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UBC Theses and Dissertations

Semiconductor device modelling using the multigrid method Adams, Stephen E. 1988

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S E M I C O N D U C T O R D E V I C E M O D E L L I N G U S I N G T H E M U L T I G R I D M E T H O D B y S t e p h e n E. A d a m s B. M a t h ( A p p l i e d M a t h e m a t i c s a n d C o m p u t e r Sc ience) U n i v e r s i t y of W a t e r l o o A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S I N S T I T U T E O F A P P L I E D M A T H E M A T I C S W e accept th i s thes is as c o n f o r m i n g to the r equ i r ed s t a n d a r d T H E U N I V E R S I T Y O F BRITISH C O L U M B I A J u n e 1988 © S tephen E. A d a m s , 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. 1 further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract T h i s thesis examine s the a p p l i c a t i o n of the m u l t i g r i d m e t h o d to the s em i conduc to r equat ions . A n ove rv iew of s em i conduc to r dev ice m o d e l l i n g i n presented, a n d the m u l t i -g r i d m e t h o d is de sc r ibed . Several mod i f i c a t i on s to the bas ic m u l t i g r i d a l g o r i t h m are e va l ua ted based o n t he i r pe r f o rmance for a one d i m e n s i o n a l m o d e l p r o b l e m . It was f o u n d t h a t u s ing a s y m m e t r i c Gaus s - Se ide l r e l a x a t i o n scheme, a spec ia l p r o l onga t i o n based on the d i sc rete equat ions , a n d l o ca l r e l a x a t i o n sweeps near the p n - j u n c t i o n s p ro -duced a robust, a n d eff icient code. T h i s mod i f i ed a l g o r i t h m is also successfu l for a w ide va r ie ty of cases, a n d i t s pe r f o rmance compares f a vou r ab l y w i t h o the r m u l t i g r i d a l go r i t hms tha t have been app l i ed t o the s em i conduc to r equat ions . 11 Table of Contents Abstract ii List of Tables vii List of Figures viii Acknowledgement ix 1 Introduction 1 2 Introduction to the Model l ing of Semiconductor Devices 5 2.1 B a s i c O p e r a t i o n of S e m i c o n d u c t o r Dev i ce s 5 2.2 D e r i v a t i o n of the S e m i c o n d u c t o r Dev i ce E q u a t i o n s 8 2.2.1 Po i s son ' s E q u a t i o n 8 2.2.2 T h e C o n t i n u i t y E q u a t i o n s 9 2.2.3 T h e C a r r i e r T r a n s p o r t E q u a t i o n s 10 2.3 T h e S e m i c o n d u c t o r E q u a t i o n s 12 2.3.1 B o u n d a r y C o n d i t i o n s 13 2.3.2 S i ngu la r P e r t u r b a t i o n S ca l i n g 14 2.3.3 T r an s f o rma t i on s of Dependen t Va r i ab l e s 16 2.4 P a r a m e t e r M o d e l s 18 2.4.1 T h e D o p i n g P r o f i l e 18 2.4.2 T h e R e c o m b i n a t i o n - G e n e r a t i o n F u n c t i o n 19 2.4.3 T h e C a r r i e r M o b i l i t i e s 20 i i i 2.5 T h e N u m e r i c a l S o l u t i o n of the S e m i c o n d u c t o r E q u a t i o n s 21 2.5.1 D i s c r e t i z i n g the E q u a t i o n s 21 2.5.2 S o l v i n g N o n l i n e a r Sy s tems of E q u a t i o n s 24 3 T h e Mul t igr id M e t h o d 27 3.1 I n t r o d u c t i o n t o the M u l t i g r i d M e t h o d 28 3.2 C o m p o n e n t s of the M u l t i g r i d A l g o r i t h m 33 3.2.1 R e l a x a t i o n O p e r a t o r s 33 3.2.2 R e s t r i c t i o n O p e r a t o r s 35 3.2.3 Coa r se G r i d O p e r a t o r s 35 3.2.4 P r o l o n g a t i o n O p e r a t o r s 36 3.3 E x t e n s i o n s t o N o n l i n e a r Sy s tems of E q u a t i o n s 37 3.4 S o l v i n g the S e m i c o n d u c t o r E q u a t i o n s u s ing the M u l t i g r i d M e t h o d . . . 39 3.4.1 A n t i c i p a t e d D i f f i cu l t i e s 39 3.4.2 D e s c r i p t i o n of the M o d e l P r o b l e m 41 3.4.3 A p p l y i n g S t a n d a r d M u l t i g r i d Techn iques 42 4 A n Improved Mult igr id Implementation 49 4.1 M o d i f i c a t i o n s t o the B a s i c C o d e 49 4.1.1 Improvement s t o the R e l a x a t i o n Scheme 50 4.1.2 M o d i f i e d P r o l o n g a t i o n s 57 4.1.3 A l t e r n a t i v e R e s t r i c t i o n O p e r a t o r s 65 4.2 S u m m a r y of Successfu l M o d i f i c a t i o n s 67 4.3 F u r t h e r T e s t i n g of SC-1 68 4.4 C o m p a r i s o n of SC-1 a n d H e m k e r ' s A l g o r i t h m 74 4.4.1 D e s c r i p t i o n of H e m k e r ' s A l g o r i t h m 74 4.4.2 C o m p a r i s o n Resu l t s 77 iv 5 Discussion 81 5.1 S u m m a r y 81 5.2 Suggest ions for F u t u r e S t u d y 82 Bibliography 85 v List of Tables 2.1 T y p i c a l S ca l i n g Va lue s 15 2.2 R a n g e of Dependen t Va r i ab l e s 17 2.3 Va l ue s of pa r amete r s o c c u r i n g i n R 20 3.4 Conve rgence D a t a for Ba s i c C o d e ( F A S ) 44 3.5 Conve rgence D a t a fo r Ba s i c C o d e ( F M G ) 46 4.6 E f fec t of u s ing l o c a l r e l axa t i on s ( F M G ) 51 4.7 E f f ec t of u s ing l o c a l r e l axa t i on s ( F A S ) 52 4.8 E f f ec t of i n t r o d u c i n g l oca l r e l axa t i on s on f iner levels 52 4.9 E f f ec t of a S y m m e t r i c Gaus s - Se ide l R e l a x a t i o n ( F A S ) 53 4.10 E f fec t of a S y m m e t r i c Gaus s - Se ide l R e l a x a t i o n ( F M G ) 54 4.11 Effect, of a S y m m e t r i c Gauss -Se ide l R e l a x a t i o n , w i t h l o ca l re laxat ions ( F A S ) 54 4.12 E f fec t of a S y m m e t r i c Gaus s - Se ide l R e l a x a t i o n , w i t h l oca l re l axa t ions ( F M G ) 55 4.13 E f fec t of a G a u s s - S e i d e l - G u m m e l R e l a x a t i o n ( F A S ) 56 4.14 E f fec t of a G a u s s - S e i d e l - G u m m e l R e l a x a t i o n , w i t h l o c a l r e l axa t i on s ( F A S ) 56 4.15 E f fec t of p r o l o n g a t i o n P R O L - 1 ( F A S ) 60 4.16 E f fec t of p r o l o n g a t i o n P R O L - 1 ( F M G ) 60 4.17 E f fec t of an a p p r o x i m a t i o n of P R O L - 1 ( F M G ) 61 4.18 E f fec t of p r o l o n g a t i o n P R O L - 2 , w i t h o u t l o ca l r e l axa t i on s ( F A S ) 62 4.19 E f fec t of p r o l o n g a t i o n P R O L - 2 , w i t h o u t l o ca l r e l axa t i on s ( F M G ) . . . . 63 vi 4.20 E f fec t of p r o l o n g a t i o n P R O L - 2 , w i t h l o c a l r e l axa t i on s ( F A S ) 63 4.21 E f fec t of p r o l o n g a t i o n P R O L - 2 , w i t h l o c a l r e l axa t i on s ( F M G ) 63 4.22 E f fec t of p r o l o n g a t i o n P R O L - 3 , w i t h l o c a l r e l axa t i on s ( F A S ) 65 4.23 E f fec t of p r o l o n g a t i o n P R O L - 3 , w i t h l o c a l r e l axa t i on s ( F M G ) 65 4.24 Conve rgence D a t a fo r Re s t i c t i o n s ( F A S ) 66 4.25 Conve rgence D a t a for Re s t i c t i on s ( F M G ) 67 4.26 Conve rgence D a t a for Test 3 ( F o r w a r d B i a s ) 73 4.27 Conve rgence D a t a for Test 3 (Reverse B i a s ) 73 v i i List of Figures 2.1 A pn-junction 7 3.2 T y p i c a l S i m u l a t i o n D o m a i n 40 3.3 P o t e n t i a l ip w i t h V0 = 0 (Test 1) 44 3.4 C a r r i e r concent ra t i on s w i t h V0 = 0 (Test 1) 45 3.5 P o t e n t i a l ip w i t h V0 = 30 (Test 1) 46 3.6 C a r r i e r concent ra t i on s w i t h V0 = 30 (Test 1) 47 4.7 Convergence of S C - 1 , V0 = - 2 0 0 , - 4 0 0 , Nf = 64 (Test 1) 69 4.8 Conve rgence of S C - 1 , V0 = - 2 0 0 , Nf = 16,32,64 (Test 1) 70 4.9 Convergence of S C - 1 , V0 = 200, Nj = 8 ,16,32,64 (Test 1) 70 4.10 Convergence of SC - 1 , V0 = 200,400, Nf = 64 (Test 1) 71 4.11 Convergence of S C - 1 , V0 = 200, Nj = 8 ,16,32,64 (Test 2) 71 4.12 Convergence of S C - 1 , V0 = - 4 0 , Nf = 8 ,16,32,64 (Test 2) 72 4.13 Convergence of S C - 1 , V0 = - 2 0 0 , Nf = 16,32,64 (Test 2) 72 4.14 E s t i m a t i n g Jn and Jv o n non-staggered gr ids 75 4.15 E s t i m a t i n g Jn a n d Jp o n staggered gr ids 76 4.16 Convergence of SC -2 , V0 = - 4 0 , Nf = 8 ,16,32,64 (Test 2) 78 4.17 Convergence of SC -2 , V0 = - 2 0 0 , Nf = 8 ,16,32 (Test 2) 79 4.18 Convergence of SC -2 , V0 = 200,400, Nf - 64 (Test 2) 79 4.19 Convergence of SC -2 , V0 = 200, Nf = 8 ,16,32,64 (Test 2) 80 4.20 Convergence of SC -2 , V0 = - 2 0 0 , Nf = 8 ,16,32,64 (Test 1) 80 v i i i Acknowledgement I w o u l d l i ke t o t h a n k m y superv i so r , D r . U r i A s che r , for suggest ing m y thes is t op i c , a n d p r o v i d i n g i n va l uab l e adv ice. I w o u l d also l i ke t o t hank m y parents , w h o have a lways been s u p p o r t i v e and encou rag ing , even w h e n t hey weren ' t sure wha t I was do ing , or why. A n d I a m p a r t i c u l a r l y i n d e b t e d t o Sa l l y Ross , for reasons t h a t she s t i l l doesn ' t u n d e r s t a n d . IX Chapter 1 Introduction T h e c o n s t r u c t i o n of s em i conduc to r devices is a c o m p l e x process. W h i l e t he mass p ro -d u c t i o n of semiconduc to r s is i nexpens i ve , t he c r ea t i on of a few p r o t o t ype s c an be p ro -h i b i t i v e l y cos t l y and t i m e con suming . B y u s i ng a m a t h e m a t i c a l m o d e l of the devices deve lopers can inves t i gate the p rope r t i e s of h y p o t h e t i c a l dev ices before a c t u a l l y con-s t r u c t i n g one. T h e use of s imu la t i on s reduces the n u m b e r of p r o t o t y p e s con s t ruc ted , w i t h a co r r e spond i ng r e d u c t i o n of costs a n d t i m e c o m m i t m e n t s . T h i s thes i s examines the a p p l i c a t i o n of the m u l t i g r i d m e t h o d t o the n u m e r i c a l s i m u l a t i o n of s em iconduc to r devices. T h e equat ions w h i c h govern the b e h a v i o u r of s em iconduc to r s were f i r s t deve loped by v an Roo sb roeck i n the ear ly 1950's [38]. T h e bas ic s em i conduc to r dev ice equat ions are a s y s tem of non l i nea r p a r t i a l d i f fe rent ia l equat ions w h i c h descr ibe the e lec t ro s ta t i c p o t e n t i a l , cur rent flow, and car r ie r concent ra t ions . These equat i on s are s i ngu l a r l y per -t u r b e d , and are subject to b o t h D i r i c h l e t and N e u m a n n b o u n d a r y cond i t i on s . In terna l b o u n d a r y layers, where the so lu t i on var ies r ap id l y , arise i n the i n te r i o r of the d o m a i n . E a r l y s tudies of the equat ions e m p l o y e d s i m p l i f y i n g , a n d somet imes d ras t i c , as-s umpt i on s t o make the mode l mo re access ible t o a n a l y t i c techn iques [34]. M o r e re-cent ly, a s y m p t o t i c ana lys i s has been used t o p rov ide qua l i t a t i v e i n f o r m a t i o n about the so l u t i on s t ruc tu re [26]. These a n a l y t i c approaches o f ten invo lve a s sumpt i on s wh i ch l i m i t t h e i r a p p l i c a b i l i t y i n techno log i ca l l y i n te re s t ing cases. 1 Chapter 1. Introduction 2 T h e need for so lu t ions t o rea l i s t i c p rob lems has led t o the use of n u m e r i c a l s imu -l a t i on s based d i r e c t l y on the f u n d a m e n t a l equat ions . T h e f i r s t n u m e r i c a l m o d e l of a one d i m e n s i o n a l b i p o l a r t r an s i s t o r was deve loped by G u m m e l i n 1964 [17], a n d a m o d -i f ied a p p r o a c h was l a te r a p p l i e d t o a p n - j u n c t i o n by D e M a r i [12]. S l o t b o o m [35] f i rst used a f u l l two d i m e n s i o n a l m o d e l of a b i p o l a r t r an s i s t o r i n 1969, a n d two d imen s i ona l s imu l a t i on s of t he s t a t i c equa t i on s are now a lmos t s t a n d a r d i n t he deve lopment of p r o to -types [31]. T h r e e d i m e n s i o n a l s imu l a t i on s have been p e r f o r m e d i n recent years [11], b u t t he i r usefulness is o f ten l i m i t e d due t o the in tens i ve c o m p u t a t i o n s requ i red . So lu t i on s of the t i m e dependent p r o b l e m have been r e p o r t e d by M o c k [26] a n d Y a m a g u c h i [39]. Severa l surveys of n u m e r i c a l s imu l a t i on s have been p u b l i s h e d , i n c l u d i n g [31], [30] and [6]-T h e n u m e r i c a l s o l u t i on of the s em i conduc to r equat i on s is a n i n te re s t i ng and cha l -l eng ing p r o b l e m . A n o n s t a n d a r d d i s c r e t i z a t i on of the equat i on s mus t be used, and the equat i on s are p o o r l y sca led. T h e sharp i n t e r na l layers ( i n t e rna l b o u n d a r y layers) re-qu i re t ha t a h i g h l y n o n u n i f o r m m e s h be used to p r o v i de a u n i f o r m l y accurate re so lu t i on of the s o l u t i on eff ic iently. T h e i t e r a t i v e p rocedu re w h i c h is needed t o dea l w i t h the non -l i near i t i e s mus t be b o t h robus t a n d eff ic ient, s ince s imu l a t i on s are u sua l l y p e r f o r m e d for a wide_ range of o p e r a t i n g cond i t i on s . T h e m u l t i g r i d m e t h o d [9] is an efficient n u m e r i c a l a l g o r i t h m for so l v ing the alge-b r a i c equat i on s w h i c h arise f r o m the d i s c r e t i z a t i on of d i f fe rent ia l equat ions . R e l a x a t i o n sweeps are c o m b i n e d w i t h coarse g r i d co r r ec t i on steps t o p r o d u c e a scheme w h i c h is, t heo re t i ca l l y , o p t i m a l l y eff ic ient. T h e m e t h o d has been successfu l ly used for m a n y p rob -lems, but i t is not a lways obv iou s how it s hou ld be i m p l e m e n t e d t o a t t a i n the m a x i m u m eff ic iency for a new p r o b l e m . P r e v i o u s a t t e m p t s to app l y the m u l t i g r i d m e t h o d t o the s e m i c o n d u c t o r equat ions ([7], [15] a n d [19]) have met w i t h l i m i t e d success, a n d do not a lways appea r t o be as efficient as poss ib le. T h i s thesis eva luates several mod i f i c a t i on s Chapter 1. Introduction 3 t o t he bas i c m u l t i g r i d a l g o r i t h m , a n d demons t ra te s t h a t the m u l t i g r i d m e t h o d can be success fu l l y used for s e m i c o n d u c t o r dev ice m o d e l l i n g i n one space d imens i on . A n ove rv i ew of s em i conduc to r dev ice m o d e l l i n g is p resented i n C h a p t e r 2. §2.1 con ta in s a s imp le i n t r o d u c t i o n t o s em i conduc to r devices, a n d a d e r i v a t i o n of the gov-e r n i n g equat ions is p resented i n §2.2. Some a n a l y t i c aspects of the p r o b l e m , i n c l u d i n g the b o u n d a r y cond i t i on s , a n a p p r o p r i a t e sca l ing , a n d a r e f o r m u l a t i o n o f the p r ob l em i n t e rms of new dependent var iab les , are d i scussed i n §2.3. T h e f o r m of the phy s i ca l p a r a m e t e r s o c c u r r i n g i n the equat ions is de ta i l ed i n §2.4. §2.5 out l i nes ce r t a i n aspects of t he n u m e r i c a l s o l u t i o n of the p r o b l e m , p re sent ing a s tab le d i s c r e t i z a t i on of the equa-t i on s a n d de sc r i b i ng the two method s w h i c h are c o m m o n l y used t o solve the non l i near equat ions . C h a p t e r 3 conta in s a b r i e f i n t r o d u c t i o n to the m u l t i g r i d m e t h o d . §3.1 discusses the m u l t i g r i d i d e a a n d presents t he f u n d a m e n t a l a l go r i t hms . T h e component s of the a l g o r i t h m are d i scussed i n more de t a i l i n §3.2, wh i l e t he ex tens i on of the bas ic m e t h o d to sy s tems of non l i nea r equat i on s is o u t l i n e d i n §3.3. §3.4 discusses t he app l i c a t i on o f the m u l t i g r i d m e t h o d t o the s em i conduc to r equat ions . T h e p rob lems wh i ch are e x p e c t e d t o arise are o u t l i n e d , a n d the behav i ou r of a s t r a i gh t f o rwa rd i m p l e m e n t a t i o n is de sc r i bed . Improvement s t o the bas ic a l g o r i t h m are desc r ibed i n C h a p t e r 4. S y m m e t r i c re lax-a t i o n i t e ra t i on s , p r o l onga t i on s based on the d i s c re t i zed equat ions , and l o c a l r e l axa t i on sweeps a l l s i gn i f i can t l y i m p r o v e the convergence of the m e t h o d . These mod i f i ca t i on s , a n d t he i r effects, are desc r ibed i n § 4 . 1 . T h e most successfu l va r iant s are co l lected in to one p r o g r a m (SC-1 ) . §4.3 demons t ra te s tha t SC-1 is successfu l for a va r ie ty of p rob -lems, i n c l u d i n g the m o d e l l i n g of a t h y r i s t o r . §4.4 compare s SC-1 t o an a d a p t a t i o n of a m u l t i g r i d i m p l e m e n t a t i o n desc r ibed by H e m k e r [19]. Fo r the p rob lems cons idered, our p r o g r a m is mo re eff ic ient a n d more robust t h a n H e m k e r ' s vers ion. Chapter 1. Introduction 4 C h a p t e r 5 s ummar i z e s the findings of th i s thesis, a n d conta in s some suggest ions for f u t u r e work i n th i s area. Chapter 2 Introduction to the Model l ing of Semiconductor Devices A comp le te d i scus s ion of the behav i ou r of s em i conduc to r dev ices w o u l d requ i re a k n o w l -edge of so l i d s tate phys ics a n d q u a n t u m mechan ic s , bu t for our purposes an i n t u i t i v e u n d e r s t a n d i n g of t he i r o p e r a t i o n w i l l be suff ic ient. §2.1 presents a s imp l i f i ed v i ew of the phys i c s of s em i conduc to r dev ices a n d in t roduces the p h e n o m e n a that make semiconduc -tor s t e chno l og i c a l l y i n te res t ing . A comp le te d i scuss ion of the phys ic s of s em i conduc to r dev ices c a n be f ound i n [31] and [34]. T h e r e m a i n d e r of th i s chapte r conta ins an overv iew of s em i conduc to r dev ice m o d -e l l i ng . §2.2 a n d §2.3 d iscuss the m o d e l i t se l f — the non l i nea r equat ions w h i c h govern s em i conduc to r s are deve loped i n §2.2, and §2.3 out l i nes the b o u n d a r y cond i t i on s , and some a l t e rna t i v e f o rmu l a t i on s of the p r o b l e m . §2.4 descr ibes the f o r m of the p a r a m -eters w h i c h o c cu r i n the bas i c equat ions . §2.5 discusses the n u m e r i c a l s o l u t i on of the equat ions . 2.1 Basic Operation of Semiconductor Devices T r a d i t i o n a l l y ma te r i a l s have been class i f ied acco rd i ng t o t he i r e l ec t r i ca l p roper t ie s as e i ther c onduc to r s or in su la to r s : under n o r m a l cond i t i on s conduc to r s conduct e l ec t r i c i t y eas i ly w h i l e i n su l a to r s resist t he current f low. T h i s c las s i f i cat ion has been supp lemented w i t h ano the r category, semiconducto r s , w h i c h a l low current t o flow more eas i ly t h a n i n su l a to r s do, bu t are s t i l l r e l a t i ve l y poo r conducto r s . T h e e l e c t r i c a l p roper t ie s of a m a t e r i a l can be exp l a i ned i n terms of the i n t r i n s i c 5 Chapter 2. Introduction to the Modelling of Semiconductor Devices 6 c o n c e n t r a t i o n of c o n d u c t i o n e lect rons i n the so l i d [34]. ( C o n d u c t i o n e lect rons are those e lect rons w h i c h are i n an e lec t ron b a n d so far r emoved f r o m the nuc leus t h a t they can move about f reely; va lence e lect rons are e lect rons w h i c h occupy energy band s nearer the nuc leus a n d are respons ib le for c hem i ca l bond ings . ) A t y p i c a l s em i conduc to r l i ke s i l i c on w i l l have 1 0 1 0 c o n d u c t i o n e lect rons pe r c m 3 ; for a m e t a l there w i l l be r ough l y 1 0 2 2 per c m 3 wh i l e for an i n s u l a t o r there w i l l be j u s t a few t h o u s a n d c o n d u c t i o n e lectrons pe r cm3. Sem i conduc to r s are i m p o r t a n t because a s l ight increase i n the energy leve l (caused by a p p l y i n g an e lec t r i c f ie ld or r a i s i ng the t e m p e r a t u r e ) w i l l convert la rge numbe r s of va lence e lect rons t o c o n d u c t i o n e lectrons. O b v i o u s l y th i s w i l l create gaps i n the valence b a n d where the va lence e lect rons p rev i ou s l y were. The se gaps, w h i c h are ca l l ed holes, c an be t r e a t ed as po s i t i ve l y charged carr iers . T h e movement of the holes a n d e lect rons creates a cur rent . A s l ight change i n the energy leve l changes the cha rac te r of the m a t e r i a l , a n d the s em i conduc to r becomes a m u c h be t t e r conduc to r . In t h e r m a l e q u i l i b r i u m the n u m b e r of holes is equa l t o the n u m b e r of c o n d u c t i o n e lect rons ( th i s e q u i l i b r i u m concen t r a t i on is deno ted n t ) . T h e e l e c t r i c a l p roper t i e s of the dev ice c an be changed by a process k n o w n as dop ing . T h e r e are two po s s i b i l i -t ies: e i the r acceptor a toms, w h i c h can .accept e lect rons a n d thus c reate holes, or donor a toms w h i c h can p r oduce excess c o n d u c t i o n e lectrons , can be i m p l a n t e d i n t o the m a -t e r i a l . T h e concen t r a t i on of the i m p l a n t e d carr iers can be severa l o rders of m a g n i t u d e la rger t h a n the i n t r i n s i c leve l so d o p i n g changes the c o n d u c t i v i t y s i gn i f i cant ly . It is t he a r rangement of the i m p u r i t i e s ( the d o p i n g prof i le ) w h i c h determines the behav i ou r of the s em iconduc to r dev ice. T h e s implest e x amp l e of a p r a c t i c a l d op i n g pro f i le is the p n - j u n c t i o n i l l u s t r a t e d i n F i g u r e 2.1. T h e net d o p i n g leaves the left (p-side) reg ion w i t h an excess of acceptors wh i l e t he r ight (n-s ide) is d o m i n a t e d by donor a toms. T h e inter face be tween the n-s ide Chapter 2. Introduction to the Modelling of Semiconductor Devices anode acceptor concentration cathode 10* cm"' donor concentration F i g u r e 2.1: A p n - j u n c t i o n a n d the p-side is k n o w n as a p n - j u n c t i o n . If a f o r w a r d b ias is a p p l i e d t o the d i ode (i.e., the vo l tage at t he p-s ide is m a d e po s i t i v e w i t h respect t o the vo l tage at the n-s ide) a la rge cu r ren t resu l t s , even for a s m a l l b ias. B u t f o r a reverse b ias ( w i t h the vo l tage at t he p-side less t h a n at the n-s ide) the current f low is neg l i g ib le ; even for a large negat ive vo l tage on l y a s m a l l ( leakage) cu r ren t results. T h e p n - j u n c t i o n can be cons idered as a s w i t c h , o r va lve, w h i c h a l lows a cur rent to flow i n one d i r e c t i on only. T h e p n - j u n c t i o n is t he s imp les t b u i l d i n g b l ock of s e m i c o n d u c t o r technology, bu t i t s b e h a v i o u r i l l u s t r a te s the i m p o r t a n c e of the d o p i n g prof i le . A d i s cu s s i on of the p n -j u n c t i o n a n d o the r m o r e s oph i s t i c a t ed dev ices can be f ound i n severa l t ex t s , i n c l u d i n g [23] a n d [34]. Chapter 2. Introduction to the Modelling of Semiconductor Devices 8 2.2 Derivation of the Semiconductor Device Equations T h e b e h a v i o u r of a s em iconduc to r dev ice can be de sc r i bed i n te rms of the e lec t ro s ta t i c p o t e n t i a l , the ca r r i e r concent ra t i on s , a n d the cur rent f low. A de ta i l ed de r i v a t i on of the gove rn ing equat i on s can be f o u n d i n [31]. T h e br iefer d i scus s ion w h i c h fo l lows is based on the deve lopment f o u n d i n [24]. W e beg i n w i t h M a x w e l l ' s equat ions : c u r l t f c u r l £ d i v D d i v S where E a n d D are the e lec t romagnet i c and d i sp lacement vectors , H a n d B are the m a g n e t i c f i e ld a n d i n d u c t i o n vectors , J is the c o n d u c t i o n cu r rent dens i t y a n d p is t he space charge density. T h e e lect r i c field a n d the d i sp lacement can be re l a ted by D = eE where e, the p e r m i t t i v i t y of the m e d i u m , is genera l l y a 3x3 m a t r i x . Fo r most c o m m o n app l i c a t i on s of s em i conduc to r devices i t is reasonab le t o assume t ha t the m e d i u m is i s o t r op i c so t ha t e can be t rea ted as a scalar. 2.2.1 Poisson's Equation T h e t h i r d of M a x w e l l ' s equat ions (2.3) can be r e w r i t t e n as Po i s son ' s e q u a t i o n by r e l a t i n g the e lec t r i c field vec to r E t o the e lec t ro s ta t i c p o t e n t i a l ip. E q u a t i o n (2.4) can be sat i s f ied by i n t r o d u c i n g a vec to r field A so t ha t B = c u r l A S u b s t i t u t i n g th i s r e l a t i o n i n to (2.2) gives dA c u r l ( £ + — ) = 0. (2.5) F o r a s imp l y connected d o m a i n , th i s imp l i e s t ha t E c an be expressed as: E=-?£- VV-. (2.6) T d D _dB ~ dt P 0 (2.1) (2.2) (2.3) (2-4). Chapter 2. Introduction to the Modelling of Semiconductor Devices 9 C o m b i n i n g th i s r e l a t i o n w i t h (2.3) gives (for homogeneous e) V(eVV>) = -p, (2.7) w h i c h is t he u sua l f o r m of Po i s s on ' s equa t i on . T h e space charge dens i t y p c an be expressed i n te rms of t he e l ementa r y charge q t imes the s u m of the p o s i t i v e l y charged hole c oncen t r a t i on p, the negat i ve l y charged e l ec t ron c o n c e n t r a t i o n n a n d a n d t he d o p i n g pro f i le C: p = q(p-n + C). (2.8) T h e d o p i n g pro f i le C is a f u n c t i o n of po s i t i on only. W h i l e th i s s u b s t i t u t i o n can be v iewed as a m a t h e m a t i c a l s u b s t i t u t i o n , i t is phy s i c a l l y reasonab le on l y i f some cond i -t ions o n n a n d p h o l d [31]. E q u a t i o n (2.7) becomes (for cons tant e) AV> = * ( n - p - C ) . (2.9) 2.2.2 T h e Continuity Equations T h e p r i n c i p l e of conse rva t ion of charge leads t o two con t i nu i t y equat ions . A p p l y i n g the d ivergence ope r a t o r to the first of M a x w e l l ' s equat ions (2.1) gives dp d i v ( c u r L t f ) = d i v J + = 0. (2.10) T h e c o n d u c t i o n cur rent dens i t y J can be sp l i t i n t o two componen t s Jn a n d Jp : J = Jn + Jp. Jn is t he current c rea ted by the c o n d u c t i o n of e lectrons, w h i l e Jp is t he cu r ren t caused by t he holes. T h e d o p i n g prof i le C is t i m e i nva r i an t , so t h a t (2.10) becomes dp dn - d i v J p - q— = d i v J n - q —. (2.11) T h i s r e l a t i o n can be r e w r i t t e n as two c o n t i n u i t y equat ions by f o r m a l l y i n t r o d u c i n g a f u n c t i o n R, so t h a t dp divJp + q-£ = -qR (2.12) Chapter 2. Introduction to the Modelling of Semiconductor Devices 10 d i v J n -q~^ = qR- (2.13) N o new i n f o r m a t i o n is ga ined by r e w r i t i n g (2.11) as t w o equat ions , bu t th i s f o rmu -l a t i o n leads t o a s imp le , phy s i c a l i n t e r p r e t a t i o n of t he f u n c t i o n R. R c an be unde r s t ood as t he d i f ference be tween the ra te at w h i c h e lec t ron -ho le pa i r s comb ine a n d the ra te at w h i c h t h e y are generated. F o r th i s reason R is u sua l l y ca l l ed the r e c o m b i n a t i o n -genera t i on f u n c t i o n . R > 0 means t h a t net r e c o m b i n a t i o n is o c cu r i n g , w h i l e negat ive R means t h a t pa i r s are b e i n g generated. R can be m o d e l l e d p h y s i c a l l y u s i ng t he methods of s t a t i s t i c a l phys ics ; c o m m o n l y used mode l s are m e n t i o n e d be low. 2.2.3 T h e Carr ier Transport Equations E x p r e s s i n g the cu r ren t dens i t ies Jn a n d Jp i n t e rms of the e lect r i c f ie ld a n d the car r ier c oncen t r a t i on s is a c umbe r s ome task. B o l t z m a n n ' s t r a n s p o r t equa t i on a n d the methods of s t a t i s t i c a l phys ics are used t o der ive, after l eng thy ca l cu l a t i on s , f a i r l y s imp le results. A m o r e i n t u i t i v e a r gument is used here t o deve lop the same re la t ions ; a r i gorous devel -o p m e n t , w h i c h discusses the necessary a p p r o x i m a t i o n s a n d analyses t he errors i nvo lved i n m a k i n g these a s sumpt i on s , is g i ven i n [31]. C u r r e n t f low i n s e m i c o n d u c t o r dev ices has two m a i n sources: the d i f fus ion of elec-t ron s a n d holes due t o the v a r i a t i o n of t he i r concent ra t i on s , a n d the d r i f t of the carr iers due t o the v a r i a t i o n of the e lect r i c field. T h e e lec t ron a n d ho le cur rent flows are deter-m i n e d by the s u m of the d i f fu s ion a n d dr i f t t e rms , so t ha t Jn = Jnji + J?'1* (2.14) a n d Jv = Jf" + JW\ _ (2.15) where J ^ J f , J*1** are the cur rent dens i t ies due to d i f fu s i on , and J*™**, J*7"1** are due to the d r i f t . Chapter 2. Introduction to the Modelling of Semiconductor Devices 11 T h e car r ie r s diffuse i n the d i r e c t i o n of the g rad ient of the c oncen t r a t i on f r o m regions of h i gher c o n c e n t r a t i o n t o regions of lower concen t r a t i on . T h e t o t a l f l u x is p r o p o r t i o n a l t o th i s g rad ient a n d the charge pe r pa r t i c l e q, so J * » = qDnVn (2.16) a n d J?" = -qDvVp. (2.17) ( T h e s igns of the r ight h a n d sides are chosen so t ha t t he d i f fu s ion coeff ic ients Dn and Dp are pos i t i ve . ) T h e d r i f t cu r rent dens i t ies are s imp l y the p r o d u c t s of the charge pe r pa r t i c l e , t he ca r r i e r concen t ra t i on s , a n d the average dr i ft ve loc i t ie s , denoted by a n d for elec-t rons a n d holes respect ive ly . T h u s , Jn%it = qnvdn (2.18) a n d Krxjt = wi- (2-19) F o r m o d e r a t e field s t rengths t he dr i f t ve loc i t ies are p r o p o r t i o n a l t o the e lect r i c field, so t h a t i / = finE a n d i/f = fipE, where fip a n d \in are the hole a n d e lec t ron mob i l i t i e s , respect ive ly . The se re la t ions for the d i f fu s ion a n d dr i f t component s of the cur rent a l l ow us to w r i t e : Jn = qDnVn + qpnnE (2.20) Jp = -qDpVp + q^pPE. (2.21) F i n a l l y , E i n s t e i n ' s re la t ions Dn = Vj\in and Dp = UTP~P are used t o re la te the hT d i f fu s ion coeff ic ients t o the mob i l i t i e s . Here Uj = is the t h e r m a l vo l tage and fcf, Chapter 2. Introduction to the Modelling of Semiconductor Devices 12 is B o l t z m a n n ' s cons tant . E i n s t e i n ' s r e l a t i on s are a p p r o x i m a t i o n s w i t h an accep tab l y s m a l l e r ror i f the s e m i c o n d u c t o r dev ice is nondegenerate a n d the dev ice t e m p e r a t u r e T is s pa t i a l l y homogeneous [31]. These cu r ren t re l a t i on s c a n be w r i t t e n i n terms of rj} by a s suming t h a t t he i n d u c t i o n vec to r B is i ndependen t of t i m e , a n d s u b s t i t u t i n g E = -Vi/>. A m o r e r igorous deve lopment p roduce s t he same resu l t s , b u t p rov ides mo re ins ight i n t o t he phy s i c a l a s sumpt i on s w h i c h are necessary fo r (2.20) a n d (2.21) t o ho l d . These equat i on s neglect any cur rent not caused by d i f fus ion o r the e lec t r i c f i e ld . In pa r t i c u l a r , t e m p e r a t u r e effects a n d the v a r i a t i o n of t he i n t r i n s i c ca r r i e r c oncen t r a t i o n , w h i c h m a y be i m p o r t a n t fo r some app l i c a t i on s , are neg lected. A l s o , t he s upe rpo s i t i o n of terms i n (2.20) a n d (2.21) is convenient bu t no t necessar i ly cor rect . 2.3 The Semiconductor Equations T h e bas ic s em i conduc to r equat ions w h i c h w i l l be used t h r o u g h o u t t he r e m a i n d e r of th i s thes is are co l l ec ted i n th i s sect ion. T h e b o u n d a r y c ond i t i on s w h i c h supp lement these equat ions are d i scussed i n §2.3 .1 , a n d the equat ions are expres sed i n te rms of n o n d i m e n s i o n a l var iab les i n §2.3.2. §2.3.3 presents two a l t e rna t i v e f o r m u l a t i o n s of the equat ions . T h e m a j o r a s sumpt i on s w h i c h are necessary fo r these bas ic equat i on s to be va l i d are: t h a t the cu r rent flow is caused on l y by d i f fu s ion a n d the d r i f t caused by the e lectr ic f i e ld ; t h a t E i n s t e i n ' s r e l a t i on s ho ld ; t h a t t he dev ice t e m p e r a t u r e T is cons tant ; a n d tha t the i n d u c t i o n vec to r B is i ndependent of t ime . These a s sumpt i on s h o l d for the m a j o r i t y of app l i ca t i on s . T h e f o l l ow i ng s y s tem of p a r t i a l d i f ferent ia] equat ions is ob t a i ned : eA</> = q(n-p-C) (2.22) d i v J n - 9 ^ = qR (2.23) Chapter 2. Introduction to the Modelling of Semiconductor Devices 13 d i v J p + g ^ = -qR (2.24) Jn — qDnVn — qfinn\7il; (2.25) J p = -qDpVP- g/XppVV'. (2.26) S u b s t i t u t i n g (2.25,2.26) i n t o (2.23,2.24) resu l t s i n a s y s tem of th ree non l i nea r p a r t i a l d i f fe rent ia l equa t i on s fo r rp,n a n d p. These equa t i on s were f i rst presented by V a n Roo sb roeck [38] i n v i r t u a l l y the same f o r m i n 1950. W e w i l l cons ider on l y t he s teady s tate s e m i c o n d u c t o r p r o b l e m , o b t a i n e d b y se t t i ng t he t i m e der i va t i ves t o zero. 2.3.1 Boundary Conditions S e m i c o n d u c t o r dev ices are i n t r i n s i c a l l y three d i m e n s i o n a l bu t the bas ic equat ions can also be posed i n one or two d imens ions . Fo r a one d i m e n s i o n a l p r o b l e m the b o u n d a r y of the dev ice is t r i v i a l l y represented by ju s t two po in t s ; fo r m u l t i - d i m e n s i o n a l p r ob l ems the b o u n d a r y 3D of the d o m a i n D is cons idered t o be p iecewise s m o o t h . 3D can be sp l i t i n t o two par t s : 3D — 3DP U 3Da, where 3Dp co r responds t o the ac tua l , p h y s i c a l dev ice bounda r i e s , a n d 3Da cons ists of those pa r t s of the b o u n d a r y w h i c h are a r t i f i c i a l l y i n t r o d u c e d t o l i m i t the size of the d o m a i n . T h e phy s i c a l pa r t of the b o u n d a r y cons ists o f i n s u l a t i n g bounda r i e s a n d contact s . A t i n s u l a t i n g a n d a r t i f i c i a l bounda r i e s t he componen t s of the e lect r i c field and cur rent dens i t ies n o r m a l t o the b o u n d a r y van i sh : E • r\dDN = Jn • r\dDN = Jp • r\dDN = 0 (2.27) where r is the u n i t o u t w a r d n o r m a l t o 3D, a n d 3D^ is the un i on of t he i n s u l a t i n g a n d a r t i f i c i a l bounda r i e s . Homogeneous N e u m a n n cond i t i on s h o l d on 3D^\ Chapter 2. Introduction to the Modelling of Semiconductor Devices 14 A t an O h m i c con tac t t h e r m a l e q u i l i b r i u m a n d a van i sh ing space charge are as sumed to h o l d , so np = n 2 a n d p = n — p — C = 0. ip is a s sumed t o be the s um of the b u i l t -i n p o t e n t i a l w h i c h is p r o d u c e d b y the d o p i n g a n d the a p p l i e d p o t e n t i a l Vo- These a s sumpt i on s l e ad t o the D i r i c h l e t c ond i t i on s n\dDo = l / 2 ( C + V /C2 + 4n t 2 ) (2.29) p\dDo = l/2(-C + yJc* + 4n2) (2.30) ( C + JC2 + An2) 4>\dDo = UT\nC- X- -) + V0. (2.31) Here dDo is the u n i o n of a l l contact s i n the boundary . In genera l Vo is a f u n c t i o n of t i m e , b u t for the s t a t i ona r y p r o b l e m it is t r e a t ed as a constant . O t h e r , less c o m m o n , b o u n d a r y cond i t i on s arise w h e n sem iconduc to r - ox ide i n te r -faces a n d S c h o t t k y con tac t s are cons idered. A t ho r ough d i scuss ion of the b o u n d a r y c ond i t i on s c an be f o u n d i n [31]. 2.3.2 Singular Perturbation Scaling T h e b e h a v i o u r of the dependent var iab les becomes more manageab le i f a su i tab le sca l i ng is used, a n d the govern ing equat ions are expressed i n te rms of d imens ion less quant i t ie s . T h e r e are, of course, several ways t o do th i s [30]. T h e sca l ing w h i c h best i so lates the s t r u c tu re of the equat ion s is the s ingu la r p e r t u r b a t i o n sca l i ng [24]. W i t h th i s sca l ing , a l l l engths are sca led b y a cha rac te r i s t i c dev ice l eng th I, a n d the po tent i a l s are scaled by the t h e r m a l vo l tage UT- T h e d o p i n g pro f i le C ( z ) , a n d the car r ie r concent ra t ions , are sca led by C m , t he m a x i m u m m a g n i t u d e of C(x). F i n a l l y the car r ie r mob i l i t i e s pn a n d / i p are sca led t o have order of m a g n i t u d e 1. T y p i c a l s ca l i ng values are shown i n T a b l e 2.1 [24]. T h e r e s u l t i n g scaled equat ions , w i t h the scaled quant i t i e s denoted by the same Chapter 2. Introduction to the Modelling of Semiconductor Devices 15 V a r i a b l e Sca l i ng F a c t o r X 5 x 1 0 ~ 3 c m v> 0.0259V n 1 0 1 7 c m - 3 P 1 01 7 c m - 3 1 0 1 7 c m - 3 l O O O c m 2 ! / - 1 ^ - 1 l O O O c m 2 ^ - 1 ^ - 1 T a b l e 2.1: T y p i c a l S ca l i n g Va lues s ymbo l s as the unsea led var iab les were, are: A2AV> d i vJp .+ — d i v J ^ -dp dt dn ~dt (n-p-C) -R R fj,n(Vn — nVt/») - / x p ( V p + pVV>) where A IJUT_ qCmP (2.32) (2.33) (2.34) (2.35) (2.36) is a ve ry s m a l l va lue , t y p i c a l l y of the order of m a g n i t u d e 1 0 ~ 3 t o 1 0 - 5 . A n o t h e r s m a l l p a r a m e t e r 62 = nx/Cm arises w h e n the b o u n d a r y cond i t i on s are co r re spond ing l y sca led bu t does not appea r i n the d i f fe rent ia l equat ions. T h e equat i on s are s ingu la r l y p e r t u r b e d since a sma l l p a r amete r A 2 mu l t i p l i e s the h ighest de r i va t i ve i n Po i s son ' s equa t i on . T h e so lu t i on component s are expec ted t o have layers across j u n c t i o n s of the dop i n g prof i le , since where tp is s lowly v a r y i n g n—p ~ C ( x ) (as A —• 0) a n d C(x) has d i s cont inu i t i e s at the j u n c t i o n s (see be low) . A n u m e r i c a l s o l u t i o n of t he equat ions shou ld account for th i s behav iou r : a fine d i s c r e t i z a t i on w i l l Chapter 2. Introduction to the Modelling of Semiconductor Devices 16 be needed near the layers i f t he s o l u t i on is t o be a ccu ra te l y a p p r o x i m a t e d there, bu t a coarse mesh w i l l suffice elsewhere. 2.3.3 Transformations of Dependent Variables Fo r b o t h a n a l y t i c a n d c o m p u t a t i o n a l work there are poss ib le advantages t o u s i ng sets of dependent var iab les o the r t h a n tp, n a n d p [30]. O n e t r a n s f o r m a t i o n i n c o m m o n use replaces t he unsea led ca r r i e r concen t ra t i on s n a n d p by the qua s i - F e rm i levels <f>n a n d (j>v as fo l lows: n — e i'-4>n (2.37) p = t**~*. (2.38) ( In t e rms of the sca led var iab les , th i s t r a n s f o r m a t i o n is: n = 6 2 e ^ - 1 * " , p = 6 2 e * p _ ^ . ) T h e bas ic (unsealed) equa t i on s (2.22 - 2.26) are t r a n s f o r m e d t o : eAV> - q(e*-*» - e*>~* - C(x)) = 0 (2.39) d i v ( / i n e ^ - * » V ^ n ) + /2 = 0 (2.40) d i v ( M p e ^ V > p ) - R = 0, (2.41) so the c o n t i n u i t y equat i on s are now i n d ivergence f o r m . T h e r e is a phy s i ca l i n te rp re -t a t i o n of <f>n a n d <j)p, b u t fo r ou r purposes i t is reasonab le t o t reat the t r a n s f o r m a t i o n s i m p l y as a change of var iab les . T h e second t r a n s f o r m a t i o n t o the S l o t b o o m var iab le s u a n d v reads: n = e^u (2.42) p = e ~ V (2.43) or, equ i va len t l y : u = e ~ * " (2.44) Chapter 2. Introduction to the Modelling of Semiconductor Devices 17 v = e**. (2.45) ( T h e co r r e spond i n g t r a n s f o r m a t i o n i n t e rms of the sca led var iab les is: n = 82e^u, p = 82e~^v.) T h e r e s u l t i n g (unsealed) equat i on s are: eAV> - qie^u - e^'v - C(x)) = 0 (2.46) d i v ( / z B e * V u ) - R = 0 (2.47) d i v ( / i p e ~ * V u ) - R = 0. (2.48) B o t h t r a n s f o r m a t i o n s " r e g u l a r i z e " the p r o b l e m , i n t ha t <f>n, <f>p, u a n d v are a l l s low var iab les (they va ry s lowly t h r o u g h the j u n c t i o n s ) , a n d the equat ions are no longer s ingu la r , s i n gu l a r l y p e r t u r b e d . T h e S l o t b o o m var iab le s are a t t r a c t i v e m a t h e m a t i c a l l y because (2.47) a n d (2.48) are l i near i n u and v, w h i l e (2.40) and (2.41) are quas i l i near i n 4>n a n d <pp; f u r t h e r (2.47) and (2.48) are self ad jo in t for g iven values of tp a n d R. T h e d y n a m i c ranges of (<f>n,<f)p) a n d (n,p) are m o d e r a t e , and successful n u m e r i c a l c o m p u -t a t i o n s have been p e r f o r m e d u s i ng b o t h sets of va r iab les [30]. T h e S l o t b o o m var iab les are p o o r l y sca led; t h i s f o r m u l a t i o n is genera l l y o n l y usefu l for a na l y t i c invest igat ions . T h e t y p i c a l va lues w h i c h the sca led var iab les t ake on are s u m m a r i z e d i n Tab le 2.2 [30]. Q u a n t i t y R a n g e n [ i o - 2 0 , i o ] P [ i o -2 0 , i o ] [0,193.4] <t>v [0,193.4] U [ i , i o 8 4 ] V [ i o - 8 4 , i ] T a b l e 2.2: Range of Dependent Va r i ab le s Chapter 2. Introduction to the Modelling of Semiconductor Devices 18 2.4 Parameter Models T h e f o r m of the pa ramete r s a r i s i ng i n the deve lopment of the s em i conduc to r equat ions needs t o be spec i f i ed before f u r t he r ana ly s i s c an be per fo rmed. T h e m o d e l l i n g of these p h y s i c a l p a r amete r s is a d i f f i cu l t task i n v o l v i n g research i n so l i d s tate phys ic s [31]. P r a c t i c a l l y t he pa ramete r s are k n o w n on l y as the (numer i ca l ) s o l u t i on t o ano the r set of p a r t i a l d i f f e ren t i a l equat ions , o r resu l t f r o m f i t t i n g mea su red d a t a t o t heo re t i c a l mode l s . T h e f unc t i on s w h i c h are de sc r i bed be low ( in terms of the unsea led var iab les ) are c o m m o n l y used, bu t there is s t i l l some unce r t a i n t y i n these mode l s . A comp le te d i scus s ion of the p a r a m e t e r mode l s is c on ta i ned i n [31]. 2.4.1 T h e Doping Profile T h e b e h a v i o u r of a s em iconduc to r dev ice is m a i n l y de te rm ined by i t s d o p i n g prof i le , the c o n c e n t r a t i o n of dono r a n d accepto r i m p u r i t i e s i n the dev ice. T h e c rea t i on of the d o p i n g p ro f i l e is a c o m p l i c a t e d process i n v o l v i n g ion i m p l a n t a t i o n , d i f fus ion a n d t h e r m a l o x i d a t i o n . T h e r e s u l t i n g dop i n g pro f i le C(x) is the net difference of e l ec t r i ca l l y ac t i ve donors a n d acceptors . Reg ion s i n w h i c h C(x) > 0 (i.e., the concen t ra t i on of donors exceeds t he c o n c e n t r a t i o n of acceptor s ) are t e r m e d n -doma ins , wh i l e regions where C(x) < 0 are ca l l ed p -doma ins . T h e b o u n d a r y be tween an n -doma in a n d a p -doma in is ca l l ed a p n - j u n c t i o n . T h e dop i n g pro f i le var ies e x t r eme l y r ap i d l y near pn - j unc t i on s bu t is re l a t i ve l y cons tant away f r o m t he j unc t i on s . Fo r m a t h e m a t i c a l convenience j unc t i o n s are o f ten m o d e l l e d by a b r u p t t r an s i t i on s , so t ha t C(x) has j u m p d i scont inu i t ie s . T h e d i s con t i nu i t i e s are nonphy s i c a l b u t they p r o v i de a good a p p r o x i m a t i o n t o m a n y rea l i s t i c prof i les , and are convenient for m a t h e m a t i c a l analys i s . T h e d o p i n g f u n c t i o n can also be m o d e l l e d us ing er ror f unc t i on s and decay i ng exponent ia l s . Chapter 2. Introduction to the Modelling of Semiconductor Devices 19 C(x) t y p i c a l l y takes on values i n the range ( — 1.0 x 10 1 8 , 1 . 0 x 1 0 1 8 ) . 2.4.2 T h e Recombination-Generation Function T h e d e r i v a t i o n of the s em i conduc to r dev ice equat ions i n t r o d u c e d the r e c o m b i n a t i o n -genera t i on f u n c t i o n R i n a pu re l y f o r m a l way. T h i s f u n c t i o n has a phy s i c a l i n t e r p r e t a -t i o n as the net d i f ference of the rates of r e c o m b i n a t i o n a n d gene ra t i on of e l ec t ron - ho le pa i r s . R e c o m b i n a t i o n descr ibes the n e u t r a l i z a t i o n of a ho le w h e n a c o n d u c t i o n e l ec t ron becomes a va lence e lec t ron , wh i l e genera t i on refers t o the oppo s i t e process i n w h i c h a valence e lec t ron becomes a c o n d u c t i o n e lec t ron . In t h e r m a l e q u i l i b r i u m the concen t ra t i on s of e lect rons a n d holes w i l l fluctuate b u t , ove ra l l , there w i l l be a d y n a m i c ba l ance so R = 0. T h e a p p l i c a t i o n of an e x t e r n a l s t imu lu s w i l l d i s t u r b the e q u i l i b r i u m , a n d var ious processes o c cu r w h i c h a t t e m p t t o create a new steady-s tate. If excess car r ie r s have been c rea ted t h e n r e c o m b i n a t i o n processes p reva i l (R > 0) wh i l e generat ive processes d o m i n a t e (R < 0) i f carr iers were removed. T h e r e c o m b i n a t i o n and generat ion is affected by severa l energy t r a n s i t i o n processes. E a c h of these c omponen t processes can be ana ly sed u s i ng techn iques f r o m q u a n t u m mechan i c s t o p r oduce an overa l l m o d e l of R [31]. T h e s imples t energy t r a n s i t i o n process, the two pa r t i c l e t r a n s i t i o n , is m o d e l l e d by the S c h o c k l e y - R e a d - H a l l t e r m RSRH = , n P ~ n ; " , , (2.49) wh i l e t he A u g e r r e c o m b i n a t i o n - generat ion t e r m RAU = {Cfn + CAUp)(np - n 2 ) (2.50) mode l s t he three pa r t i c l e t r an s i t i o n . He re T p and rln d epend o n the d o p i n g prof i le a n d represent t he hole a n d e lect ron l i f e t imes . T y p i c a l values for rln a n d r^, as we l l as for the constants C*v a n d CAV are g i ven i n Tab l e 2.3. Chapter 2. Introduction to the Modelling of Semiconductor Devices 20 P a r a m e t e r T y p i c a l V a l u e < 1CT 6 5 < 1 0 - 6 5 1 0 1 0 c m - 3 2.8 x 1 0 " 3 1 cm6s-1 CAV 9.9 x 1 0 _ 3 2 c m 6 s - 1 T a b l e 2.3: Va lue s of pa ramete r s o c c u r i n g i n R O t h e r r e c o m b i n a t i o n a n d gene ra t i on processes w h i c h are i m p o r t a n t on l y i n spec ia l s i t ua t i on s , or whose c o n t r i b u t i o n is sma l l , are o m i t t e d f r o m th i s d i scuss ion b u t are desc r ibed i n [31]. M a n y mode l s s i m p l y use R — RSRH- F o r some compu te r mode l s , a n d for m u c h a n a l y t i c wo rk , i t is a s sumed t ha t R = 0. S t r i c t l y th i s is v a l i d o n l y fo r t h e r m a l e q u i l i b r i u m , bu t i t is also reasonab le for p r ob l ems ' n ea r ' e q u i l i b r i u m . 2.4.3 T h e Carr ier Mobilit ies T h e car r ie r m o b i l i t i e s fin a n d \iv o c cu r as m u l t i p l i c a t i v e factor s i n the express ions for the cur rent c omponen t s Jn a n d Jp, so accu ra te m o d e l l i n g of the m o b i l i t i e s is i m p o r t a n t . U n f o r t u n a t e l y , t he phy s i c a l mechan i sms i n vo l ved are d i f f i cu l t t o m o d e l exact ly , so m a n y c o m m o n l y used mode l s are based on l y on e x p e r i m e n t a l d a t a [24]. T h e car r ie r mob i l i t i e s are r e l a ted t o the e lec t ron a n d ho le r e l a x a t i o n t imes a n d TTV by = ( 2- 5 1) mn »v = —. (2-52) where m * a n d m * are the effect ive e lec t ron and ho le masses respect ive ly . T h e r e l a x a t i o n t imes descr ibe the average t i m e between s ca t te r i ng events for the carr iers . T h e car r ie r s are scat tered by t h e r m a l l a t t i ce v i b r a t i on s , i on i zed a n d n e u t r a l i m p u r i t i e s , surfaces and other carr iers . M o d e l s fo r each effect are desc r ibed i n [31]; the c o m b i n e d effect can be Chapter 2. Introduction to the Modelling of Semiconductor Devices 21 f o u n d by an a p p r o p r i a t e averag ing of the i n d i v i d u a l events. T h e car r ie r mob i l i t i e s are f u r t h e r r educed by the s a t u r a t i o n of the d r i f t ve l oc i t y due t o l a t t i c e v i b r a t i o n . T h e e l e c t r on m o b i l i t y genera l l y var ies be tween 50 cm2V~1s~1 and 1500 cm2V~1s~1, wh i l e the ho le m o b i l i t y fip ranges be tween 50 cm2V~1s~1 a n d 500 c m 2 V - 1 s _ 1 . 2.5 T h e Numerical Solution of the Semiconductor Equations T h e p rev i ou s sect ions have deve loped a n d d i scussed a m a t h e m a t i c a l m o d e l of semi-c o n d u c t o r devices. T h e m o d e l cons ists of a s y s tem of non l i nea r p a r t i a l d i f fe rent ia l equa t i on s w h i c h cannot be solved ana l y t i c a l l y , i n genera l . T o generate so lu t ions for a r b i t r a r y c ond i t i on s n u m e r i c a l techn iques mus t be used. T h i s sect ion addresses t he two m a j o r issues a r i s i ng f r o m the n u m e r i c a l p r ob l em. F i r s t , a s tab le d i s c r e t i z a t i o n of the bas ic equat ions is de r i ved us ing a finite difference scheme. T h e techn iques w h i c h are c o m m o n l y used t o solve the non l i nea r equat i on s are o u t l i n e d i n §2.5.2. W e cons ider the bas ic equat ions i n t he i r sca led f o r m (2.32 - 2.36) t h r o u g h o u t t he r e m a i n d e r of th i s thes is. 2.5.1 Discretizing the Equations T h e r e are severa l c las s ica l me thod s w h i c h can be used t o d i sc re t i ze the sem iconduc to r equa t i on s [6]. F i n i t e e lements , finite differences and the finite box m e t h o d (a gener-a l i z a t i o n of the finite d i f ference m e t h o d ) have a l l been successful ly used t o generate d i s c re t i z a t i on s . T h e most c o m m o n l y used app roach is a spec ia l i zed finite difference scheme k n o w n as the S c h a r f e t t e r - G u m m e l d i s c r e t i z a t i o n [32]. T h i s a pp r oach w i l l be d e t a i l e d be l ow for the one d imen s i ona l case. W e cons ider a set of mesh po in t s { i i } , = o,i..w w i t h xo' = 0 a n d x # = 1 the b o u n d a r y po in t s . A g a i n reca l l t h a t t he s i ngu la r l y p e r t u r b e d na t u r e of the equat ions d i c ta te s that Chapter 2. Introduction to the Modelling of Semiconductor Devices 22 t he mesh be h i gh l y n o n u n i f o r m : away f r o m the d o p i n g prof i le j u n c t i o n s the so lu t i on is s m o o t h l y v a r y i n g a n d so a coarse g r i d is suff ic ient, bu t the s o l u t i on changes r a p i d l y t h r o u g h the j u n c t i o n layers so a f ine g r i d is r equ i red i n these regions. However, for an eff ic ient s o l u t i on (espec ia l l y i n the m u l t i g r i d con tex t ) i t is i m p o r t a n t t o have a s tab le d i s c r e t i z a t i o n even w h e n the g r i d is coarse at the j unc t i on s . T h e d i s c r e t i z a t i o n of Po i s son ' s equa t i on w i t h a s m a l l p a r amete r is a s t anda rd p rob -l e m w h i c h is t r e a t ed i n m a n y pub l i c a t i o n s , for i n s tance [29]. W e use the s t anda rd three po i n t scheme for Po i s son ' s e q u a t i o n on a n o n u n i f o r m g r i d t o g ive 2 A 2 2A 2 (j>i+i - ijfi) - -—— — r T ^ t - ^ t - i ) = nl-pi- C(xi) (2.53) hi(hi-i + hi) hi-i(hi-i + hi) where hi = Xi+i — Xi. T h e e r ro r i n vo l ved i n m a k i n g the d i s c r e t i z a t i o n is l i nea r l y p r o p o r t i o n a l ( w i t h a very sma l l p r o p o r t i o n a l i t y cons tan t ) to the mesh spac ing and the t h i r d p a r t i a l de r i va t i ve s of the p o t e n t i a l . N o t e t h a t as A —• 0 the reduced so lu t i on away f r o m the j u n c t i o n s is r ep roduced . T h e t r e a t m e n t of the c on t i nu i t y equat ions is less s t anda rd . A l t h o u g h the sma l l p a r a m e t e r A 2 appear s e x p l i c i t l y on l y i n Po i s son ' s equa t i on , the ent i re s y s tem is s ingu-l a r l y p e r t u r b e d . (It is, i n fact , s ingu lar , s i ngu la r l y p e r t u r b e d , i.e. the reduced sy s tem o b t a i n e d by s e t t i ng A = 0 cannot be so lved d i r ec t l y fo r the reduced p o t e n t i a l s ince tp does not appea r i n the a lgebra ic equa t i on 0 = n — p—C(x); see, for e x amp l e [3].) U s i n g s t a n d a r d s y m m e t r i c differences p roduces a scheme w h i c h is not suf f ic ient ly numer i c a l l y s tab le i n the sense t ha t Vt/> is large near j unc t i on s , m a k i n g the o f f -d iagona l t e rms i n t h e s y m m e t r i c d i s c r e t i z a t i o n large. T h e t r a n s f o r m a t i o n t o the S l o t b o o m var iables (u , v) is r e gu l a r i z i n g , b u t these var iab les are poo r l y sca led a n d hence n u m e r i c a l l y unusab le. However , i t is usefu l for genera t i ng a d i s c re t i za t i on . In one d i m e n s i o n Jn = 62nne*— (2.54) Chapter 2. Introduction to the Modelling of Semiconductor Devices 23 a n d d i s c r e t i z i n g the cur rent conse rva t ion r e l a t i o n u s i ng a s y m m e t r i c scheme we have Jn,i+l/2 ~ Jn,i-\/2 l/2 ( f c i + fti-i) = R(xi). (2.55) du A s s u m i n g t ha t p,n, Jn a n d —— are constant o n the i n t e r v a l [xi,Xi+i], (2.54) gives da; d u _ Jne~+ dx 82p,n w h i c h can be i n teg ra ted exac t l y : (2.56) J X, 1 Jn,i+l/2 r<5-^|x, + ] (e-^+> - c - * ' ) - (2-57) ^ n , l + l / 2 ( f ) 1 + 1 / 2 i e Jn,i+l/2hj . ^ _ ^ T h i s dete rmines Jn,i+i/2 m t e rms of n i + 1 , r i j , tpi+1 a n d ipi. S u b s t i t u t i n g th i s i n t o (2.55) we have: 2^n,i+l /2 ,hi(hi-i + ht 2/^71,7-1/2 / i ,_ i (/ i ; _ i -I- hi) T where B(x) = ( £ ( ^ + i - V^K + i - Btyi - ipl+1)nl) -( S ( ^ - ^ - i ) n i - 5 ( V - i - i - ^ ) K - i = * ( n » P i ) , (2.58) ex - 1 A s im i l a r t r ea tmen t of the con t i nu i t y equa t i on for holes gives 2/ip,i-l /2 ( B ( V \ - i - ^ ) f t - W ' , - ^ - i ) ) / > t - i = Rin^p,). (2.59) hi-i{hi--i + hi) T h e l o c a l t r u n c a t i o n e r ro r for the d i s c re t i zed equat ions is p r o p o r t i o n a l t o the g r i d spac ing . T h i s c o m m o n l y used d i s c r e t i z a t i on is k n o w n as the S c h a r f e t t e r - G u m m e l scheme [32], a n d is an in s tance of e xponen t i a l fitting [4]. Chapter 2. Introduction to the Modelling of Semiconductor Devices 24 2.5.2 Solving Nonlinear Systems of Equations T h e bas ic s e m i c o n d u c t o r equat ions are of the f o r m F(u) = 0 (2.60) where / F1(u1,u2,u3)\ F(u) = (2.61) F2{u1,u2,u3) \ F 3 ( U 1 , U 2 , ' U 3 ) / Here , u2, u3) represent the dependent var iab les (T/>, n,p), (ip, <f)n, <f>p) o r (if>, u,v). T h e so l u t i on of non l i nea r sys tems of equat ions is a s t a n d a r d p r o b l e m , d i scussed i n de ta i l i n [28], for e xamp le . T h e two approaches w h i c h are c o m m o n l y used for the s em i conduc to r equa t i on s are N e w t o n ' s m e t h o d a n d G u m m e l ' s i t e r a t i o n . N e w t o n ' s m e t h o d is a c las s ica l i t e r a t i v e techn ique for so l v i ng non l i n ea r systems of equat ions . It generates a sequence of a p p r o x i m a t e so lut ions {uk} s t a r t i n g f r o m an i n i t i a l po i n t u° i n the f o l l ow i ng manne r : uk + 1 := u k + du (2.62) where du satisf ies t he equat ions J(uh)du = -F(uk) (2.63) w i t h J(u) b e i n g the J a c o b i a n m a t r i x of (2.60). E q u a t i o n (2.62) de te rm ine s a N e w t o n co r r e c t i on d i r e c t i o n du a n d (2.63) takes a un i t s tep i n th i s d i r e c t i on t o def ine the new i t e r a t e uk+1. N e w t o n ' s m e t h o d is q u a d r a t i c a l l y convergent near the so l u t i on of the non l i near equat ions . Howeve r , i t is gua ranteed t o converge on l y l o c a l l y so t ha t a g o o d s t a r t i ng p o i n t is necessary t o ensure convergence. F u r t h e r , each i t e r a t i o n can be expens ive s ince Chapter 2. Introduction to the Modelling of Semiconductor Devices 25 i t invo lves t he e va l ua t i on of the J a c o b i a n m a t r i x , a n d the s o l u t i on of a set of (poss ib ly i l l - c o n d i t i o n e d ) equat ions . M o d i f i e d N e w t o n m e t h o d s a t t e m p t t o i m p r o v e t he convergence of the bas ic algo-r i t h m b y chang i n g the d i r e c t i o n of the N e w t o n co r r ec t i on a n d the leng th of the step t a k e n i n t h i s d i r e c t i on . Q u a s i - N e w t o n m e t h o d s reduce t he cost of each i t e r a t i o n by u s i ng a p p r o x i m a t i o n s t o the J a c o b i a n m a t r i x w h i c h are eas i ly c o m p u t a b l e , a n d inver t -ib le . T h e a p p l i c a t i o n of these techn iques t o the s em i conduc to r equat ions is de ta i l ed i n [5]-O n e p r o b l e m w h i c h arises w h e n a p p l y i n g N e w t o n ' s m e t h o d t o the semiconduc to r equa t i on s . i s t h a t the J a c o b i a n mat r i ce s can be i l l - c ond i t i o ned . T o overcome th i s i l l -c o n d i t i o n i n g the co r rec t i on t r a n s f o r m a t i o n techn ique has been suggested [30]. T h e co r r e c t i on t r a n s f o r m a t i o n techn ique combines two sets of var iab les by u s ing the quas i -F e r m i va r i ab le s <f>n a n d <f>p fo r ca l cu l a t i on s wh i l e i m p l i c i t l y l i n e a r i z i n g i n te rms of the ca r r i e r c oncen t r a t i on s n a n d p. A de ta i l ed ana lys i s of the c o n d i t i o n i n g can be f ound i n [3]-N e w t o n ' s m e t h o d t rea t s t he three non l i nea r equat ions s imu l taneous ly , bu t i f the te rms w h i c h coup le the equat ions are not large t h e n these equat ions can be solved sequent i a l l y u s i n g a non l i nea r Gaus s - Se ide l i t e r a t i on . T h i s i t e r a t i o n was first app l i ed t o the s e m i c o n d u c t o r equat i on s b y G u m m e l [17], a n d has become k n o w n as G u m m e l ' s m e t h o d . G u m m e l ' s m e t h o d work s as fo l lows. G i v e n (tpk, nk,pk) at i t e r a t i o n k the nex t i te ra te t j ) k + 1 is d e t e r m i n e d by so l v i ng Po i s son ' s equa t i on : A2AV>*+1 =nk-pk + C(x). (2.64) U s i n g t{>k+1 t he values of (nk+1 ,pk+1) are f o u n d by so l v ing the con t i nu i t y equat ions : divJn{4>k+1,nk+1)-R(xl>k+1,nk,pk) = 0 (2.65) Chapter 2. Introduction to the Modelling o f Semiconductor Devices 26 divJp(r(>k+1,pk+1) + R{4>k+\nk,pk) = 0. (2.66) The se equat ions are l i nea r because R is e va l ua ted u s i ng the values f r o m the prev ious i t e r a t i o n . In th i s o r i g i n a l f o r m , G u m m e l ' s m e t h o d can be shown to d iverge i n some cases [24]. However , by l i n e a r i z i n g R a n d e va l u a t i n g R at the new i terates these p rob lems c an be avo ided . G u m m e l ' s m e t h o d is s imp le , b u t has been ve ry va l uab le i n p r a c t i c e . T h e i t e r a t i on w i l l u sua l l y converge, even s t a r t i n g f r o m a f a i r l y p o o r i n i t i a l va lue [31]. Its m a i n d i sadvantage is t ha t i ts convergence ra te is u sua l l y no b e t t e r t h a n l i near , a n d it becomes e x t r e m e l y slow w h e n the equat i on s are s t rong l y l i n ked , e.g., w h e n there is a h i gh app l i ed p o t e n t i a l , a n d the ca r r i e r concent ra t i on s are large. Severa l m o d i f i c a t i o n s have been suggested t o i m p r o v e the speed of the G u m m e l i ter -a t i o n . H w a n g et al [20] suggest a b lock i t e r a t i o n w h i c h is s t rong l y re l a ted t o G u m m e l ' s m e t h o d , b u t w h i c h a t t e m p t s t o i nc l ude more of the l inkages between equat ions . Several au tho r s have suggested u s i ng G u m m e l ' s m e t h o d t o generate a good i n i t i a l e s t imate for N e w t o n ' s m e t h o d [1]. Chapter 3 The Mult igr id Method T h e n u m e r i c a l s o l u t i on of d i f fe rent ia l equat ions u sua l l y invo lves the d i s c r e t i z a t i o n of the equa t i on s u s i ng a t echn ique such as f in i te differences or f in i te e lements. T h e m u l t i g r i d m e t h o d is an efficient m e t h o d for so l v i ng the a lgebra ic equat i on s w h i c h resu l t f r o m th i s d i s c r e t i z a t i o n . T h e m u l t i g r i d a l g o r i t h m is, theoret i ca l l y , an o p t i m a l l y eff ic ient m e t h o d i n t ha t the a m o u n t of c o m p u t a t i o n a l work r equ i r ed to achieve a f i xed accu racy is p r o p o r t i o n a l t o t he n u m b e r of unknowns . T h i s t heo re t i c a l eff ic iency can be rea l i zed i n a p r a c t i c a l i m p l e m e n t a t i o n , as was first demon s t r a t ed by B r a n d t [8]. T h e m e t h o d has been used ef fect ive ly fo r m a n y p rob lems , bu t i t is not a lways obv ious how it shou ld be i m p l e m e n t e d for a d i f f i cu l t p r o b l e m . E x t e n s i v e d i scuss ions about the m u l t i g r i d m e t h o d a n d i ts a p p l i c a t i o n s can be f o u n d i n [18], [10] a n d [36]. T h i s chapte r p rov ides an e lementa ry i n t r o d u c t i o n t o the m u l t i g r i d m e t h o d . T h e first sect ion mot i va te s ' the m u l t i g r i d idea, and presents t he bas ic a l go r i t hms . T h e componen t s of the m u l t i g r i d a l g o r i t h m are discussed i n §3.2, and the ex tens i on of the bas ic m e t h o d t o non l i nea r systems of equat ions is o u t l i n e d i n §3.3. §3.4 discusses the a p p l i c a t i o n of the m u l t i g r i d m e t h o d t o the sem iconduc to r p r o b l e m . Some of the d i f f i cu l t ie s wh i ch are expec ted t o arise are discussed, a n d an i m p l e m e n t a t i o n based on s t a n d a r d m u l t i g r i d techn iques is desc r ibed. 27 Chapter 3. The Multigrid Method 28 3.1 Introduction to the Mul t igr id M e t h o d T h e m u l t i g r i d i d e a can be m o t i v a t e d by e x a m i n i n g the c la s s i ca l me thod s for so l v i ng the sys tems of equa t i on s w h i c h ar ise f r o m d i f fe rent i a l equat ions . Fo r def in iteness, we cons ider a p r o b l e m Lu — f o n some d o m a i n D, where L is a l i nea r d i f fe rent ia l ope ra to r w h i c h i n co rpo ra te s the b o u n d a r y cond i t i on s a n d / is a f u n c t i o n w h i c h does not depend on t he u n k n o w n u. T h e d i s c r e t i z a t i o n p roduces a s y s t e m of l i nea r a lgebra ic equat ions of t he f o r m L^u^ — fh, for a m a t r i x Lh a n d vector s Uh a n d fh- ( T h e subsc r ip t h refers t o the d i s tance be tween mesh po i n t s i n the d i s c re t i zed d o m a i n , a s sumed t o be un i f o rm. ) Lh is a large, sparse m a t r i x , t y p i c a l l y w i t h a b a n d e d s t r uc tu re . T h e so lu t i on of t h i s l i near s y s tem can be one of the mos t expens ive pa r t s of s o l v i ng the d i f fe rent ia l p r o b l e m . T h e m e t h o d s of s o l u t i on fo r systems of l i nea r equa t i on s are genera l l y c lass i f ied as e i ther d i rect or i t e r a t i v e m e t h o d s [16]. D i r e c t me thod s , such as G a u s s i a n E l i m i n a t i o n , f ac to r the coeff ic ient m a t r i x a n d p roduce the so l u t i on i n a finite n u m b e r of steps. I te rat i ve m e t h o d s p r oduce a series of a p p r o x i m a t e so lu t ions w h i c h , idea l ly , converge a s y m p t o t i c a l l y t o the exact s o l u t i on . I te ra t i ve ( r e l axa t i on ) me thod s are o f ten be t t e r s u i t ed for large sparse p rob lems because they do not suffer f r o m the p r o b l e m of fill-in. I te ra t i ve m e t h o d s generate a sequence of a p p r o x i m a t e so lu t ions {Ul,j = 0 ,1,2, . . . } , s t a r t i n g f r o m an i n i t i a l va lue U°, u s ing a fixed po in t i t e r a t i o n Ut1 :=*(U'h), J = 0 , 1 , 2 , . . . where the s o l u t i on of the l i near equat ions is t he on l y fixed po i n t of O n e class of i t e r a t i v e me thod s uses the i t e r a t i o n $(U) = U-B-1LhU + B-1fh Chapter 3. The Multigrid Method 29 where B is an eas i l y i n ve r t ed a p p r o x i m a t i o n of L^. T h e e r ro r at i t e r a t i o n k, vk = Uk-uh, satisf ies t he r e l a t i o n vk+1 = {I-B-lLh)vk where I is t he i d e n t i t y m a t r i x . T h e e r ro r approaches zero, a n d the m e t h o d converges, i f t he s p e c t r a l r ad i u s of the i t e r a t i o n m a t r i x (7 — B~lLh) is less t h a n 1, i.e., p{I-B-lLh)<\. M a n y c o m m o n i t e ra t i ve techn iques are based on a s p l i t t i n g of the coeff ic ient m a t r i x Lh as Lh = D + L -f U, where L a n d U are lower a n d uppe r t r i a n g u l a r mat r i ce s re spect i ve ly , a n d D is a d i agona l m a t r i x . T h e J a c o b i m e t h o d is t he s implest i t e r a t i o n , w i t h B = D. O t h e r sp l i t t i ng s i n c l ude the Gau s s - Se i de l m e t h o d , w i t h B = L + D, and the Success ive O v e r r e l a x a t i o n M e t h o d ( S O R ) w h i c h uses B = D 4- where w is a sca la r p a r amete r . T h e best r e l a x a t i o n me thod s are s t i l l expens ive. U s u a l l y the error, mea su red by u s i ng the r e s i dua l rk = LhUfc — fh, is q u i c k l y r educed d u r i n g the first few i te ra t i on s , bu t subsequent i t e r a t i on s reduce rk by on l y a s m a l l amoun t . T h e r e s i dua l is m u c h smoo the r after a few re l axa t i on s have been p e r f o r m e d ( there are fewer changes of s ign, a n d the co r r e l a t i on between ne i ghbou r i n g po in t s is s t ronger) . T h e s m o o t h re s idua l suggests t h a t the e r ro r i t se l f is a lso smoo th . T h e effect of the r e l a xa t i on s can be u n d e r s t o o d by decompos i n g the error i n t e rms of i ts F o u r i e r c ompo -nents . T h e r e l axa t i on s reduce the h i gh f requency component s bu t do l i t t l e for the lower f requenc ies . T h e convergence of a r e l a x a t i o n scheme can be acce lerated by e l i m i n a t i n g these s m o o t h componen t s f r o m the error. Chapter 3. The Multigrid Method 30 T h e m u l t i g r i d m e t h o d uses r e l axa t i on s t o e l i m i n a t e the h i gh f requency component s of the e r ro r and a coarse g r i d co r r ec t i on ( C G C ) t o dea l w i t h the l ow f requency compo -nents. L e t Vk be the e r ro r i n the cu r ren t a p p r o x i m a t i o n Uh, so Uh + Vh = Uh- T h e er ror Vh satisf ies LhVh — fh- A f t e r a n u m b e r of r e l a x a t i o n sweeps have been pe r f o rmed , Vh a n d rh are s m o o t h f unc t i on s a n d c an be w e l l represented on a coarser g r i d . A n a p p r o x i m a -t i o n Vh of Vh c an be f o u n d as the i n t e r p o l a t i o n of Vfj, where VJJ satisf ies LHVH = r # on a coarse g r i d . A new a p p r o x i m a t i o n of the s o l u t i on c an t hen be f o u n d as Uh '•= Uh + Vh-Here LJJ a n d r # are coarse g r i d a p p r o x i m a t i o n s of Lh and respect ive ly . T h e coarse g r i d a n d fine g r i d are u sua l l y r e l a ted by H = 2h ( s t anda rd coarsen ing) . T h e coarse g r i d equat ions L^VH — c a n D e so lved i n a s i m i l a r manne r : h i gh f requency er ror c omponen t s are e l i m i n a t e d by u s i ng re l axa t i on s , a n d the r e m a i n i n g s m o o t h e r ro r is f o u n d a p p r o x i m a t e l y by u s ing a s t i l l coarser g r i d . A d i rect so lu t i on p r o c e d u r e m a y be used on the coarsest g r i d . T h e f o l l ow i ng no ta t i on s w i l l be used t o descr ibe the m u l t i g r i d a l g o r i t h m more con -cisely. L e t Gk be a d i s c r e t i z a t i on of the d o m a i n u s i ng a mesh size hk. O n g r i d Gk the d i s c r e t i z ed p r o b l e m is LkUk = fk (3.67) w i t h Lk a m a t r i x , fk a g iven vector a n d Uk t he exact so lu t i on vector . T h e exact c o r r e c t i on Vk satisf ies the re s i dua l equa t i on LkVk =rk, • (3.68) where r*. = LkUk — fk-, a n d Uk is an a p p r o x i m a t i o n of w i t h Uk + Vk — u^. Vk is an a p p r o x i m a t i o n of Vk-T h e m u l t i g r i d m e t h o d uses a sequence of m g r ids G 1 , G 2 , . . . G m w h i c h are succes-s ive ly finer, h1 > h2 > ... > hm. T y p i c a l l y hm = 2hm+1. In a d d i t i o n , two operators are needed t o t rans fe r i n f o r m a t i o n between the gr ids. T h e p ro l onga t i on opera to r Chapter 3. The Multigrid Method 31 acts on vector s w h i c h a p p r o x i m a t e mesh f unc t i on s on Gk~1 t o p r oduce vectors o n Gk u s i n g some f o r m of i n t e r p o l a t i o n . T h e r e s t r i c t i on o p e r a t o r has the oppos i te ef-fect , t r an s f e r r i n g vectors f r o m Gk t o Gk~1. W h e n d i scus s ing o n l y one or two gr ids, the m o r e suggest ive no ta t i on s Lh, LH, Uh, UJJ etc. w i l l be used, w i t h r e s t r i c t i on 7jf a n d p r o l o n g a t i o n IH. T h e bas ic m u l t i g r i d a l g o r i t h m can be o u t l i n e d as fo l lows: Procedure MG(Lh,Uh,fh, uuu2) Step 1 S m o o t h the errors: R e l a x L^Uh = fh V\ t imes . Step 2 C a l c u l a t e a n d res t r i c t the re s idua l : rH := Ih(fk ~ LhUh). Step 3 So lve the coarse g r i d equat ions : LHvH = rH. Step 4 P r o l o n g a t e the co r r e c t i on a n d u p d a t e the a p p r o x i m a t i o n : UH:=UH + I*vH. Step 5 S m o o t h the r e m a i n i n g errors: R e l a x LhUh. — fh vi t imes . Step 6 R e t u r n t o Step 1 a n d repeat u n t i l convergence. Chapter 3. The Multigrid Method 32 If S tep 3 solves t he coarse g r i d equat i on s d i r e c t l y t h e n th i s is a two -g r i d m e t h o d . A t r ue m u l t i g r i d m e t h o d can be o b t a i n e d by a p p r o x i m a t e l y so l v ing the coarse g r id equa t i on s u s i n g a recur s i ve i n v o c a t i o n of the m u l t i g r i d p rocedure : F o r count :— 1 t o 7 D o M G ( I ^ , ^ , r ^ , ^1 ,^2) T h e p a r a m e t e r 7 de te rm ine s the accu racy of the s o l u t i o n t o the coarse g r i d equat ions a n d specif ies the f o r m of the i t e r a t i o n : 7 = 1 is k n o w n as a F - c y c l e , wh i l e 7 = 2 is a W-cyc le . Ui a n d u2 d e t e rm ine the amoun t of s m o o t h i n g pe r f o rmed on each g r i d ; t y p i c a l l y V\ a n d u2 are not mo re t h a n two. It is c o m m o n to t reat 7, v-i and v2 a s constants , bu t t he i r va lues can also be m o d i f i e d d y n a m i c a l l y t o i m p r o v e the eff ic iency of the i t e r a t i on . T h e coarse gr ids w h i c h are used i n the s o l u t i on process can also be used to generate an accu ra te i n i t i a l guess on the next f iner leve l . T h i s process, ca l led F u l l M u l t i g r i d ( F M G ) [9], beg ins w i t h an a r b i t r a r y i n i t i a l guess on a coarse g r i d . A n a p p r o x i m a t e s o l u t i on is f o u n d on th i s g r i d , a n d the s o l u t i on is i n t e r p o l a t e d t o the next finer g r i d to act as the s t a r t i n g va lue fo r the i t e r a t i o n at t h a t level . T h e bas ic m u l t i g r i d a l g o r i t h m is used t o solve the p r o b l e m on the new leve l , and th i s s o l u t i on is t rans fe r red t o a s t i l l finer leve l , etc. M u l t i g r i d m e t h o d s are u sua l l y ana ly sed by con s i de r i ng a m o d e l p r ob l em or by us ing L o c a l M o d e A n a l y s i s ( L M A ) (see [9], [18] a n d [36]). T h e m o d e l p r o b l e m app roach p roduce s m a t h e m a t i c a l l y r i gorous p red i c t i on s of the convergence rate by e x a m i n i n g the e igenvectors of the c omp le te m u l t i g r i d process, but these est imates are not sharp, a n d the a p p r o a c h is a p p l i c a b l e on ly t o a sma l l class of p rob lems . T h e mo re p r a c t i c a l L M A app l ies Fou r i e r m o d e ana lys i s , a s suming t ha t the m u l t i g r i d component s ( such as t he r e l a x a t i o n ope ra to r ) act on l y loca l ly . ( T h i s a s s u m p t i o n requi res va r i ab le coeff ic ients t o be fixed, and doma in s to be t r ea ted as unbounded. ) L M A is general ly not r igorous b u t it leads t o a heur i s t i c u n d e r s t a n d i n g of m u l t i g r i d processes i n te rms of h i gh and Chapter 3. The Multigrid Method 33 low f requency errors. T h e r e l a x a t i o n sweeps a n d the coarse g r i d c o r r e c t i o n ( C G C ) affect d i f ferent c ompo -nents of the er ror . H i g h f requency errors cannot be adequate l y reso lved o n the coarse g r i d a n d so need t o be e l i m i n a t e d d u r i n g the re l axa t i on s . These h i g h f requency errors are una f fec ted by the C G C since they l ie i n the n u l l space of t he r e s t r i c t i o n ope ra -tor . Converse ly , s m o o t h errors, w h i c h can be we l l represented on the coarse g r i d , are e l i m i n a t e d by the C G C . 3.2 Components of the Mult igr id Algor i thm T h e effectiveness of the m u l t i g r i d a l g o r i t h m is de te rm ined by the deta i l s o f the i m p l e -m e n t a t i o n . T h e cho ice of the component s — the r e l a xa t i on ope ra to r , t he r e s t r i c t i on a n d p r o l o n g a t i o n , a n d the coarse g r i d ope ra to r — is of ten p r o b l e m dependent . [18] and [9] c o n t a i n extens i ve d iscuss ions of the factor s w h i c h affect th i s choice. In p a r t i c u l a r , [9] p rov ide s severa l p r a c t i c a l examples. T h i s sect ion rev iews the techn iques w h i c h have been successfu l ly used for s t anda rd p rob lems . 3.2.1 Relaxation Operators A n y i t e r a t i v e s o l u t i on p rocedure can be used as a r e l a x a t i o n opera to r . T h e c r i t e r i on for a g o o d r e l a x a t i o n scheme is not necessar i ly t ha t i t converge qu ick l y , b u t t ha t i t s m o o t h the errors eff ic ient ly. N o one re l axa t i on scheme is app rop r i a t e for a l l s i tua t ions ; several cand ida te s are m e n t i o n e d here. O n e of the most c o m m o n , a n d successful, r e l a x a t i o n schemes is the Gaus s - Se ide l i t e r a t i o n . E a c h mesh po i n t of the d i s c re t i zed d o m a i n is scanned i n some p re sc r i bed o rde r a n d the va lue o f the u n k n o w n at th i s p o i n t is changed i n o rder t o sat i s fy t he Chapter 3. The Multigrid Method 34 c o r r e s p o n d i n g equa t i on . F o r a non l i nea r p r o b l e m the d iscrete equat ions m a y be non -l inear . It is u sua l l y suff ic ient t o use one step of N e w t o n ' s m e t h o d to a p p r o x i m a t e l y solve the non l i nea r equa t i on . T h e l i n e a r i z a t i o n of the equat ions takes p lace on a l o ca l leve l ( abou t a p o i n t ) and does not necessar i ly co r re spond to a g l oba l l i n ea r i z a t i on of the p r o b l e m . T h e m e t h o d of Success ive O v e r R e l a x a t i o n ( S O R ) i n t roduces a sca lar p a r amete r u> w h i c h m u l t i p l i e s t he Gaus s - Se ide l c o r r ec t i on . W i t h an a p p r o p r i a t e choice of to the convergence rate of S O R is m u c h be t te r t h a n for o r d i n a r y Gaus s - Se ide l . However , the best s m o o t h i n g r a te is o b t a i n e d w i t h u> = 1, so the u sua l Gaus s - Se ide l i t e r a t i o n is as effect ive as S O R , a n d is cheaper [9]. T h e J a c o b i m e t h o d re laxes a l l of the equa t i on s s imu l taneous ly , so t ha t the correc-t i ons are c a l c u l a t e d i n t e rms of the o r i g i na l va lues only. D a m p e d J a c o b i methods , i n w h i c h t he J a c o b i c o r r e c t i on is m u l t i p l i e d by a sca lar a; < 1, have bet te r s m o o t h i n g p rope r t i e s . In p r a c t i c e the J a c o b i r e l axa t i on s are in fer ior t o Gaus s - Se ide l i t e ra t i on s s ince t h e y requ i re mo re storage, more work , a n d do not s m o o t h the res idua l s as we l l [9]. Howeve r , the J a c o b i i t e r a t i o n is o f ten be t t e r su i ted for the ana lys i s of m u l t i g r i d method s . T h e o rder i n w h i c h the Gaus s - Se ide l r e l a x a t i o n encounters the mesh po in t s has a n i m p o r t a n t effect on the s m o o t h i n g rate. T h e u sua l o rde r i ng is l ex i cog raph i c — a po i n t w i t h ind ices ( z a , i 2 , . . . , z m ) is encounte red before a po i n t l abe l l ed (ji,j2, • • • ,jm) i f ik < jk a n d ii = ji for I < k. A s y m m e t r i c Gaus s - Se ide l r e l a xa t i on consists of a Gau s s - Se i de l sweep i n l ex i cog raph i c order, fo l l owed by a sweep i n the oppos i te order. A red -b lack o r de r i n g d i v ides the mesh po in t s i n t o two d i s t i nc t sets ( red and b l ack ) i n t he same p a t t e r n as a checke rboa rd (a po in t is ' r e d ' i f t he s u m of i t s ind ices is even, a n d ' b l a c k ' o therwi se ) . E a c h set of po in t s is scanned independent l y , u sua l l y i n l e x i c o g r a p h i c order. O rde r i ng s w h i c h use m o r e ' co lou r s ' , a n d w h i c h g roup the po in t s Chapter 3. The Multigrid Method 35 i n different, ways, are also poss ib le [18]. 3.2.2 Restriction Operators T h e r e s t r i c t i o n ope r a t o r t ransfers t he res idua l s f r o m a f iner g r i d Gk t o a coarser g r i d Gk~l by u s i n g some f o r m of i n t e r p o l a t i o n . T h e s implest scheme, i n j e c t i on , d i r e c t l y t ransfers i n f o r m a t i o n between the gr ids at c o m m o n po in t s . F u l l we i gh t i ng averages the values of t he f u n c t i o n at ne i ghbou r i n g po in t s of the fine g r i d , and at taches th i s averaged va lue t o the coarser g r i d po in t . T h e l a t t e r u sua l l y has a mo re sound t heo re t i c a l j u s t i f i c a t i o n . Fo r a one d imen s i ona l p r o b l e m , us ing a s t anda rd coarsen ing, {Ikk-lU% = (Uk)2j (3.69) w h e n i n j e c t i o n is used, wh i l e (lt'Uk): = l/A(Uk0_r + 2Uk: + UkJ+1) (3.70) for a f u l l we i gh t i ng . It is somet imes advantageous to choose the re s t r i c t i on ope ra to r t o be the ad jo in t of the p r o l o n g a t i o n , so Ik~l = (Ik-iY [18]. L i n e a r i n t e r p o l a t i o n and the f u l l we i gh t i n g r e s t r i c t i on o p e r a t o r are re l a ted i n th i s way. U s u a l l y th i s r e l a t i on is used t o p re sc r i be the f o r m of the r e s t r i c t i on opera to r once the p ro l onga t i on has been chosen. 3.2.3 Coarse G r i d Operators T h e r e are two c o m m o n ways to define the coarse g r i d o p e r a t o r LH once Lh has been f o r m e d f r o m t he cont inuous ope ra to r L. T h e mos t obv ious m e t h o d is to use the same d i s c r e t i z a t i o n m e t h o d on the coarse g r id w i t h a mesh size of H r a ther t h a n h\ th i s p roduces a coarse g r i d ope ra to r wh i ch is ana logous to the fine g r i d operator . Chapter 3. The Multigrid Method 36 A less obv iou s m e t h o d is t o def ine LH i n t e rms of L^, t he r e s t r i c t i on If? and the i n t e r p o l a t i o n IH, as T o eva luate LH, t he so l u t i on is m o v e d t o the fine g r i d , t he f ine g r i d o p e r a t o r is app l i ed , a n d the result is r e s t r i c t ed back t o the coarse g r i d . T h i s ope ra to r , somet imes ca l led t he G a l e r k i n f o r m , is o f ten used for ana ly s i s of the m u l t i g r i d m e t h o d , bu t i t s usefulness is somet imes l i m i t e d i n p rac t i ce [18]. 3.2.4 Prolongation Operators T h e p r o l o n g a t i o n Ik_: t ransfers t he co r r e c t i on (coarse g r i d s o l u t i on ) f r o m a coarser g r i d G f c _ 1 t o a finer g r i d Gk. T h e most c o m m o n , a n d s imp les t , p ro l onga t i on s are p iecewise l i near i n te rpo l a t i on s . These assume t ha t t he coarse g r i d s o l u t i o n is l i nea r between mesh po in t s , so a s imp le averag ing is used t o find the i n t e r p o l a t e d va lues at the fine g r i d po int s . F o r a one -d imens i ona l p r o b l e m , u s ing l i near i n t e r p o l a t i o n w i t h a s t a n d a r d coarsen ing, O t h e r p ro l onga t i on s use a more accu ra te scheme such as p iecewise cub i c i n t e rpo -l a t i o n . Fo r second order d i f fe rent ia l p r ob l ems the e x t r a cost of these mo re accu ra te schemes does not seem to be wo r t hwh i l e , bu t they mus t be used for h i gher o rder p r ob -A n o t h e r m e t h o d of p r o l onga t i on is m o t i v a t e d by cons ide ra t ions of the n u l l space a n d range of the opera to r s [25]. A n efficient m e t h o d is p r o d u c e d i f t he error after the r e l a xa t i on sweep lies in the range of the p r o l onga t i o n opera to r . W h e n comb ined LH •= IhLhIH (3.71) lems [9]. Chapter 3. The Multigrid Method 37 w i t h a G a l e r k i n f o r m of t he coarse g r i d ope ra to r th i s ensures t h a t the l ow f requency c o m p o n e n t s of the e r ro r are comp le te l y e l i m i n a t e d by the coarse g r i d co r rec t i on step; i n spec ia l cases t h i s p roduce s a d i rect solver. Fo r a red -b l ack Gau s s - Se i de l r e l a xa t i on , the p r o l o n g a t i o n s hou l d sat i s fy at a l l b l ack g r i d po in t s . 3.3 Extensions to Nonlinear Systems of Equations T h e above d i scus s ion has focused on the p r o b l e m of a sca lar , l i nea r d i f fe rent i a l equa t i on , b u t t he m u l t i g r i d m e t h o d can be a p p l i e d to m u c h m o r e genera l p rob lems . These genera l i za t i on s w i l l be o u t l i n e d here. Fo r a non l i nea r p r o b l e m NhUh — 0, there are two approaches w h i c h use the m u l t i g r i d m e t h o d . One a p p r o a c h app l ies t he l i nea r m u l t i g r i d a l g o r i t h m to a l i n e a r i z a t i o n of the p r o b l e m , o b t a i n e d u s i n g N e w t o n ' s m e t h o d , or qua s i - l i nea r i z a t i on . T h i s app roach has the advantage of m o d u l a r i t y , s ince the new, non l i nea r p r o b l e m is r educed to a s imp ler , l i nea r p r o b l e m , b u t th i s can be a cumber some app roach w h i c h is expens i ve i n terms of storage. A n o t h e r way to e x t e n d the m u l t i g r i d a l g o r i t h m t o the non l i nea r case is t h rough the use of the F A S a l g o r i t h m i n t r o d u c e d by Brandt. [9]. T h e co r rec t i on Vh = Uh — Uh w i l l sat i s fy the re s i dua l e q u a t i o n where i n genera l Nh ^ Nh- A new i te ra te c an be o b t a i n e d as Uh := Uh -f V/,. "where Vh fc-i = 0 Nhvh = Nh(Uh + vh)- NhUh = rh (3.72) a p p r o x i m a t e l y solves Nhvk = rh-Chapter 3. The Multigrid Method 38 T h e m u l t i g r i d a l g o r i t h m can be ex tended i n a s t r a i g h t f o rwa rd manne r so tha t i t c an be used t o solve th i s defect equa t i on . N o n l i n e a r r e l a x a t i o n schemes ana logous to the l i nea r m e t h o d s de sc r i bed above can be used t o s m o o t h the errors, and a coarse g r i d co r r ec t i on step can be def ined as fo l lows. W e a p p r o x i m a t e Nhvh — rh o n t he coarser g r i d as NHVH = RH, a n d t h e n p ro l onga te the co r r e c t i o n VH t o the finer g r i d a n d u p d a t e t he cu r ren t i te ra te . NH is def ined as before, bu t now on the coarser g r i d , so NHvH = NH(UH + vH) - N H U H = rH. (3.73) UH is some e s t imate of Uh, e.g., UH = IhUh- S ince (3.73) can be so lved as NHWH = RH w i t h WH — UH + VH, t h i s coarse g r i d equa t i on c an be so lved by successive app l i c a t i on s of t he m u l t i g r i d m e t h o d . N e w t o n ' s m e t h o d , o r a s im i l a r non l i nea r i t e r a t i on , can be used t o solve (3.73) on the coarsest level . N o n l i n e a r m u l t i g r i d a l g o r i t h m s are not u sua l l y ana l y sed d i r ec t l y ; i n s tead, l i nea r i zed vers ions are cons idered. However , t he concepts of h i gh a n d low f requency er rors a p p l y equa l l y we l l t o t he l i nea r a n d non l i nea r p rob lems . T h e m u l t i g r i d a l g o r i t h m can be eas i ly e x tended t o i n c l ude systems of d i f fe rent ia l equat ions . T h e de s c r i p t i on of the a l g o r i t h m , w i t h t he scalar f un c t i on u r ep laced by a vec to r f u n c t i o n u, is t he same. T h e r e s t r i c t i o n a n d p r o l o n g a t i o n operator s behave as before, b u t now there is added f l e x i b i l i t y i n the i r i m p l e m e n t a t i o n . One app roach is t o re s t r i c t a n d i n t e rpo l a t e each so lu t i on c omponen t separately, u s ing the operator s as de s c r i bed for the scalar s i t u a t i o n . A n a l t e r na t i v e is t o use comb ina t i on s o f the so l u t i on componen t s w h i c h are smoothe r t h a n the so l u t i on itself. T h i s is u se fu l when the i n d i v i d u a l s o l u t i on componen t s are r a p i d l y v a r y i n g , bu t a t r a n s f o r m a t i o n of the var iab le s is be t t e r behaved. T h e r e l a x a t i o n opera to r s can also be e x t ended t o dea l w i t h systems of equat ions [9]. Chapter 3. The Multigrid Method 39 If t he equat ions are not s tong ly l i n k e d a n d each equa t i on c an be i den t i f i ed w i t h one u n -k n o w n , t h e n a Gau s s - Se i de l i t e r a t i o n can be used at each po in t . O t h e r w i s e a co l lect i ve r e l a x a t i o n , i n w h i c h a l l of the equat ions at a mesh po i n t are sat i s f ied s imu l taneous ly , s h o u l d be used. D i s t r i b u t e d r e l a x a t i o n schemes sat i s fy the equat ion s by i m p l i c i t l y con-s i de r i n g a l i nea r c o m b i n a t i o n of unknowns , a n d are usefu l w h e n the o r i g i n a l equat ions are i l l - c o n d i t i o n e d a n d w h e n co l lect i ve r e l axa t i on s are avo ided fo r ef f ic iency reasons. 3.4 Solving the Semiconductor Equations using the Mul t igr id M e t h o d T h e n u m e r i c a l s o l u t i on of the s em i conduc to r equat ions is a c o m p l e x p r o b l e m , so the a p p l i c a t i o n of the m u l t i g r i d m e t h o d is e xpec ted t o encounte r m a n y d i f f i cu l t ie s . T h i s sec t i on discusses some of the p rob lems w h i c h are expec ted t o ar ise a n d the app l i c a t i o n of s t a n d a r d m u l t i g r i d techn iques to the p r o b l e m . W h i l e th i s s t r a i g h t f o r w a r d app roach was no t e xpec ted t o be successful , i t p rov ide s a usefu l reference po i n t fo r the d i scuss ion i n t he f o l l ow ing chapte r , w h i c h descr ibes a series of mod i f i c a t i on s that g rea t l y improve t he convergence of the a l g o r i t h m . 3.4.1 Anticipated Difficulties T h e s i m u l a t i o n of a rea l i s t i c s em i conduc to r dev ice invo lves a n u m b e r of d i f f i cu l t ies . A t y p i c a l two d imen s i ona l s imu l a t i on d o m a i n is shown i n F i g u r e 3.2. In the i n su l a to r B C D E B the e lec t ro s ta t i c p o t e n t i a l ip satisfies L ap l a ce ' s equa t i on Aip = 0, a n d n = p = 0. T h e bas ic s em i conduc to r equat ions need t o be solved i n the rest, of the d o m a i n , A B E F G H A . D i r i c h l e t b o u n d a r y cond i t i on s h o l d a long A B , B C , C D , D E a n d E F , wh i l e N e u m a n n cond i t i on s are specif ied a long the interface between the i n s u l a t o r a n d the s e m i c o n d u c t o r ( B E ) , a n d also o n the a r t i f i c i a l bounda r i e s F G , G H a n d H A . M a t c h i n g c o n d i t i o n s are also r equ i r ed a long B E . Chapter 3. The Multigrid Method 40 c D A B Insulator E F n n p Bulk Ff T3 F i g u r e 3.2: T y p i c a l S i m u l a t i o n D o m a i n T h e d o p i n g pro f i le is d i s con t i nuous at the pn - j unc t i on s . T h e s o l u t i on componen t s va ry r a p i d l y nea r these j u n c t i o n s , so a very fine d i s c r e t i z a t i o n is needed i n these regions. A w a y f r o m the j u n c t i o n s the s o l u t i o n componen t s are s lowly v a r y i n g a n d so a f a i r l y coarse mesh c a n be used t h r o u g h mos t of the d o m a i n . T h e s o l u t i on p rocedu re m u s t cope w i t h a h i g h l y n o n u n i f o r m mesh . T h e s i n gu l a r l y p e r t u r b e d n a t u r e of the govern ing equat ions d i c ta tes t h a t a n o n -s t a n d a r d d i s c r e t i z a t i o n be used, as d i scussed i n §2.5. T h e r e su l t i n g a l gebra ic equat ions are n o n s y m m e t r i c a n d m a y be i l l - c o n d i t i o n e d a n d p o o r l y scaled. A reasonab le i n i t i a l guess is not a lways ava i l ab le fo r an i t e r a t i v e s o l u t i on p rocedure . A c omp le te dev ice s i m u l a t i o n w i l l i nves t i ga te the b e h a v i o u r of a dev ice over a w i d e range of o p e r a t i n g cond i t i on s , so t he gove rn ing equat ions w i l l need t o be so lved re-peated ly . A l a rge n u m b e r of m e s h po i n t s is needed t o ensure a suf f ic ient ly accu ra te s o l u t i on , so t h e s o l u t i on p rocedu re needs t o be b o t h robus t and eff ic ient. W h i l e the s t a n d a r d m u l t i g r i d m e t h o d has been h i gh l y successful for some p rob lems , Chapter 3. The Multigrid Method 41 there is l i t t l e reason t o expect t h a t i t w i l l overcome the d i f f i cu l t ie s o u t l i n e d above. M a n y author s s t u d y i n g the use of the m u l t i g r i d m e t h o d cons ider on l y " n i c e " p r ob l ems w h i c h are l inear , w e l l - c o n d i t i o n e d a n d s t rong l y e l l i p t i c . In p a r t i c u l a r , t he m o d e l p r o b l e m ana lys i s w h i c h shows the m u l t i g r i d m e t h o d t o be o p t i m a l l y eff ic ient appl ies on l y t o Po i s son ' s e q u a t i o n on a square d o m a i n . In a d d i t i o n t o the d i f f i cu l t ie s w h i c h most m e t h o d s are e x p e c t e d t o encounter there are several p r ob l ems w h i c h are speci f ic t o the m u l t i g r i d m e t h o d . A n o n u n i f o r m mesh can be generated by con s i de r i ng the u n i o n of u n i f o r m gr ids w h i c h do no t cover the ent i re d o m a i n [9], bu t t h i s is a cumbe r s ome app roach t o i m p l e m e n t . T h e s i ngu la r p e r t u r b a t i o n na t u r e of the equat ions , a n d the sha rp i n t e r n a l layers, requ i re n o n s t a n d a r d re s t r i c t i ons a n d p ro l onga t i on s to t rans fe r i n f o r m a t i o n be tween gr ids. [21] a n d [22] discuss t he use of the m u l t i g r i d m e t h o d for s i ngu l a r l y p e r t u r b e d p rob l ems , a n d [2] cons iders p r ob l ems w i t h d i s con t i nuous coeff ic ients. F i n a l l y , t he r e l a x a t i o n scheme must cope w i t h a s y s tem of non l i nea r equat i on s whose l i n e a r i z a t i o n m a y f a i l t o be d i agona l l y dom inan t . O t h e r a t t e m p t s t o a p p l y t he m u l t i g r i d m e t h o d t o the s em i conduc to r equat ions have been r e p o r t e d [7], [15]. These i m p l e m e n t a t i o n s were successfu l on l y w h e n the coarsest g r i d used was re l a t i ve l y f ine. H e m k e r ' s a l g o r i t h m , de sc r i bed i n [19], successfu l ly uses very coarse gr ids , bu t i t does not a lways appea r t o be as eff ic ient as poss ib le. H e m k e r ' s i m p l e m e n t a t i o n (wh i ch has s i gn i f i cant l y i n f l uenced ours ) is d i scussed in §4.4, a n d compar i son s w i t h ou r a l g o r i t h m are r epo r t ed . 3.4.2 Description of the M o d e l Problem T o eva luate va r i ou s mod i f i c a t i on s of the m u l t i g r i d m e t h o d a s imp le one d imen s i ona l m o d e l p r o b l e m has been used. T h e two endpo in t s are a s sumed t o be O h m i c contact s , so t ha t on ly D i r i c h l e t b o u n d a r y cond i t i on s arise. Fu r t he r , on l y u n i f o r m gr ids are used i n the so l u t i on process. Chapter 3. The Multigrid Method 42 T h e test p r o b l e m used i n th i s sec t ion cons ists of a one d imen s i ona l np -d i ode on the i n t e r v a l [—1,1] (Test 1). T h e (scaled) d o p i n g pro f i le has an n p - j u n c t i o n at x = 0: C(x) ' 1 - 1 < x < 0 - 1 0 " 6 1 > x > 0 w i t h A 2 = 0.4 x 1 0 ~ 6 a n d <52 = 0.1 x 1 0 ~ 6 . R e c o m b i n a t i o n a n d genera t i on are a s sumed to be neg l i g ib le , so R = 0. C o m p u t a t i o n s were p e r f o r m e d for the e q u i l i b r i u m case (i.e., Vo = 0) .and for V0 = 30 ( f o rwa rd b ias ) . ( N o t e t h a t i n te rms of the scaled var iab les , 1 V o l t co r re sponds t o V 0 ~ 40.) D i r i c h l e t c ond i t i o n s h o l d at the bounda r i e s x — — 1 a n d x = 1, as de sc r i bed i n §2.3. T h e i n i t i a l e s t ima te used by the m u l t i g r i d p rocedu re was f u ( - l ) x<0 u(x) = < { u ( l ) x > 0 for u = (tp,n,p). T h i s i n i t i a l e s t imate satisf ies t he b o u n d a r y cond i t i on s . T h e var ious m o d i f i c a t i o n s were e va l ua ted ba sed on the n u m b e r of i te ra t i on s r equ i r ed to reach convergence. A n i t e r a t i o n was t e r m i n a t e d w h e n two successive i terates uk+1 a n d uk sat i s f ied \uk+1 -uk\ < 1 0 - 3 | u J + 1 | (3.74) for each c o m p o n e n t uk+1. 3.4.3 A p p l y i n g Standard Mult igr id Techniques T h i s sect ion descr ibes two bas ic m u l t i g r i d codes w h i c h use the s t a n d a r d F A S a n d F M G m u l t i g r i d i m p l e m e n t a t i o n s t o solve the s em i conduc to r equat ions . T h e a p p l i c a t i o n of the bas ic m u l t i g r i d ideas is usefu l for severa l reasons. M o s t i m p o r t a n t l y , the behav i ou r of these s t r a i g h t f o rwa rd i m p l e m e n t a t i o n s m a y i nd i ca te w h i c h component s of the a l g o r i t h m C h a p t e r 3. The Multigrid Method 43 need t o be i m p r o v e d . F u r t h e r , the pe r f o rmance of m o d i f i e d vers ions of the code can be mea su r ed by c o m p a r i n g to the pe r f o rmance of the bas ic vers ions. T h e bas ic codes can also act as a f r a m e w o r k w h i c h a l lows m o d i f i c a t i o n s to be i m p l e m e n t e d i ndependen t l y of one another . T h e bas i c p r og rams use a non l i nea r ver s ion of the Gaus s - Se ide l r e l a xa t i on . A r ed -b l ack o r d e r i n g of the g r i d po in t s is u sed, w i t h t he even n u m b e r e d po i n t s v i s i t ed first ( i n l e x i c o g r a p h i c o rde r ) , f o l l owed by the o d d n u m b e r e d po in t s i n l e x i c og r aph i c order. T h e non l i n ea r equa t i on s are a p p r o x i m a t e l y so lved by a p p l y i n g one step of N e w t o n ' s m e t h o d at each po i n t i n regions where the s o l u t i o n is we l l behaved. Nea r j unc t i on s , where the s o l u t i o n changes r ap id l y , two steps of N e w t o n ' s m e t h o d are used to ob t a i n a mo re a c cu r a te s o l u t i o n of the r e l a x a t i o n equat ions . T h e J a c o b i a n of the d i s c re t i zed equat ions is m a d e d i a gona l l y d o m i n a n t i n l e ad i n g te rms by i m p l i c i t l y s o l v i ng fo r cor rect ions t o the new var iab les (dip, dh, dp) = (dip, dn — ndip,dp + pdip) i n s tead o f the u sua l cor rect ions (dtp, dn, dp). In s u m m a r y , F A S a n d F M G m u l t i g r i d codes have been i m p l e m e n t e d . T h e bas ic codes consist of a (mod i f i ed ) co l l ec t i ve Gau s s - S e i de l -Newton r e l a x a t i o n scheme, l i near i n t e r p o l a t i o n for the p r o l o n g a t i o n , a n d a f u l l we i gh t i ng r e s t r i c t i o n scheme. L i n e a r i n t e r p o l a t i o n is u sed t o t rans fer the so lu t i on f r o m a coarser g r i d t o a finer g r id i n the F M G code. T o accu ra te l y resolve the i n te r i o r layer near x = 0 th i s scheme avoids i n t e r p o l a t i o n across the j u n c t i o n . B o t h the F A S and F M G vers ions of the code were a p p l i e d t o t he test p r o b l e m de sc r i bed i n §3.4.2. T h e p rog rams use a series of u n i f o r m gr ids t o d i s c re t i ze the d o m a i n , w i t h Nf = 64 i n te r i o r po in t s used on the finest level a n d Nc — 2 po i n t s on the coarsest level . T h e gr ids are nested one w i t h i n the nex t , w i t h the mesh s ize hm d o u b l e d at each success ive g r i d , so hm-1 = 2hm for m = 2,3,..., 6. T h e o p e r a t o r on each g r i d is the resu l t of a p p l y i n g the d i s c re t i za t i on p rocedu re of Sec t i on 2.5 w i t h the a p p r o p r i a t e g r id size. Chapter 3. The Multigrid Method 44 I I I I I I I I I I I I I I I I I M I I I I I I I I I I I I I I I I I I I M I t l l t l l l l l l l l l M I I I I I I I I I -1 1 F i g u r e 3.3: P o t e n t i a l V> w i t h V 0 = 0 (Test 1) T h e s o l u t i on componen t s for b o t h the e q u i l i b r i u m a n d f o r w a r d b i a sed p rob lems are shown i n F i g u r e s 3.3 - 3.6. Tab le s 3.4 a n d 3.5 show the n u m b e r of i t e ra t i on s r equ i r ed for the two i m p l e m e n t a -t i on s t o reach convergence for the e q u i l i b r i u m a n d f o r w a r d b i a sed cases: T h e F M G code converged for a l l cases, b u t the F A S ver s ion d i d not converge w i t h (7, i/2) = (2 ,1 ,1 ) . T h e convergence rates are not impres s i ve , a n d t y p i c a l l y s lowed d o w n as the i t e r a t i o n p roceeded. (7 , ^1 ,^2 ) N u m b e r of I te ra t ions V o = 0 Vo = 30 (1 ,1 ,1 ) 11 34 (1 ,2 ,2 ) 6 21 (2 ,1 ,1 ) 7 ov. (2 ,2 ,2 ) 6 7 T a b l e 3.4: N u m b e r of i t e ra t i on s r equ i r ed for Test 1, u s i ng the s t a n d a r d F A S i m p l e m e n -t a t i o n , ov. means t h a t the i t e r a t i o n fa i l ed due t o n u m e r i c a l overf low. Chapter 3. The Multigrid Method 45 1.0 0.9 0.8* 0-7. 0.6. 0.5 0.4 0-3 0.2. 0.1. 0.0 llllllllllllllllllllllllllllllMlkWWMWWWWWWH -1 • P F i g u r e 3.4: C a r r i e r concen t r a t i on s w i t h V0 = 0 (Test 1) Chapter 3. The Multigrid Method 46 F i g u r e 3.5: P o t e n t i a l xp w i t h V0 = 30 (Test 1) (7 , ^1 , ^2 ) N u m b e r of I te ra t ions V0 = 0 V o = 30 ( 1 ,1 ,1 ) 7 18 ( 1 ,2 ,2 ) 5 18 ( 2 ,1 ,1 ) 6 6 ( 2 ,2 ,2 ) 4 5 T a b l e 3.5: N u m b e r of i t e ra t i on s r equ i r ed for Test 1, u s i ng the s t a n d a r d F M G i m p l e -m e n t a t i o n . Chapter 3. The Multigrid Method 47 0.010. 0.009 0.008* 0.007' 0.006. 0.005. 0.004. 0.003' 0.002. 0.001. 0.000 -1 F i g u r e 3.6: C a r r i e r concen t ra t i on s w i t h V0 = 30 (Test 1) Chapter 3. The Multigrid Method 48 T h e f o l l o w i n g chap te r discusses the causes of the p o o r pe r f o rmance of the a l g o r i t hm a n d presents m o d i f i c a t i o n s w h i c h g reat l y i m p r o v e the convergence behav i ou r . Chapter 4 A n Improved Mult igr id Implementation M o d i f i c a t i o n s t o the bas ic codes desc r ibed i n §3.4.3 re su l ted i n d r a m a t i c improvement s i n t he convergence b e h a v i o u r of the a l g o r i t h m . Changes t o the r e l a x a t i o n p rocedu re a n d the p r o l o n g a t i o n o p e r a t o r were p a r t i c u l a r l y effect ive. §4.1 descr ibes these m o d i f i -ca t i on s a n d t he i r effect on t he a l g o r i t h m ' s convergence for a m o d e l p r o b l e m . T h e most successfu l m o d i f i c a t i o n s are co l l ec ted i n one m u l t i g r i d code, S C - 1 , desc r ibed i n §4.2. T h e i m p r o v e d convergence behav i ou r is not l i m i t e d to th i s s imp le p r o b l e m . §4.3 dis-cusses the a p p l i c a t i o n of SC-1 t o p rob l ems of a mo re p r a c t i c a l interest. W i d e ranges of a p p l i e d vo l tages are cons idered fo r several d o p i n g prof i les, i n c l u d i n g t ha t of a t hy r i s t o r . §4.4 compare s th i s i m p l e m e n t a t i o n of the m u l t i g r i d m e t h o d t o an a d a p t a t i o n of a m u l t i g r i d a l g o r i t h m de s c r i bed by H e m k e r [19]. 4.1 Modifications to the Basic Code T h e convergence p rope r t i e s o f the s t r a i gh t f o rwa rd i m p l e m e n t a t i o n of the m u l t i g r i d m e t h o d de sc r i bed i n §3.4.3 c a n be s i gn i f i cant l y i m p r o v e d by s u i t ab l y m o d i f y i n g the componen t s of the a l g o r i t h m . T h i s sect ion descr ibes the successfu l mod i f i c a t i on s and the r e su l t i n g imp rovemen t s . T h e changes were e va l ua ted based on the i r pe r f o rmance on the m o d e l p r o b l e m de sc r i bed i n §3.4.2. A l l m o d i f i c a t i o n s were m a d e i ndependen t l y of one another , w i t h the r e m a i n d e r of the code unchanged ; the bas ic codes desc r ibed i n §3.4.3 were used as a f r amewo rk for these mod i f i c a t i on s . 49 Chapter 4. An Improved Multigrid Implementation 50 §4.1.1 descr ibes t he r e l a x a t i o n operator s w h i c h successfu l ly reduced the n u m b e r of i t e r a t i on s r equ i r ed t o reach convergence. It was f ound t ha t the benef i ts of f u l l r e l a xa t i on i t e r a t i on s can be l a rge l y r ea l i zed w i t h l o ca l r e l axa t i on s , con f ined t o the n e i g h b o u r h o o d of t he j u n c t i o n s , a n d t h a t a co l lec t i ve Gaus s - Se ide l scheme was the mos t effect ive smoothe r . A n i t e r a t i v e p rocedu re can also be used at each po in t i n the r e l a x a t i o n , i n s t e a d of the co l l ec t i ve scheme. T h e p r o l o n g a t i o n opera to r s w h i c h i m p r o v e d the a l g o r i t h m are de sc r i bed i n §4.1.2. T h e mos t successfu l p r o l o n g a t i o n , P R O L - 1 , is based on an extens ion of the ideas o f A l g e b r a i c M u l t i g r i d . A less expens ive vers ion of P R O L - 1 ( P R O L - 2 ) a l so achieves s ig-n i f i c an t imp rovement s . P R O L - 3 , w h i c h i m p l i c i t l y p ro longates the co r rec t i on s t o the s m o o t h e r qua s i - F e rm i var iab les , is an effective ope r a t o r w h i c h can be e x t e n d e d t o m u l -t i d i m e n s i o n a l p rob lems . §4.1.3 descr ibes t he effects of two a l t e rna t i ve r e s t r i c t i on schemes, t he s imp le in jec -t i o n ope ra to r , a n d a r e s t r i c t i o n w h i c h is the t ranspose of the p r o l onga t i o n P R O L - 2 . 4.1.1 Improvements to the Relaxation Scheme T h e r e l a x a t i o n o p e r a t o r p lays a m a j o r role i n the so lu t i on p rocedure . T h e scheme s hou l d be efficient a n d inexpens i ve , b u t must smoo th the er ro r i n ju s t a few i te rat ions . M o d e l p r o b l e m ana ly s i s [36] shows tha t the r e l a x a t i o n p rocedure mus t ba lance two c o m p e t i n g goals: i t s hou l d reduce the h i gh f requency errors bu t also mu s t avo id ex-c i t i n g any s m o o t h , l ow f requency component s w h i c h cannot be qu i c k l y e l i m i n a t e d by re l axa t i on s . S i m i l a r l y t he coarse g r i d co r rec t i on step shou ld e l im i na te low f requency er ror s w i t h o u t c r ea t i n g any n o n s m o o t h errors. T h e behav i ou r of the bas ic code i l l u s t ra tes these concepts . Tab l e 3.4 shows tha t ( w i t h 7 = 1) e x t r a r e l axa t i on s are he lp fu l , bu t they i n t r o d u c e low f requency errors w h i c h are best e l i m i n a t e d u s ing coarse g r i d co r rec t i on steps (7 = 2). T h e F M G vers ion Chapter 4. An Improved Multigrid Implementation 51 behaves i n a s i m i l a r m a n n e r ( Tab l e 3.5), w i t h the benef i ts of the e x t r a s m o o t h i n g i t e r a t i on s lost due t o the low f requency errors t h a t are exc i ted w h e n 7 = 1. T h e r a p i d changes of the s o l u t i on c omponen t s near the n p - j u n c t i o n i n t r o d u c e h i gh f requency er ror s i n the i n t e r i o r layer. The se er rors can be d a m p e d by r e l axa t i on s , but s ince the s o l u t i o n is on l y s lowly v a r y i n g t h r o u g h o u t the rest o f the d o m a i n the r e l a x a t i o n sweeps are needed on l y near the j u n c t i o n . L o c a l r e l a x a t i o n sweeps were i n t r o d u c e d w h i c h p e r f o r m e d the u sua l red -b lack Gau s s -Se ide l r e l a x a t i o n on the three mesh po in t s n e i g h b o u r i n g the j u n c t i o n , w i t h the s o l u t i on t r ea ted as fixed at a l l o ther po int s . T h e l o ca l sweeps were p e r f o r m e d on a l l gr ids w h i c h con ta i ned enough po int s . T h e effect of the l o c a l r e l a x a t i o n sweeps for the F M G i m p l e m e n t a t i o n was r ema rk -able, p a r t i c u l a r l y fo r the f o r w a r d b ias case ( Tab l e 4.6). A l t h o u g h the added expense was m i n i m a l s ince the e x t r a wo r k was done on j u s t a few mesh po in t s , the n u m b e r of i t e ra t i on s r e q u i r e d was r educed by a f ac to r of one-ha l f for some cases. T h e l o ca l r e l ax -a t i ons m a d e t he F A S code mo re robus t , bu t d i d not i m p r o v e t he speed of convergence (Tab le 4.7). (7 , ^1 ,^2 ) N u m b e r of I te ra t ions V o = 0 V 0 = 30 (1 ,1 ,1 ) 6(7) 8(18) (1 ,2 ,2 ) 5(5) 17(18) (2 ,1 ,1 ) 3(6) 3(6) (2 ,2 ,2 ) 4(4) 3(5) Tab l e 4.6: N u m b e r of i t e ra t i on s requ i red for Test 1, u s ing the s t a n d a r d F M G i m p l e -m e n t a t i o n , w i t h l o c a l r e l axa t i on s pe r f o rmed near the n p - j u n c t i o n . B r a c k e t e d number s g ive the n u m b e r of i t e ra t i on s t a ken by the bas ic code. T h e l o ca l r e l a x a t i o n sweeps were most effective w h e n t hey were pe r f o rmed on a l l Chapter 4. An Improved Multigrid Implementation 52 (7 , ^1 , ^2 ) N u m b e r of I te ra t ions V o = 0 y 0 = 30 (1 ,1 ,1 ) 7(11) 38(34) ( 1 ,2 ,2 ) 6(6) 22(21) ( 2 ,1 ,1 ) 6(7) 7(") (2 ,2 ,2 ) 5(6) 6(7) T a b l e 4.7: N u m b e r of i t e ra t i on s r equ i r ed fo r Test 1, u s i n g the s t a n d a r d F A S i m p l e m e n -t a t i o n , w i t h l o c a l r e l a xa t i on s p e r f o r m e d near the n p - j u n c t i o n . B r a c k e t e d numbe r s g ive t he n u m b e r of i t e r a t i on s t aken by the bas ic code. suf f ic ient ly f ine gr ids . U s i n g l o c a l r e l a xa t i on s on j u s t t he finest gr ids, w i t h (7 , ^1 ,^2) fixed at ( 1 ,1 ,1 ) , t he n u m b e r of i t e r a t i on s r e m a i n e d abou t the same as f o r the basic, a l g o r i t h m w i t h o u t l o c a l r e l axa t i on s ( Tab l e 4.8). Coarsest leve l where N u m b e r of I te ra t ions l o c a l r e l a xa t i on s are used V 0 = 0 V 0 = 30 3 6 8 4 6 14 5 6 14 6 7 18 T a b l e 4.8: N u m b e r of i te ra t i on s r equ i r ed for Test 1, u s i n g the s t a n d a r d F M G i m p l e -m e n t a t i o n , w i t h the l o c a l r e l axa t i on s p e r f o r m e d on l y w i t h level > 3 ,4 ,5 ,6 A s e x p e c t e d the l o c a l r e l a x a t i o n sweeps were most benef i c i a l w h e n jus t one f u l l r e l a x a t i o n step was p e r f o r m e d at each step (u^ — v2 = 1). W i t h = v2 = 2 the errors are a l r eady suf f ic ient ly s m o o t h e d so the e x t r a re l axa t i on s are not usefu l . Fu r the r , t he l o c a l r e l a xa t i on s do not avo id the p r o b l e m of e x c i t i n g low f requency errors , and can cause the pe r f o rmance t o degenerate i n some instances. T h e order i n w h i c h the g r i d po in t s are v i s i t ed is i m p o r t a n t t o the per fo rmance of Chapter 4. An Improved Multigrid Implementation 53 the a l g o r i t h m . It is p a r t i c u l a r l y i m p o r t a n t for s i ngu l a r l y p e r t u r b e d p rob l ems t ha t the r e l a x a t i o n sweep ref lect t he cha rac te r i s t i c d i rec t i on s def ined by the p r o b l e m [9]. T h e ca r r i e r c o n t i n u i t y equa t i on s depend o n b o t h +V^> a n d — Vi/> so i n f o r m a t i o n is be i n g t r a n s m i t t e d i n b o t h d i r ec t i on s , a n d a s y m m e t r i c r e l a x a t i o n scheme is a p p r o p r i a t e [19]. T h e s y m m e t r i c Gau s s - Se i de l r e l a x a t i o n ( S Y M G S ) v i s i t s t he m e s h po in t s f i rst i n l e x i c o g r a p h i c o rder , a n d t h e n i n t he reverse order. S Y M G S reduced the n u m b e r of i t -e ra t i on s r e q u i r e d fo r the test p r ob l ems (Tab les 4.9 a n d 4.10) for b o t h t he F A S a n d F M G i m p l e m e n t a t i o n s . T h e s y m m e t r i c r e l a x a t i o n invo lves tw i ce as m u c h wo rk as a red -b lack scheme s ince each p o i n t is v i s i t e d tw i ce d u r i n g a sweep, b u t the p e r f o r m a n c e imp rove -men t is not due o n l y t o th i s e x t r a work : F M G u s i ng S Y M G S requi res 12 i t e ra t i on s w i t h u-i = u2 = I, w h i l e u s i ng the red-b lack scheme w i t h v1 = v2 = 2 ( w h i c h pe r f o rms the same amoun t of w o r k ) takes 18 i t e ra t i on s . T h e r e l a x a t i o n o r de r i n g i m p r o v e d the convergence of the F M G i m p l e m e n t a t i o n mo re t h a n the F A S ver s ion , suggest ing t h a t t he F A S e r ro r is d o m i n a t e d by low f requency errors w h i c h are e l i m i n a t e d by the F M G p r o cedu re on the coarser g r ids before t he finest level is reached. N u m b e r of I te rat ions Vo = 0 V 0 = 30 5(11) 23(34) (1 ,2 ,2 ) 4(6) 9(21) (2 ,1 ,1 ) 5(7) 7 ( - ) ( 2 ,2 ,2 ) 4(6) 6(7) T a b l e 4.9: N u m b e r o f i t e ra t i on s r equ i r ed for Test 1, u s ing the s t a n d a r d F A S i m p l e m e n -t a t i o n w i t h a s y m m e t r i c Gaus s - Se ide l r e l a x a t i o n . B r a c k e t e d numbe r s g ive the n u m b e r o f i t e r a t i on s t a ken by the bas ic code. L o c a l r e l a x a t i o n sweeps near t he n p - j u n c t i o n f u r t he r r educed the n u m b e r of i t -e ra t i on s needed t o reach convergence u s i ng S Y M G S (Tab l e 4.11). T h e effect for Chapter 4. An Improved Multigrid Implementation 54 (7, vx,v2) N u m b e r of I te ra t ions V o = 0 V0 = 30 (1 ,1 .1 ) 5(7) 12(18) (1 ,2 ,2 ) 4(5) 10(18) (2 ,1 ,1 ) 4(6) 5(6) (2 ,2 ,2 ) 4(4) 4(5) T a b l e 4.10: N u m b e r of i t e ra t i on s requ i red for Test 1, u s ing t he s t a n d a r d F M G i m -p l e m e n t a t i o n w i t h a s y m m e t r i c Gaus s - Se ide l r e l a x a t i o n . B r a c k e t e d numbe r s give the n u m b e r of i t e r a t i on s t aken by the bas ic code. N u m b e r of I te ra t ions V0 = 0 VQ = 30 (1,1,-1) 5(11) 13(34) (1 ,2 ,2 ) 4(6) 12(21) (2 ,1 ,1 ) 4(7) 6 ( - ) (2 ,2,2) 4(6) 5(7) T a b l e 4.11: N u m b e r of i te ra t ions r equ i r ed for Test 1, u s ing the s t a n d a r d F A S i m p l e -m e n t a t i o n w i t h a s y m m e t r i c Gaus s - Se ide l r e l a x a t i o n and l o c a l r e l axa t i on s pe r f o rmed nea r the j u n c t i o n . B r a c k e t e d number s g ive the n u m b e r of i t e ra t i on s t a k e n by the bas ic code. Chapter 4. An Improved Multigrid Implementation 55 (7 , ^1 , ^2 ) = (1 ,1 ,1 ) is mos t no t i ceab le — the l oca l r e l a xa t i on s have m u c h the same effect as the f u l l r e l a x a t i o n sweeps v\ — v2 = 2. A l s o , the n u m b e r of i t e ra t i on s needed for t he F A S code u s i n g the s y m m e t r i c r e l a x a t i o n w i t h l o c a l sweeps is c o m p a r a b l e t o the n u m b e r r equ i r ed by t he F M G code. (7 ,^1 , ^2) N u m b e r of I te ra t ions Vo = 0 V 0 = 30 (1 ,1 ,1 ) 4(7) 10(18) (1 ,2 ,2 ) 4(6) 10(18) (2 ,1 ,1 ) 4(6) 4(6) (2 ,2 ,2 ) 4(4) 4(5) T a b l e 4.12: N u m b e r of i t e ra t i on s r equ i r ed for Test 1, u s i n g the s t a n d a r d F M G i m -p l e m e n t a t i o n w i t h a s y m m e t r i c Gaus s - Se ide l r e l a x a t i o n , a n d l o c a l r e l a x a t i o n sweeps p e r f o r m e d near the j u n c t i o n . B r a c k e t e d number s give the n u m b e r of i t e r a t i on s t aken by the bas ic code. T h e co l l ec t i ve Gau s s - Se i de l r e l a x a t i o n is very effect ive, b u t i t can be cos t l y s ince at each po i n t th ree non l i nea r equat ions need t o be so lved s imu l taneous l y . N e w t o n ' s m e t h o d can be used at each po in t , bu t th i s invo lves s o l v i ng a (3 x 3) s y s tem of (poss ib ly i l l - c o n d i t i o n e d ) l i nea r equat ions at each i t e r a t i o n . A m o r e a t t r a c t i v e , a n d less e xpen -sive, a p p r o a c h is t o use a n i t e r a t i v e p r o cedu re t o dea l w i t h the non l i nea r equat ions at each po i n t . T h e non l i nea r Gau s s - Se i de l i t e r a t i o n k n o w n as G u m m e l ' s m e t h o d (desc r ibed i n §2.5) was used t o solve t he non l i nea r equat ions at each po in t . F o u r i te ra t i on s of G u m m e l ' s m e t h o d were used, w i t h the po in t s v i s i t ed i n a s y m m e t r i c sweep. T h e n u m b e r of m u l t i -g r i d i t e r a t i on s r equ i r ed is s i m i l a r to the s y m m e t r i c ( co l lect i ve ) Gaus s - Se ide l r e l a xa t i on for a l l cases ( c o m p a r i n g Tab les 4.13 a n d 4.14 to Tab les 4.9 a n d 4.11). T h i s d e m o n -st rates t h a t an i t e r a t i v e p rocedu re can rep lace the m a t r i x i nver s ion used by N e w t o n ' s m e t h o d . ( U s i n g j u s t one i t e r a t i o n of G u m m e l ' s m e t h o d at each po in t was not useful.) Chapter 4. An Improved Multigrid Implementation 56 (7 , ^1 ,^2 ) N u m b e r of I te rat ions V 0 = 0 V 0 = 30 (1 ,1 ,1 ) 5(11) 23(34) ( 1 ,2 ,2 ) 4(6) 11(21) (2 ,1 ,1 ) 5(7) 7 ( " ) ( 2 ,2 ,2 ) 4(6) 6(7) T a b l e 4.13: N u m b e r of i t e ra t i on s r equ i r ed fo r Test 1, u s ing the s t a n d a r d F A S i m p l e -m e n t a t i o n w i t h a G u m m e l i t e r a t i o n u sed at each po in t . B r a c k e t e d numbe r s give the n u m b e r of i t e r a t i on s t a ken by the bas i c code. (7 , ^1 ,^2 ) N u m b e r of I te rat ions V o = 0 V 0 = 30 (1 ,1 ,1 ) 5(11) 14(34) (1 ,2 ,2 ) 4(6) 11(21) (2 ,1 ,1 ) 4(7) 6 ( - ) ( 2 ,2 ,2 ) 4(6) 5(7) T a b l e 4.14: N u m b e r of i t e ra t i on s r equ i r ed fo r Test 1, u s ing the s t a n d a r d F A S i m p l e -m e n t a t i o n w i t h a G u m m e l i t e r a t i o n used at each po i n t , a n d l o c a l r e l a xa t i on sweeps p e r f o r m e d near the j u n c t i o n . B r a c k e t e d number s g ive the n u m b e r of i t e ra t i on s t aken by the bas ic code. Chapter 4. An Improved Multigrid Implementation 57 4.1.2 Modified Prolongations T h e sha rp i n te r i o r layer w h i c h fo rms near the n p - j u n c t i o n creates d i f f i cu l t ie s for the p r o l o n g a t i o n ope ra to r . T h e errors are e x p e c t e d t o va ry r a p i d l y near the j u n c t i o n , a n d the s imp le l i near i n t e r p o l a t i o n scheme used by the bas ic codes is no t ab le to resolve these changes. T h i s sect ion descr ibes three effect ive p r o l o n g a t i o n operato r s . T h e f i rst p r o l o n g a t i o n , P R O L - 1 , is m o t i v a t e d by the concepts of A l g e b r a i c M u l t i g r i d a n d is based on the d i s c re t i zed equat ions themse lves . A second ver s ion , P R O L - 2 , is an a p p r o x i m a t i o n of P R O L - 1 . B o t h P R O L - 1 a n d P R O L - 2 s i gn i f i can t l y speed up the convergence of the code. P R O L - 3 hand les the j u n c t i o n layer by i m p l i c i t l y p r o l o n g a t i n g the co r rec t i on s i n te rms of t he we l l - behaved q u a s i - F e r m i var iab les . P R O L - 3 is of spec ia l interest because it uses a po i n tw i s e t r a n s f o r m a t i o n w h i c h can be eas i ly e x tended t o m u l t i d i m e n s i o n a l p rob lems. A l g e b r a i c M u l t i g r i d ( A M G ) [25] a t t e m p t s t o e x t e n d the u sua l geomet r i c m u l t i g r i d m e t h o d t o p rob l ems w h i c h are no t de r i ved f r o m d i f fe rent i a l equat ions . A M G usu -a l l y cons iders on l y l i near p r ob l ems and so is not d i r e c t l y app l i c ab l e to the non l i nea r s e m i c o n d u c t o r equat ions . T h e A M G ideas are based on an ana lys i s of the range a n d nu l l space of the m u l t i g r i d opera to r s w h i c h suggests t h a t , after " smoothing, the er ror s hou ld l ie i n the range of the p r o l onga t i on . F o l l o w i n g [21] we cons ider a l i near p r o b l e m LhUh — fh o n a g r i d Gh, w i t h Sh t he space of g r i d f unc t i on s on Gh. T h e ana logous coarse g r i d p r o b l e m is po sed on g r i d GH. Sh can be w r i t t e n as Sh = RangeIH 0 Nullspacelf?Lh (4.75) (see [22]). A f t e r t he a p p l i c a t i o n of the r e l a xa t i on o p e r a t o r the er ror t\ = u^. — Uh can be Chapter 4. An Improved Multigrid Implementation 58 w r i t t e n as vh = rj + IHwH (4.76) where 77 E Nullspacelf?Lh a n d WH € <S'H. T h e n the r e s t r i c t i on of the r e s i dua l gives rH = tfLhVh = tfLhIhHwH. (4.77) T h e coarse g r i d co r r ec t i on VJJ t h e n solves LHvH = rH = I^LJhHwH, (4.78) so i f t he G a l e r k i n f o r m of the coarse g r i d ope ra to r is used {LH = I^Lhljj) t h e n VJJ = wji- T h i s c o r r ec t i on is i n t e r po l a t ed , a n d the er ror a f te r the p r o l o n g a t i o n is: Vh •= vh- IHwH := r? + IHwH - IHwH := V- (4.79) If t he e r ro r after the r e l a xa t i on is i n the range of SH (so n = 0) t h e n the er ror after t he coarse g r i d c o r r e c t i on is zero. M o r e general ly, t he coarse g r id co r r ec t i on reduces the e r ro r by a fac to r of pc — WvW/Wv + IHWH\\-T h e A M G concepts suggest several des ign c r i t e r i a for the choice of the p r o l onga t i on , r e s t r i c t i o n and coarse g r i d ope ra to r (e.g., t ha t the G a l e r k i n f o r m of the coarse g r i d o p e r a t o r be used) bu t it appears t h a t t he most i m p o r t a n t f ac to r is t he choice of the p r o l o n g a t i o n ([18], [2], [21]). ( W h e n the G a l e r k i n f o r m of the coarse g r i d ope ra to r is rep l aced by the more p r a c t i c a l f in i te difference ana log of Lh the convergence rate differs f r o m pa on l y by a t e r m of order h? [21].) T h e des ign of the p ro l onga t i on is gu i ded by the d i screte opera to r Lh (see §3.4.4 a n d §10.3 in [18], a n d [2]). F o r a s tanda rd coarsen ing we choose the p r o l onga t i on IH so tha t Chapter 4. An Improved Multigrid Implementation 59 the p r o l onga ted co r r e c t i on := IfjVH satisf ies t he homogeneous re s idua l equa t i on at a l l b l ack ( odd ) po i n t s , so (Lhvh)i = 0, i o d d . ( R e c a l l t h a t , u s i n g a s t a n d a r d coarsen ing, t he b lack (odd) po i n t s are those po i n t s w h i c h appea r on the fine g r i d bu t not on the coarser gr id.) A t t he red po in t s ( w h i c h are c o m m o n t o b o t h gr ids ) t he co r r e c t i on is t r an s fe r red d i rec t l y . W i t h a red -b lack Gau s s - Se i de l r e l a x a t i o n t h i s ensures t h a t t he s m o o t h e d e r ro r l ies i n the range of the p r o l o n g a t i o n , so 77 = 0 i n (4.79). F u r t h e r , t he re s i dua l at the o d d po in t s , w h i c h is zero after the c o m p l e t i o n of the b lack phase of the r e l a x a t i o n , r ema in s zero after the coarse g r i d c o r r e c t i o n step, so the p r o l onga t i o n ef fect ive ly pe r fo rms a p a r t i a l s m o o t h i n g step [18]. W h i l e the above d i scus s ion is not d i r e c t l y app l i c ab l e t o the non l i nea r p r o b l e m NhUh = 0, the p r o l o n g a t i o n can s t i l l be chosen so t ha t t he p r o l onga ted co r r ec t i on satisf ies t he homogeneous r e s i dua l equa t i on at a l l o d d po in t s , so (Nhvh)i = 0, i o d d (see §3.3). T h e p r o l o n g a t i o n P R O L - 1 i m p l e m e n t s th i s by accu ra te l y so l v ing (us ing N e w t o n ' s m e t h o d ) t he f o l l o w i n g equat ions fo r the new values n " a n d p" , at each o d d po i n t i: /T2(V>J+1 - 2 C + rki-i) + < - p? - C(Xi) = 0, (4.80) B ( ^ _ a - # > i - i = 0, (4-81) a n d B(W-rl>i+i)pi+i-( B ( ^ + 1 - V > r ) + C ) ) P ? + B W - 1>i-i)Pi-i = 0 , (4.82) Chapter 4. An Improved Multigrid Implementation 60 w i t h u p d a t e d va lues used at the ne i g hbou r i n g po i n t s i ± 1. P R O L - 1 g rea t l y r educed the n u m b e r of i t e ra t i on s r equ i r ed t o reach convergence, c o m p a r e d t o the bas i c code (Tab les 4.15 a n d 4.16). W i t h (7, Ui, v2) = (1 ,1 ,1 ) P R O L - 1 reduces t he n u m b e r of i t e ra t i on s r equ i r ed by a f ac to r o f three. T h i s p r o l o n g a t i o n does not seem t o exc i t e l ow f requency errors , so the results for 7 = 1 a n d 7 = 2 are m u c h the same. F u r t h e r , the F M G i m p l e m e n t a t i o n p rov ides no s ign i f icant i m p r o v e m e n t over t he F A S ve r s ion , bu t F A S does not requ i re the e x t r a work t o generate a n i n i t i a l guess on the finest level . (However , for mo re d i f f i cu l t p rob lems the F M G ver s i on m i gh t be e x p e c t e d t o be more robust since it p roduces a be t te r s t a r t i n g po i n t on the finest gr id. ) (7 , ^1 ,^2) N u m b e r of I te ra t ions Vo = 0 V0 = 30 (1,1,1) 7(11) 11(34) (1 ,2,2) 4(6) 7(21) (2 ,1,1) 5(7) 7 ( - ) (2,2,2) 4(6) 6(7) T a b l e 4.15: N u m b e r of i te ra t ions requ i red for Test 1 u s ing the s t a n d a r d F A S i m p l e m e n -t a t i o n , a n d the p r o l onga t i o n P R O L - 1 . B r a c k e t e d number s g ive the n u m b e r of i t e ra t i on s t a ken by the bas ic code. (7 , ^1 ,^2) N u m b e r of I te rat ions Vo = 0 Vo = 30 (1,1,1) 6(7) 12(18) (1,2,2) 4(5) 7(18) (2,1,1) 5(6) 5(6) (2,2,2) 4(4) 4(5) T a b l e 4.16: N u m b e r of i te ra t ions r equ i r ed for Test 1 u s ing the s t a n d a r d F M G i m -p l e m e n t a t i o n , and the p ro l onga t i on P R O L - 1 . B r a c k e t e d number s give the n u m b e r of i t e r a t i on s t a k e n by the bas ic code. Chapter 4. An Improved Multigrid Implementation 61 W h i l e P R O L - 1 is effect ive, i t can also be expens i ve since at each o d d po in t encoun -te red by the p r o l o n g a t i o n a non l i nea r sy s tem of equat ions needs to be solved. T h e cost can be r educed by u s i ng j u s t one i t e r a t i o n of N e w t o n ' s m e t h o d t o a p p r o x i m a t e l y solve these equat ions . T h i s a p p r o x i m a t i o n is h i gh l y successfu l fo r the F M G ver s ion ( Tab l e 4.17), w i t h on l y 4 - 6 i t e r a t i on s requ i red for Test -1 . ( T h e pe r f o rmance of the F A S ve r s i on worsened s l i gh t l y w h e n the equat ions were not so lved accurate ly. ) (7 , ^1 , ^2 ) N u m b e r of I te rat ions v 0 = o Vo = 30 (1,1.1) 5(7) 5(18) (1/2,2) 4(5) 6(18) (2 ,1 ,1 ) 4(6) 5(6) (2 ,2 ,2 ) 4(4) 5(5) T a b l e 4.17: N u m b e r of i t e ra t i on s requ i red for Test 1.using the s t a n d a r d F M G i m p l e -m e n t a t i o n , a n d the p r o l o n g a t i o n P R O L - 1 , w i t h the non l i nea r equat ions solved app rox -ima te l y . B r a c k e t e d number s g ive the n u m b e r of i t e ra t i on s t a ken by the bas ic code. A n o t h e r i nexpens i ve p r o l onga t i on , P R O L - 2 , is, l ike P R O L - 1 , r e l a ted t o the d i f fer-ence opera to r . A l i nea r i n t e r p o l a t i o n of the co r rec t i on t o ip is used, w h i c h corresponds t o the l i nea r i n t e r p o l a t i o n of tp used i n the de r i v a t i on of the S c h a r f e t t e r - G u m m e l d i s -c r e t i z a t i o n . Once t he k n e w values of rj>* are k n o w n , tp can be t r ea ted as f i xed i n the c o n t i n u i t y equat ions . T h i s a l lows the cor rect ions dn^ and dpi t o be ca l cu l a ted so t h a t they co r r e spond t o cor rect ions t o the currents Jn a n d Jp: w r + i - V-D^+I - ( w - v > r + i ) + Bw-w_1))dni + B(W-i - V'D^-i = 0 (4.83) a n d Chapter 4. An Improved Multigrid Implementation 62 w r - c . j d p . - _ a = 0. ( 4 - 8 4 ) T h e supe r s c r i p t n emphas i zes t h a t the new (cor rected) s o l u t i o n values are used at the n e i g h b o u r i n g po in t s . A n i m p o r t a n t feature of P R O L - 2 is t ha t it uses the s m o o t h var iab les Jn a n d J p , w h i c h va ry s l ow ly t h r o u g h the j u n c t i o n s , un l i ke n a n d p. W i t h o u t u s i ng l o c a l r e l a xa t i on s the effects of P R O L - 2 are m i x e d . T h e n u m b e r of i t e r a t i o n s r equ i r ed for the test p r o b l e m w i t h V 0 = 0 are r o u g h l y the same as w h e n l i near i n t e r p o l a t i o n is used, for b o t h the F A S a n d F M G i m p l e m e n t a t i o n s (Tab le s 4.18,4.19). F o r the f o r w a r d b i a sed p r o b l e m the n u m b e r of i t e ra t i on s is g rea t l y r educed w h e n suff i-c ient s m o o t h i n g is done, bu t p rob l ems w i t h n u m e r i c a l over f low are encoun te red when vx = u2 = 1. ( 7 , " 1 , " 2 ) N u m b e r of I te ra t ions Vb = 0 V0 = 30 (1 ,1 .1 ) 9(11) o-.(34) (1 ,2 ,2 ) 6(6) 12(21) (2 ,1,1) 7(7) ov.(-) (2 ,2,2) 5(6) 7(7) T a b l e 4.18: N u m b e r of i t e ra t i on s r equ i r ed for Test 1, u s i n g the s t a n d a r d F A S i m p l e -m e n t a t i o n , w i t h the p r o l onga t i o n P R O L - 2 . B r a c k e t e d numbe r s g ive t he n u m b e r of i t e r a t i on s t aken by the bas ic code, (ov. denotes n u m e r i c a l overf low) T h e a d d i t i o n a l s m o o t h i n g does not need t o be p e r f o r m e d t h r oughou t the ent i re d o m a i n (Tab les 4.20 a n d 4.21). W i t h l oca l r e l a xa t i on sweeps p e r f o r m e d on l y i n t he n e i g h b o u r h o o d of the n p - j u n c t i o n the i m p l e m e n t a t i o n is m u c h more robus t . T h e i n -creased ra te of convergence is p a r t i c u l a r l y no t i ceab le for the f o rwa rd b i a sed p r o b l e m . W i t h F A S , and (7, v2) — (1,1,1) the n u m b e r of i t e ra t i on s r equ i r ed is less t h a n ha l f t h a t r equ i r ed by the s t a n d a r d i m p l e m e n t a t i o n us ing a l i nea r i n t e r p o l a t i o n (Tab le 4.20), w h i l e fo r F M G the n u m b e r of i t e ra t i on s is reduced by a f a c t o r of three ( Tab l e 4.21). Chapter 4. An Improved Multigrid Implementation 63 (7 , ^1 ,^2 ) N u m b e r of I te rat ions V"o = 0 Vo = 30 (1 ,1 ,1 ) 7(7) ov.(18) (1 ,2 ,2 ) 4(5) 6(18) ( 2 ,1 ,1 ) 5(6) ov.(6) (2 ,2 ,2 ) 5(4) 5(5) T a b l e 4.19: N u m b e r o f i t e ra t i on s r equ i r ed for Test 1, u s i n g the s t a n d a r d F M G i m -p l e m e n t a t i o n , w i t h t he p r o l o n g a t i o n P R O L - 2 . B r a c k e t e d numbe r s g ive the n u m b e r of i t e r a t i on s t aken by the bas ic code. (ov. denotes n u m e r i c a l over f low) (7 , ^1 ,^2 ) N u m b e r of I te rat ions V o = 0 Vo = 30 (1 ,1 ,1 ) 5(11) 16(34) (1 ,2 ,2 ) 5(6) 11(21) (2 ,1 ,1 ) 5(7) 8 ( - ) (2 ,2 ,2 ) 5(6) 6(7) T a b l e 4.20: N u m b e r of i t e ra t i on s r equ i r ed for Test P r o b l e m 1, u s i ng the s t a n d a r d F A S i m p l e m e n t a t i o n , w i t h the p r o l o n g a t i o n P R O L - 2 a n d l o c a l r e l axa t i on s p e r f o r m e d near the j u n c t i o n . B r a c k e t e d numbe r s g ive the n u m b e r of i t e ra t i on s t a ken by the bas ic code. (7 , ^1 ,^2 ) N u m b e r of I terat ions V 0 = 0 V 0 = 30 (1 ,1 ,1 ) 5(7) 6(18) (1 ,2 ,2 ) 5(5) 7(18) (2 ,1 ,1 ) 5(6) 5(6) (2 ,2 ,2 ) 5(4) 4(5) T a b l e 4.21: N u m b e r of i t e ra t i on s r equ i r ed for Test 1, u s ing the s t a n d a r d F M G i m p l e -m e n t a t i o n , w i t h the p r o l o n g a t i o n P R O L - 2 a n d l o ca l r e l axa t i on s p e r f o r m e d near the j u n c t i o n . B r a c k e t e d numbe r s give the n u m b e r of i t e ra t i on s t a ken by the bas ic code. Chapter 4. An Improved Multigrid Implementation 64 P R O L - 1 a n d P R O L - 2 b o t h i nvo l ve the s o l u t i on of equat i on s w h i c h depend on values at t he ne i g hbou r i n g po in t s , a n d no t j u s t at the po i n t of interest. In m u l t i d i m e n s i o n a l p r ob l ems th i s c an l i m i t t he i r a p p l i c a b i l i t y unless s t a n d a r d coar sen ing is u sed [18], [2]. T h e p r o l o n g a t i o n P R O L - 3 uses the po i n tw i se t r a n s f o r m a t i o n t o the s m o o t h quas i -F e r m i var iab les <pn and 4>p to avo id the p rob l ems w h i c h m a y ar ise i n mo re t h a n one d imens i on . T h e sca led var iab les n a n d <f>n are r e l a t ed b y n = 62e^~^n, so t h a t <f>n = ip — l n (n i 5~ 2 ) . D e n o t i n g the unco r r e c t ed values as n° and <pn, a n d the co r rec ted values by n " and ^ = « - « = ^ A . > ( 4 " 8 5 ) where d<f>n a n d dn are the cor rect ions t o <f>n a n d n respect ive ly . T h e co r rec t ions to (pn are i n t e r p o l a t e d l i nea r l y at an o d d po i n t i, so d<f>m = l/2(d<f>ni_1 + d(f>n = l / 2 1 n ( ^ _ 2 = i ) . (4.86) T h i s change i n <f>n p roduces a new value of n,: T h e o r i g i na l va lue of nl is sca led by a geometr i c average of the r a t i o of new to o l d values at n e i g hbou r i n g po in t s . A s im i l a r de r i v a t i on p roduces the f o l l ow ing i n t e r p o l a t i o n for p V P t + i P i - i S ince n a n d p are s m o o t h l y v a r y i n g away f r o m the j unc t i o n s , P R O L - 3 uses the s imp le r l i nea r i n t e r p o l a t i o n th roughout most of the d o m a i n , w i t h the above re la t ions used on l y i n the ne i ghbou rhood of the j u n c t i o n . Un less l o ca l r e l a x a t i o n sweeps were used near the p n - j u n c t i o n to p rov ide e x t r a s m o o t h i n g of the s o l u t i on , n u m e r i c a l overf low occu red w h e n P R O L - 3 was used. W i t h Chapter 4. An Improved Multigrid Implementation 65 l o c a l r e l a xa t i on s the resu l t s were m u c h the same as for the s t a n d a r d i m p l e m e n t a t i o n w h e n u s i n g l o c a l r e l axa t i on s (Tab le 4.22). P R O L - 3 reduces the n u m b e r of i t e ra t i on s needed for the f o r w a r d b i a sed p r o b l e m u s i ng F A S , bu t there is no m a r k e d dif ference for e i t he r F A S or F M G ( c o m p a r i n g Tab les 4.22 a n d 4.23 t o 4.7 a n d 4.6). ( 7 , ^ 1 , ^ 2 ) N u m b e r of I te rat ions v 0 = o V 0 = 30 (1 ,1 .1 ) 8(11) 18(34) (1,2,2) 6(6) 17(21) (2,1,1) 5(7) 7 ( " ) (2 ,2 ,2 ) 4(6) 6(7) T a b l e 4.22: N u m b e r of i t e ra t i on s r equ i r ed fo r Test 1, us ing the s t a n d a r d F A S i m p l e -m e n t a t i o n , a n d the p r o l o n g a t i o n P R O L - 3 , w i t h l o c a l r e l axa t i on s p e r f o r m e d near t he j u n c t i o n . B r a c k e t e d numbe r s g ive the n u m b e r of i t e ra t i on s t a ken by the bas ic code. ( 7 , ^ 1 , ^ 2 ) N u m b e r of I te rat ions Vo = 0 Vo = 30 (1 .1 ,1 ) 6(7) 16(18) (1 ,2 ,2 ) 5(5) 11(18) (2 ,1 ,1 ) 4(6) 6(6) (2 ,2 ,2 ) 4(4) 4(5) T a b l e 4.23: N u m b e r of i t e ra t i on s r equ i r ed for Test 1, us ing the s t a n d a r d F M G i m p l e -m e n t a t i o n , a n d the p r o l o n g a t i o n P R O L - 3 w i t h l o c a l r e l axa t i on s pe r f o rmed near the j u n c t i o n . B r a c k e t e d numbe r s give the n u m b e r of i t e ra t i on s t a ken by the bas ic code. 4.1.3 Alternative Restriction Operators T h e r e are two c o m m o n l y used r e s t r i c t i o n opera to r s , f u l l we i gh t i ng and i n j ec t i on . These, a n d ano the r r e s t r i c t i on m o t i v a t e d by the p r i n c i p l e s of A l g e b r a i c M u l t i g r i d ( A M G ) , were Chapter 4. An Improved Multigrid Implementation 66 i m p l e m e n t e d a n d tes ted. T h e opera to r s , a n d the resu l t s of these tests, are desc r ibed here. T h e r e s t r i c t i o n o p e r a t o r t ransfers the re s i dua l f r o m a f ine g r i d t o a coarser g r i d . It s hou l d be s imp le a n d effect ive, a n d the resu l t of the r e s t r i c t i o n shou ld a ccu ra te l y ref lect the o r i g i n a l r e s i dua l , w i t h o u t any n o n s m o o t h errors be i n g i n t r o d u c e d . T h e f u l l -we i gh t i n g r e s t r i c t i o n o p e r a t o r is used i n the s t a n d a r d code desc r ibed i n Sec t i on 3.4.3. T h e s imp le r i n j e c t i o n ope ra to r was also i m p l e m e n t e d , but i t was not effect ive. U s i n g i n j e c t i o n , the i t e r a t i o n converged for the e q u i l i b r i u m case w h e n F M G was used w i t h ( 7 , ^ 1 , ^ 2 ) = (2 ,2 ,2 ) . Fo r a l l o ther cases the i t e r a t i o n f a i l ed due t o n u m e r i c a l overf low. A t h i r d r e s t r i c t i o n ope r a t o r was f o r m e d as the ad jo in t of the i n t e r p o l a t i o n P R O L - 2 . T h e success of th i s r e s t r i c t i o n was l i m i t e d . W h e n used i n c o m b i n a t i o n w i t h P R O L - 2 a n d l o c a l r e l a x a t i o n sweeps the convergence for the e q u i l i b r i u m case was sat i s factory, bu t for the f o r w a r d b i a sed p r o b l e m the p r o g r a m converged for on l y two cases (Tables 4.24 a n d 4.25). ( W h e n a l i nea r i n t e r p o l a t i o n is u sed the convergence is s lower, a n d the p r o g r a m diverges for the same values of ( 7 ,1 /1 , z^))-( 7 , ^ 1 , ^ 2 ) N u m b e r of I te ra t ions V 0 = 0 V 0 = 30 (1 ,1 ,1 ) 5(11) ov(34) (1 ,2 ,2 ) 6(6) 13(21) (2 ,1 ,1 ) 8(7) 8 ( - ) (2 ,2 ,2 ) 5(6) ov(7) T a b l e 4.24: N u m b e r of i t e ra t i on s requ i red for Test 1 ( F A S ) , u s ing a r e s t r i c t i on w h i c h is the ad jo in t of P R O L - 2 , a n d l o c a l r e l a xa t i on sweeps. B r a c k e t e d numbe r s give the n u m b e r of i t e r a t i on s t a k e n by the bas ic code, (ov means t h a t n u m e r i c a l overf low occur red . ) Chapter 4. An Improved Multigrid Implementation 67 ( 7 , ^ 1 , ^ 2 ) N u m b e r of I te rat ions V 0 = 0 V 0 = 30 (1 ,1 ,1 ) 8(7) ou(18) (1 ,2 ,2 ) 5(5) ou(18) (2 ,1 ,1 ) 6(6) cw(6) (2 ,2 ,2 ) 5(4) ov(5) T a b l e 4.25: N u m b e r of i t e ra t i on s r equ i r ed for Test 1 ( F M G ) , u s ing a r e s t r i c t i on w h i c h is t he adjo in t of P R O L - 2 , a n d l o c a l r e l a xa t i on sweeps. B r a c k e t e d numbe r s g ive the n u m b e r of i t e r a t i on s t a k e n by the bas ic code, (ov means tha t n u m e r i c a l overf low occur red . ) 4.2 Summary of Successful Modifications T h e code S C - 1 , w h i c h w i l l be referred to t h r o u g h o u t the r ema i nde r of th i s thes is , i n co rpo ra te s t he mos t successful mod i f i c a t i on s de sc r i bed i n the p rev ious sect ion. T h e m a i n features of t he code are l i s t ed be low. 1. T h e F M G p rocedu re is used t o generate an i n i t i a l guess on the finest leve l . A l i near i n t e r p o l a t i o n w h i c h avoids i n t e r p o l a t i n g across j u n c t i o n s is used to t rans fe r the s o l u t i o n to the nex t finer level . 2. A s y m m e t r i c , co l lec t i ve Gaus s - Se ide l r e l a x a t i o n p rocedure is used. One i t e r a t i o n of N e w t o n ' s m e t h o d is used to solve the non l i nea r equat ions , except near j u n c t i o n s where two i te ra t i on s are used. 3. L o c a l r e l a x a t i o n sweeps are used i n the ne i ghbou rhood of the j unc t i on s . 4. T h e p r o l o n g a t i o n P R O L - 2 is used. W h i l e P R O L - 1 appears t o be more effect ive, it is a l so ve ry expens ive, so the more efficient vers ion has been chosen. 5. T h e r e s t r i c t i o n ope ra to r is a f u l l we ight ing scheme. Chapter 4. An Improved Multigrid Implementation 68 4 .3 F u r t h e r T e s t i n g o f S C - 1 T h e d r a m a t i c i m p r o v e m e n t s i n the convergence rates de s c r i bed i n the p rev ious sect ion are not l i m i t e d t o the spec ia l m o d e l p r o b l e m cons ide red there. T h i s sect ion d e m o n -strates t h a t SC-1 can be successfu l ly a p p l i e d to p r o b l e m s of a mo re p r a c t i c a l interest , w i t h a w ide range of a p p l i e d po ten t i a l s a n d di f ferent d o p i n g prof i les. T h e convergence of SC-1 was s t ud i ed fo r three different p rob l ems , Test 1 (desc r ibed i n §3.4.2), Test 2, con s i s t i ng of the s t a n d a r d d i ode test p r o b l e m presented i n [30], a n d Test 3, m o d e l l i n g a t h y r i s t o r . In sca led f o r m , the d o p i n g pro f i le of the s t a n d a r d d i ode is: - 1 - l < x < 0 D(x) = { 0 x = 0 + 1 0 < x < 1. N o t e t h a t , i n cont ra s t t o Test 1, th i s represents a n p n - j u n c t i o n , so a p p l y i n g a po s i t i ve p o t e n t i a l at x = 1 is a reverse b i a s i ng , wh i l e a nega t i ve p o t e n t i a l is a f o rwa rd bias. T h e d o p i n g pro f i le w h i c h mode l s t he t h y r i s t o r is D{x) = < + 1 - 1 < x < - 0.5 - 1 0 ~ 3 - 0 . 5 < x <0 1 0 ~ 5 0 < x <0.5 - 1 0 . 5 < x < l . T h e a p p l i e d po ten t i a l s r anged f r o m - 4 0 0 to +400 w i t h A 2 = 1.67 x 1 0 - 7 , 82 = 1.22 x 1 0 ~ 8 a n d R = 0. T h e convergence c r i t e r i on u sed here is s t r i c te r t h a n t ha t used i n § 4 . 1 . T h e so l u t i on was j udged t o be converged w h e n each c omponen t u3 of two success ive i terates uk+1 a n d uk sat i s f ied \uk+1 -uk\ < 1.0 x 1 0 ~ 6 | 4 + 1 | . Chapter 4. An Improved Multigrid Implementation 69 1. 0 -1. log || residual || -_ -3. -4 -5. -6. -7_ -8 -9 " 1 ' i " i i i i i i i i 1 2 3 4 5 6 7 8 9 10 Iteration • V=-200 • V=-400 F i gu r e 4.7: Convergence of S C - 1 , V"0 = - 2 0 0 , - 4 0 0 , Nf = 64 (Test 1) F o r Test 1, w i t h reverse biases of V 0 = —200 a n d Vo = —400, SC-1 converges r a p i d l y ( F i g u r e 4.7). T h e convergence ra te does not s i gn i f i cant l y change as the size of the mesh changes ( F i g u r e 4.8). T h e convergence ra te is also i ndependen t of t he mesh size for f o r w a r d b i a s i n g w i t h V 0 = 200 ( F i g u r e 4.9). T h i s convergence ra te is a p p r o x i m a t e l y the same w h e n a la rger p o t e n t i a l of V 0 = 400 is used ( F i g u r e 4.10). F o r the reverse b iased s t anda rd d iode (Test 2) w i t h V 0 = 200 the convergence of S C -1 is a lmost i m m e d i a t e , for a l l g r i d spac ings ( F i g u r e 4.11). SC-1 also converges qu i c k l y for very large reverse biases, w i t h V 0 = 4000 ( r ough l y 100 V o l t s ) . F o r a s m a l l f o rwa rd b i a s i ng , ( V 0 = —40) there is s t i l l r a p i d convergence but the convergence is slower on t he finer gr ids ( F i g u r e 4.12). T h i s t r e n d is mo re apparent w h e n the larger f o rwa rd b ias V 0 = - 2 0 0 is used ( F i g u r e 4.13). T h e t h y r i s t o r of Test 3 presents spec ia l d i f f i cu l t ie s s ince the p r o b l e m is not a lways we l l - c ond i t i o ned [3] and the so lu t i on is not un i que — for a g iven vo l tage, there are Chapter 4. An Improved Multigrid Implementation -o. -1.. -2 _ -3 log 11residual 11 1 2 3 A 5 6 7 8 9 10 11 12 Iteration • N - 1 6 o N - 3 2 • N - 6 4 F i g u r e 4.8: Convergence of S C - 1 , V0 = - 2 0 0 , N} = 16,32,64 (Test 1) F i g u r e 4.9: Convergence of SC - 1 , V0 = 200, Nf = 8 ,16,32,64 (Test 1) Chapter 4. An Improved Multigrid Implementation F i g u r e 4.10: Convergence of S C - 1 , V0 = 200,400, Nf = 64 (Test 1) log 11 residual || _ 2-6. 1.3 0.0. 1.3. 2.6. 3.9. 5.2. 6.5. 7.6' 9.1 • N - 8 oN - 16 i t 6 7 Iteration • N - 32 i 8 i 9 IN - 64 i 10 i 11 12 F i g u r e 4.11: Convergence of SC - 1 , V0 = 200, Nf = 8 ,16,32,64 (Test 2) Chapter 4. An Improved Multigrid Implementation log || residual || T i l 1 2 3 A 5 6 7 8 9 10 11 12 Iteration • N « 8 n N - 1 6 *N = 32 _ N « 6 4 F i g u r e 4.12: Convergence of SC - 1 , V0 = - 4 0 , N} = 8 ,16,32,64 (Test 2) log 11 residual 11 _ ' i i i i i i i i • i i i i i i i i i i i i i i i 1 2 3 4 5 6 7 8 9 10 12 14 1617 19 21 23 25 Iteration • N - 1 6 D N - 3 2 • N - 6 4 F i g u r e 4.13: Convergence of SC -1 , V0 = - 2 0 0 , Nf = 16,32,64 (Test 2) Chapter 4. An Improved Multigrid Implementation 73 severa l poss ib le cu r ren t s w h i c h m a y resu l t . T h i s nonun iqueness a l lows for a phy s i c a l p h e n o m o n o n k n o w n as l a t c h - u p [37]. T o find a l l s o l u t i on b ranches w o u l d requ i re tha t c o n t i n u a t i o n be used, b u t so lu t ions can be f o u n d for r e l a t i ve l y sma l l a p p l i e d potent ia l s . W i t h a f o r w a r d b ias of V 0 = 40 SC-1 converges very q u i c k l y ( Tab l e 4.26), b u t for V 0 = 0 the convergence is very s low unless e x t r a coarse g r i d c o r r e c t i o n steps are used ( 7 = 2). T h e F M G p r o cedu re d i d not converge d i r e c t l y fo r a reverse b i a sed p r o b l e m (Vo = —40) because a su f f i c ient ly a ccu ra te i n i t i a l guess was not used. T h e code converged p r ope r l y w h e n be t te r s t a r t i n g values were o b t a i n e d by u s i ng a s imp l e c o n t i n u a t i o n i n t e rms of the a p p l i e d p o t e n t i a l ( T ab l e 4.27). (7,^1,1/2) N u m b e r of I te ra t ions V o = 0 V 0 = 40 (1 ,1 ,1 ) 39 6 (1 ,2 ,2 ) 30 4 (2 ,1 ,1 ) 10 5 (2 ,2 ,2 ) 8 4 T a b l e 4.26: N u m b e r of i t e r a t i on s requ i red for Test 3, u s ing SC -1 . V0 N u m b e r of I terat ions 0 10 - 1 0 10 - 2 0 9 - 5 0 10 T a b l e 4.27: N u m b e r of i t e r a t i on s requ i red for Test 3, u s i ng a c o n t i n u a t i o n i n t e r m s of V 0 , s t a r t i n g at V 0 = 0 a n d progres s ing t o V 0 = —50. Chapter 4. An Improved Multigrid Implementation 74 4.4 Comparison of SC-1 and Hemker's A lgor i thm T h e o n l y a p p l i c a t i o n of the m u l t i g r i d m e t h o d t o the s e m i c o n d u c t o r equat ions t ha t we are aware of w h i c h has success fu l ly used very coarse gr ids in the s o l u t i on p rocedure is de s c r i bed b y H e m k e r [19]. A m u l t i g r i d code based on H e m k e r ' s ideas has been i m p l e m e n t e d a n d i t s p e r f o rmance is c o m p a r e d t o the p r o g r a m S C - 1 . T h i s sect ion descr ibes H e m k e r ' s a l g o r i t h m , a n d the compar i sons . 4.4.1 Description of Hemker's A lgor i thm H e m k e r uses the m u l t i g r i d m e t h o d to solve t he s e m i c o n d u c t o r equat ions i n t e rms of the unsea led q u a s i - F e r m i va r i ab le s (equat ions 2.39 - 2.41), w i t h R = 0. T h e p r o l onga t i on a n d r e s t r i c t i o n opera to r s are based on i n t e rpo l a t i on s of the cur rent s Jn a n d Jp. T o f a c i l i t a t e t he c o n s t r u c t i o n of a p p r o p r i a t e ope ra to r s a h i e ra rchy of staggered gr ids is used, as de sc r i bed be low. T h e s t a n d a r d d i s c r e t i z a t i o n of the ca r r i e r c o n t i n u i t y equat i on s uses a s y m m e t r i c scheme t o a p p r o x i m a t e d i v J n : (as i n §2.5). E xp re s s i on s fo r J„,,+i /2, a n d s i m i l a r l y Jp,,-+i/2, are t h e n f o u n d i n te rms of t he u n k n o w n s at the g r i d po in t s i a n d i + 1. T h u s , the cur rent s Jn a n d Jp are k n o w n on l y at the m i d p o i n t s of the interva l s be tween mesh ' po in t s , u s i ng f u n c t i o n values at t he g r i d po in t s . W h e n the m u l t i g r i d m e t h o d uses s t anda rd coar sen ing ( d o u b l i n g the mesh size on the coarser levels ) t he po s i t i on s where Jn a n d Jp c an be e s t i m a t e d change, wh i l e the u n k n o w n s are c a l c u l a t ed at co r re spond ing pos i t i on s on coarser levels ( F i gu re 4.14). T o f a c i l i t a t e the use of p r o l onga t i on a n d r e s t r i c t i on opera to r s w h i c h are based on Chapter 4. An Improved Multigrid Implementation 75 h 2 i-l / />/ j+2 Fine Grid O O D O D O D O D Coarse Grid F i g u r e 4.14: E s t i m a t i n g Jn a n d Jp o n non-s taggered gr ids. Jn a n d J p are k n o w n at the po s i t i on s •; (ip,n,p) are k n o w n at the g r i d po in t s • . Jn a n d Jp, H e m k e r uses a series of staggered gr ids so t h a t Jn a n d Jp are e s t i m a t e d at c o r r e s p o n d i n g po in t s on the coarse a n d fine gr ids. T h e u n k n o w n s are c a l c u l a t e d at t he m i d p o i n t s of a u n i f o r m g r i d , a l l ow ing J„ a n d Jp t o be e s t i m a t e d at the g r i d po in t s . In th i s ve r s i on the u n k n o w n s are ca l cu l a ted at different pos i t i on s on the d i f ferent gr ids , a l l o w i n g the cu r rent s to be k n o w n at fixed po in t s ( F i g u r e 4.15). T h e i n t e r p o l a t i o n a n d r e s t r i c t i on opera to r s are des igned t o conserve the e s t imates of Jn a n d Jp w h e n t r an s f e r r i n g between levels. T h i s leads to a p iecewise e x p o n e n t i a l i n t e r p o l a t i o n of <f>n and <j>p, wh i l e l i near i n t e r p o l a t i o n is used for p r o l onga t i n g co r rec t ions t o V'- T h e coarse g r i d ope ra to r NH is t he finite difference ana log of Nh, bu t due t o the cho ice of p r o l o n g a t i o n a n d re s t r i c t i on operato r s , NH is also the G a l e r k i n f o r m of N^: NH = IffNJk. A s y m m e t r i c , co l l ec t i ve Gaus s - Se ide l r e l a x a t i o n scheme is used, w i t h N e w t o n ' s m e t h o d e m p l o y e d t o solve the non l i nea r equat ions w h i c h ar ise at each po i n t . T h e Chapter 4. An Improved Multigrid Implementation 76 Fine Grid n •. • -o . • e • e • Coarse Grid F i g u r e 4.15: E s t i m a t i n g Jn a n d Jp on s taggered gr ids . Jn a n d Jp are k n o w n at the g r i d po in t s • ; (tl\n,p) are k n o w n at the po in t s •. co r rec t i on t r a n s f o r m a t i o n t echn ique desc r ibed i n § 2 .5 is a lso used: w h i l e the c a l c u l a -t i on s use <f>n a n d <f>p, the l i n e a r i z a t i o n is p e r f o r m e d i n t e rms of n a n d p. T h e F M G p rocedu re is used t o generate an i n i t i a l guess on the finest level . S o l u -t i ons are f o u n d on the coarsest level u s ing a c o m b i n a t i o n of techn iques . T h e s o l u t i on s t ra tegy uses a c o n t i n u a t i o n process i n te rms of the a p p l i e d p o t e n t i a l V 0 , s t a r t i n g f r o m e q u i l i b r i u m a n d m o v i n g i n steps t o the des i red vo l tage. E a c h stage of the c o n t i n u a t i o n process uses a c o m b i n a t i o n of N e w t o n ' s m e t h o d a n d Gaus s - Se ide l re l axa t i on s . H e m k e r presents convergence resu l t s for a s t a n d a r d d i ode [30] w i t h a s y m m e t r i c d o p i n g prof i le . F o r a sma l l f o r w a r d b ias there is a cons tant decrease of the re s i dua l n o r m at each i t e r a t i o n , i ndependen t of the mesh size. Fo r a l a rger f o rwa rd b ias the convergence slows down after the first few i t e r a t i on s , t h o u g h th i s is somewhat a l l e v i a ted by u s i n g a VF-cyc le ( 7 = 2 ) t o generate mo re accu ra te coarse g r i d so lut ions . P r o b l e m s w i t h a reverse b ias have a r a p i d convergence rate. Chapter 4. An Improved Multigrid Implementation 77 4.4.2 Comparison Results T o c o m p a r e H e m k e r ' s a l g o r i t h m t o the code SC-1 ano the r code ( SC -2 ) , based on H e m k e r ' s suggest ions, was p repa red . SC - 2 pe r fo rms ca l cu l a t i on s i n te rms of the car-r ier concen t ra t i on s n a n d p i n s t ead of (j>n a n d <f>p. T h e g r i d t rans fe r ope ra to r s were changed t o co r re spond t o th i s f o r m u l a t i o n , so t ha t the new p r o l o n g a t i o n , r e s t r i c t i on a n d d i f fe rent ia l ope ra to r s sat i s f ied NH = IHNh%- (4-9°) T h i s ver s ion removed the necess ity ( exper ienced w i t h the a l g o r i t h m of [19]) fo r c on t i n -u a t i o n on the coarsest leve l for the range of pa ramete r s used. SC -2 was tes ted on the p rob l ems de sc r i bed i n §4.3. T h e tests i n c l u d e d the s t a n d a r d s y m m e t r i c d iode a n d the n o n s y m m e t r i c d o p i n g pro f i le of §3.4.2. T h e (scaled) a p p l i e d po ten t i a l s r anged f r o m a reverse bias of —400 t o a f o r w a r d b ias of +400, w i t h A 2 = 1.67 x 1 0 - 7 , 82 = 1.22 x 1 0 - 8 a n d R = 0. B o t h p rog rams used the F M G p rocedu re t o generate the s t a r t i n g values on the coarsest gr ids. T h e y are c o m p a r e d on the bas is of the r e d u c t i o n of the re s idua l o n the finest leve l only. T h e p r o g r a m pa ramete r s ( 7 , ^ 1 , ^ 2 ) were fixed at (1 ,1,1) t h r oughou t the tes t ing . Fo r t he s y m m e t r i c d o p i n g prof i le (Test 2) w i t h a sma l l f o r w a r d bias ( V 0 = —40) SC-1 a n d SC - 2 behaved i n m u c h the same manne r , w i t h the n o r m of the re s idua l r educed by a constant f a c to r at each i t e r a t i o n ( c o m p a r i n g F i gu re s 4.16 a n d 4.12). T h i s convergence ra te is i ndependent of the mesh size. A s the m a g n i t u d e of the app l i ed p o t e n t i a l increased the pe r f o rmance of b o t h SC-1 a n d SC -2 de te r i o r a ted as the mesh spac ing decreased ( F i gu re s 4.13 a n d 4.17). SC-1 p roved to be t he more robus t code, converg ing for b o t h V 0 = —200 and V0 = —400 w i t h 7 = 1. S C - 2 d iverged for b o t h cases w i t h 7 = 1; w i t h the more accu ra te coarse g r i d so lu t ions o b t a i n e d u s ing 7 = 2 , Chapter 4. An Improved Multigrid Implementation 78 2 1. 0. -1_ log || residual || " 2 . - 3 - 4 - 5 " 6 - 7 - 8 i i i i t i i i i i , , 1 2 3 4 5 6 7 8 9 10 11 12 Iteration • N « 8 D N - 1 6 ^ N - 3 2 « N « 6 4 F i g u r e 4.16: Convergence of SC -2 , Vo = - 4 0 , Nf = 8,16,32,64 (Test 2) SC - 2 converged for V 0 = —200 ( F i gu re 4.17) bu t not for V 0 = —400. Convergence for t he reverse b ias cases ( V 0 = 200,400) was a lmost i m m e d i a t e for b o t h SC-1 a n d SC-2 ( F i gu re s 4.11 a n d 4.18), and the convergence rates were i ndependent of the g r id size ( F i g u re s 4.11 a n d 4.19). T h e pe r f o rmance of SC - 2 was very dif ferent w h e n the n o n s y m m e t r i c d o p i n g p ro -f i le was used. W i t h a reverse b ias ( V 0 = —200) SC -2 cons i s tent ly converged but the b e h a v i o u r was e r r a t i c , w i t h the n o r m of the re s i dua l sharp ly inc reas ing at the f irst i t e r a t i o n ( F i g u r e 4.20). A s im i l a r b e h a v i o u r was observed for V 0 = —400. SC-1 had a m u c h be t t e r convergence p a t t e r n ( F i gu re s 4.7 a n d 4.8). A t convergence the res idua l n o r m is m u c h sma l le r w h e n SC-1 is used, c o m p a r e d t o SC-2 . SC-1 is mo re successfu l t h a n SC - 2 for f o rwa rd b ias p rob lems. Fo r V 0 = 200 SC-2 converges on l y when the finest g r i d is e x t r e m e l y coarse ( Nj = 4), bu t SC-1 demon-s t ra tes a s teady convergence ra te for a l l mesh sizes ( F i gu re 4.9). SC-1 exh ib i t s the same s teady convergence ra te for V 0 = 400 as we l l , bu t SC -2 does not converge at a l l . Chapter 4. An Improved Multigrid Implementation 3 log || residual || 'i i i i i i i • i i i i i i i i i i i 1 2 3 4 5 6 7 8 910 12 14 1617 19 21 23 25 Iteration • N = 8 ON - 16 4 N - 3 2 F i g u r e 4.17: Conve rgence of SC -2 , V0 = - 2 0 0 , Nf = 8,16,32 (Test 2) log || residual || 1 1 ' » I I i 1 2 3 4 5 6 7 Iteration • V-200 D V=400 F i g u r e 4.18: Convergence of SC -2 , V 0 = 200,400, Nf = 64 (Test 2) Chapter 4. An Improved Multigrid Implementation 2 • • i i i i i 1 2 3 A 5 6 7 8 9 10 11 12 Iteration • N - 8 n N - 1 6 *N = 32 _N = 64 F i g u r e 4.19: Convergence of SC -2 , V 0 = 200, Nf = 8,16, 32, 64 (Test 2) F i g u r e 4.20: Convergence of SC -2 , V0 = - 2 0 0 , Nj = 8 ,16,32,64 (Test 1) Chapter 5 Discussion T h i s thes i s has d e m o n s t r a t e d t ha t the m u l t i g r i d m e t h o d can be successful ly used t o solve the s t a t i o n a r y s e m i c o n d u c t o r equat ions i n one space d imens i on . T h e m a j o r resu l t s of the thes is are s u m m a r i z e d i n § 5 . 1 , wh i l e §5.2 discusses some of the ways i n w h i c h th i s wo rk c an be ex tended . 5.1 Summary W h i l e s t r a i g h t f o r w a r d m u l t i g r i d i m p l e m e n t a t i o n s are not usefu l for the sem iconduc to r equat ions , m o d i f i c a t i o n s t o the bas ic a l g o r i t h m p r oduce a successful a n d efficient p r o -g r am. Change s t o the r e l a x a t i o n a n d p r o l o n g a t i o n operator s h a d the greatest effect on the convergence rate. T h e mos t successfu l r e l a x a t i o n scheme is a co l l ec t i ve Gaus s - Se ide l i t e r a t i on , w i t h the po in t s v i s i t e d i n a s y m m e t r i c order. L o c a l r e l a xa t i on sweeps, conf ined t o the n e i g h b o u r h o o d of j u n c t i o n s , are effective a n d inexpens i ve , a n d have m u c h the same effect as f u l l i t e r a t i on s do. O n e i t e r a t i o n of N e w t o n ' s m e t h o d c an be used to l i nea r i ze the equa t i on s t h r o u g h o u t mos t of the d o m a i n , bu t two i te ra t ions s hou ld be used near j u n c t i o n s where the so lu t i on is r a p i d l y va r y i ng . P R O L - 1 , w h i c h is based on the ideas of A l g e b r a i c M u l t i g r i d a n d the discrete equa -t ions , is t he mos t effective p r o l onga t i o n ope ra to r , bu t P R O L - 2 , w h i c h is a cheaper a p p r o x i m a t i o n of P R O L - 1 , is a more p r a c t i c a l i m p l e m e n t a t i o n . W h e n comb ined w i t h l o c a l r e l a x a t i o n sweeps, P R O L - 2 is robust , and g rea t l y improves the convergence of the Chapter 5. Discussion 82 a l g o r i t h m , c o m p a r e d t o l i near i n t e r p o l a t i o n . T h e m o d i f i e d a l g o r i t h m can be successfu l ly app l i ed t o a va r i e t y of p rob l ems w i t h a w i d e range of a p p l i e d potent ia l s . Its pe r f o rmance compare s f a vou r ab l y w i t h other m u l t i g r i d a l go r i t hms . T h e resu l t s of th i s thesis i nd i ca te t h a t t he dop i n g pro f i le j u n c t i o n s cause t he slow convergence of s t a n d a r d m u l t i g r i d a l go r i t hms . A w a y f r o m the j u n c t i o n s t he so l u t i on is w e l l behaved , so the n o r m a l m u l t i g r i d techn iques can be expec ted to be successful , bu t the i n t e r i o r layers requ i re spec ia l t r e a tmen t , i n c l u d i n g e x t r a smoo th i ng . 5.2 Suggestions for Future Study W h i l e th i s thes is has cons idered on l y some s imp le one d imen s i ona l p r ob l ems , th i s wo rk c an f o r m the bas is for more extens ive studies. Severa l areas of interest are l i s t ed be low. 1. Fo r m a n y i n te re s t i ng dop i ng prof i les, e.g., thy r i s to r s , the so l u t i on is not un i que a n d c o n t i n u a t i o n mus t be used to ca l cu l a te a l l s o l u t i on branches. C o n t i n u a t i o n techn iques are desc r ibed i n [4], for examp le , and the i r use i n the m u l t i g r i d contex t is d i scussed i n [18] and [9]. 2. S o l v i n g t he sem i conduc to r equat ions i n one d imen s i on is "useful t o i so late t he issues i n vo l ved i n a p p l y i n g the m u l t i g r i d m e t h o d t o the p r o b l e m , bu t i n p rac t i ce the equat i on s are u sua l l y posed i n two or three d imens ions . M u c h of the work of th i s thes is c an be d i rec t l y e x t ended t o m u l t i d i m e n s i o n a l p rob lems , bu t some d i f f i cu l t ie s are expec ted . These i nc l ude : • T h e s y m m e t r i c r e l a xa t i on scheme is successful because i n f o r m a t i o n is t ran s -m i t t e d i n the d i rec t i on of b o t h -f and — Vt/>, a n d i n one d i m e n s i o n these d i rec t i on s cor respond to e i ther inc reas ing or decreas ing x. In two or more Chapter 5. Discussion 83 d imens i on s Vi/> is not necessar i l y p a r a l l e l t o a c oo rd i na te axis , so the p rope r r e l a x a t i o n o r de r i n g is not obv ious . • T h e l o c a l r e l a x a t i o n sweeps encounte r j u s t a few mesh po in t s near the j u n c -t i ons i n one d i m e n s i o n , bu t i n l a rger p r ob l ems these sweeps w i l l i nvo l ve la rger regions. B l o c k r e l a xa t i on s m a y need t o be used, w i t h the u n k n o w n s a l ong a g r i d l i ne r e l a x e d s imu l taneous l y . • It is not obv ious h o w to e x t e n d the p ro l onga t i on s P R O L - 1 a n d P R O L - 2 t o m u l t i d i m e n s i o n a l p rob lems . F u r t h e r , the a l g o r i t h m must be e x tended t o dea l w i t h c e r t a i n aspects of the m u l t i d i m e n s i o n a l p r o b l e m (e.g., N e u m a n n b o u n d a r y cond i t i on s ) w h i c h are no t present i n the one d imen s i ona l case. 3. To adequa te l y resolve the r a p i d s o l u t i on changes near the j u n c t i o n s , a n o n u n i f o r m mesh shou ld be used. B r a n d t [9] descr ibes a m e t h o d of i n c o r p o r a t i n g n o n u n i f o r m gr ids i n t o the m u l t i g r i d a l g o r i t h m by u s ing a h i e ra r chy of u n i f o r m gr ids w h i c h do not cover the ent i re d o m a i n . T h e mesh re f inement can be done adapt i ve l y , or m a y fo l l ow a p re s c r i bed p a t t e r n . Some of the issues w h i c h m a y ar ise are d i scussed i n [13]. 4. W h i l e o n l y the s t a t i ona r y s em i conduc to r equat i on s have been cons idered here the t rans ien t p r o b l e m is a l so of interest . T h e s o l u t i on of the t i m e dependent equat ions invo lves s o l v i ng a series of c lose ly r e l a t ed s t a t i ona r y p rob lems ; i n th i s way, the t rans ient p r o b l e m is c lose ly r e l a ted t o c on t i nua t i on . M u l t i g r i d a l go r i t hms for t i m e dependent p r ob l ems are d i scussed i n [18] a n d [9]. 5. M o r e i n s i gh t i n t o the b e h a v i o u r of t he m u l t i g r i d a l g o r i t h m can be ga ined by a n a l y z i n g the behav i ou r of the errors. Severa l a s sumpt i on s mus t be made t o a l l ow Chapter 5. Discussion 84 L o c a l M o d e A n a l y s i s ( L M A ) t o be app l i ed — t he equa t i on s mus t be l i nea r i zed , a n d t he coeff ic ients mus t be cons idered t o be fixed or s lowly va ry i ng . W h i l e these a s sumpt i on s are re s t r i c t i ve , the heu r i s t i c ana ly s i s m a y p rov ide gu ide l ines for f u t u re deve lopment s . Bibliography [1] D.E. A k c a s u , " Conve r gence p rope r t i e s of N e w t o n ' s m e t h o d fo r the so l u t i on of the s e m i c o n d u c t o r t r a n s p o r t equa t i on s a n d h y b r i d s o l u t i on techn iques for m u l t i -d i m e n s i o n a l s i m u l a t i o n of V L S I dev i ce s " , Solid State Electron. 27 (1984), 319 -328. [2] R .E. A l cou f f e , A . B r a n d t , J . E . D e n d y a n d J . W . P a i n t e r , " T h e m u l t i g r i d m e t h o d for the D i f f u s i on E q u a t i o n s w i t h s t r ong l y d i s cont inuous coef f i c ient s " , SIAM J. Sci. Stat. Comput. 2 (1981), 430 - 454. [3] U .A sche r , P.A M a r k o w i c h , C . Schmeiser , H . S te in ruck , a n d R. Weis s " C o n d i t i o n -i n g of t he S teady S t a te S e m i c o n d u c t o r D e v i c e P r o b l e m " , SIAM J. Appl. Math ( i n press). [4] U .A sche r , R . M . M . M a t t h e i j a n d R.D. Ru s se l l , Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, P r e n t i c e H a l l , 1988. [5] R. E. B a n k and D. J . Rose, " P a r a m e t e r se lect ion for N e w t o n l ike me thod s ap-p l i c a b l e t o N o n l i n e a r P a r t i a l D i f f e r en t i a l E q u a t i o n s " , SIAM J. Num. Anal. 17 (1980), 806 - 822. [6] R. E. B a n k , D. J . Rose a n d W . F i c h t n e r , " N u m e r i c a l m e t h o d s for s em i conduc to r dev ice s i m u l a t i o n " , SIAM J. Sci. Stat. Compution 4 (1981), 416 - 435. [7] R. E. B a n k , J . W . J e r o m e and D. J . Rose, " A n a l y t i c a l a n d n u m e r i c a l aspects of s em i conduc to r dev ice m o d e l l i n g " , i n Computing Methods in Applied Sciences and Engineering, R. G l o w i n s k i a n d J . L . L i o n s ed i tors , N o r t h - H o l l a n d (1982). Bibliography 86 [8] A . B r a n d t , " M u l t i - l e v e l a d a p t i v e techn ique ( M L A T ) for fast n u m e r i c a l s o l u t i on t o b o u n d a r y va lue p r o b l e m s " , i n Proceedings Third International Conference on Numerical Methods in Fluid Mechanics, Paris 1972, H . Cabanne s a n d R. T e m a n ed i to r s , Sp r i n ge r -Ve r l a g , B e r l i n , (1973). [9] A . B r a n d t , " G u i d e t o M u l t i g r i d D e v e l o p m e n t " , i n Multigrid Methods, W . H a c k -bu s ch a n d U . T r o t t e n b e r g ed i to r s , Sp r i n ge r -Ve r l a g , B e r l i n , 1982. [10] W . B r i g g s a n d S. M c C o r m i c k , " I n t r o d u c t i o n " , i n Multigrid Methods, S. M c -C o r m i c k , ed i to r , S I A M , 1987. [11] E. M . B u t u r l a , P. E. C o t t r e l l , B . M . G r o s s m a n , K .A . Sa l sburg , M . B . L a w l o r , C . T . M c M u l l e n , " T h r e e D i m e n s i o n a l F i n i t e E l e m e n t S i m u l a t i o n of S em i conduc to r D e v i c e s " , Proc. Int. Solid-State Circuits Conf. (1980), 76 - 77. [12] A . D e M a r i , " A n A c c u r a t e N u m e r i c a l S t eady - S t a t e One D i m e n s i o n a l S o l u t i on of t he P N - J u n c t i o n " , Solid-State Electron. 11 (1968), 33 - 58. [13] J . E . D e n d y , " A p r i o r i L o c a l G r i d Re f i nement i n the m u l t i g r i d m e t h o d " , in Elliptic Problem Solvers II, G . B i r k h o f f a n d A . Schoens tad t , ed i tors , A c a d e m i c Pres s , 1984. [14] W . F i c h t n e r a n d D. J . Rose, " O n the n u m e r i c a l so lu t i on of non l i nea r e l l i p t i c P D E ' s a r i s i n g f r o m sem iconduc to r dev ice m o d e l l i n g " , i n Elliptic Problem Solvers, A c a d e m i c Press , 1981. [15] S.P. G a u r a n d A . B r a n d t , " N u m e r i c a l S o l u t i o n of s em iconduc to r t ranspor t equa-t i on s i n two d imens ions by the m u l t i g r i d m e t h o d " , i n Advances in Computer Methods for Partial Differential Equations II R. V i c hneve t s k y ed i to r , 1977. Bibliography 87 [16] G . G o l u b a n d C . F . van L o a n , Matrix Computations, J ohn s H o p k i n s U n i v e r s i t y P res s , 1983. [17] H . K. G u m m e l , " A Sel f -cons i stent i t e r a t i v e scheme for one- d i m e n s i o n a l s teady s ta te t r an s i s t o r ca l cu l a t i on s " , IEEE Trans. Elect. Dev. ED-11 (1964), 455 - 465. [18] W . H a c k b u s c h , Multigrid Methods and Applications, S p r i n ge r -Ve r l a g , 1985. [19] P .W. H e m k e r " A N o n l i n e a r M u l t i g r i d M e t h o d for one d imen s i ona l s em i conduc to r dev ice s i m u l a t i o n : T h e D i o d e " , manu s c r i p t , 1987. [20] K. H w a n g , D. H. N a v o n , T . W . T a n g a n d M A . O s m a n , " I m p r o v e d convergence of n u m e r i c a l dev ice s i m u l a t i o n i t e r a t i v e a l g o r i t h m s " , IEEE Trans. Elect. Dev. 3 2 (1985), 1143 - 1145. [21] D. K a m o w i t z , " M u l t i g r i d A p p l i e d to S i ngu l a r P e r t u r b a t i o n P r o b l e m s " , ICASE report 87-1 (1987). [22] D. K a m o w i t z a n d S.V. Pa r t e r , " A S t udy of Some M u l t i g r i d Ideas " , A p p l i e d M a t h , a n d C o m p . 17 (1985) 153 - 184. [23] F. K. Mana s se , Semiconductor Electronics Design , P r e n t i c e - H a l l , E n g l e w o o d C l i f f s , 1977. [24] P.A. M a r k o w i c h , The Stationary Semiconductor Equations, Sp r inger , W i e n N e w Y o r k , 1986. [25] S.F. M c C o r m i c k , " A n A l g e b r a i c I n t e rp re ta t i on of M u l t i g r i d M e t h o d s " , SIAM J. Numer. Anal. 19 (1982), 548 - 560. [26] M.S. M o c k , " A T i m e - D e p e n d e n t N u m e r i c a l M o d e l of the I n s u l a t ed -Ga te F i e l d -E f fect T r a n s i s t o r " , Solid State Electron 2 4 (1981), 959 - 966. Bibliography 88 [27] M.S. M o c k , " A n e x a m p l e of nonun iqueness of s t a t i o n a r y so lu t ions i n semicon-d u c t o r m o d e l s " , COMPEL 2 (1983), 165 - 174. [28] J . M . O r t e g a a n d W . C . R h e i n b o l d t , I t e ra t i ve Solution of Nonlinear Equations in Several Variables, A c a d e m i c Pres s , 1970. [29] A . R . M i t c h e l l a n d D.F. G r i f f i t h s , The Finite Difference Method in Partial Differ-ential Equations, W i l e y , 1980. [30] S. J . P o l a k , C. D e n He i j e r , W . H . A . Sch i lder s a n d P. A . M a r k o w i c h , " S em i con -d u c t o r M o d e l l i n g f r o m the N u m e r i c a l P o i n t of V i e w " , Int. J. Num. Meth. Eng. 2 4 (1987), 763 - 838. [31] S. Se lberher r , Analysis and Simulation of Semiconductor Devices, Spr inger, W i e n N e w Y o r k , 1984. [32] D.L. Scha r fe t te r a n d H.K. G u m m e l , " L a r g e Scale S i gna l A n a l y s i s of a S i l i c on R e a d D i o d e O s c i l l a t o r " , IEEE Trans. Electron Devices E D - 1 6 (1969), 64 - 77. [33] T . I. S e i d m a n , " S t e a d y - s t a t e so lut ions of d i f fu s i on r eac t i on sys tems w i t h e lec t ro -s ta t i c c o n v e c t i o n " , Nonlinear Analysis 4 (1980), 623 - 637. [34] R.A. S m i t h , Semiconductors, Second Edition , C a m b r i d g e U n i v e r s i t y Press, L o n -don , 1978. [35] J . W . S l o t b o o m , " I t e r a t i v e Scheme for 1- a n d 2 -D imen s i ona l D .C . -T rans i s to r S i m -u l a t i o n " , Electron. Lett. 5 (1969), 677 - 678. [36] K. S t u b e n a n d U . T r o t t e n b u r g , " M u l t i g r i d M e t h o d s : F u n d a m e n t a l A l g o r i t h m s , M o d e l P r o b l e m A n a l y s i s a n d A p p l i c a t i o n s " , i n Multigrid Methods, W . Hackbu s ch a n d U . T r o t t e n b u r g ed i to r s , Sp r i nge r -Ve r l ag , 1982. Bibliography 89 [37] S .M. Sze, Physics of Semiconductor Devices, W i l e y , 1981. [38] W . V . van Roo sb roeck , " T h e o r y of f low of e lect rons a n d holes i n G e r m a n i u m and o ther s e m i c o n d u c t o r s " , Bell Syst. Techn. J. 29 (1950), 560 - 607. [39] K. Y a m a g u c h i , " A T i m e - D e p e n d e n t a n d T w o - D i m e n s i o n a l N u m e r i c a l M o d e l for M O S F E T D e v i c e O p e r a t i o n " , Solid State Electron.26 (1983), 907 - 916. 

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