Ka, U. Ravaioli, C.-W. Shu, ad R. Dutto, A imroved eergy trasort model icludig oarabolicity ad o-maxwellia