UBC Theses and Dissertations

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UBC Theses and Dissertations

Modelling the DC performance of GAAS Homojunction bipolar transistors Lee, Soon Peng 1985

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c M O D E L L I N G T H E DC PERFORMANCE OF GAAS HOMOJUNCTION B I P O L A R T R A N S I S T O R S by SOON PENG L E E 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 M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF MASTER OF A P P L I E D S C I E N C E i n T H E F A C U L T Y OF GRADUATE S T U D I E S DEPARTMENT OF E L E C T R I C A L E N G I N E E R I N G We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y OF B R I T I S H COLUMBIA NOVEMBER, 1985 © SOON PENG L E E , 1985 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i 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 an a d v a n c e d d e g r e e a t t h e T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . DEPARTMENT OF E L E C T R I C A L ENGINEERING T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 D a t e : NOVEMBER. 1985 A b s t r a c t Two models, one a n a l y t i c a l and one n u m e r i c a l , have been developed to p r e d i c t the dc performance of GaAs homojunction b i p o l a r t r a n s i s t o r s . In each case the m i n o r i t y c a r r i e r p r o p e r t i e s of l i f e t i m e and m o b i l i t y have been d e s c r i b e d by polynomial f i t s t o recent d a t a . Bandgap narrowing i n the e m i t t e r and base r e g i o n s has a l s o been taken i n t o account. The a n a l y t i c a l model assumes uniform doping i n the three r e g i o n s of the t r a n s i s t o r and i s thus a p p r o p r i a t e to p r e d i c t i n g the performance of d e v i c e s f a b r i c a t e d u s i n g e p i t a x i a l t e c h n o l o g i e s . T h i s model i s a l s o u s e f u l f o r c a r r y i n g out s e n s i t i v i t y a n a l y s e s . The importance of parameters such as r e g i o n a l widths and doping d e n s i t i e s , m i n o r i t y c a r r i e r l i f e t i m e s and s u r f a c e recombination v e l o c i t y i s examined here. The numerical model i s u s e f u l f o r d e s c r i b i n g the performance of ion-implanted d e v i c e s . Good agreement i s obtained between r e s u l t s from the model and recent experimental data from p r o t o t y p e d e v i c e s . i i Table of Contents 1. I n t r o d u c t i o n 1 2. The a n a l y t i c a l model 9 2.1 I n t r o d u c t i o n 9 2.2 The t r a n s i s t o r model 10 2.3 The model parameters ....19 2.4 The s o l u t i o n procedure 32 2.5 R e s u l t s and d i s c u s s i o n 38 3. The numerical model 55 3.1 I n t r o d u c t i o n 55 3.2 Model d e s c r i p t i o n ....58 3.2.1 B a s i c equations and boundary c o n d i t i o n s ...58 3.2.2 The p h y s i c a l parameters 61 3.2.3 Ion i m p l a n t a t i o n parameters 62 3.3 S o l u t i o n procedure u s i n g SEDAN 70 3.4 R e s u l t s and d i s c u s s i o n - 72 4. C o n c l u s i o n s 81 REFERENCES . 83 APPENDIX A 89 111 L i s t of f i g u r e s F i g u r e Page 2.1. Schematic of c u r r e n t ' f l o w s i n a homojunction b i p o l a r t r a n s i s t o r 11 2.2. Experimental data and best curve f i t f o r the dependence of e l e c t r o n H a l l m o b i l i t y on m a j o r i t y c a r r i e r c o n c e n t r a t i o n 26 2.3. Experimental data and best curve f i t f o r the dependence of ho l e H a l l m o b i l i t y on m a j o r i t y c a r r i e r c o n c e n t r a t i o n 27 2.4. R a t i o of e l e c t r o n m o b i l i t y i n p-type GaAs, M^, t o the e l e c t r o n m o b i l i t y i n n-type GaAs, M", as a f u n c t i o n of e l e c t r o n c o n c e n t r a t i o n 29 n 2.5. Experimental data and best curve f i t f o r the dependence of e l e c t r o n l i f e t i m e on h o l e c o n c e n t r a t i o n 30 2.6. Experimental data and best curve f i t f o r the dependence of hole l i f e t i m e on e l e c t r o n c o n c e n t r a t i o n .... 31 2.7. Flow c h a r t f o r the s o l u t i o n procedure of the a n a l y t i c a l model ...36 2.8. C o l l e c t o r c u r r e n t d e n s i t y as a f u n c t i o n of e m i t t e r - c o l l e c t o r v o l t a g e with the base-emitter v o l t a g e as a parameter 39 2.9. P l o t of base c u r r e n t d e n s i t y , c o l l e c t o r c u r r e n t d e n s i t y versus base-emitter v o l t a g e . n i s the i d e a l i t y f a c t o r 40 i v L i s t of F i g u r e s F i g u r e Page 2.10. The e f f e c t of e l e c t r o n l i f e t i m e i n the base on g a i n , as p r e d i c t e d by the a n a l y t i c a l model u s i n g the parameters l i s t e d i n T a b l e 2.3 41 2.11. The e f f e c t of basewidth on g a i n , as p r e d i c t e d by the a n a l y t i c a l model u s i n g the parameters [ other than Wfi ] l i s t e d i n T a b l e 2.3 42 2.12. The e f f e c t of hole l i f e t i m e i n the e m i t t e r on g a i n , as p r e d i c t e d by the a n a l y t i c a l model using the parameters l i s t e d i n Tabl e 2.3 45 2.13. The e f f e c t of e m i t t e r width on g a i n , as p r e d i c t e d by the a n a l y t i c a l model u s i n g the parameters [ other than WE ] l i s t e d i n T a b l e 2.3 47 2.14. The e f f e c t of s u r f a c e recombination v e l o c i t y on g a i n , as p r e d i c t e d by the a n a l y t i c a l model using the parameters [ other than s p ] l i s t e d i n Table 2.3 48 2.15. The e f f e c t of e m i t t e r doping on g a i n , as p r e d i c t e d by the a n a l y t i c a l model u s i n g the parameters [ other than N £ ] l i s t e d i n T a b l e 2.3 50 2.16. The e f f e c t of c o l l e c t o r doping on g a i n , as p r e d i c t e d by the a n a l y t i c a l model u s i n g the parameters [ other than ] l i s t e d i n T a b l e 2.3 ..52 v L i s t of F i g u r e s F i g u r e Page 2.17. The gain p r e d i c t e d by the a n a l y t i c a l model u s i n g data f o r d e v i c e s given by B a i l b e et a l . [35], Tan and M i l n e s [5] and Nuese et a l . [12] ....53 3.1. P r o j e c t e d range versus implant energy f o r selenium, s i l i c o n and b e r y l l i u m i n GaAs 67 3.2. Standard d e v i a t i o n versus implant energy f o r selenium, s i l i c o n and b e r y l l i u m i n GaAs ..68 3.3. The computed doping p r o f i l e f o r the Hughes d e v i c e s t r u c t u r e [14], showing the e f f e c t of a c t i v a t i o n e f f i c i e n c y and i n - d i f f u s i o n f o r the implanted S i s p e c i e s 74 3.4. The e f f e c t of s i l i c o n a c t i v a t i o n e f f i c i e n c y , s i l i c o n i n - d i f f u s i o n and bandgap narrowing (BGN) on g a i n f o r the Hughes d e v i c e s t r u c t u r e [14], as p r e d i c t e d by the numerical model 75 3.5. The computed doping p r o f i l e f o r the Texas Instrument d e v i c e s t r u c t u r e s of R e f s . [15,17], assuming 70% a c t i v a t i o n of the implanted selenium and 100% a c t i v a t i o n of the implanted b e r y l l i u m . The r e d i s t r i b u t i o n of i m p u r i t i e s i n the e p i t a x i a l c o l l e c t o r of the d e v i c e [17] i s r e p r e s e n t e d by an abrupt p r o f i l e •• 77 3.6. The Computed g a i n f o r the Texas Instrument d e v i c e s t r u c t u r e s of R e f s . [15,17], assuming 70% a c t i v a t i o n of the implanted selenium and 100% a c t i v a t i o n of the implanted b e r y l l i u m 80 v i L i s t of T a b l e s Table Page 2.1. C o e f f i c i e n t s f o r M h i n equation (2.47) 33 2.2. C o e f f i c i e n t s f o r m i n o r i t y c a r r i e r l i f e t i m e s i n equation (2.51) 33 2.3. Input data used i n the modeling program 34 3.1. Implantation schedule used i n Refs. [14,15,17] f o r f a b r i c a t i n g GaAs n-p-n b i p o l a r t r a n s i s t o r s 65 3.2. C o e f f i c i e n t s f o r R p i n equation (3.13) 66 3.3. C o e f f i c i e n t s f o r AR p i n eq u a t i o n (3.14) 66 v i i A c k n o w l e d g e m e n t I w o u l d l i k e t o t h a n k my s u p e r v i s o r , D r . D. L . P u l f r e y , f o r h i s c o n t i n u a l g u i d a n c e , s u p p o r t a n d e n c o u r a g e m e n t d u r i n g t h e c o u r s e o f t h i s w o r k . I w o u l d a l s o l i k e t o t h a n k D r . P. V a n H a l e n f o r h i s h e l p i n d e v e l o p i n g t h e a n a l y t i c a l m o d e l a n d M r . D. H u i f o r h i s h e l p i n u s i n g t h e c u r v e f i t t i n g r o u t i n e s on t h e HP 9 8 1 6 . T h a n k s a r e e x t e n d e d t o M r . D. S. C a m p o r e s e a n d M r . W. T a n g f o r t h e i r many u s e f u l d i s c u s s i o n s . * • • vixi 1. INTRODUCTION In recent y e a r s , there has been a growing i n t e r e s t i n the use of g a l l i u m a r s e n i d e f o r i n t e g r a t e d c i r c u i t a p p l i c a t i o n s . Because of i t s energy band s t r u c t u r e , g a l l i u m a r s e n i d e e x h i b i t s some m a t e r i a l p r o p e r t i e s which are s u p e r i o r to t h a t of other semiconductors such as s i l i c o n and germanium. I t s e l e c t r o n m o b i l i t y i s about 5-6 times higher than t h a t of s i l i c o n , while i t s energy bandgap i s about 1.43 eV, compared to 1.11 eV of s i l i c o n at room temperature. In a d d i t i o n , GaAs s u b s t r a t e s are a v a i l a b l e i n a s e m i - i n s u l a t i n g form which s u b s t a n t i a l l y reduces the p a r a s i t i c c a p a c i t a n c e s of e l e c t r o n i c c i r c u i t s , r e s u l t i n g i n f a s t e r d e v i c e speeds. T h e r e f o r e , GaAs i s s u i t a b l e f o r a number of h i g h frequency, h i g h temperature a p p l i c a t i o n s , such as u l t r a h i g h speed l o g i c and s i g n a l p r o c e s s i n g , not p r e s e n t l y r e a l i s a b l e by s i l i c o n d e v i c e s . To date, most of the g a l l i u m a r s e n i d e c i r c u i t s have been f a b r i c a t e d u s i n g MESFET (Metal Semiconductor F i e l d E f f e c t T r a n s i s t o r ) technology. Complex l a r g e s c a l e i n t e g r a t e d c i r c u i t s with a few thousand gates have been made. However, as the l e v e l of i n t e g r a t i o n of GaAs c i r c u i t s i n c r e a s e s , some d i f f i c u l t y i s encounterd w i t h the FET t e c h n o l o g y . L o g i c c i r c u i t s f a b r i c a t e d u s i n g depletion-mode MESFETs cannot c o n t a i n more than 10,000 gates due to the l a r g e l o g i c swing and 'normally on' c h a r a c t e r i s t i c s of the D-MESFETs [ 1 ] . The l a t t e r c o u l d cause severe power d i s s i p a t i o n problems. Although the enhancement-type MESFET 1 2 i s normally o f f , i t s low l o g i c swing (about 0.5 V) makes i t d i f f i c u l t to c o n t r o l the d e v i c e process parameters such as t h i c k n e s s and doping l e v e l i n the t r a n s i s t o r channel l a y e r t o the degree necessary to ma i n t a i n a uniform p i n c h - o f f v o l t a g e . To o b t a i n a good y i e l d of d e v i c e s at the VLSI l e v e l , the t r a n s i s t o r ' s t h r e s h o l d v o l t a g e should not d e v i a t e more t h a t f i v e percent [ 2 ] ; t h i s would be very d i f f i c u l t t o ach i e v e with the E-MESFETs, s i n c e that t r a n s l a t e s i n t o a p i n c h - o f f v o l t a g e v a r i a t i o n of no more than 25 mV. In view of these problems with MESFETs, one would l i k e t o c o n s i d e r the p o s s i b i l i t y of u s i n g b i p o l a r technology f o r the f a b r i c a t i o n of GaAs VLSI c i r c u i t s . I t i s known that b i p o l a r t r a n s i s t o r s o f f e r s e v e r a l advantages over f i e l d e f f e c t t r a n s i s t o r s . The t h r e s h o l d v o l t a g e (the emitter-base b i a s v o l t a g e f o r some f i x e d c o l l e c t o r c u r r e n t ) of a b i p o l a r t r a n s i s t o r i s r e l a t i v e l y c onstant f o r a given b i a s , s i n c e i t i s mainly determined by the energy bandgaps of the e m i t t e r and base r e g i o n of the d e v i c e , r a t h e r than by the process parameters. A t h r e s h o l d v a r i a t i o n of a few m i l l i v o l t s can be e a s i l y o b t a i n e d i n b i p o l a r i n t e g r a t e d c i r c u i t s . B e s i d e s , b i p o l a r s are a l s o known t o have l a r g e r d r i v i n g c a p a b i l i t y than f i e l d e f f e c t t r a n s i s t o r s , t h e r e f o r e they are more s u i t a b l e f o r power d e v i c e a p p l i c a t i o n s , or c i r c u i t s with l a r g e f a n - o u t s . One of the p o s s i b l e c h o i c e s f o r GaAs b i p o l a r VLSI a p p l i c a t i o n s i s the h e t e r o j u n c t i o n b i p o l a r t r a n s i s t o r . Kroemer [3] has p r e d i c t e d t h a t the h e t e r o j u n c t i o n b i p o l a r s 3 e x h i b i t a frequency response which i s s u p e r i o r t o that of FETs. A comparison by Eden [4] esti m a t e s t h a t the gain-bandwidth product of a GaAs h e t e r o j u n c t i o n b i p o l a r t r a n s i s t o r can be as hig h as 100-200 Ghz, while a MESFET with a gate l e n g t h of 0.5 micron can o n l y a c h i e v e about 30 Ghz. D e s p i t e t h i s performance advantage enjoyed by the h e t e r o j u n c t i o n b i p o l a r t r a n s i s t o r s oyer the MESFETs, the complexity of t h e i r f a b r i c a t i o n , which i n v o l v e s s o p h i s t i c a t e d M o l e c u l a r Beam E p i t a x y (MBE) techniques t o grow the h e t e r o j u n c t i o n , the use of i o n - i m p l a n t a t i o n f o r e l e c t r i c a l i s o l a t i o n , the requirements of an extremely t h i n base l a y e r ( l e s s than 100 nanometers) and sm a l l c o n t a c t a r e a s , has made the r e a l i s a t i o n of a l a r g e s c a l e i n t e g r a t e d c i r c u i t r a t h e r d i f f i c u l t at the moment. The GaAs homojunction b i p o l a r s , on the other hand, seem t o o f f e r a b e t t e r p e r s p e c t i v e . The homojunction t r a n s i s t o r f a b r i c a t i o n p rocess i s simpler as only i o n - i m p l a n t a t i o n s are employed t o c r e a t e the v a r i o u s doped l a y e r s as w e l l as the i s o l a t i o n r e g i o n s i n the t r a n s i s t o r s t r u c t u r e . The ease of f a b r i c a t i o n i m p l i e s t h a t homojunction b i p o l a r s should e x h i b i t a high e r y i e l d i n comparison t o h e t e r o j u n c t i o n b i p o l a r s . GaAs homojunction b i p o l a r t r a n s i s t o r s are g e n e r a l l y thought t o s u f f e r an in h e r e n t design disadvantage t h a t make them r a t h e r unfavourable i n frequency response compared t o the h e t e r o j u n c t i o n b i p o l a r s . In order t o mai n t a i n a h i g h e m i t t e r i n j e c t i o n e f f i c i e n c y i n the homojunction b i p o l a r , the base doping l e v e l has to be made o n l y a smal l f r a c t i o n 4 of the e m i t t e r doping l e v e l . The base r e g i o n i n a h e t e r o j u n c t i o n b i p o l a r , however, can be comparable i n doping or even more h e a v i l y doped than the e m i t t e r r e g i o n without a f f e c t i n g the i n j e c t i o n e f f i c i e n c y , due to the h e t e r o j u n c t i o n e m i t t e r that p r o v i d e s a l a r g e energy b a r r i e r t o the e m i t t e r b a c k - i n j e c t e d c u r r e n t . I t i s g e n e r a l l y agreed t h a t a heavily-doped base r e g i o n reduces the base r e s i s t a n c e , l e a d i n g t o s u b s t a n t i a l l y improved t r a n s i s t o r frequency response. Re c e n t l y , Tan and M i l n e s [5] have shown by t h e i r numerical study of the frequency response of the GaAs homojunction and h e t e r o j u n c t i o n npn t r a n s i s t o r s t h a t , i f the p a r a s i t i c base r e s i s t a n c e i s minimized i n the homojunction b i p o l a r t r a n s i s t o r d e s i g n , i t i s p o s s i b l e t o ach i e v e h i g h speed o p e r a t i o n i n a GaAs homojunction b i p o l a r t h a t i s comparable t o t h a t of a GaAs h e t e r o j u n c t i o n b i p o l a r . T h e r e f o r e , i t i s deemed important t o e v a l u a t e and i n v e s t i g a t e the o p e r a t i o n and performance of the GaAs homojunction b i p o l a r t r a n s i s t o r i n view of i t s v a s t p o t e n t i a l . The work on GaAs homojunction b i p o l a r t r a n s i s t o r s began i n the e a r l y 1960's. The f i r s t t r a n s i s t o r was made by a l l o y i n g a t i n e m i t t e r t o a d i f f u s e d p-type base [ 6 ] . Succeeding d e v i c e s were f a b r i c a t e d by u s i n g t w o - d i f f u s i o n p r o c e s s e s [7-11], or by employing vapour phase e p i t a x y methods [ 12]. The experimental r e s u l t s of these e a r l y d e v i c e s were r a t h e r d i s c o u r a g i n g . In g e n e r a l , the s t a t i c c u r r e n t g a i n s o b t a i n e d were q u i t e low, and the frequency 5 gain-bandwidth products were t y p i c a l l y i n the range of a few hundred Mhz. In some cas e s , when d e v i c e s w i t h h i g h g a i n were made, e i t h e r they possessed very poor c u t o f f frequency [10], or they degraded w i t h i n a s h o r t p e r i o d of t h e i r f a b r i c a t i o n [ 1 2 ] . The y i e l d of these t r a n s i s t o r s was very low, due to the d i f f i c u l t y i n c o n t r o l l i n g the r e q u i r e d submicron base width by d i f f u s i o n or e p i t a x i a l t e c h n i q u e s . Not s u r p r i s i n g l y , r e s e a r c h i n t o GaAs homojunction b i p o l a r d e v i c e s was d i s c o n t i n u e d f o r some time a f t e r these i n i t i a l e f f o r t s . The use of i o n - i m p l a n t a t i o n i n semiconductor d e v i c e f a b r i c a t i o n and p r o c e s s i n g has brought a r e p r i e v e f o r the GaAs homojunction b i p o l a r t r a n s i s t o r s . P r e c i s e c o n t r o l over the j u n c t i o n depth and the doping l e v e l i n the v a r i o u s t r a n s i s t o r r e g i o n s can be a c h i e v e d by a d j u s t i n g the implant dose and energy. R e c e n t l y , a number of i o n - i m p l a n t e d GaAs homojunction b i p o l a r s have been made [13-19]. The s u c c e s s f u l development of p l a n a r v e r t i c a l npn d e v i c e s and l a t e r a l pnp t r a n s i s t o r s i n d i c a t e the p o t e n t i a l a p p l i c a t i o n of t h i s b i p o l a r technology i n the I n t e g r a t e d - I n j e c t i o n L o g i c type of d i g i t a l c i r c u i t s . The c u r r e n t gains measured i n these d e v i c e s were t y p i c a l l y i n the range of 7-35; these were a t t r i b u t e d t o u n s p e c i f i e d leakage c u r r e n t s a t the s u r f a c e . I t has been suggested t h a t t h i s s u r f a c e leakage e f f e c t can be reduced by i n c o r p o r a t i n g a guard r i n g s t r u c t u r e i n the d e v i c e design [20], thereby y i e l d i n g h i g h e r t r a n s i s t o r g a i n s . 6 In view of these new experimental r e s u l t s , i t was f e l t t h a t a t h e o r e t i c a l m o d e l l i n g of the GaAs homojunction b i p o l a r would be very u s e f u l . The s i m u l a t i o n of b i p o l a r t r a n s i s t o r behaviour should p r o v i d e i n s i g h t f u l i n f o r m a t i o n on the f a c t o r s a f f e c t i n g the o p e r a t i o n of an a c t u a l d e v i c e . In a d d i t i o n , by s t u d y i n g the e f f e c t of v a r y i n g d e v i c e design parameters such as the doping l e v e l s i n the e m i t t e r , base and c o l l e c t o r r e g i o n s , i t should be p o s s i b l e to come up with a d e v i c e design that p r o v i d e s optimum performance. A number of problems are encountered i n m o d e l l i n g GaAs b i p o l a r d e v i c e s . At present, there i s a l a c k of knowledge of some p r o p e r t i e s of GaAs such as the doping dependence of m o b i l i t y and recombination, the e f f e c t s i n t r o d u c e d by heavy doping and by s u r f a c e s t a t e s . In a d d i t i o n , the number of d e f e c t s i n GaAs are higher than t h a t f o r elemental s i l i c o n , where a n t i s i t e d e f e c t s do not have to be c o n s i d e r e d . The presence of a l a r g e number of d e f e c t s , together with the f a c t t h a t the recombination mechanisms i n b i p o l a r d e v i c e s are a f f e c t e d by the growth pr o c e s s of the s u b s t r a t e used, imply t h a t the t r a n s i s t o r c h a r a c t e r i s t i c s w i l l be m a t e r i a l dependent. In the case of ion-implanted t r a n s i s t o r s , the e f f e c t s of the implant mask, of the d i f f u s i o n of dopants d u r i n g thermal a n n e a l i n g , and the a c t u a l a c t i v a t i o n e f f i c i e n c y of implanted s p e c i e s on the c a r r i e r c o n c e n t r a t i o n p r o f i l e have to be c o n s i d e r e d as w e l l . In view of these u n c e r t a i n t i e s , i t i s very d i f f i c u l t to come up w i t h a model t h a t w i l l g i v e an exact p r e d i c t i o n of the d e v i c e behaviour 7 f o r a g i v e n o p e r a t i n g c o n d i t i o n . T h e a p p r o a c h t h a t h a s b e e n t a k e n i n t h e work d e s c r i b e d i n t h i s t h e s i s s e e k s t o i n c o r p o r a t e t h e p a r a m e t e r s p e r t a i n i n g t o t h e s e e f f e c t s i n t o t h e m o d e l , a n d by m a k i n g t h e a p p r o p r i a t e a s s u m p t i o n s , t o p r e d i c t t h e o r e t i c a l l y t h e c h a r a c t e r i s t i c s o f p a r t i c u l a r b i p o l a r t r a n s i s t o r s , a n d t o c o r r e l a t e t h e s e t o t h o s e o b s e r v e d e x p e r i m e n t a l l y . In t h i s work we r e s t r i c t o u r s e l v e s t o a s t u d y o f t h e dc p e r f o r m a n c e o f GaAs h o m o j u n c t i o n b i p o l a r t r a n s i s t o r s . Two d i f f e r e n t m o d e l s a r e d e v e l o p e d t o s t u d y t h e dc b e h a v i o u r o f GaAs b i p o l a r h o m o j u n c t i o n t r a n s i s t o r s . An a n a l y t i c a l m o d e l , w h i c h r e q u i r e s t h e a s s u m p t i o n o f u n i f o r m d o p i n g i n t h e v a r i o u s t r a n s i s t o r r e g i o n s , i s d e s c r i b e d i n C h a p t e r 2 . A n u m e r i c a l m o d e l , w h i c h i s b a s e d on t h e w e l l - k n o w n SEDAN p r o g r a m [ 2 1 ] , s u i t a b l y m o d i f i e d t o accommodate t h e p a r a m e t e r s a n d p r o p e r t i e s o f GaAs i s d e s c r i b e d i n C h a p t e r 3 . I t i s a s s u m e d i n b o t h m o d e l s t h a t t h e t r a n s i s t o r i s o p e r a t i n g u n d e r low i n j e c t i o n c o n d i t i o n s . A l s o , o n l y o n e - d i m e n s i o n a l d e v i c e s t r u c t u r e s a r e c o n s i d e r e d , s i n c e m u l t i - d i m e n s i o n a l e f f e c t s a r e n o t p r o f o u n d when t h e d r i v i n g c u r r e n t s a r e l o w . T h e u s e o f two d i f f e r e n t m o d e l s a r i s e s f r o m t h e n e e d t o c h a r a c t e r i z e t r a n s i s t o r s t r u c t u r e s b u i l t by d i f f e r e n t f a b r i c a t i o n t e c h n i q u e s . In a d d i t i o n , t h e a n a l y t i c a l m o d e l i s u s e f u l i n p e r f o r m i n g a s e n s i t i v i t y a n a l y s i s on d e s i g n p a r a m e t e r s w h i c h a f f e c t d e v i c e b e h a v i o u r , w h i l e t h e n u m e r i c a l m o d e l i s u s e d t o s i m u l a t e t h e o u t p u t s o f d e v i c e s w i t h n o n - u n i f o r m d o p i n g , s u c h a s t h o s e t r a n s i s t o r s 8 r e c e n t l y f a b r i c a t e d b y i o n - i m p l a n t a t i o n [ 1 3 - 1 9 ] . 2. T H E A N A L Y T I C A L M O D E L 2.1 I N T R O D U C T I O N I n - t h i s c h a p t e r , a n a n a l y t i c a l m o d e l f o r t h e c a l c u l a t i o n o f t h e D C c h a r a c t e r i s t i c s o f a G a A s h o m o j u n c t i o n n p n b i p o l a r t r a n s i s t o r i s d e s c r i b e d . T h i s m o d e l h a s t h e a d v a n t a g e o f u s i n g s i m p l e a n a l y t i c a l , c l o s e d f o r m e x p r e s s i o n s t o r e l a t e t h e t r a n s i s t o r c u r r e n t c o m p o n e n t s t o t h e b i a s c o n d i t i o n s . I t i s u s e d a s a s e n s i t i v i t y t e s t f o r s t u d y i n g t h e e f f e c t o f t h e v a r i o u s d e v i c e p a r a m e t e r s s u c h a s m i n o r i t y c a r r i e r l i f e t i m e s , s u r f a c e r e c o m b i n a t i o n v e l o c i t y , r e g i o n a l d o p i n g a n d p h y s i c a l d i m e n s i o n s o n t h e p e r f o r m a n c e o f t h e b i p o l a r t r a n s i s t o r . T h e d o p i n g l e v e l s i n t h e e m i t t e r , b a s e a n d c o l l e c t o r r e g i o n s o f t h e t r a n s i s t o r a r e a s s u m e d t o b e u n i f o r m , t h e r e f o r e t h i s m o d e l i s m o s t s u i t a b l e f o r c h a r a c t e r i s i n g G a A s h o m o j u n c t i o n b i p o l a r t r a n s i s t o r s b u i l t b y e p i t a x i a l t e c h n i q u e s s u c h a s L P E ( L i q u i d P h a s e E p i t a x y ) , V P E ( V a p o r P h a s e E p i t a x y ) , o r M B E ( M o l e c u l a r B e a m E p i t a x y ) . T h e a n a l y t i c a l m o d e l i s a o n e - d i m e n s i o n a l t r a n s i s t o r m o d e l a n d t h e m o d e l p a r a m e t e r s a r e t h o s e i n v o l v i n g t h e p h y s i c a l p r o p e r t i e s o f G a A s n e e d e d i n t h e c o m p u t a t i o n s o f t h e c u r r e n t c o m p o n e n t s r e s u l t i n g f r o m a c e r t a i n a p p l i e d b i a s . T h e d e s c r i p t i o n a n d f o r m u l a t i o n o f t h e t r a n s i s t o r m o d e l a n d t h e m o d e l p a r a m e t e r s , t o g e t h e r w i t h t h e r e s u l t s a n d d i s c u s s i o n s a r e g i v e n i n t h e s u b s e q u e n t s e c t i o n s o f t h i s c h a p t e r . 9 10 2 . 2 T H E T R A N S I S T O R M O D E L F o r a o n e - d i m e n s i o n a l , u n i f o r m l y d o p e d b i p o l a r t r a n s i s t o r , t h e d e v i c e a n a l y s i s c a n b e p e r f o r m e d b y t h e c l a s s i c a l S h o c k l e y r e g i o n a l a p p r o a c h , i n w h i c h t h e t r a n s i s t o r s t r u c t u r e i s d i v i d e d i n t o t w o d e p l e t i o n r e g i o n s , a n d t h r e e c h a r g e - n e u t r a l r e g i o n s . T h e v a r i o u s D C c u r r e n t d e n s i t y c o m p o n e n t s i n t h e d e v i c e a r e c a l c u l a t e d b y s o l v i n g t h e c o n t i n u i t y e q u a t i o n s a n d t h e c u r r e n t t r a n s p o r t e q u a t i o n s f o r a g i v e n s e t o f b o u n d a r y c o n d i t i o n s . T h e c u r r e n t f l o w s i n t h e o n e - d i m e n s i o n a l t r a n s i s t o r s t r u c t u r e a r e s h o w n i n F i g u r e 2 . 1 , w h e r e W g , W f i a n d a r e t h e w i d t h s o f t h e n e u t r a l r e g i o n s o f t h e e m i t t e r , b a s e a n d c o l l e c t o r r e s p e c t i v e l y . ( - X E ~ W E ) a n d ( W c + X c ) r e p r e s e n t t h e e m i t t e r a n d c o l l e c t o r e n d s r e s p e c t i v e l y . T h e s t e a d y - s t a t e c o n t i n u i t y e q u a t i o n s f o r e l e c t r o n s a n d h o l e s w i t h n o c a r r i e r g e n e r a t i o n a r e a s f o l l o w s : - p - ^ = o d x 2 D p r p n - n° d z n p . - — c = 0 ( 2 . 2 ) d x 2 V n T h e c u r r e n t t r a n s p o r t e q u a t i o n s i n t h e n e u t r a l r e g i o n s a r e g i v e n b y : d n J n " ^ ( 2 . 3 ) 11 Collector (n) wc+xc F i g . 2 .1 S c h e m a t i c o f c u r r e n t f l o w s i n a h o m o j u n c t i o n b i p o l a r t r a n s i s t o r . 12 JP • -"DP t (2-4) T h e c u r r e n t s c o n s i s t o f t h e d i f f u s i o n c o m p o n e n t o n l y , s i n c e t h e r e i s n o p o t e n t i a l v a r i a t i o n i n a c o n s t a n t l y d o p e d s e m i c o n d u c t o r r e g i o n t o g i v e r i s e t o t h e d r i f t t e r m . B y s o l v i n g e q u a t i o n s ( 2 . 1 ) - ( 2 . 4 ) , t h e g e n e r a l s o l u t i o n s a r e d e r i v e d a s : D J n = q [ A . e x p ( ) + B . e x p ( f 2 1 ) ] ( 2 . 5 ) L n L n L n D J = q r 2 [ C . e x p ( ) + C . e x p ( ~ ) ] ( 2 . 6 ) P P P w h e r e q i s t h e e l e c t r o n i c c h a r g e , D a n d D , r a n d T , L ^ 3 n p ' n p n a n d L p a r e t h e d i f f u s i o n c o n s t a n t s , l i f e t i m e s a n d d i f f u s i o n l e n g t h s , r e s p e c t i v e l y . J n , J p a r e t h e e l e c t r o n a n d h o l e c u r r e n t s , w h i l e p ° a n d • n ° a r e t h e e q u i l i b r i u m h o l e a n d e l e c t r o n c o n c e n t r a t i o n s i n n - t y p e a n d p - t y p e m a t e r i a l r e s p e c t i v e l y . T h e b o u n d a r y c o n d i t i o n s a r e d e t e r m i n e d b y t h e m i n o r i t y c a r r i e r c o n c e n t r a t i o n a t t h e j u n c t i o n d e p l e t i o n e d g e s a n d a t t h e s u r f a c e s ( i . e . t h e e m i t t e r a n d c o l l e c t o r c o n t a c t e n d s ) . I n c o m p u t i n g t h e d e v i c e c h a r a c t e r i s t i c s , i t i s a s s u m e d t h a t t h e t r a n s i s t o r i s o p e r a t i n g i n t h e n o r m a l , a c t i v e m o d e i n a c o m m o n e m i t t e r c o n f i g u r a t i o n , w i t h a f i x e d V C E a n d a f o r w a r d b i a s V g E . T h e b o u n d a r y c o n d i t i o n s a t t h e s p a c e c h a r g e e d g e s , o r t h e j u n c t i o n l a w [ 2 2 ] , a r e e x p r e s s e d a s : 13 P " PS n - n ° n - n B p - P; x = - X r x = 0 x = X B x = - X , q V f i E = p | [ e x p ( ) - 1 ] ( 2 . 7 ) 9 V B E = n j [ e x p ( -ZZT ) ~ 1 ] ( 2 . 8 ) k T = n ° [ e x p ( q V B C k T q V - Pgf e x p ( k T B C ) - 1 ] ( 2 . 9 ) ) - 1 ] ( 2 . 1 0 ) w h i l e t h e s u r f a c e b o u n d a r y c o n d i t i o n s a r e g i v e n a s : ) & E d x x - " V X E x = W c + X c = s p ( p - p | = 0 ( 2 . 1 1 ) ( 2 . 1 2 ) w h e r e p E , n | a n d p £ a r e t h e e q u i l i b r i u m m i n o r i t y c a r r i e r c o n c e n t r a t i o n s i n t h e e m i t t e r , b a s e a n d c o l l e c t o r r e s p e c t i v e l y . V g ^ i s t h e b a s e - c o l l e c t o r b i a s , k i s B o l t z m a n n ' s c o n s t a n t , T t h e r o o m t e m p e r a t u r e , D E t h e e m i t t e r m i n o r i t y c a r r i e r d i f f u s i o n c o n s t a n t , a n d s F i s t h e s u r f a c e r e c o m b i n a t i o n v e l o c i t y a t t h e e m i t t e r c o n t a c t . I t i s a s s u m e d t h a t t h e c o l l e c t o r e n d h a s a n o h m i c c o n t a c t , t h e r e f o r e t h e s u r f a c e r e c o m b i n a t i o n v e l o c i t y t h e r e i s i n f i n i t e . T h e v a r i o u s c u r r e n t c o m p o n e n t s i n t h e t r a n s i s t o r a r e a s s h o w n i n F i g u r e 2 . 1 . T h e e l e c t r o n c u r r e n t d e n s i t i e s a t t h e t w o n e u t r a l e d g e s a r e d e n o t e d b y J n ( 0 ) a n d J n ( x B ) « T n e h o l e 1 4 c u r r e n t d e n s i t y i n j e c t e d i n t o t h e e m i t t e r f r o m t h e b a s e i s r e p r e s e n t e d b y J ( - X „ ) f w h i l e t h e c o l l e c t o r h o l e l e a k a g e P ^ c u r r e n t d e n s i t y i s i n d i c a t e d b y J p ( X c ) . T h e s e c u r r e n t c o m p o n e n t s a r e c a l c u l a t e d b y i m p o s i n g t h e b o u n d a r y c o n d i t i o n s ( 2 . 7 ) - ( 2 . 1 2 ) o n t h e g e n e r a l c u r r e n t d e n s i t y e x p r e s s i o n s ( 2 . 5 ) - ( 2 . 6 ) . T h e r e s u l t s a r e a s f o l l o w s : q D E p E L E q V B E { e X p ( ~ k T ~ ) " 1 1 S „ L „ 2 W .{ - - r — . [ 1 + e x p ( — * ) J 2 . W + [ e x p ( —± ) - 1] } . S F L E n . 2 ' W E . , 1 — — . [ 1 - e x p ( — — ) ] 2 . W + [ e x p ( — - ) + 1] }" 1 ( 2 . 1 3 ) J n ( 0 ) q D n ° 2 . W n { e x p ( — * ) - 1 } - ' L B L B q V B E .{ [ e x p ( l m ) - 1 ] . [ e x p ( k T 2 . W T B ) + 1] - 2 . e x p ( ^ ) . [ e x p ( ^ S £ ) - }] } B ( 2 . 1 4 ) 15 q D n ° 2 . W n ^ B C 2 ' W B . { - [ e x p ( ) e x p ( - j - 5 ) +1 ] B W B <3 V BE + 2 . e x P ( ) A E X P ( " T T } ~ 1 ] } B ( 2 . 1 5 ) S D cPc ^ B C J p ( X c ) = [ e x p ( ) - 1 ] ( 2 . 1 6 ) I n t h e a b o v e e q u a t i o n s D g , D c a r e t h e m i n o r i t y c a r r i e r d i f f u s i o n c o n s t a n t s i n t h e b a s e , c o l l e c t o r ; a n d L „ , L _ , L n a r e t h e m i n o r i t y c a r r i e r d i f f u s i o n l e n g t h s i n t h e e m i t t e r , b a s e , c o l l e c t o r r e s p e c t i v e l y . T h e c u r r e n t c o m p o n e n t s d u e t o t h e g e n e r a t i o n a n d r e c o m b i n a t i o n o f c a r r i e r s i n t h e j u n c t i o n s p a c e c h a r g e r e g i o n s a r e g i v e n b e l o w . T h e r e c o m b i n a t i o n c u r r e n t i n t h e f o r w a r d b i a s e d e m i t t e r - b a s e j u n c t i o n , J R E C » i s e x p r e s s e d b y C h o o ' s e q u a t i o n s f o r a n a s y m m e t r i c a l l y d o p e d s t e p j u n c t i o n , n a m e l y [ 2 3 ] : 3 V B E q n . W B E 2 . s i n h ( — ) ^ ' R E C • < r B . r E > ' q / k T . ( V f a i E - V B E ) (2.17) 16 w h e r e f ( b ) = / ( i ^ b 2 ) ' t a n " 1 [ j ' ( 1 " b 2 } ] f o r b < 1 ( 2 . 1 8 a ) f(b> = TTT2 T T • t a n h " 1 [ - / ( b 2 - 1 ) ] v ( b^ - 1 ) 7 f o r b > 1 ( 2 . 1 8 b ) f ( b ) = - f o r b = 1 - ( 2 . 1 8 c ) a n d <3vBE E t " E i b = E X P ( j ^ T ) • c o s h [ — — 1 TE + r l n ( — ) ] ( 2 . 1 9 ) 2 T B r q ( V b i E - V B E ) , K = 2 . s i n h [ — ] ( 2 . 2 0 ) K • 1 T N T N 7 = / ( — — ) + / ( — — ) T E E T B B q( v h . - v ) + 2 b . c o s h [ — 5 £ i — ] ( 2 . 2 1 ) K • i. k T N E N B V b i E " X • l n [ -rTT 1 < 2- 2 2 ) 17 2 c V h i E N E + N B W _ E = • ( ^ . ) ( 2 . 2 3 ) B E q N N T h e g e n e r a t i o n c u r r e n t i n t h e r e v e r s e - b i a s e d c o l l e c t o r - b a s e j u n c t i o n , J Q E N e q u a t i o n [ 2 4 ] : J G E N r i s g i v e n b y t h e g e n e r a l S a h - N o y c e - S h o c k l e y q n ^ ^ E ^ - E B C . c o s h [ t 1 w h e r e ' G E N 2 . / ( T B . T c ) * 1 k T 1 T C + ^ . l n ( — ) ] " 1 ( 2 . 2 4 ) 2 T B 2 e V , . N +N k T N C N B V , = ^ . l n [ ] ( 2 . 2 6 ) b i C q n ? V b i E a n d V b i C a r e t ^ l e t > u i I t — i n v o l t a g e s a t t h e b a s e - e m i t t e r a n d b a s e - c o l l e c t o r j u n c t i o n s w h i l e W__ a n d W _ „ a r e t h e e m i t t e r a n d c o l l e c t o r d e p l e t i o n w i d t h s , r e s p e c t i v e l y . N E , N _ , N-, a r e t h e d o p i n g l e v e l s i n t h e e m i t t e r , b a s e a n d c o l l e c t o r r e s p e c t i v e l y , n ^ , e a n d E ^ a r e , r e s p e c t i v e l y , t h e i n t r i n s i c c a r r i e r c o n c e n t r a t i o n , p e r m i t t i v i t y , a n d i n t r i n s i c F e r m i l e v e l i n G a A s , w h i l e E f c i s t h e e n e r g y l e v e l o f t h e r e c o m b i n a t i o n c e n t e r s p r e s e n t i n t h e m a t e r i a l . I n u s i n g t h e e x p r e s s i o n s f o r J R E C a n d J G E N ' * t * s a s s u m e d t h a t t h e r e a r e s i n g l e - l e v e l , u n i f o r m l y d i s t r i b u t e d r e c o m b i n a t i o n - g e n e r a t i o n ' c e n t e r s l o c a t e d a t o r n e a r t h e 18 i n t r i n s i c F e r m i l e v e l , t h e r e f o r e E f c i s a p p r o x i m a t e l y e q u a l t o E ^ . I t i s f u r t h e r a s s u m e d t h a t 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 m e c h a n i s m i n v o l v e d i s o f t h e S h o c k l e y - R e a d - H a l l t y p e . T h i s a s s u m p t i o n i s s u p p o r t e d b y m i n o r i t y c a r r i e r d i f f u s i o n l e n g t h s t u d i e s i n G a A s b y v a r i o u s a u t h o r s . S e k e l a e t a l . [ 2 5 ] s h o w e d i n t h e i r s t u d i e s o f h o l e d i f f u s i o n l e n g t h i n n - t y p e G a A s t h a t t h e l a r g e s t m e a s u r e d i s a b o u t o n e t h i r d t h a t o f t h e t h e o r e t i c a l v a l u e p r e d i c t e d b y R y a n a n d E b e r h a r d t [ 2 6 ] f o r r a d i a t i v e r e c o m b i n a t i o n , w h i c h i m p l i e s t h a t t h e h o l e l i f e t i m e d u e t o i n d i r e c t r e c o m b i n a t i o n i s t y p i c a l l y a n o r d e r o f m a g n i t u d e l o w e r t h a n t h a t d u e t o d i r e c t r a d i a t i v e r e c o m b i n a t i o n . T h e s t u d i e s o n e l e c t r o n d i f f u s i o n l e n g t h i n p - t y p e G a A s b y C a s e y e t a l . [ 2 7 , 2 8 ] a l s o i n d i c a t e d t h a t t h e e l e c t r o n l i f e t i m e i s d o m i n a t e d b y n o n r a d i a t i v e r e c o m b i n a t i o n f o r h o l e c o n c e n t r a t i o n s l e s s t h a n 1 X 1 0 1 6 c m " 3 . T h i s i n s i g n i f i c a n t c o n t r i b u t i o n o f r a d i a t i v e r e c o m b i n a t i o n t o t h e t o t a l r e c o m b i n a t i o n m e c h a n i s m c a n b e a t t r i b u t e d t o t h e p r e s e n c e o f a l a r g e n u m b e r o f r e c o m b i n a t i o n c e n t e r s a n d d e f e c t s i n G a A s [ 2 9 ] , d e s p i t e t h e f a c t t h a t G a A s i s a d i r e c t b a n d g a p s e m i c o n d u c t o r . T h e S h o c k l e y - R e a d - H a l l r e c o m b i n a t i o n i s a s s u m e d t o b e t h e d o m i n a n t c o m p o n e n t o f t h e i n d i r e c t r e c o m b i n a t i o n m e c h a n i s m s i n c e i t i s b e l i e v e d t h a t A u g e r r e c o m b i n a t i o n i s i m p o r t a n t i n G a A s o n l y a t v e r y h i g h d o p i n g d e n s i t i e s ( g r e a t e r t h a n 1 0 1 9 c m " 3 ) [ 3 0 ] . T h e t e r m i n a l c u r r e n t d e n s i t i e s a r e c a l c u l a t e d b y s u m m i n g u p t h e c u r r e n t c o m p o n e n t s i n e q u a t i o n s 19 ( 2 . 1 3 - 2 . 1 7 , 2 . 2 4 ) . T h e t o t a l b a s e c u r r e n t d e n s i t y i s g i v e n b y : J B = J p ( ~ V + J n ( 0 ) " W " J p ( X C ) + J R E C " J G E N ( 2 ' 2 7 ) A n d t h e c o l l e c t o r c u r r e n t d e n s i t y i s g i v e n a s : J c = J n ( X B ) + J p ( X c ) + J G E N ( 2 . 2 8 ) T h e s t a t i c c u r r e n t g a i n , 0 o r h f e , i s e x p r e s s e d a s t h e r a t i o o f t h e c o l l e c t o r a n d b a s e c u r r e n t d e n s i t i e s : JC 0 = ( 2 . 2 9 ) 2 . 3 T H E M O D E L P A R A M E T E R S I n t h e m o d e l , t h e t h e r m a l e q u i l i b r i u m c a r r i e r c o n c e n t r a t i o n s a r e c o m p u t e d u s i n g F e r m i - D i r a c s t a t i s t i c s t o a c c o u n t f o r c a r r i e r d e g e n e r a c y . T h e d e n s i t i e s o f e l e c t r o n s a n d h o l e s i n t h e c o n d u c t i o n a n d v a l e n c e b a n d s a r e g i v e n b y t h e c o n v e n t i o n a l f o r m s : n ( c m ~ 3 ) = N , F ] / 2 ( V ) ( 2 . 3 0 ) p ( c m ~ 3 ) = N V ( 2 . 3 1 ) w h e r e 20 V = k T a n d ( 2 . 3 2 ) - E _ f k T ( 2 . 3 3 ) I n t h e s e e q u a t i o n s , n , p a r e t h e e l e c t r o n a n d h o l e c o n c e n t r a t i o n s r e s p e c t i v e l y , Fj/2(x) i s t h e F e r m i - D i r a c I n t e g r a l o f o r d e r 1 / 2 , a p p r o x i m a t e d b y a c c u r a t e , l o w o r d e r p o l y n o m i a l s [ 3 1 ] , E r i s t h e F e r m i e n e r g y , a n d E i s t h e w i d t h o f t h e f o r b i d d e n e n e r g y b a n d g a p i n e V . T h e z e r o r e f e r e n c e f o r t h e e n e r g y l e v e l s i s t a k e n a t t h e v a l e n c e b a n d e d g e . T h e e f f e c t i v e d e n s i t i e s o f s t a t e s , i n t h e a p p r o p r i a t e b a n d s , a r e e x p r e s s e d a s : N c ( c m - 3 ) = 2 . [ * 2irm k T e , 3 / 2 10 ' ( 2 . 3 4 ) N y ( c m - 3 ) 2 . [ * 2 7 r m h k T 1 3 / 2 10 - 6 ( 2 . 3 5 ) w h e r e h i s P l a n c k ' s c o n s t a n t . T h e e f f e c t i v e m a s s o f * e l e c t r o n s , m e i n t h e c o n d u c t i o n b a n d s y s t e m , i s g i v e n b y B l a k e m o r e [ 3 2 ] a s a n a v e r a g i n g f u n c t i o n o f t h e e f f e c t i v e m a s s e s o f e l e c t r o n s i n t h e l o w e s t c o n d u c t i o n b a n d m i n i m a , T 6 , a n d t h e t w o h i g h e r e n e r g y v a l l e y s , L 6 a n d X 6 . T h e n o n - p a r a b o l i c e f f e c t o f t h e l o w e s t b a n d i s a l s o i n c l u d e d b y a n a d d i t i o n a l t e r m i n t h e e x p r e s s i o n f o r t h e T 6 m a s s 21 p a r a m e t e r : • , m 3/2 , , ^ ! F S , 2 ™ . m e " « "co -I 1 ' 4 E g • F ^ T 1 + m 3/2 . e X P ' " " O > . X F / / 2 ( r ? ) + m 3 / 2 f ^ P j ^ " kT? } . . 2 / 3 r , m L • [ ] } ( 2 - 3 6 ) T h e s c a l a r e f f e c t i v e m a s s o f t h e r 6 b a n d , m i s e q u a l t o c o 0 . 0 6 3 2 m o ; w h e r e m 0 i s t h e f r e e e l e c t r o n m a s s ( 9 . 1 0 9 5 x 1 0 " 3 1 k g ) . T h e n o n - p a r a b o l i c c o e f f i c i e n t a i s e q u a l t o - 0 . 8 2 4 . m x , m L a r e t h e e f f e c t i v e m a s s e s o f t h e t w o h i g h e r e n e r g y m i n i m a , w h i c h a r e a s s i g n e d v a l u e s o f 0 . 5 5 m o a n d 0 . 8 5 m o r e s p e c t i v e l y . A r - = 0 . 2 8 4 e V , A r x = 0 . 4 7 6 e V a r e t h e e n e r g y s e p a r a t i o n s b e t w e e n t h e T - L a n d T - X b a n d s . F. , ( x ) t h e F e r m i - D i r a c I n t e g r a l o f o r d e r 3 / 2 , c o m e s f r o m t h e k" t e r m i n t h e e n e r g y - w a v e v e c t o r r e l a t i o n o f t h e T 6 b a n d . T h i s i n t e g r a l i s c o m p u t e d u s i n g s h o r t s e r i e s a p p r o x i m a t i o n s [ 3 1 ] , F o r n - t y p e , * w e a k l y - d o p e d G a A s a t r o o m t e m p e r a t u r e , m g c a n b e a p p r o x i m a t e d b y t h e f i r s t t e r m o f e q u a t i o n ( 2 . 3 6 ) , s i n c e t h e f r a c t i o n s o f e l e c t r o n s i n t h e L 6 , X 6 b a n d s a r e r e l a t i v e l y s m a l l . H o w e v e r , f o r a h e a v i l y d o p e d m a t e r i a l s u c h t h a t rj £ 0 , t h e c o n t r i b u t i o n s f r o m t h e e l e c t r o n p o p u l a t i o n i n t h e t w o u p p e r b a n d s a r e n o n - n e g l i g i b l e a n d l e a d t o a n i n c r e a s e d e l e c t r o n e f f e c t i v e m a s s . 22 T h e h o l e e f f e c t i v e m a s s i s a l s o c a l c u l a t e d f r o m t h e d e t a i l e d e q u a t i o n o f t h e l i g h t h o l e a n d h e a v y h o l e b a n d s i n t h e v a l e n c e b a n d s y s t e m [ 3 2 ] : * / ? 1 5 B K T F U ) m h = { m ^ 2 + mp.[ 1 - — ] ^  ( 2 . 3 7 ) w h e r e m ^ , m L a r e t h e h e a v y h o l e a n d l i g h t h o l e e f f e c t i v e m a s s e s , t a k e n a s 0 . 5 m o a n d 0 . 0 8 8 m o r e s p e c t i v e l y . T h e n o n - p a r a b o l i c c o e f f i c i e n t B o f t h e l i g h t h o l e b a n d i s e q u a l t o - 3 . 8 0 . T o c o m p u t e t h e c a r r i e r e f f e c t i v e m a s s e s , a n d h e n c e t h e c a r r i e r c o n c e n t r a t i o n s , t h e a c t u a l b a n d g a p a n d t h e e x a c t l o c a t i o n o f t h e F e r m i l e v e l a r e r e q u i r e d . A t h i g h d o p i n g l e v e l s , t h e e n e r g y b a n d g a p i s e f f e c t i v e l y r e d u c e d , d u e t o t h e o v e r l a p p i n g o f i m p u r i t y b a n d s w i t h t h e t a i l s t a t e s o f t h e c o n d u c t i o n a n d v a l e n c e b a n d s y s t e m . T h i s b a n d g a p n a r r o w i n g e f f e c t c a n b e r e l a t e d t o t h e e f f e c t i v e i n t r i n s i c c a r r i e r c o n c e n t r a t i o n s u c h t h a t [ 3 2 , 3 3 ] : E ( e V ) = 1 . 4 2 2 4 8 - A E ( 2 . 3 8 ) 9 9 a n d n i A E ( e V ) = k T . l n [ r ] ( 2 . 3 9 ) 9 n i l T h e v a l u e o f t h e i n t r i n s i c c a r r i e r c o n c e n t r a t i o n n^ a t r o o m t e m p e r a t u r e h a s b e e n d e t e r m i n e d b y v a r i o u s e x p e r i m e n t s a n d 23 s h o w n t o v a r y f r o m 1 . 8 X 1 0 6 c m " 3 t o 3 . 0 x 1 0 s c m " 3 [ 3 4 ] . I t i s t a k e n a s 2 x 1 0 s c m " 3 i n t h e m o d e l . T h e v a r i a t i o n o f t h e e f f e c t i v e i n t r i n s i c c a r r i e r c o n c e n t r a t i o n n ^ g w i t h d o p i n g d e n s i t y i s e x p r e s s e d b y t h e e m p i r i c a l r e l a t i o n s o f B a i l b e e t a l . [ 3 5 ] , n a m e l y : f o r n - t y p e m a t e r i a l , n . ( c m " 3 ) = 9 x 1 0 ' 5 + 3 . 3 8 X 1 0 " 3 . / ( N ^ ) i e D N D - 3 . 4 7 X 1 0 " 5 . l n [ J^JJ ] ( 2 . 4 0 ) a n d f o r p - t y p e m a t e r i a l , n i e ( c m " 3 ) = 9 x 1 0 s + 3 . 3 8 x l 0 " 3 . / ( N ^ ) N A - 6 . 7 2 x 1 0 " 5 . l n [ jfprr ] ( 2 . 4 1 ) T h e F e r m i l e v e l i s c o m p u t e d b y a c h i e v i n g t h e c h a r g e n e u t r a l i t y c o n d i t i o n , i n w h i c h t h e t o t a l n e g a t i v e c h a r g e s ( e l e c t r o n s a n d i o n i z e d a c c e p t o r s ) a r e e q u a l t o t h e t o t a l p o s i t i v e c h a r g e s ( h o l e s a n d i o n i z e d d o n o r s ) : n + N A - ( p + N p ) = 0 ( 2 . 4 2 ) T h e n u m b e r o f i o n i z e d i m p u r i t i e s , i s g i v e n b y : N t N . ( c m " 3 ) = — 7 - 7 ( 2 . 4 3 ) i 1 + g . e x p ( A ) 24 w h e r e N F C i s t h e t o t a l d e n s i t y o f i m p u r i t y d o p a n t i n c m - 3 , g i s t h e d e g e n e r a c y f a c t o r (2 f o r a c c e p t o r s , 4 f o r d o n o r s ) , a n d A = ( E f - E g + E ^ ) / k T f o r d o n o r s t a t e s a n d ( E A - E f ) / k T f o r a c c e p t o r s t a t e s . T h e i m p u r i t y a c t i v a t i o n e n e r g y r e l a t i v e t o t h e a p p r o p r i a t e b a n d i s d e s c r i b e d b y t h e e m p i r i c a l r e l a t i o n [ 3 6 ] : E A ( e V ) = E ° - A . N J / 3 ( 2 . 4 4 ) w h e r e E ° i s t h e a c t i v a t i o n e n e r g y f o r i n f i n i t e d i l u t i o n . T h e c o e f f i c i e n t A i s t a k e n a s 1 . 9 X 1 0 " 8 c m . e V f o r n - t y p e d o p a n t [ 3 7 ] , a n d 2 . 3 x 1 0 " * c m . e V f o r p - t y p e d o p a n t [ 3 8 ] . T h e N ^ 3 d e p e n d e n c e c o m e s f r o m t h e r e l a t i o n b e t w e e n t h e i m p u r i t y a t o m s p a c i n g a n d c o n c e n t r a t i o n . I n t h e d i f f u s i o n c u r r e n t c a l c u l a t i o n s , m i n o r i t y c a r r i e r p a r a m e t e r s s u c h a s d i f f u s i o n c o n s t a n t s a n d l i f e t i m e s a r e n e e d e d b e s i d e s t h e e q u i l i b r i u m c a r r i e r c o n c e n t r a t i o n s . T h e e l e c t r o n a n d h o l e d i f f u s i o n c o n s t a n t s a r e e x p r e s s e d b y t h e g e n e r a l i z e d E i n s t e i n r e l a t i o n , w r i t t e n a s a r a p i d l y c o n v e r g i n g s e r i e s [ 3 9 ] : D ( c m 2 s e c - 1 ) = u [ 1 + 0 . 3 5 3 5 5 ) n n q N R - 9 . 9 X 1 0 " 3 ( £ ) 2  N C + 4 . 4 5 x 1 0 - " (^ ) 3 ] ( 2 . 4 5 ) N C 25 D ( c m 2 s e c - 1 ) = u — [ 1 + 0 . 3 5 3 5 5 ( £ ) P P q N Y - 9 . 9 X 1 0 " 3 (J; ) 2  N V + 4 . 4 5 x 1 0 - " ( £ ) 3 ] ( 2 . 4 6 ) N V w h e r e un, M p a r e t h e m i n o r i t y c a r r i e r m o b i l i t i e s f q r e l e c t r o n s a n d h o l e s . T o m o d e l t h e c a r r i e r m o b i l i t i e s , d a t a f r o m H a l l e f f e c t m e a s u r e m e n t s f o r e l e c t r o n m o b i l i t y i n n - t y p e G a A s [ 3 7 , 4 0 - 4 5 ] , a n d h o l e m o b i l i t y i n p - t y p e G a A s [ 3 8 , 4 0 - 4 1 , 4 6 - 4 9 ] a r e g a t h e r e d a n d f i t t e d w i t h a f i f t h o r d e r l e a s t s q u a r e p o l y n o m i a l a g a i n s t t h e m a j o r i t y c a r r i e r c o n c e n t r a t i o n : 5 i M u ( c m 2 V " ' s e c " 1 ) = Z A . . x ( 2 . 4 7 ) H i = 0 1 w h e r e x = L o g , 0 ( n / c m " 3 ) f o r n - t y p e m a t e r i a l a n d L o g , 0 ( p / c n r 3 ) f o r p - t y p e m a t e r i a l . T h e c o e f f i c i e n t s A ^ f o r b o t h e l e c t r o n a n d h o l e H a l l m o b i l i t y a r e l i s t e d i n T a b l e 2 . 1 . T h e e x p e r i m e n t a l d a t a f o r e l e c t r o n s a n d h o l e s t o g e t h e r w i t h t h e f i t t e d c u r v e f r o m e q u a t i o n ( 2 . 4 7 ) a r e p l o t t e d i n F i g u r e s 2 . 2 a n d 2 . 3 r e s p e c t i v e l y . I t i s a s s u m e d t h a t t h e v a l u e s o f H a l l m o b i l i t y h a v e b e e n m e a s u r e d u n d e r l o w m a g n e t i c f i e l d i n t e n s i t i e s s o t h a t t h e m a j o r i t y c a r r i e r d r i f t m o b i l i t y c a n b e o b t a i n e d b y d i v i d i n g t h e H a l l m o b i l i t y b y t h e w e a k f i e l d H a l l f a c t o r R „ : 26 10000 -sec 8000 'Volt-E o 6000-Mobil 4000-Hall [ron Eled 2000-0-+ 10' A Emel'yanenko et al. [41] X Hill [40] • Dvoryankin et al. [37] Katoda & Sugano [42] Walukiewicz et al. [43] X Cox & DiLorenzo [44] Ashen et al. [45] computer fit 10 10" 10" 10' Electron Concentration [ cm - 3 ] F i g . 2 . 2 E x p e r i m e n t a l d a t a a n d b e s t c u r v e f i t f o r t h e d e p e n d e n c e o f e l e c t r o n H a l l m o b i l i t y o n m a j o r i t y c a r r i e r c o n c e n t r a t i o n . 27 F i g . 2 . 3 E x p e r i m e n t a l d a t a a n d b e s t c u r v e f i t f o r t h e d e p e n d e n c e o f h o l e H a l l m o b i l i t y o n m a j o r i t y c a r r i e r c o n c e n t r a t i o n . 28 M H u(cm2V 1 s e c " 1 ) = — ( 2 . 4 8 ) R H w h e r e R H i s t a k e n a s 1 . 1 7 5 a n d 1 . 2 5 f o r e l e c t r o n a n d h o l e H a l l m o b i l i t y r e s p e c t i v e l y [ 3 2 ] . F o r e l e c t r o n s , t h e r a t i o o f m i n o r i t y c a r r i e r t o m a j o r i t y c a r r i e r m o b i l i t y i s d e s c r i b e d b y : ^ ( c n ^ V - 1 s e c " 1 ) = f ( n ) . M " ( 2 . 4 9 ) w h e r e uP, nn a r e t h e r e s p e c t i v e e l e c t r o n d r i f t m o b i l i t i e s i n p - t y p e a n d n - t y p e m a t e r i a l . f ( n ) r e p r e s e n t s a 4 t h o r d e r p o l y n o m i a l f i t t e d t o t h e d a t a o f r e f e r e n c e [ 5 0 ] : > f ( n ) = 6 8 4 . 3 2 7 - I 6 8 . 1 7 5 y + I 5 . 4 9 7 y 2 - 0 . 6 3 3 y 3 + C O O g y " ( 2 . 5 0 ) w h e r e y = L o g , 0 ( n / c n r 3 ) . T h e d e p e n d e n c e o f f ( n ) o n e l e c t r o n c o n c e n t r a t i o n i s p l o t t e d i n F i g u r e 2 . 4 . F o r h o l e s , a s i m i l a r e x p r e s s i o n r e l a t i n g t h e m a j o r i t y c a r r i e r t o t h e m i n o r i t y c a r r i e r m o b i l i t y d o e s n o t e x i s t . T h e r e f o r e t h e m o b i l i t y r a t i o f o r h o l e s i s a s s u m e d t o b e u n i t y . C u r v e f i t t i n g s a r e a l s o p e r f o r m e d o n d a t a f o r e l e c t r o n [ 4 9 , 5 1 - 5 4 ] a n d h o l e [ 5 1 , 5 3 - 5 8 ] m i n o r i t y c a r r i e r l i f e t i m e s . T h e s e d a t a a r e p l o t t e d i n F i g u r e s 2 . 5 a n d 2 . 6 . T h e f i t t e d c u r v e s a r e o f t h e f o r m : 2 9 F i g . 2 . 4 R a t i o o f e l e c t r o n m o b i l i t y i n p - t y p e G a A s , / i P , t o t h e e l e c t r o n m o b i l i t y i n n - t y p e G a A s , M",, a s a f u n c t i o n o f e l e c t r o n c o n c e n t r a t i o n . 30 10000 1000 0.0H 10" 10'8 Hole Concentration [ cm"3 ] F i g . 2 . 5 E x p e r i m e n t a l d a t a a n d b e s t c u r v e f i t f o r t h e d e p e n d e n c e o f e l e c t r o n l i f e t i m e o n h o l e c o n c e n t r a t i o n . 31 F i g . 2 . 6 E x p e r i m e n t a l d a t a a n d b e s t c u r v e f i t f o r t h e d e p e n d e n c e o f h o l e l i f e t i m e o n e l e c t r o n c o n c e n t r a t i o n . 32 i n . L o g 1 0 ( r / s e c ) = E B - . x ( 2 . 5 1 ) i = 0 1 w h e r e r d e n o t e s t h e m i n o r i t y c a r r i e r l i f e t i m e . I n t h i s e q u a t i o n , x = L o g , 0 ( p / c n r 3 ) , m=5 f o r e l e c t r o n l i f e t i m e T , a n d x = L o g , 0 ( n / c m " 3 ) , m=3 f o r h o l e l i f e t i m e T . T h e c o e f f i c i e n t s B^ a r e g i v e n i n T a b l e 2 . 2 . T o c o m p l e t e t h e s e t o f p a r a m e t e r s n e e d e d i n t h e m o d e l c a l c u l a t i o n s , t h e r e l a t i v e p e r m i t t i v i t y o f G a A s i s c h o s e n a s 1 3 . 1 . T h e e m i t t e r s u r f a c e r e c o m b i n a t i o n v e l o c i t y i s t a k e n a s 2 x 1 0 s c m ' 3 [ 5 7 ] e x c e p t w h e n i t i s s p e c i f i e d a s a n i n p u t p a r a m e t e r . 2 . 4 T H E S O L U T I O N P R O C E D U R E T o s p e c i f y t h e m o d e l c o m p l e t e l y , i n p u t p a r a m e t e r s s u c h a s s e m i c o n d u c t o r l a y e r t h i c k n e s s e s a n d w i d t h s , a n d t h e b i a s v o l t a g e s a r e r e q u i r e d . T h e s e p a r a m e t e r s a r e c h o s e n f o r a n i d e a l , u n i f o r m l y d o p e d b i p o l a r t r a n s i s t o r o p e r a t i n g i n t h e c o m m o n e m i t t e r c o n f i g u r a t i o n , w i t h p r o f i l e a n d w i d t h s s i m i l a r t o t h a t o f t h e d e v i c e r e p o r t e d i n r e f e r e n c e [ 1 7 ] , T h e i n p u t d a t a a r e l i s t e d i n T a b l e 2 . 3 . I n t h e c o m m o n e m i t t e r m o d e , t h e b i p o l a r t r a n s i s t o r i s n o r m a l l y b i a s e d b y a c o n s t a n t b a s e c u r r e n t s o u r c e . I n t h i s c a s e , a c o n s t a n t v o l t a g e s o u r c e , V " B E i s u s e d d u e t o t h e e a s e o f i t s i m p l e m e n t a t i o n i n t h e m o d e l . B y d o i n g s o , i t i s a s s u m e d t h e r e i s n o t e m p e r a t u r e v a r i a t i o n , s o t h a t t h e e x p o n e n t i a l 33 E l e c t r o n H a l l M o b i l i t y H o l e H a l l M o b i l i t y ( c m 2 V 1 s e c " 1 ) ( c m 2 V " 1 s e c " 1 ) A 0 0 . 0 5 3 5 0 2 5 1 1 . 2 6 8 A , - 6 9 7 8 3 . 1 6 0 - 1 5 6 1 0 7 . 0 0 0 A 2 1 7 3 4 8 . 3 2 8 1 9 2 0 5 . 6 0 0 A 3 - 1 5 7 9 . 5 9 9 - 1 1 6 8 . 9 2 6 A « 6 3 . 0 3 6 3 5 . 2 0 9 A 5 - 0 . 9 3 5 - 0 . 4 2 0 T a b l e 2 . 1 C o e f f i c i e n t s f o r M H i n e q u a t i o n ( 2 . 4 7 ) . L o 9 i o < T n / S e c ) L o g 1 0 ( T p / s e c ) B 0 - 7 . 6 3 2 X 1 0 -6 5 4 5 . 0 7 5 B , - 6 7 8 . 5 7 2 - 9 9 . 5 7 4 B 2 1 5 7 . 9 3 5 5 . 9 6 7 B 3 - 1 3 . 7 3 8 - 0 . 1 1 9 B f t 0 . 5 3 1 B 5 - 0 . 0 0 7 T a b l e 2 . 2 C o e f f i c i e n t s f o r m i n o r i t y c a r r i e r l i f e t i m e s i n e q u a t i o n ( 2 . 5 1 ) . 34 N E E m i t t e r d o p i n g d e n s i t y 1 x 1 0 1 8 c m " 3 N B B a s e d o p i n g d e n s i t y I x l O 1 7 c m - 3 N C C o l l e c t o r d o p i n g d e n s i t y 1 x 1 0 1 6 c m - 3 W E E m i t t e r l a y e r t h i c k n e s s 0 . 2 5 um W B B a s e l a y e r t h i c k n e s s 0 . 4 0 Aim w c C o l l e c t o r l a y e r t h i c k n e s s 2 . 0 0 Mm S F E m i t t e r s u r f a c e r e c o m b i n a t i o n v e l o c i t y 2 x 1 0 6 c m s e c " 1 V C E E m i t t e r - c o l l e c t o r r e v e r s e b i a s v o l t a g e 5 . 0 V V B E E m i t t e r - b a s e f o r w a r d b i a s v o l t a g e 0 . 6 - 1 . 3 0 V T a b l e 2 . 3 I n p u t d a t a u s e d i n t h e m o d e l i n g p r o g r a m . 35 t e r m i n v o l v i n g q V B E / k T , w h i c h d e t e r m i n e s t h e b a s e c u r r e n t , i s t h e s a m e f o r a g i v e n V g E . T h e f l o w c h a r t f o r t h e m o d e l i n g p r o g r a m i s g i v e n i n F i g u r e 2 . 7 . I n t h i s a n a l y t i c a l a n a l y s i s , t h e e f f e c t o f c h a n g e s i n t h e w i d t h s , d o p i n g d e n s i t i e s a n d m i n o r i t y c a r r i e r l i f e t i m e s o f t h e e m i t t e r a n d b a s e r e g i o n s , t h e c o l l e c t o r d o p i n g d e n s i t y a n d t h e e m i t t e r s u r f a c e r e c o m b i n a t i o n v e l o c i t y a r e e x a m i n e d b y c o m p u t i n g t h e v a r i a t i o n o f t h e D C g a i n v e r s u s c o l l e c t o r c u r r e n t d e n s i t y . - T h e i n p u t a n d o u t p u t c h a r a c t e r i s t i c s a r e a l s o d e t e r m i n e d . T h e s o l u t i o n p r o c e d u r e i s a s f o l l o w s : ( 1 ) S p e c i f y t h e d e v i c e p a r a m e t e r s a n d o p e r a t i n g c o n d i t i o n s . ( 2 ) R e a d i n t h e r e q u i r e d s e n s i t i v i t y p a r a m e t e r . ( 3 ) C a l c u l a t e t h e a c t i v a t i o n e n e r g y a n d b a n d g a p o f t h e e m i t t e r r e g i o n a c c o r d i n g t o e q u a t i o n s (2 . 3 8 ) - ( 2 . 4 1 ) , ( 2 . 4 4 ) . ( 4 ) C o m p u t e t h e F e r m i e n e r g y , t h e e f f e c t i v e m a s s e s " , d e n s i t i e s o f s t a t e s a n d t h e c a r r i e r c o n c e n t r a t i o n s b y s o l v i n g t h e n o n - l i n e a r e q u a t i o n ( 2 . 4 2 ) u s i n g t h e s u b r o u t i n e Z E R O . ( 5 ) d e t e r m i n e t h e m i n o r i t y c a r r i e r p a r a m e t e r s : d i f f u s i o n c o n s t a n t s , m o b i l i t i e s a n d l i f e t i m e s a c c o r d i n g t o e q u a t i o n s ( 2 . 4 5 ) ~ ( 2 . 5 1 ) . ( 6 ) R e p e a t s t e p s ( 3 ) - ( 5 ) f o r t h e b a s e a n d c o l l e c t o r r e g i o n s . 36 Data input for emitter, base and collector NE« NB« NC» W E » VWC" V C E ' SF» 9 « E A Read in sensitivity parameter Calculate E . E A q' A Compute E f , m e , m^, N c , Ny, n, p for one region No Compute minority carrier mobilities, lifetimes and related parameters Compute J B , J c , 0 for one V g E value Increase V r No Yes Fig. 2.7 Flow chart for the solution procedure of the analytical model. 37 ( 7 ) C a l c u l a t e t h e b a s e , c o l l e c t o r c u r r e n t d e n s i t i e s a n d D C g a i n u s i n g e q u a t i o n s ( 2 . 1 3 ) - ( 2 . 1 7 ) , ( 2 . 2 4 ) , ( 2 . 2 7 ) - ( 2 . 2 9 ) f o r a n i n i t i a l V " B E . ( 8 ) I n c r e a s e V B E a n d r e p e a t s t e p ( 7 ) u n t i l t h e f i n a l V " B E i s r e a c h e d . T h e p r o g r a m e x e c u t i o n s t o p s w h e n a l l t h e s e n s i t i v i t y p a r a m e t e r s h a v e b e e n r e a d , o t h e r w i s e s t e p s ( 2 ) - ( 8 ) a r e r e p e a t e d . T h e s a m e p r o c e d u r e i s u s e d t o c o m p u t e t h e d e v i c e o u t p u t c h a r a c t e r i s t i c s , e x c e p t t h e b i a s v o l t a g e i s now v__ a n d t h e c a l c u l a t i o n s a r e c o m p u t e d f o r a f i x e d v" B E. I n t h e s u b r o u t i n e Z E R O , t h e s o l u t i o n o f a n o n - l i n e a r e q u a t i o n i s o b t a i n e d b y a c o m b i n a t i o n o f t h e b i s e c t i o n a n d s e c a n t m e t h o d s [ 5 9 ] , L e t t h e n e t c h a r g e f u n c t i o n i n e q u a t i o n ( 2 . 4 2 ) b e F ( x ) . A n i n i t i a l i n t e r v a l [ B , C ] i s t h e n c h o s e n s u c h t h a t F ( B ) . F ( C ) ^ 0 . I f F ( B ) i s n o t e q u a l t o z e r o , a n i t e r a t i o n i s p e r f o r m e d t o f i n d new v a l u e s o f B a n d C b y s h r i n k i n g [ B , C ] , s u b j e c t t o t h e c o n d i t i o n | F ( B ) | < | F ( C ) | . T h i s i t e r a t i o n s t o p s w h e n t h e c r i t e r i o n j B—C j ^ 2 ( r e l a t i v e e r r o r . | B | + a b s o l u t e e r r o r ) i s r e a c h e d . B i s t h e n t h e r e q u i r e d s o l u t i o n . T h e a b s o l u t e a n d i n i t i a l r e l a t i v e e r r o r i s c h o s e n t o b e 1 x 1 0 " 6 i n t h e p r o g r a m . T h e a b s o l u t e e r r o r i s n e e d e d i n t h e e v e n t t h a t t h e s o l u t i o n i s 0 . T h e m o d e l l i n g p r o g r a m i s l i s t e d i n A p p e n d i x A . 38 2 . 5 R E S U L T S A N D D I S C U S S I O N T o d e m o n s t r a t e t h e w o r k i n g o f t h e a n a l y t i c a l m o d e l , t h e t r a n s i s t o r o u t p u t c h a r a c t e r i s t i c s a r e g i v e n i n F i g u r e 2 . 8 . T h e c u r v e s a r e p l o t t e d f o r v a l u e s o f V " B E r a n g i n g f r o m 1 . 1 0 V • to 1 . 2 2 V a s i n d i c a t e d . T h e i n p u t c h a r a c t e r i s t i c s a n d t h e J c - V * B E r e l a t i o n s h i p a r e p r e s e n t e d i n F i g u r e 2 . 9 . A s s e e n f r o m t h i s f i g u r e , i n t h e l o w c u r r e n t r a n g e , J D s h o w s a v o l t a g e d e p e n d e n c e o f ( q V B E ) / ( n k T ) , w h e r e t h e i d e a l i t y f a c t o r n = 1 . 8 9 . T h i s v a l u e i s c l o s e t o t h e t h e o r e t i c a l v a l u e o f 2 w h i c h w o u l d i n d i c a t e t h a t t h e b a s e c u r r e n t d e n s i t y i s d o m i n a t e d b y J - - . , t h e s p a c e c h a r g e r e c o m b i n a t i o n c u r r e n t , s e e e q u a t i o n ( 2 . 1 7 ) . A s V " B E i n c r e a s e s , J R E C b e c o m e s r e l a t i v e l y l e s s i m p o r t a n t , a n d t h e v a l u e o f n i s a p p r o x i m a t e l y e q u a l t o o n e , i n d i c a t i n g t h e d o m i n a n t c o m p o n e n t i n t h e b a s e c u r r e n t i s t h a t o f t h e i n j e c t e d c u r r e n t . F o r t h e c o l l e c t o r c u r r e n t d e n s i t y J c , t h e i d e a l i t y f a c t o r i s 1 . 0 1 4 t h r o u g h o u t t h e c u r r e n t r a n g e . T h i s s h o w s t h a t J c i s p r e d o m i n a n t l y J n ^ x g ^ ' t * i e i n j e c t e d e l e c t r o n c u r r e n t f r o m t h e e m i t t e r w h i c h d i f f u s e s t o t h e b a s e - c o l l e c t o r j u n c t i o n e d g e . T h i s i s a s e x p e c t e d s i n c e t h e g e n e r a t i o n c u r r e n t i n t h e r e v e r s e - b i a s e d j u n c t i o n , J ^ E N , a n d t h e c o l l e c t o r h o l e l e a k a g e c u r r e n t J ( X _ ) a r e g e n e r a l l y P *~ s m a l l w h e n c o m p a r e d t o t h e d i f f u s i o n c u r r e n t . T o s t u d y t h e e f f e c t s o f b a s e p a r a m e t e r v a r i a t i o n o n t h e t r a n s i s t o r p e r f o r m a n c e , p l o t s o f d c g a i n v e r s u s t h e c o l l e c t o r c u r r e n t d e n s i t y u s i n g b a s e l i f e t i m e a n d b a s e w i d t h a s p a r a m e t e r s a r e s h o w n i n F i g u r e s 2 . 1 0 a n d 2 . 1 1 . T h e 39 6000 - r V„ = 1.22 V 3 VCE [ Volts ] F i g . 2 . 8 C o l l e c t o r c u r r e n t d e n s i t y a s a f u n c t i o n o f e m i t t e r - c o l l e c t o r v o l t a g e w i t h t h e b a s e - e m i t t e r v o l t a g e a s a p a r a m e t e r 4 0 Emitter Base Voltage, V B E [ Volts ] F i g . 2 . 9 P l o t o f b a s e c u r r e n t d e n s i t y , c o l l e c t o r c u r r e n t d e n s i t y v e r s u s b a s e - e m i t t e r v o l t a g e , n i s t h e i d e a l i t y f a c t o r . 41 F i g . 2 . 1 0 T h e e f f e c t o f e l e c t r o n l i f e t i m e i n t h e b a s e o n g a i n , a s p r e d i c t e d b y t h e a n a l y t i c a l m o d e l u s i n g t h e p a r a m e t e r s l i s t e d i n T a b l e 2 . 3 . 42 F i g . 2 . 1 1 T h e e f f e c t o f b a s e w i d t h o n g a i n , a s p r e d i c t e d b y t h e a n a l y t i c a l m o d e l u s i n g t h e p a r a m e t e r s [ o t h e r t h a n W D ] ' l i s t e d i n T a b l e 2 . 3 . 43 e l e c t r o n l i f e t i m e i n p - t y p e G a A s i s k n o w n t o b e e x t r e m e l y s h o r t . S u c h a l o w m i n o r i t y c a r r i e r l i f e t i m e i n t h e b a s e r e g i o n o f a n p n b i p o l a r t r a n s i s t o r h a s a p r o f o u n d e f f e c t o n t h e d c g a i n . B e c a u s e o f t h e i r s h o r t e r d i f f u s i o n l e n g t h , m o r e i n j e c t e d e l e c t r o n s f r o m t h e e m i t t e r w i l l r e c o m b i n e w i t h h o l e s p r e s e n t i n t h e n e u t r a l b a s e . T h e i n c r e a s e i n b a s e r e c o m b i n a t i o n b r i n g s a b o u t a d e c r e a s e o f t h e c o l l e c t o r c u r r e n t , w h i l e a t t h e s a m e t i m e , c a u s e s a n i n c r e a s e o f t h e b a s e c u r r e n t . T h i s w i l l r e s u l t i n a s u b s t a n t i a l r e d u c t i o n i n t h e t r a n s i s t o r g a i n . Y u a n e t a l . [ 1 5 ] e s t i m a t e d a n e l e c t r o n l i f e t i m e i n t h e b a s e o f t h e i r G a A s h o m o j u n c t i o n b i p o l a r t r a n s i s t o r t o b e 1 0 " 1 0 s e c a n d h e l d t h i s r e s p o n s i b l e f o r t h e l o w v a l u e o f h ^ e o f 8 w h i c h t h e y m e a s u r e d i n t h e i r e x p e r i m e n t a l d e v i c e . F r o m F i g u r e 2 . 1 0 , i t c a n b e o b s e r v e d t h a t f o r s u c h a l o w b a s e l i f e t i m e , g a i n s e x c e e d i n g 2 0 c a n n o t b e a t t a i n e d w i t h a b a s e w i d t h o f 0 . 4 a m . F o r a b a s e d o p i n g d e n s i t y o f 1 0 1 7 c m - 3 , t h e l o n g e s t l i f e t i m e e v e r m e a s u r e d i n G a A s i s c l o s e t o 5 X 1 0 " 8 s e c , a s s e e n i n F i g u r e 2 . 5 . E v e n f o r t h i s v a l u e o f l i f e t i m e , t h e g a i n o b t a i n e d i s o n l y i n t h e n e i g h b o u r h o o d o f 1 0 0 . I n v i e w o f t h i s , a f u r t h e r i n c r e a s e i n t r a n s i s t o r g a i n w i l l d e m a n d a n a r r o w e r b a s e w i d t h . F r o m F i g u r e 2 . 1 1 , f o r a b a s e w i d t h v a r i a t i o n o f 0 . 8 urn t o 0 . 2 urn, t h e m a x i m u m g a i n i n c r e a s e s f r o m 15 t o a b o u t 4 0 0 . T h e r e f o r e , a t r a n s i s t o r g a i n o f a r o u n d 5 0 0 w o u l d r e q u i r e a b a s e w i d t h o f l e s s t h a n 0 . 2 urn, w h i c h i s n a r r o w e r t h a n t h a t i n a n y o f t h e d e v i c e s r e p o r t e d s o f a r . F o r a • d e v i c e w i t h g o o d b a s e p r o p e r t i e s , t h e b a s e t r a n s p o r t f a c t o r w i l l b e v e r y c l o s e t o 44 o n e . I n f a c t , i n t h i s c a s e , a c o m b i n a t i o n o f h i g h b a s e l i f e t i m e o f 5 x l 0 - 8 s e c a n d a b a s e w i d t h o f 0 . 2 y i e l d s a b a s e t r a n s p o r t f a c t o r o f 0 . 9 9 9 9 . F o r s u c h a h i g h v a l u e o f b a s e t r a n s p o r t f a c t o r , t h e c o m m o n e m i t t e r g a i n w i l l t h e n b e l i m i t e d b y t h e e m i t t e r i n j e c t i o n e f f i c i e n c y , w h i c h i s g r e a t l y i n f l u e n c e d b y t h e e m i t t e r b a c k - i n j e c t e d h o l e c u r r e n t , J ( - X _ ) . P Hi I f t h e b a s e p r o p e r t i e s c o u l d b e i m p r o v e d s u c h t h a t t h e g a i n i s o n l y l i m i t e d b y t h e e m i t t e r b a c k - i n j e c t e d h o l e c u r r e n t , t h e n t h e e f f e c t o f l i f e t i m e s , w i d t h s , d o p i n g d e n s i t i e s a n d s u r f a c e r e c o m b i n a t i o n i n t h e e m i t t e r w o u l d h a v e t o b e e x a m i n e d . T h e i n f l u e n c e o f e m i t t e r l i f e t i m e o n d c g a i n i s s h o w n i n F i g u r e 2 . 1 2 . F o r l o w c o l l e c t o r c u r r e n t s , t h e e f f e c t o f T e i s t h e s a m e a s t h a t o f T F I, d u e t o t h e d o m i n a n c e o f c a r r i e r r e c o m b i n a t i o n i n t h e e m i t t e r - b a s e j u n c t i o n d e p l e t i o n r e g i o n , a s i n d i c a t e d b y J - - , , i n e q u a t i o n ( 2 . 1 7 ) . A t h i g h e r c u r r e n t s , t h e c h a n g e i n m a x i m u m g a i n o b t a i n e d i s o n l y f r o m 5 0 t o 90 a s t h e l i f e t i m e v a r i e s f r o m a v a l u e o f 1 0 " 1 0 s e c t o 1 0 " 7 s e c . T h i s i n s e n s i t i v i t y o f g a i n t o t h e c h a n g e i n T £ c a n b e a t t r i b u t e d t o t h e l a c k o f d e p e n d e n c e o f t h e e m i t t e r G u m m e l n u m b e r o n T g . F o r i n s t a n c e , a s T £ c h a n g e s f r o m 1 0 ' 1 0 s e c t o 1 0 " 7 s e c , t h e G u m m e l n u m b e r o n l y i n c r e a s e s f r o m 2 . 4 X 1 0 1 2 s e c c m " " t o 4 . 7 4 X 1 0 1 2 s e c c m ' " a t a c o l l e c t o r c u r r e n t d e n s i t y o f 1 x 1 0 ° Amp c m - 2 . I n a d d i t i o n , t h e r e l a t i v e c h a n g e i n e m i t t e r a n d b a s e w i d t h s a l s o b r i n g s a b o u t t h e s a t u r a t i o n o f t h e g a i n i n c r e a s e . A s V _ _ i n c r e a s e s , b o t h W a n d W b e c o m e s l a r g e r , w i t h 4 5 100 Collector Current Density [ A m p / c m 2 ] F i g . 2 . 1 2 T h e e f f e c t o f h o l e l i f e t i m e i n t h e e m i t t e r o n g a i n , a s p r e d i c t e d b y t h e a n l y t i c a l m o d e l u s i n g t h e p a r a m e t e r s l i s t e d i n T a b l e 2 . 3 . 46 t h e i n c r e m e n t a l c h a n g e i n W b e i n g c o m p a r a t i v e l y m o r e t h a n f o r W E d u e t o t h e l a r g e r s h r i n k a g e i n t h e b a s e - c o l l e c t o r d e p l e t i o n r e g i o n . A t a s u r f a c e r e c o m b i n a t i o n v e l o c i t y o f 2 x 1 0 6 c m / s e c , t h e e m i t t e r c o n t a c t i s e s s e n t i a l l y a n o h m i c o n e . T h e i n c r e a s e i n e m i t t e r w i d t h w i l l r e d u c e t h e b a c k - i n j e c t e d h o l e c u r r e n t , a s J p ( X E ) v a r i e s a s c o t h ( W E / L E ) f o r l a r g e Sp. A s s e e n f r o m F i g u r e 2 . 1 3 , t h e g a i n i n c r e a s e s f r o m a b o u t 40 t o 120 f o r a v a r i a t i o n o f W_ f r o m 0 . 1 nm t o 0 . 5 nm. On t h e o t h e r h a n d , t h e l a r g e r b a s e w i d t h a l s o r e d u c e s t h e g a i n d u e t o h i g h e r c a r r i e r r e c o m b i n a t i o n i n t h e n e u t r a l b a s e . T h e c o m b i n e d e f f e c t o f t h e c h a n g e i n t h e w i d t h s o f t h e b a s e a n d e m i t t e r n e u t r a l r e g i o n s i s t o r e d u c e t h e a m o u n t o f i n c r e a s e i n g a i n . T h e e f f e c t o f s u r f a c e r e c o m b i n a t i o n v e l o c i t y o n t h e t r a n s i s t o r p e r f o r m a n c e i s s h o w n i n F i g u r e 2 . 1 4 . F o r t h e m e a s u r e d v a l u e o f s u r f a c e r e c o m b i n a t i o n v e l o c i t y o f 2 x l 0 6 c m / s e c [ 5 7 ] a t t h e e m i t t e r e n d , t h e t r a n s i s t o r b e h a v i o u r i s e f f e c t i v e l y t h e s a m e a s f o r a n o h m i c c o n t a c t w i t h a n e m i t t e r w i d t h o f 0 . 2 5 nm. A s a s i g n i f i c a n t p o r t i o n o f t h e b a c k i n j e c t e d h o l e c u r r e n t i s d u e t o t h e r e c o m b i n a t i o n o f h o l e s a t t h e e m i t t e r s u r f a c e , b o t h a r e d u c t i o n i n s F a n d W E w o u l d g r e a t l y i m p r o v e t h e e m i t t e r i n j e c t i o n e f f i c i e n c y . I n f a c t , f o r a n e m i t t e r w i d t h o f 0 . 2 5 nm, a r e d u c t i o n o f s F t o 100 c m / s e c w o u l d i n c r e a s e t h e e m i t t e r i n j e c t i o n e f f i c i e n c y t o 0 . 9 9 9 8 . F u r t h e r r e d u c t i o n i n W E c o u l d a l s o i m p r o v e t h e i n j e c t i o n e f f i c i e n c y . H o w e v e r , s u c h a l o w v a l u e o f s p h a s n o t y e t b e e n r e a l i s e d i n G a A s 47 1000 F i g . 2 . 1 3 T h e e f f e c t o f e m i t t e r w i d t h o n g a i n , a s p r e d i c t e d b y t h e a n a l y t i c a l m o d e l u s i n g t h e p a r a m e t e r s [ o t h e r t h a n W_ ] l i s t e d i n T a b l e 2 . 3 . 48 F i g . 2 . 1 4 T h e e f f e c t o f s u r f a c e r e c o m b i n a t i o n v e l o c i t y o n g a i n , a s p r e d i c t e d b y t h e a n a l y t i c a l m o d e l u s i n g t h e p a r a m e t e r s [ o t h e r t h a n s „ ] l i s t e d i n T a b l e 2 . 3 . 49 h o m o j u n c t i o n b i p o l a r d e v i c e s . T o a c h i e v e v a l u e s c o m p a r a b l e t o t h e b e s t r e p o r t e d f o r S i d e v i c e s ( 15 c m / s e c f o r p o l y s i l i c o n c o n t a c t s t o s i l i c o n e m i t t e r s [ 6 0 ] ) w o u l d r e q u i r e i m p r o v e d p a s s i v a t i o n o f t h e s u r f a c e , o r t h e u s e o f a h e t e r o j u n c t i o n e m i t t e r . B o t h t h e s e m e a s u r e s w o u l d s e r v e t o s u p p r e s s t h e b a c k i n j e c t e d h o l e c u r r e n t . A n o t h e r w a y o f i m p r o v i n g t h e e m i t t e r i n j e c t i o n e f f i c i e n c y w o u l d b e t o u s e a m o r e h e a v i l y d o p e d e m i t t e r . H o w e v e r , a s t h e e m i t t e r d o p i n g b e c o m e s v e r y h i g h , t h e e f f e c t s o f c a r r i e r d e g e n e r a c y , a n d b a n d g a p n a r r o w i n g d e s c r i b e d b y e q u a t i o n s (2 . 3 8 ) - ( 2 . 4 1 ) , - c a u s e a r e d u c t i o n o f h ^ g . T h e e f f e c t o f e m i t t e r d o p i n g o n g a i n i s s h o w n i n F i g u r e 2 . 1 5 . I n t h e l o w c o l l e c t o r c u r r e n t r a n g e , h e a v y e m i t t e r d o p i n g r e s u l t s i n a d r o p o f h ^ g . T h i s i s b e c a u s e t h e s h r i n k a g e o f t h e e n e r g y b a n d g a p i n t h e e m i t t e r d u e t o h e a v y d o p i n g r e s u l t s - i n a n i n c r e a s e d e f f e c t i v e i n t r i n s i c c a r r i e r c o n c e n t r a t i o n , w h i c h i n t u r n e n h a n c e s t h e s p a c e c h a r g e r e c o m b i n a t i o n c u r r e n t , J R E C « A s t h e b a s e c u r r e n t d e n s i t y i s d o m i n a t e d b y J R E Q i n t h e l o w c u r r e n t r a n g e , a h i g h e r d o p i n g l e v e l i n t h e e m i t t e r w i l l i n c r e a s e J G m u c h m o r e t h a n J C , t h e r e b y r e d u c i n g B. I n t h e h i g h e r c o l l e c t o r c u r r e n t r a n g e , a B o f a b o u t 2 7 0 i s o b t a i n e d f o r ' a n e m i t t e r d o p i n g o f max 1 X 1 0 1 9 c m - 3 a s c o m p a r e d t o a B o f a b o u t 90 f o r max N E = 1 x 1 0 1 8 c m - 3 . T h e h i g h e r e m i t t e r d o p i n g r e d u c e s t h e n u m b e r o f h o l e s p r e s e n t i n t h e e m i t t e r , h e n c e t h e e m i t t e r i n j e c t i o n e f f i c i e n c y i s g r e a t l y i m p r o v e d , d u e t o t h e s m a l l e r b a c k i n j e c t e d h o l e c u r r e n t J ( - X ) , a s w o u l d b e e x p e c t e d . 50 1000-1 Collector Current Density [ Amp/cm2 ] F i g . 2 . 1 5 T h e e f f e c t o f e m i t t e r d o p i n g o n g a i n , a s p r e d i c t e d b y t h e a n a l y t i c a l m o d e l u s i n g t h e p a r a m e t e r s [ o t h e r t h a n N_ ] l i s t e d i n T a b l e 2 . 3 . 51 F o r a h e a v i l y d o p e d e m i t t e r , t h e b a n d g a p n a r r o w i n g s h o u l d c h a n g e t h e e f f e c t i v e e m i t t e r d o p i n g d e n s i t y b y a f a c t o r o f ( n ? / n . 2 ) , w h e r e n . a n d n . a r e t h e d i l u t e a n d e f f e c t i v e 1 i e 1 i e i n t r i n s i c c a r r i e r c o n c e n t r a t i o n s , r e s p e c t i v e l y . H o w e v e r , i n t h e c a s e s u n d e r d i s c u s s i o n , t h i s c o r r e c t i o n i s n o t s i g n i f i c a n t a n d h a s l i t t l e e f f e c t o n t h e i n j e c t i o n e f f i c i e n c y i n t h e h i g h c u r r e n t r a n g e . T h e e f f e c t o f c o l l e c t o r d o p i n g o n 0 i s s h o w n i n F i g u r e 2 . 1 6 . T h e h i g h e r c o l l e c t o r d o p i n g h a s t h e s a m e r e s u l t a s w o u l d b e a c h i e v e d b y e f f e c t i v e l y r e d u c i n g t h e n e u t r a l b a s e w i d t h . A s t h e c o l l e c t o r d o p i n g i s i n c r e a s e d , f o r a g i v e n b a s e - c o l l e c t o r r e v e r s e b i a s , m o s t o f t h e d e p l e t i o n r e g i o n i n t h e b a s e - c o l l e c t o r j u n c t i o n i s e x t e n d e d i n t o t h e b a s e , r e s u l t i n g i n a s m a l l e r n e u t r a l b a s e . F o r i n s t a n c e , a t a r e v e r s e b i a s V B C o f 4 V o l t s , t h e n e u t r a l b a s e w i d t h i s 0 . 3 2 2 um f o r N c = 1 0 1 * c m " 3 , a n d 0 . 0 6 um f o r a c o l l e c t o r d o p i n g o f 8 x 1 0 1 7 c m " 3 . T h e r e f o r e , i t w o u l d b e e x p e c t e d t h a t p'max w i l l b e g r e a t e r f o r h i g h e r c o l l e c t o r d o p i n g d e n s i t i e s . T h i s i s b o r n e o u t b y t h e r e s u l t s s h o w n i n F i g u r e 2 . 1 6 . T h e m o d e l r e s u l t s f o r t h e d e v i c e s s t u d i e d b y T a n a n d M i l n e s [ 5 ] a r e s h o w n i n F i g u r e 2 . 1 7 . T h e i r d e v i c e s w e r e r e p r e s e n t a t i v e o f t r a n s i s t o r s f a b r i c a t e d b y M B E . T h i s s h o u l d l e a d t o u n i f o r m l y d o p e d r e g i o n s w h i c h a r e a p p r o p r i a t e f o r a n a l y s i n g b y t h e p r e s e n t m o d e l . T h e m o d e l p r e d i c t s a m a x i m u m g a i n o f 4 0 0 , w h i c h i s i n g o o d a g r e e m e n t w i t h t h e v a l u e s o f " s o m e h u n d r e d " e s t i m a t e d b y t h e a u t h o r s i n R e f e r e n c e [ 5 ] , T h e s i m u l a t i o n r e s u l t s f o r t h e d e v i c e s o f B a i l b e e t a l . 52 1000-1 N c = 8 x 1 0 " c m Collector Current Density [ A m p / c m 2 ] F i g . 2 . 1 6 T h e e f f e c t o f c o l l e c t o r d o p i n g o n g a i n , a s p r e d i c t e d b y t h e a n a l y t i c a l m o d e l u s i n g t h e p a r a m e t e r s [ o t h e r t h a n N_ ] l i s t e d i n T a b l e 2 . 3 . 53 F i g . 2 . 1 7 T h e g a i n p r e d i c t e d b y t h e a n a l y t i c a l m o d e l u s i n g d a t a f o r d e v i c e s g i v e n b y B a i l b e e t a l . [ 3 5 ] , T a n a n d M i l n e s [ 5 ] a n d N u e s e e t a l . [ 1 2 ] . 54 [ 3 5 ] , p r e p a r e d u s i n g L P E , a n d o f N u e s e e t a l . [ 1 2 ] , p r e p a r e d u s i n g V P E , a r e a l s o s h o w n i n F i g u r e 2 . 1 7 . T h e s e d e v i c e s a l s o s h o u l d p o s s e s s u n i f o r m l y d o p e d s e m i c o n d u c t o r r e g i o n s . T h e m o d e l p r e d i c t s a g a i n o f 2 5 f o r N u e s e ' s d e v i c e s , w h i c h i s i n a c c o r d a n c e w i t h t h e i r m e a s u r e d v a l u e s o f 3 0 - 9 0 . I n t h e c a s e o f B a i l b e ' s d e v i c e s , t h e m e a s u r e d g a i n s a r e i n t h e r a n g e o f 1 2 - 2 5 , w h i c h i s s o m e w h a t h i g h e r t h a n t h e v a l u e s o f 7 - 1 0 p r e d i c t e d b y t h e m o d e l i n F i g u r e 2 . 1 7 . 3 . T H E N U M E R I C A L M O D E L 3 . 1 I N T R O D U C T I O N T h e t r a n s i s t o r m o d e l u s e d i n t h e a n a l y t i c a l a n a l y s i s i n C h a p t e r 2 a s s u m e s u n i f o r m l y - d o p e d e m i t t e r , b a s e a n d c o l l e c t o r r e g i o n s . T h i s m o d e l c a n o n l y b e a p p l i e d t o d e v i c e s b u i l t b y e p i t a x i a l p r o c e s s e s . M o s t o f t h e p o t e n t i a l l y p r a c t i c a l G a A s h o m o j u n c t i o n b i p o l a r t r a n s i s t o r s h a v e b e e n m a d e s o f a r b y s o l i d s t a t e d i f f u s i o n o r i o n - i m p l a n t a t i o n t e c h n i q u e s , w h i c h l e a d t o n o n - u n i f o r m s p a t i a l d o p i n g d e n s i t i e s . T h e r e f o r e t h e a n a l y t i c a l a p p r o a c h w i l l o n l y p r o v i d e a n a p p r o x i m a t e p r e d i c t i o n o f t h e D C c h a r a c t e r i s t i c s o f t h e s e d e v i c e s . I n a d d i t i o n , t h e b o u n d a r y c o n d i t i o n s a t b o t h t h e b a s e e n d s , t h e s o - c a l l e d j u n c t i o n l a w i n t h e r e g i o n a l a n a l y s i s [ 2 2 ] , a r e k n o w n t o b e c o m e i n a c c u r a t e u n d e r m e d i u m o r h i g h i n j e c t i o n c o n d i t i o n s . F o r a t r a n s i s t o r w i t h a h i g h r e s i s t i v i t y c o l l e c t o r , t h e b a s e b o u n d a r y c a n b e p u s h e d b e y o n d t h e m e t a l l u r g i c a l b a s e - c o l l e c t o r j u n c t i o n p o i n t i n t o t h e c o l l e c t o r r e g i o n u n d e r h i g h i n j e c t i o n , r e s u l t i n g i n a d r a s t i c d e c r e a s e o f c u t o f f f r e q u e n c y w i t h i n c r e a s i n g c o l l e c t o r D C c u r r e n t ( t h e K i r k E f f e c t ) . A l t h o u g h o n l y b i p o l a r s o p e r a t i n g u n d e r l o w i n j e c t i o n a r e c o n s i d e r e d i n t h i s w o r k , t h e l i m i t a t i o n s i m p o s e d b y t h e a n a l y t i c a l a p p r o a c h p o i n t t o t h e d e s i r a b i l i t y o f u s i n g n u m e r i c a l m e t h o d s f o r a m o r e a c c u r a t e s i m u l a t i o n o f d e v i c e b e h a v i o u r . T h e f u l l n u m e r i c a l m o d e l i n g o f a s e m i c o n d u c t o r d e v i c e , b a s e d o n t h e f i v e b a s i c p a r t i a l d i f f e r e n t i a l e q u a t i o n s , t h e 5 5 56 c o n t i n u i t y e q u a t i o n s , t h e c u r r e n t t r a n s p o r t e q u a t i o n s a n d P o i s s o n ' s e q u a t i o n w a s f i r s t s u g g e s t e d b y G u m m e l [ 6 1 ] i n 1 9 6 4 f o r c a l c u l a t i n g t h e , D C c h a r a c t e r i s t i c s o f a o n e - d i m e n s i o n a l b i p o l a r t r a n s i s t o r . H i s a p p r o a c h w a s f u r t h e r d e v e l o p e d a n d a p p l i e d t o a p n j u n c t i o n u n d e r b o t h D C a n d t r a n s i e n t c o n d i t i o n s b y D e M a r i [ 6 2 ] [ 6 3 ] , a n d t o I M P A T T d i o d e s b y S c h a f e t t e r a n d G u m m e l [ 6 4 ] . A t w o d i m e n s i o n a l a n a l y s i s o f a b i p o l a r t r a n s i s t o r w a s p r e s e n t e d b y S l o t b o o m i n 1 9 6 9 [ 6 5 ] b y s o l v i n g P o i s s o n ' s e q u a t i o n a n d t h e c o n t i n u i t y e q u a t i o n s . A t a b o u t t h e s a m e t i m e , t w o - d i m e n s i o n a l s o l u t i o n s o f P o i s s o n ' s e q u a t i o n f o r a MOS s t r u c t u r e w e r e c a l c u l a t e d b y L o e b e t a l . [ 6 6 ] a n d S c h r o e d e r a n d M u l l e r [ 6 7 ] , S i n c e t h e n , t w o d i m e n s i o n a l s t e a d y s t a t e a n d t r a n s i e n t m o d e l i n g h a s b e e n w i d e l y a p p l i e d t o v a r i o u s s e m i c o n d u c t o r d e v i c e s s u c h a s J F E T ' s , M O S F E T ' s , M E S F E T ' s a n d t h y r i s t o r s . R e c e n t l y , t h r e e d i m e n s i o n a l s t a t i c m o d e l i n g h a s a l s o b e e n a t t e m p t e d o n s i l i c o n M O S F E T ' s [ 6 8 - 6 9 ] . I n t h i s a n a l y s i s , t h e n u m e r i c a l m o d e l u s e d i s t h e o n e - d i m e n s i o n a l m o d e l S E D A N ( S e m i c o n d u c t o r D e v i c e A n a l y s i s , S t a n f o r d U n i v e r s i t y , J a n u a r y 1 9 8 0 V e r s i o n ) [ 2 1 ] . S E D A N w a s p r i m a r i l y w r i t t e n f o r a p p l i c a t i o n t o s i l i c o n d e v i c e s . I n t h i s w o r k , t h e m o d e l i s a p p l i e d t o G a A s b y m a k i n g t h e a p p r o p r i a t e c h a n g e s t o t h e m o d e l p a r a m e t e r s s u c h a s t h e d o p i n g d e p e n d e n c e o f m o b i l i t y , l i f e t i m e s a n d e n e r g y b a n d g a p s h r i n k a g e p e r t i n e n t t o G a A s . T h e s i m u l a t i o n s f o c u s o n f a c t o r s w h i c h a r e r e l e v a n t t o t h e r e c e n t l y p u b l i s h e d r e s u l t s f o r " i o n - i m p l a n t e d G a A s h o m o j u n c t i o n b i p o l a r t r a n s i s t o r s 5 7 [ 1 4 , 1 5 , 1 7 ] . I n u s i n g i o n - i m p l a n t a t i o n f o r d e v i c e f a b r i c a t i o n , t h e r e a r e s o m e u n c e r t a i n t i e s c o n c e r n i n g t h e a c t u a l d i s t r i b u t i o n s o f t h e c a r r i e r s i n t h e d e v i c e . T h e e l e c t r i c a l a c t i v a t i o n o f t h e i m p l a n t e d s p e c i e s v a r i e s w i t h t h e d o p a n t , t h e s u b s t r a t e m a t e r i a l , t h e i m p l a n t d o s e a n d e n e r g y , t h e e n c a p s u l a n t u s e d t o p r o t e c t t h e G a A s s u r f a c e d u r i n g t h e r m a l a n n e a l i n g a n d t h e i m p l a n t t e m p e r a t u r e . T h e i n d i f f u s i o n o f i m p u r i t i e s d u r i n g h i g h t e m p e r a t u r e a n n e a l i n g a n d t h e t h i c k n e s s o f t h e m a s k i n g l a y e r a l s o a f f e c t t h e f i n a l c a r r i e r p r o f i l e . T h e r e f o r e , a s i n t h e c a s e o f t h e a n a l y t i c a l a p p r o a c h , s o m e r e a s o n a b l e a s s u m p t i o n s p e r t a i n i n g t o t h e s e p r o c e s s - d e p e n d e n t p a r a m e t e r s h a v e t o b e m a d e . A l t h o u g h i t i s d i f f i c u l t t o p e r f o r m a s e n s i t i v i t y a n a l y s i s o n t h e i n f l u e n c e o f e a c h p h y s i c a l p a r a m e t e r o n t h e d e v i c e b e h a v i o u r i n S E D A N , a s t u d y o f t h e D C t r a n s i s t o r g a i n w i t h d i f f e r e n t d e g r e e s o f d o p a n t a c t i v a t i o n , t h e i n f l u e n c e o f t h e r m a l d o p a n t d i f f u s i o n a n d b a n d g a p n a r r o w i n g i s p e r f o r m e d t o e s t a b l i s h t h e i m p o r t a n c e o f t h e s e p a r a m e t e r s o n d e v i c e p e r f o r m a n c e . T h e s i m u l a t i o n r e s u l t s o f t h e i o n - i m p l a n t e d t r a n s i s t o r s [ 1 4 , 1 5 , 1 7 ] a r e a l s o c o m p a r e d t o t h e m e a s u r e d v a l u e s t o s h o w t h e a c c u r a c y o f t h i s m o d e l i n p r e d i c t i n g t r a n s i s t o r s t a t i c c h a r a c t e r i s t i c s . T h e s u c c e e d i n g s e c t i o n s o f t h i s c h a p t e r d e s c r i b e t h e n u m e r i c a l m o d e l a n d i t s r e l a t e d p a r a m e t e r s n e e d e d i n t h e c o m p u t a t i o n s p e r f o r m e d w i t h S E D A N , a n d a l s o p r e s e n t t h e r e s u l t s a n d d i s c u s s i o n o f t h e s e s i m u l a t i o n s . 58 3 . 2 M O D E L D E S C R I P T I O N 3 . 2 . 1 B A S I C E Q U A T I O N S A N D B O U N D A R Y C O N D I T I O N S S E D A N i s b a s e d o n t h e a l g o r i t h m d e v e l o p e d b y S c h a f e t t e r a n d G u m m e l [ 6 4 ] t o s o l v e s e l f - c o n s i s t e n t l y t h e f i v e f u n d a m e n t a l s e m i c o n d u c t o r d i f f e r e n t i a l e q u a t i o n s : P o i s s o n ' s e q u a t i o n , t h e c o n t i n u i t y a n d c u r r e n t e q u a t i o n s f o r e l e c t r o n s a n d h o l e s , f o r t h e s i m u l t a n e o u s s o l u t i o n o f t h e e l e c t r o s t a t i c p o t e n t i a l a n d t h e c a r r i e r c o n c e n t r a t i o n s . U n d e r n o r m a l o p e r a t i n g c o n d i t i o n s , t h e DC c h a r a c t e r i s t i c s o f a b i p o l a r t r a n s i s t o r c a n b e d e t e r m i n e d b y k n o w i n g t h e e l e c t r o s t a t i c p o t e n t i a l a n d t h e c a r r i e r c o n c e n t r a t i o n d i s t r i b u t i o n . T h e s e b a s i c e q u a t i o n s i n t h e i r o n e - d i m e n s i o n a l , s t e a d y s t a t e f o r m a r e a s f o l l o w s : P o i s s o n ' s E q u a t i o n : d E q , d x e n + N - N_) A D ( 3 . 1 ) C o n t i n u i t y E q u a t i o n s : 0 = U. P q ( 3 . 2 ) d J n d x ( 3 . 3 ) 59 A n d t h e c u r r e n t e q u a t i o n s : J p = q M p p E - k T a p g (3.4) J n = q M n n E + k T M n g (3.5) w h e r e n a n d p a r e t h e e l e c t r o n a n d h o l e c o n c e n t r a t i o n s , N ^ , N A t h e d o n o r a n d a c c e p t o r c o n c e n t r a t i o n s , e t h e p e r m i t t i v i t y f o r G a A s , q t h e e l e c t r o n i c c h a r g e a n d E t h e e l e c t r i c f i e l d i n t e n s i t y . J n a n d J p , U n a n d U p , a n d un a n d a r e t h e e l e c t r o n a n d h o l e c u r r e n t d e n s i t y , g e n e r a t i o n - r e c o m b i n a t i o n r a t e , a n d m o b i l i t y r e s p e c t i v e l y . k i s t h e w e l l - k n o w n B o l t z m a n n ' s c o n s t a n t . I t i s a s s u m e d i n S E D A N t h a t b o t h t h e e m i t t e r a n d c o l l e c t o r c o n t a c t s a r e o h m i c a n d p e r f e c t l y c o n d u c t i n g , s o t h e r e i s n o v o l t a g e d r o p a t t h e s e b o u n d a r i e s . T h e e l e c t r o s t a t i c p o t e n t i a l c a n t h e n b e g i v e n a s t h e s u m o f t h e a p p l i e d b i a s a n d t h e b u i l t - i n v o l t a g e a t t h e s e p o i n t s , w h e r e t h e p o t e n t i a l r e f e r e n c e i s t a k e n a t t h e e m i t t e r e n d . T h u s , * ( 0 ) - ^ l n [ S i O l ] (3.6) q n . * ( R ) = V p p + ^ l n [ (3.7) q n . i / / ( 0 ) , iHR) a n d n ( 0 ) , n(R) a r e t h e e l e c t r o s t a t i c p o t e n t i a l a n d e l e c t r o n c o n c e n t r a t i o n a t t h e e m i t t e r a n d c o l l e c t o r e n d s , r e s p e c t i v e l y , n ^ i s t h e i n t r i n s i c c a r r i e r 60 c o n c e n t r a t i o n o f G a A s a n d i s a s s i g n e d a v a l u e o f 2 x 1 0 6 c m " 3 . T o d e t e r m i n e t h e b o u n d a r y c o n d i t i o n s f o r t h e c a r r i e r c o n c e n t r a t i o n s , t h e r m a l e q u i l i b r i u m a n d z e r o s p a c e c h a r g e a r e a s s u m e d t o e x i s t a t t h e c o n t a c t e n d s , t h e r e f o r e : n ( 0 ) p ( 0 ) = n ? ( 3 . 8 a ) n ( R ) p ( R ) = n ? ( 3 . 8 b ) p ( 0 ) - n ( 0 ) + C ( 0 ) = 0 ( 3 . 8 c ) p ( R ) - n ( R ) + C ( R ) = 0 ( 3 . 8 d ) w h e r e C ( x ) = N ( x ) - N A ( x ) ( 3 . 8 e ) E q u a t i o n s ( 3 . 8 a ) - ( 3 . 8 d ) c a n b e r e a r r a n g e d i n t o D i r i c h l e t b o u n d a r y c o n d i t i o n s f o r e l e c t r o n s a n d h o l e s : n ( 0 ) « = ^ { C ( 0 ) 2 + 4 n ? } + C ( 0 ) 2 p ( 0 ) -= ^ { C ( 0 ) 2 + 4 n ? } - C ( 0 ) 2 n ( R ) = = ^ { C ( R ) 2 + 4 n ? } + C ( R ) ( 3 . 9 a ) ( 3 . 9 b ) ( 3 . 9 c ) 61 p ( R ) - « C ( R ) ' * 4 " i 1 - C ( R ) (3.9d) w h e r e p ( 0 ) a n d p ( R ) a r e t h e e q u i l i b r i u m h o l e c o n c e n t r a t i o n s a t t h e e m i t t e r a n d c o l l e c t o r c o n t a c t s r e s p e c t i v e l y . 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 p r o c e s s i s a s s u m e d t o b e d o m i n a t e d b y t h e S h o c k l e y - R e a d - H a l l t y p e o f m e c h a n i s m f o r s i n g l e e n e r g y l e v e l r e c o m b i n a t i o n - g e n e r a t i o n c e n t e r s , s o U a n d U a r e e x p r e s s e d a s : P p n - n ? U = U = : : : r ( 3 . 1 0 ) P r n o ( n + n , ) + r p o ( p + ? 1 ) w h e r e p 1 , n 1 a r e t h e h o l e a n d e l e c t r o n c o n c e n t r a t i o n s i n t h e c o n d u c t i o n b a n d w h e n t h e F e r m i l e v e l c o i n c i d e s w i t h t h e e n e r g y l e v e l o f 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 c e n t e r , a n d T „ ~ * T a r e t h e h o l e a n d e l e c t r o n l i f e t i m e s i n n - t y p e , n o p o r p - t y p e m a t e r i a l r e s p e c t i v e l y . 3 . 2 . 2 T H E P H Y S I C A L P A R A M E T E R S T h e p h y s i c a l p a r a m e t e r s a r e f o r m u l a t e d f o l l o w i n g a s i m i l a r a p p r o a c h t o t h a t u s e d i n t h e a n a l y t i c a l m o d e l . T h e d o p i n g d e p e n d e n c e o f t h e m i n o r i t y c a r r i e r l i f e t i m e s r p o a n d r n o i s d e s c r i b e d b y e q u a t i o n ( 2 . 5 1 ) , w i t h t h e a p p r o p r i a t e p o l y n o m i a l c o e f f i c i e n t s a n d d e p e n d e n t v a r i a b l e s g i v e n i n T a b l e 2 . 2 . T h e f i e l d - d e p e n d e n t m o b i l i t y i s e x p r e s s e d b y t h e e m p i r i c a l r e l a t i o n [ 7 0 ] : 62 * ( 3 . 1 1 ) W h e r e n i s t h e l o w f i e l d m o b i l i t y , a n d i t s d o p i n g d e p e n d e n c e i s d e r i v e d i n e q u a t i o n s ( 2 . 4 7 - 2 . 4 8 ) f o r e l e c t r o n s a n d ( 2 . 4 7 - 2 . 5 0 ) f o r h o l e s , w i t h t h e p o l y n o m i a l c o e f f i c i e n t s g i v e n i n T a b l e 2 . 1 . E i s t h e e l e c t r i c f i e l d a n d t h e s a t u r a t i o n v e l o c i t y v i s t a k e n a s I . O x l O 7 c m / s e c f o r b o t h s e l e c t r o n s a n d h o l e s [ 7 1 ] , T h e e f f e c t o f h e a v y d o p i n g o n t h e e n e r g y b a n d g a p n a r r o w i n g i s r e l a t e d t o t h e e f f e c t i v e i n t r i n s i c c a r r i e r c o n c e n t r a t i o n n j e ' w h i c h i s d e s c r i b e d i n e q u a t i o n s ( 2 . 3 8 - 2 . 4 1 ) . T h e a c t i v a t i o n e n e r g y o f t h e i m p u r i t y d o p a n t s i s a s s u m e d t o b e z e r o t o b e c o n s i s t e n t w i t h t h e s e t u p o f S E D A N a s u s e d i n s i l i c o n d e v i c e m o d e l i n g . 3 . 2 . 3 I O N I M P L A N T A T I O N P A R A M E T E R S I o n i m p l a n t a t i o n i s a d o p i n g t e c h n i q u e w h i c h i s w i d e l y u s e d i n i n t e g r a t e d c i r c u i t d e v i c e f a b r i c a t i o n . I t o f f e r s a p r e c i s e c o n t r o l o v e r t h e i m p u r i t y d e n s i t y a n d d e p t h p r o f i l e . A s s u c h , n a r r o w b a s e w i d t h s i n b i p o l a r t r a n s i s t o r s c a n b e a c h i e v e d w i t h t h i s m e t h o d . B y i m p l a n t i n g a d o p a n t i n t o G a A s , t h e l a t t i c e d a m a g e p r o d u c e d b y t h e h i g h e n e r g y i o n s r e s u l t s i n a d e g r a d a t i o n o f t h e m i n o r i t y c a r r i e r l i f e t i m e , d u e t o e x c e s s r e c o m b i n a t i o n i n t h e m a t e r i a l . I t h a s b e e n s h o w n t h a t b y u t i l i s i n g a s u i t a b l e a n n e a l i n g s c h e m e , t h e s e l a t t i c e 63 d i s o r d e r s c a n b e r e m o v e d [ 7 2 ] , h e n c e r e c o v e r i n g t h e l i f e t i m e o f t h e i m p l a n t e d l a y e r . A s d i s c u s s e d i n t h e a n a l y t i c a l m o d e l i n C h a p t e r 2 , n a r r o w b a s e w i d t h s a n d l o n g b a s e l i f e t i m e s a r e t h e p a r a m e t e r s r e q u i r e d t o a c h i e v e s u p e r i o r D C p e r f o r m a n c e i n G a A s b i p o l a r s . T h e r e f o r e , i o n - i m p l a n t a t i o n s h o u l d b e a v e r y h e l p f u l t e c h n i q u e i n t h e r e a l i s a t i o n o f h i g h g a i n d e v i c e s . T h e d o p i n g p r o f i l e d u e t o i o n - i m p l a n t a t i o n i s a p p r o x i m a t e d b y a o n e - d i m e n s i o n a l s y m m e t r i c G a u s s i a n d i s t r i b u t i o n f u n c t i o n a c c o r d i n g t o t h e t h e o r y o f L i n d h a r d , S c h a r f f , a n d S c h i ^ t t ( L S S ) [ 7 3 ] : Q ( x - R ) 2 N ( X ) - •(2 ,> . a n p 6 X p t - T S f 1 - M A ( 3 . 1 2 ) w h e r e Q 0 i s t h e i m p l a n t d o s e , N A t h e b a c k g r o u n d i m p u r i t y c o n c e n t r a t i o n , R p t h e m e a n v a l u e o r p r o j e c t e d r a n g e a n d A R p t h e s t a n d a r d d e v i a t i o n o r s t r a g g l e o f t h e n o r m a l d i s t r i b u t i o n . T h e L S S t h e o r y i s a f i r s t o r d e r a p p r o x i m a t i o n t o t h e a c t u a l d o p i n g p r o f i l e , b u t i s c o n s i d e r e d a d e q u a t e i n o u r a p p l i c a t i o n . H i g h e r o r d e r e f f e c t s s u c h a s t h e e x p o n e n t i a l d e c a y i n g t a i l s o b s e r v e d i n s o m e a c t u a l i m p l a n t p r o f i l e s d o n o t a f f e c t t h e o v e r a l l d i s t r i b u t i o n , a n d h e n c e t h e d e v i c e p e r f o r m a n c e . T o c a l c u l a t e t h e d o p i n g p r o f i l e u s i n g e q u a t i o n ( 3 . 1 2 ) , i t i s n e c e s s a r y t o h a v e k n o w l e d g e o f R p a n d A R p u n d e r g i v e n i m p l a n t c o n d i t i o n s . T a b l e s o f p r o j e c t e d r a n g e a n d s t a n d a r d d e v i a t i o n s c o m p u t e d b y G i b b o n s e t a l . [ 7 4 ] a n d R y s s e l a n d 64 R u g e [ 7 5 ] w e r e u s e d f o r s i l i c o n , s e l e n i u m , a n d b e r y l l i u m i m p l a n t s i n t o G a A s . A s t h e t a b l e s a r e i n c o n v e n i e n t t o u s e a n d d o n o t p r o v i d e a l l t h e r e q u i r e d R p ' s a n d A R p ' s u n d e r t h e i m p l a n t a t i o n s c h e m e o u t l i n e d i n T a b l e 3 . 1 , a f u n c t i o n a l f i t t o t h e t a b u l a t e d d a t a h a s b e e n p e r f o r m e d . T h e f i t t e d c u r v e s a r e l o w o r d e r , s i m p l e l e a s t s q u a r e p o l y n o m i a l s , a s s u g g e s t e d b y S e l b e r h e r r e t a l . [ 7 6 ] , a n d h a v e t h e f o r m s : n R D = I a . . E 1 ( 3 . 1 3 ) F i = 0 1 n A R D = I b . , E . ( 3 . 1 4 ) v i = 0 1 1 w h e r e a ^ , b^ a r e t h e c o e f f i c i e n t s o f t h e f i t t e d c u r v e s i n m i c r o m e t e r s a n d E i s t h e i m p l a n t e n e r g y i n k e V . T h e s e c o e f f i c i e n t s a r e l i s t e d i n T a b l e 3 . 2 a n d T a b l e 3 . 3 f o r t h e v a r i o u s e l e m e n t s . T h e n o n - v a n i s h i n g a 0 a n d b 0 a r e n e c e s s a r y i n t h e c o n s t r u c t i o n o f t h e p o l y n o m i a l f u n c t i o n s t o m i n i m i z e t h e r o o t - m e a n s q u a r e e r r o r s . T h e m a x i m u m e r r o r a p p r o x i m a t e d b y e q u a t i o n s ( 3 . 1 3 ) a n d ( 3 . 1 4 ) i s l e s s t h a n 2% w h e n c o m p a r e d t o t h e t a b u l a t e d v a l u e s i n [ 7 4 ] , [ 7 5 ] f o r a n i m p l a n t e n e r g y r a n g e o f 2 0 - 5 0 0 k e V . T h e f i t t e d c u r v e s o f R p , A R p v e r s u s i m p l a n t e n e r g y a r e s h o w n i n F i g u r e 3 . 1 a n d F i g u r e 3 . 2 . T h e h i g h t e m p e r a t u r e a n n e a l i n g s t e p s f o l l o w i n g i m p l a n t a t i o n r e s u l t i n d i f f u s i o n o f d o p a n t s i n t o t h e s u b s t r a t e . I t h a s b e e n s h o w n t h a t f o r l o w - d o s e B e , S i a n d S e i m p l a n t s ( ^ 1 X 1 0 1 0 c m ' 2 ) t h e r e i s v e r y l i m i t e d d i f f u s i o n a n d t h e c a r r i e r p r o f i l e a g r e e s w i t h t h e G a u s s i a n 6 5 R e f I o n C a p p i n g S u b s t . E n e r g y D o s e A n n e a l t e m p , ( d e p t h ) t e m p ( ° C ) ( k e V ) ( c n r 2 ) t i m e ( ° C , m i n ) [ 1 4 ] B e S i 3 N « ( 4 0 0 A ) S i 3 N f l ( 4 0 0 A ) S i 3 N „ ( 400A ) R T R T R T 2 5 0 3 X 1 0 1 2 8 5 0 , 3 0 130 2 X 1 0 1 2 8 5 0 , 3 0 90 9 X 1 0 1 1 8 5 0 , 3 0 S i S i 3 N „ ( 4 0 0 A ) R T 150 2 x 1 0 1 t t 8 5 0 , 3 0 [ 1 5 ] S e n o n e 3 5 0 3 6 0 5 X 1 0 1 3 8 5 0 , 3 0 B e n o n e R T 1 2 5 6 x 1 0 1 2 7 0 0 , 3 0 [ 1 7 ] S e S i 3 N „ S i 3 N « , 3 5 0 3 5 0 150 1 X 1 0 1 3 3 6 0 2 X 1 0 1 3 8 5 0 , 3 0 8 5 0 , 3 0 B e S i 3 N , R T 180 6 X 1 0 1 2 8 0 0 , 3 0 T a b l e 3 . 1 I m p l a n t a t i o n s c h e d u l e u s e d i n R e f s . [ 1 4 , 1 5 , 1 7 ] f o r f a b r i c a t i n g G a A s n - p - n b i p o l a r t r a n s i s t o r s . 66 D o p a n t S i l i c o n S e l e n i u m B e r y l l i u m a 0 2.76X10- 3 3 . 0 0 x 1 0 - 3 - 7 . 4 4 x 1 0 - 3 a i 7 . 5 5 x 1 0 - " 3 . 8 0 x 1 0 - " 3 . 3 1 x 1 0 " " a 2 8 . 9 9 1 X 1 0 "7 - 5 . 9 7 4 X 1 0 " 7 1 . 7 8 2 x 1 0 - 7 a 3 - 2 . 5 6 X 1 0 "9 2 . 6 0 x 1 0 - 9 - 1 . 1 4 X 1 0 - 8 a« 3 . 2 4 x 1 0 " 1 2 - 4 . 7 8 x 1 0 " 1 2 2 . 5 6 x 1 0 " 1 1 a 5 - 1 . 6 6 x 1 0 "1 5 3 . 2 6 x 1 0 " 1 5 - 1 . 9 1 x 1 0 " 1 " T a b l e 3 . 2 C o e f f i c i e n t s f o r R i n e q u a t i o n P ( 3 . 1 3 ) . D o p a n t S i l i c o n S e l e n i u m B e r y l l i u m b 0 2.84X10- 3 1 . 5 9 X 1 0 - 3 1 . 0 0 x 1 0 " 2 b , 4 . 9 5 x 1 0 - * 1 . 9 6 x 1 0 - " 2 . 0 8 x 1 0 - 3 b 2 - 1 . 0 0 5 x 1 0 " 6 - 4 . 6 2 3 X 1 0 ' 7 -1 . 0 4 x l 0 - 5 b 3 2 . 3 2 X 1 0 -9 1 . 6 0 x 1 0 " 9 3 . 2 5 X 1 0 - 8 b « - 3 . 5 4 x 1 0 " 1 2 - 2 . 8 2 x 1 0 " 1 2 - 5 . 2 8 x 1 0 " 1 1 b 5 2 . 2 7 x 1 0 - 1 5 1 . 9 0 x 1 0 " 1 5 3 . 3 9 x 1 0 - 1 " T a b l e 3 . 3 C o e f f i c i e n t s f o r A R p i n e q u a t i o n ( 3 . 1 4 ) . 67 1.4 Implant Energy [keV] F i g . 3 . 1 P r o j e c t e d r a n g e v e r s u s i m p l a n t e n e r g y f o r s e l e n i u m , s i l i c o n a n d b e r y l l i u m i n G a A s . F i g . 3 . 2 S t a n d a r d d e v i a t i o n v e r s u s i m p l a n t e n e r g y f o r s e l e n i u m , s i l i c o n a n d b e r y l l i u m i n G a A s . 69 d i s t r i b u t i o n w h e n a d j u s t e d w i t h a n a p p r o p r i a t e a c t i v a t i o n e f f i c i e n c y [ 7 7 ] . F o r h i g h - d o s e i m p l a n t s a n d a n n e a l i n g t e m p e r a t u r e s g r e a t e r t h a n 8 0 0 ° C , c o n s i d e r a b l e d i f f u s i o n t a k e s p l a c e , a n d a b r o a d e r n o r m a l d i s t r i b u t i o n i s o b t a i n e d [ 7 7 ] , T o a c c o u n t f o r t h e d i f f u s i o n e f f e c t s , t h e s t a n d a r d d e v i a t i o n i n e q u a t i o n ( 3 . 1 2 ) i s m o d i f i e d t o a n e f f e c t i v e f o r m [ 7 8 ] : n A R p e / / = ,/{ 2.1 D i . t i + A R p } ( 3 . 1 5 ) w h e r e a n d t^ a r e , r e s p e c t i v e l y , t h e a p p r o p r i a t e d i f f u s i o n c o n s t a n t a n d a n n e a l t i m e f o r t h e i t h a n n e a l s t e p . F r o m T a b l e 3 . 1 , o n l y t h e d e v i c e s f r o m H u g h e s [ 1 4 ] e m p l o y h i g h - d o s e s i l i c o n i m p l a n t s f o r t h e n + e m i t t e r , t h e r e f o r e e q u a t i o n ( 3 . 1 5 ) i s n e e d e d t o c a l c u l a t e t h e d o p i n g p r o f i l e o f t h e s e d e v i c e s . T h e d i f f u s i o n c o n s t a n t o f s i l i c o n a t 8 5 0 ° C i s t a k e n a s 3 . 3 x l 0 1 " c m 2 s e c " 1 a s d e t e r m i n e d f r o m t h e a n a l y s i s o f t h e c a r r i e r c o n c e n t r a t i o n p r o f i l e f o r a S i - i m p l a n t e d L P E b u f f e r s u b s t r a t e [ 7 9 ] . T h e c a p p i n g m a t e r i a l u s e d d u r i n g i m p l a n t a t i o n f o r t h e p r o t e c t i o n o f t h e G a A s s u r f a c e , s u c h a s t h e S i 3 ^ l a y e r u s e d i n [ 1 4 ] , c a n a f f e c t t h e p r o f i l e a s w e l l . T h i s e f f e c t c a n b e i n c l u d e d b y i n t r o d u c i n g a t r u n c a t e d G a u s s i a n d i s t r i b u t i o n , i n w h i c h R_ i s c h a n g e d t o [ 8 0 ] : RPeff = R P " tcap ( 3 . 1 6 ) 70 w h e r e t c a p i s t h e t h i c k n e s s o f t h e c a p p i n g l a y e r . F o r i o n - i m p l a n t e d m a t e r i a l , t h e a c t u a l a m o u n t o f d o p a n t t h a t g o e s i n t o s u b s t i t u t i o n a l s i t e s t o b e c o m e e l e c t r i c a l l y a c t i v e d u r i n g a n n e a l i n g d e p e n d s o n a n u m b e r o f f a c t o r s s u c h a s t h e a n n e a l t e m p e r a t u r e a n d d o s e u s e d . T h e r a t i o o f s h e e t c a r r i e r c o n c e n t r a t i o n o f a n a n n e a l e d s a m p l e a n d t h e i m p l a n t f l u e n c e u s e d i s t h e a c t i v a t i o n e f f i c i e n c y a n d i s g i v e n a s [ 8 1 ] : N V = ( 3 . 1 7 ) vie-w h e r e N g i s t h e s h e e t c a r r i e r c o n c e n t r a t i o n o f t h e s a m p l e . T h e d o p i n g f u n c t i o n u s i n g L S S t h e o r y , t a k i n g i n t o c o n s i d e r a t i o n t h e e f f e c t s o f d o p a n t d i f f u s i o n , c a p p i n g a n d c a r r i e r a c t i v a t i o n i s g i v e n , f o l l o w i n g e q u a t i o n ( 3 . 1 2 ) , a s : Qo ( x " R P e / / ) 2 N ( X ) = ^ ' A D e X P [ 7 ^ 5 3 / ( 2 7 r ) . A R p e / / 2ARPeff - N , ( 3 . 1 8 ) A s s u m i n g z e r o a c t i v a t i o n e n e r g y f o r t h e i m p l a n t e d d o p a n t s , t h e c a r r i e r p r o f i l e i s t h e n d e s c r i b e d b y e q u a t i o n ( 3 . 1 8 ) . 3 . 3 S O L U T I O N P R O C E D U R E U S I N G S E D A N T o p e r f o r m a b i p o l a r t r a n s i s t o r s i m u l a t i o n u s i n g S E D A N , a d a t a f i l e c o n t a i n i n g t h e n e c e s s a r y i n p u t i n f o r m a t i o n h a s t o b e p r o v i d e d . T h e s e i n p u t s p e c i f i c a t i o n s i n c l u d e : 71 ( 1 ) T h e d i v i d i n g o f t h e o n e - d i m e n s i o n a l t r a n s i s t o r s t r u c t u r e i n t o v a r i o u s r e g i o n s , w h e r e e a c h r e g i o n i s a s s i g n e d a u n i q u e g r i d s p a c i n g . I n t h i s c a s e , t h e r e g i o n s a r e t h e n e u t r a l w i d t h s o f t h e e m i t t e r , b a s e , c o l l e c t o r a n d t h e t w o j u n c t i o n d e p l e t i o n l a y e r s . T o s e l e c t t h e g r i d p o i n t s f o r e a c h r e g i o n , a t r i a l a n d e r r o r m e t h o d i s e m p l o y e d u n t i l a c o n v e r g e n c e o f t h e s o l u t i o n i s o b t a i n e d . I n g e n e r a l , f i n e g r i d s p a c i n g s o f a b o u t 0 . 0 1 nm t o 0 . 0 0 5 /nm a r e n e e d e d f o r v e r y s t e e p p r o f i l e s , a n d g r i d s p a c i n g s o f 0 .1 Atm a r e s u f f i c i e n t f o r c o n s t a n t o r s l o w l y v a r y i n g p r o f i l e s . F o r d e v i c e s w i t h v e r y n a r r o w b a s e s , t h e r e m u s t b e e n o u g h g r i d p o i n t s i n t h e b a s e t o e n s u r e t h e c o n v e r g e n c e a n d a c c u r a c y o f t h e s o l u t i o n . ( 2 ) T h e s e l e c t i o n o f a r e f e r e n c e p o i n t f o r t h e b a s e c o n t a c t . T h e m i d - p o i n t o f t h e m e t a l l u r g i c a l b a s e i s n o r m a l l y u s e d . H o w e v e r , i n c h o o s i n g t h e b a s e c o n t a c t p o i n t , c a r e m u s t b e t a k e n t o e n s u r e t h a t t h i s l o c a t i o n d o e s n o t e x t e n d i n t o t h e b a s e - e m i t t e r o r b a s e - c o l l e c t o r d e p l e t i o n r e g i o n s f o r a g i v e n b i a s , o r t h e s o l u t i o n w i l l f a i l t o c o n v e r g e . ( 3 ) T h e r a n g e , s t r a g g l e a n d p e a k v a l u e f o r t h e G a u s s i a n p r o f i l e f o r e a c h i m p l a n t e d d o p a n t . T h e r a n g e a n d s t r a g g l e d a t a a r e c o m p u t e d u s i n g e q u a t i o n s ( 3 . 1 3 ) - ( 3 . 1 4 ) w i t h t h e a p p r o p r i a t e p o l y n o m i a l c o e f f i c i e n t s f r o m T a b l e s 3 . 1 a n d 3 . 2 . T h e p e a k c a r r i e r c o n c e n t r a t i o n v a l u e i s o b t a i n e d f r o m e q u a t i o n ( 3 . 1 8 ) b y s e t t i n g x = R p e y y . ( 4 ) T h e b i a s v o l t a g e V „ „ , a n d i t s i n c r e m e n t a l v a l u e f o r a 72 f i x e d e m i t t e r - c o l l e c t o r v o l t a g e , V C E « T h e i n i t i a l a n d f i n a l v a l u e s o f V " B E a r e t a k e n t o b e 0 . 6 V a n d 1 . 3 0 V , r e s p e c t i v e l y , a n d V " C E i s s e t e q u a l t o 4 . 0 V . ( 5 ) T h e d e s i r e d o u t p u t s , s u c h a s p l o t s o f /3 v e r s u s J c « O n p r o v i d i n g t h e n e c e s s a r y i n f o r m a t i o n , S E D A N w i l l t h e n g e n e r a t e t h e r e q u i r e d o u t p u t s . 3 . 4 R E S U L T S A N D D I S C U S S I O N I n t h i s n u m e r i c a l a n a l y s i s , t h e s i m u l a t i o n o f t h e p e r f o r m a n c e o f t w o d e v i c e s t r u c t u r e s b u i l t b y d i f f e r e n t f a b r i c a t i o n p r o c e d u r e s i s i n v e s t i g a t e d . I n t h e H u g h e s p r o c e s s [ 1 4 ] , e v a p o r a t e d , h e a v y m e t a l m a s k s w e r e u s e d t o e n a b l e s e l e c t i v e i m p l a n t s i n t o a n n - t y p e G a A s e p i t a x i a l l a y e r g r o w n o n a n n + s u b s t r a t e . A m u l t i p l e B e i m p l a n t w a s u s e d f o r t h e b a s e f o r m a t i o n a n d a h i g h d o s e s i l i c o n i m p l a n t f o r t h e e m i t t e r . T h e a n n e a l i n g f o r b o t h i m p l a n t e d s p e c i e s w a s c a r r i e d o u t s i m u l t a n e o u s l y a t 8 5 0 ° C f o r 30 m i n u t e s . T h e i m p l a n t a t i o n p a r a m e t e r s a n d c o n d i t i o n s a r e g i v e n i n T a b l e 3 . 1 . F o r G a A s i m p l a n t e d w i t h b e r y l l i u m , f u l l e l e c t r i c a l a c t i v a t i o n o f t h e d o p a n t h a s b e e n o b t a i n e d a t a n a n n e a l i n g t e m p e r a t u r e a s l o w a s 6 0 0 ° C [ 8 2 ] . T h e r e f o r e t h e a c t i v a t i o n o f t h e b e r y l l i u m i s t a k e n t o b e 1 0 0 % . F o r t h e s i l i c o n i m p l a n t d o s e o f 2 x l 0 1 " c m - 2 , t h e v a l u e o f i t s a c t i v a t i o n e f f i c i e n c y h a s b e e n d e t e r m i n e d t o v a r y f r o m 3 . 2 % t o 30% [ 7 9 ] , a n d c o n s i d e r a b l e d i f f u s i o n o f c a r r i e r s o c c u r s a t t h e a n n e a l i n g t e m p e r a t u r e o f 8 5 0 ° C [ 7 7 ] . T h e e f f e c t o f t h e s i l i c o n a c t i v a t i o n a n d i n d i f f u s i o n , u s i n g a d i f f u s i o n 73 c o n s t a n t o f 3 . 3 x 1 0 1 B c m 2 s e c " 1 [ 7 9 ] , o n t h e d o p i n g p r o f i l e f o r t h e H u g h e s f a b r i c a t i o n r o u t i n e i s s h o w n i n F i g u r e 3 . 3 . T h e v a r i a t i o n s i n a c t i v a t i o n e f f i c i e n c y a n d d i f f u s i o n c a u s e a c h a n g e i n t h e e f f e c t i v e w i d t h s o f t h e b a s e a n d e m i t t e r r e g i o n s . F r o m F i g u r e 3 . 3 , i t c a n b e s e e n t h a t f o r a s i l i c o n a c t i v a t i o n e f f i c i e n c y o f 15%, t h e c a r r i e r i n d i f f u s i o n r e s u l t s i n a c o n s i d e r a b l e n a r r o w i n g o f t h e b a s e w i d t h o f t h e d e v i c e . T h e r e f o r e , a d e v i c e t h a t h a s a c a r r i e r p r o f i l e m o d i f i e d b y t h e d i f f u s i o n e f f e c t w o u l d b e e x p e c t e d t o e x h i b i t h i g h e r c u r r e n t g a i n . T h e r e s u l t s p r e d i c t e d b y S E D A N f o r t h e H u g h e s d e v i c e s t r u c t u r e a r e s h o w n i n F i g u r e 3 . 4 . F r o m t h e f i g u r e , t h e m a x i m u m g a i n o b t a i n e d v a r i e d o v e r t h e r a n g e o f 8 - 3 0 , d e p e n d i n g o n t h e v a l u e o f t h e a c t i v a t i o n e f f i c i e n c y a n d w h e t h e r o r n o t t h e i n d i f f u s i o n e f f e c t w a s i n c l u d e d . T h i s i s i n g o o d a g r e e m e n t w i t h t h e m e a s u r e d v a l u e s o f 7 - 2 5 f o r t h e H u g h e s e x p e r i m e n t a l d e v i c e s [ 1 4 ] , T h e e f f e c t o f b a n d g a p n a r r o w i n g o n t h e d e v i c e p e r f o r m a n c e i s a l s o s h o w n i n F i g u r e 3 . 4 . T h e e f f e c t i s s u b s t a n t i a l a n d , f o r t h e e x a m p l e s h o w n o f a s i l i c o n a c t i v a t i o n o f 1 5 % , b a n d g a p n a r r o w i n g s e r v e s t o r e d u c e t h e p r e d i c t e d P m a x v a l u e f r o m 8 0 t o 1 4 . I t i s s p e c u l a t e d i n [ 1 4 ] t h a t t h e l o w m e a s u r e d v a l u e s o f c u r r e n t g a i n f o r t h e s e e x p e r i m e n t a l d e v i c e s a r e d u e t o t h e e f f e c t o f " s u r f a c e l e a k a g e " . H o w e v e r , t h e c l o s e a g r e e m e n t b e t w e e n t h e m o d e l r e s u l t s p r e s e n t e d h e r e a n d t h e m e a s u r e d d a t a s u g g e s t t h a t t h e l o w g a i n s a t t a i n e d a r e , i n f a c t , d u e t o i n t r i n s i c p h e n o m e n a , s u c h a s d e v i c e g e o m e t r y , d o p i n g d e n s i t i e s a n d ' m a t e r i a l p r o p e r t i e s , w i t h b a n d g a p 74 Depth [ microns ] F i g . 3 . 3 T h e c o m p u t e d d o p i n g p r o f i l e f o r t h e H u g h e s d e v i c e s t r u c t u r e [ 1 4 ] , s h o w i n g t h e e f f e c t o f a c t i v a t i o n e f f i c i e n c y a n d i n - d i f f u s i o n f o r t h e i m p l a n t e d S i s p e c i e s . 7 5 100-1 c o "c i_ 3 O o Q 10- \ 30%_Si_AcU_BGN 15% S^Act^, BGN 15% Si Act., no BGN 15% Si Act., BGN & indiff. 3.2% Si Act., BGN I I M l | 10' 10 2 10 J 10 4 10= Collector Current Density [ Amp/cm2 ] F i g . 3 . 4 T h e e f f e c t o f s i l i c o n a c t i v a t i o n e f f i c i e n c y , s i l i c o n i n - d i f f u s i o n a n d b a n d g a p n a r r o w i n g ( B G N ) o n g a i n f o r t h e H u g h e s d e v i c e s t r u c t u r e [ 1 4 ] , a s p r e d i c t e d b y t h e n u m e r i c a l m o d e l . 76 n a r r o w i n g b e i n g p a r t i c u l a r l y i m p o r t a n t . I n t h e T e x a s I n s t r u m e n t f a b r i c a t i o n s e q u e n c e , u n m a s k e d i m p l a n t a t i o n s a r e u s e d f o r t h e b a s e a n d e m i t t e r r e g i o n s , a n d b o r o n i m p l a n t a t i o n i s u s e d t o a c h i e v e d e v i c e i s o l a t i o n . A s i n g l e S e i m p l a n t w a s u s e d f o r t h e e m i t t e r f o r m a t i o n i n t h e d e v i c e d e s c r i b e d i n [ 1 5 ] , w h i l e a d o u b l e S e i m p l a n t w a s e m p l o y e d i n t h e d e v i c e s c o n s i d e r e d i n [ 1 7 ] . F o r b o t h t r a n s i s t o r s , a s i n g l e B e i m p l a n t w a s u s e d f o r t h e b a s e r e g i o n . S e p a r a t e b a s e a n d e m i t t e r a n n e a l i n g w a s p e r f o r m e d a n d t h e i m p l a n t a t i o n p a r a m e t e r s a r e g i v e n i n T a b l e 3 . 1 . T h e c a l c u l a t e d d o p i n g p r o f i l e s f o r b o t h d e v i c e s t r u c t u r e s a r e s h o w n i n F i g u r e 3 . 5 . T h e d o p i n g p r o f i l e f o r t h e d o u b l e S e i m p l a n t d e v i c e [ 1 7 ] h a s b e e n c o m p u t e d b y a s s u m i n g a 70% a c t i v a t i o n o f t h e i m p l a n t e d s e l e n i u m a n d 100% a c t i v a t i o n o f t h e i m p l a n t e d b e r y l l i u m . T h i s p r o v i d e s g o o d a g r e e m e n t w i t h t h e p e a k c a r r i e r d o p i n g c o n c e n t r a t i o n , a n d t h e e m i t t e r a n d b a s e w i d t h s r e s u l t i n g f r o m t h e p r o f i l e c a l c u l a t e d b y D o e r b e c k e t a l . [ 1 7 ] , w h o u s e d t h e L S S G a u s s i a n d i s t r i b u t i o n , s u i t a b l y m o d i f i e d b y e x p e r i m e n t a l l y o b s e r v e d , b u t u n s p e c i f i e d , a c t i v a t i o n e f f i c i e n c i e s . A n a t t e m p t w a s m a d e t o m o d e l t h e r e d i s t r i b u t i o n o f t h e i m p u r i t i e s f r o m t h e n * s u b s t r a t e d u e t o o u t - d i f f u s i o n i n t o t h e u n d o p e d e p i t a x i a l l a y e r o f t h e d e v i c e i n [ 1 7 ] , T h e m e a s u r e d p r o f i l e a f t e r r e - d i s t r i b u t i o n w a s d e d u c e d f r o m C - V m e a s u r e m e n t s [ 1 7 ] . A n e r r o r f u n c t i o n p r o f i l e , a s u s e d e l s e w h e r e f o r t h e i m p u r i t y r e d i s t r i b u t i o n f r o m a b u r i e d l a y e r [ 8 3 ] , w a s e m p l o y e d , n a m e l y : 77 Texos Insf. [15] Jexas Inst. [)7J 0.1 0.2 0.3 0.4 0.5 0.6 Depth [ microns ] 0.7 0.8 0.9 F i g . 3 . 5 T h e c o m p u t e d d o p i n g p r o f i l e f o r t h e T e x a s I n s t r u m e n t d e v i c e s t r u c t u r e s o f R e f s . [ 1 5 , 1 7 ] , a s s u m i n g 70% a c t i v a t i o n o f t h e i m p l a n t e d s e l e n i u m a n d 100% a c t i v a t i o n o f t h e i m p l a n t e d b e r y l l i u m . T h e r e d i s t r i b u t i o n o f i m p u r i t i e s i n t h e e p i t a x i a l c o l l e c t o r o f t h e d e v i c e [ 1 7 ] i s r e p r e s e n t e d b y a n a b r u p t p r o f i l e . 78 2 / ( D t ) - x ] - e r f c [ x + - x 2 v / ( D t ) 2 . x + x - e r f c [ 2 . x e p i + x + e r f c [ 2 | / ( D t ) ] } ( 3 . 1 9 ) w h e r e N + i s t h e d o p i n g c o n c e n t r a t i o n o f t h e b u r i e d s u b s t r a t e , x * i s t h e t h i c k n e s s o f t h e b u r i e d l a y e r , x e p ^ i s - t h i c k n e s s o f t h e e p i t a x i a l l a y e r , a n d D a n d t a r e t h e d i f f u s i o n c o n s t a n t a n d d i f f u s i o n t i m e , r e s p e c t i v e l y . H o w e v e r , t h e p r o f i l e g e n e r a t e d f r o m e q u a t i o n ( 3 . 1 9 ) b o r e l i t t l e r e s e m b l a n c e t o t h e m e a s u r e d d i s t r i b u t i o n . T h u s , i n t h e r e s u l t s t h a t f o l l o w , a n a b r u p t p r o f i l e w i t h a d o p i n g d e n s i t y o f 8 X 1 0 1 7 c m " 3 i n t h e c o l l e c t o r i s a s s u m e d , a s s h o w n i n F i g u r e 3 . 5 . T h e d o p i n g p r o f i l e f o r t h e d e v i c e i n R e f e r e n c e [ 1 5 ] w i t h t h e s i n g l e S e i m p l a n t w a s a l s o c o m p u t e d b y a s s u m i n g 70% S e i m p l a n t a c t i v a t i o n a n d 100% b e r y l l i u m i m p l a n t a c t i v a t i o n . T h i s i s j u s t i f i a b l e a s t h e i m p l a n t c o n d i t i o n s a r e b a s i c a l l y s i m i l a r f o r b o t h d e v i c e s t r u c t u r e s [ 1 5 ] , [ 1 7 ] , a s c a n b e s e e n f r o m T a b l e 3 . 1 . T h e e p i t a x i a l l a y e r f o r t h e s i n g l e i m p l a n t d e v i c e h a d a d o p i n g d e n s i t y o f 1x 1 0 1 6 c m " 3 , w h i c h i s a s s u m e d t o b e c o n s t a n t i n t h e p r o f i l e c a l c u l a t i o n , a s i n d i c a t e d i n F i g u r e 3 . 5 . T h e c o m p u t e d r e s u l t s f o r b o t h d e v i c e s t r u c t u r e s b a s e d o n t h e c a r r i e r p r o f i l e s i n F i g u r e 3 . 5 a r e s h o w n i n 7 9 F i g u r e 3 . 6 . T h e c o m p u t e d /3 max o f t h e d e v i c e i n [ 1 5 ] i s 1 8 , w h i c h i s s o m e w h a t h i g h e r t h a n t h e m e a s u r e d v a l u e o f 8 , w h i l e [ 1 7 ] i s 3 5 , w h i c h i s i n g o o d a g r e e m e n t w i t h t h e m e a s u r e d v a l u e s o f 2 0 - 3 0 . T h e T e x a s I n s t r u m e n t d e v i c e s [ 1 5 , 1 7 ] e x h i b i t h i g h e r e m i t t e r p e a k d o p i n g d e n s i t i e s a n d n a r r o w e r b a s e w i d t h s t h a n t h e H u g h e s d e v i c e s [ 1 4 ] , H o w e v e r , t h e D C p e r f o r m a n c e s a r e n o t v e r y d i f f e r e n t . T h i s s u g g e s t s t h a t t h e e m i t t e r i s p l a y i n g a n i m p o r t a n t r o l e i n d e t e r m i n i n g t h e g a i n o f T I ' s d e v i c e s . F r o m a c o m p a r i s o n o f F i g u r e s 3 . 3 . a n d 3 . 5 , i t c a n b e a p p r e c i a t e d t h a t t h e T I p r o c e s s l e a d s t o a n a r r o w e r e m i t t e r t h a n t h a t e x h i b i t e d b y t h e H u g h e s d e v i c e s f o r w h i c h S i i n d i f f u s i o n i s s i g n i f i c a n t . T h e t w o v a l u e s o f W_ a r e a p p r o x i m a t e l y 0 . 4 Mm a n d 0 . 2 5 Mm r e s p e c t i v e l y . F i g u r e 2 . 1 3 i n d i c a t e s t h a t s u c h a d i f f e r e n c e i n t h e e m i t t e r w i d t h h a s a c o n s i d e r a b l e e f f e c t o n 0 m a x « C o m p a r i s o n o f t h e r e s u l t s f r o m F i g u r e 2 . 1 3 w i t h t h o s e f r o m t h e n u m e r i c a l m o d e l i s a p p r o p r i a t e i n t h i s c a s e a s t h e v a l u e o f s^, = 2 x 1 0 6 c m / s e c u s e d i n c o m p u t i n g F i g u r e 2 . 1 3 i s s u f f i c i e n t l y h i g h t o a d e q u a t e l y r e p r e s e n t t h e o h m i c c o n t a c t a s s u m e d i n t h e n u m e r i c a l m o d e l . T h i s s u g g e s t s t h a t a s i g n i f i c a n t i m p r o v e m e n t i n p e r f o r m a n c e o f t h e T I d e v i c e s c o u l d b e a c h i e v e d b y s e e k i n g a r e d u c t i o n i n t h e e f f e c t o f s u r f a c e r e c o m b i n a t i o n a t t h e e m i t t e r f r o n t . t h e p r e d i c t e d p\ max f o r t h e d e v i c e w i t h t h e t w o S e i m p l a n t s 8 0 100-t Gain y'' t Gain « 10-3 a o / / Texas Inst. [15] i-Texas Inst. [17] 10" 7 10" 6 10" 5 10" 4 10~ 3 10~ 2 1 0 " 1 10' 10 2 10 3 10 4 10 5 Collector Current Density [ Amp/cm2 ] F i g . 3 . 6 T h e c o m p u t e d g a i n f o r t h e T e x a s I n s t r u m e n t d e v i c e s t r u c t u r e s o f R e f s . [ 1 5 , 1 7 ] , a s s u m i n g 70% a c t i v a t i o n o f t h e i m p l a n t e d s e l e n i u m a n d 100% a c t i v a t i o n o f t h e i m p l a n t e d b e r y l l i u m . 4 . C O N C L U S I O N S . T h e a n a l y t i c a l m o d e l d e v e l o p e d f o r t h e n p n G a A s h o m o j u n c t i o n b i p o l a r t r a n s i s t o r s h a s b e e n s h o w n t o b e u s e f u l i n p r e d i c t i n g g a i n s f o r p r e v i o u s l y - r e p o r t e d d e v i c e s f a b r i c a t e d b y M B E , V P E a n d L P E t e c h n i q u e s , w h i c h a r e c o n s i d e r e d t o h a v e u n i f o r m l y - d o p e d e m i t t e r , b a s e a n d c o l l e c t o r r e g i o n s . I t h a s a l s o b e e n s h o w n t h a t t h e m e a s u r e d g a i n s o f a b o u t 1 0 - 9 0 w h i c h h a v e b e e n r e p o r t e d f o r t h e s e d e v i c e s a r e t o b e e x p e c t e d f r o m t h e d e v i c e g e o m e t r i e s a n d d o p i n g d e n s i t i e s e m p l o y e d . T h e a n a l y t i c a l m o d e l h a s a l s o p r o v e d u s e f u l i n e x a m i n i n g t h e s e n s i t i v i t y o f t h e g a i n t o t h e v a r i a t i o n o f t h e d e v i c e p a r a m e t e r s . T h e c o n c l u s i o n t h a t c a n b e d r a w n f r o m t h e a n a l y s i s , i s t h a t a h i g h g a i n t r a n s i s t o r , w i t h a /J o f t h e o r d e r o f 1 0 0 0 , w o u l d r e q u i r e a n a r r o w b a s e w i d t h (< 0 . 2 Mm), l i g h t l y d o p e d b a s e 1 0 1 7 c m - 3 ) a s w e l l a s a n a r r o w e m i t t e r w i d t h (< 0 . 2 5 nm) a n d a s u r f a c e r e c o m b i n a t i o n v e l o c i t y o f l e s s t h a n 1 0 " c m / s e c . T h e g e o m e t r i c a l a n d d o p i n g f e a t u r e s s h o u l d b e e a s i l y a t t a i n a b l e i n p r a c t i c e b u t t h e r e q u i r e d l o w s u r f a c e r e c o m b i n a t i o n v e l o c i t y m a y b e m o r e d i f f i c u l t t o a c h i e v e . T h e p a s s i v a t i o n o f G a A s s u r f a c e s i s a s u b j e c t w o r t h y o f f u r t h e r s t u d y . T h e n u m e r i c a l m o d e l d e v e l o p e d h e r e h a s b e e n s h o w n t o b e u s e f u l i n p r e d i c t i n g g a i n s f o r i o n - i m p l a n t e d G a A s t r a n s i s t o r s . T h e a g r e e m e n t w i t h e x p e r i m e n t a l d a t a i s g o o d , p r o v i d e d t h a t t h e b a n d g a p n a r r o w i n g e f f e c t i s t a k e n i n t o c o n s i d e r a t i o n . T h i s s u g g e s t s t h a t t h e D C p e r f o r m a n c e o f t h e 81 82 d e v i c e s f a b r i c a t e d t h u s f a r i s l i m i t e d b y i n t r i n s i c p h e n o m e n a , a n d n o t b y e x t r a n e o u s e f f e c t s s u c h a s s u r f a c e l e a k a g e w h i c h h a v e b e e n c l a i m e d i n t h e l i t e r a t u r e . F o r d e v i c e s w i t h g o o d b a s e p r o p e r t i e s , e m i t t e r s u r f a c e e f f e c t s a p p e a r t o l i m i t t h e p e r f o r m a n c e o f t h e d e v i c e . F u r t h e r i m p r o v e m e n t s i n g a i n w o u l d a p p e a r t o d e m a n d , a s i n t h e d e v i c e s m e n t i o n e d e a r l i e r w h i c h w e r e f a b r i c a t e d b y e p i t a x i a l t e c h n i q u e s , a r e d u c t i o n o f t h e s e e f f e c t s , p r i n c i p a l l y b y e n h a n c i n g p a s s i v a t i o n o f t h e e m i t t e r s u r f a c e . REFERENCES R. C. Eden, B. M . W e l c h , R, Z u c c a , and S. I. Long , "The p r o spec t s f o r u l t r ah i gh - speed VLSI G a A s d ig i ta l l og i c , " IEEE Trans. Electron Devices, v o l . E D - 2 6 , no. 4, pp. 299 -317 , 1979. R. C. Eden and B. M . W e l c h , "U l t ra high speed G a A s VLS I : A p p r o a c h e s , po ten t i a l and progres s , " in VLSI Electronics: Microstructure Science ( ed i ted by N. E inspruch ), pp. 109-162, v o l . 3, A c a d e m i c P re s s , N e w York , 1981. H. K roemer , "Heteros t ructure b ipo la r t r an s i s t o r s and integrated c i r cu i t s , " Proc. IEEE, v o l . 70, no. 1, pp. 13 -25, 1982. R. C. Eden, " C o m p a r i s o n o f G a A s dev i ce approaches fo r u l t r ah i gh - speed VLSI , " Proc. IEEE, v o l . 70, no. 1, pp. 5 -12, 1982. S. S. Tan and A . G. M i l n e s , " Con s i de ra t i on of the f requency p e r f o r m a n c e of G a A s h o m o j u n c t i o n and hete ro junc t i on n - p - n t r an s i s t o r s , " IEEE Trans. Electron Devices, v o l . ED - 30 , no. 10, pp. 1289-1294, 1983. G. R. A n t e l l , "Ga l l i um A r s e n i d e T ran s i s t o r s , " in Semiconductors and Semimetals ( ed i ted by R. K. W i l l a r d s o n and A . C. Beer ), v o l . 7, part A , pp. 273 -292 , A c a d e m i c P re s s , N e w York , 1971. H. Becke , H. F l a t l e y , and D. S t o l n i t z , "Doub le d i f f u s e d Ga l l i um A r s e n i d e t r an s i s to r s , " Solid-State Electron., v o l . 8, pp. 255 -265 , 1965. W. V o n Munch , H. S ta tz , and A . E. B l ake s l ee , " I so la ted G a A s t r an s i s t o r s on h i g h - r e s i s t i v i t y G a A s subst rate, " Sol id-State Electron., v o l . 9, pp. 826 -827 , 1966. W. V o n Muench and H. S ta tz , " S o l i d - t o - s o l i d d i f f u s i o n in the Ga l l i um A r s e n i d e dev i ce t echno l ogy , " Sol id-State Electron., v o l . 9, pp. 939 -942 , 1966. G. R. A n t e l l and A . P. Wh i t e , "Doub le d i f f u s e d ga l l i um arsen ide t ran s i s to r , " Int. Symp. Gallium Arsenide [ Conf. Ser.-Inst. Phys. ], pp. 201 - 205 , 1966. H. S t rack, " I r o n - d o p e d ga l l i um ar sen ide t ran s i s to r s , " Int. Symp. Gallium Arsenide [ Conf. Ser.-Inst. Phys. ] , pp. 206 -212 , 1966. C. J . Nuese, J . J . Gannon, R. H. Dean, H. F. Go s senbe rge r , and R. E. E n s t r o m , " G a A s v a p o r - g r o w n b ipo la r t r an s i s t o r s , " Solid-State Electron., v o l . 15, pp. 8 1 - 9 1 , 1972. K. V. Va idyanathan , R. A . J u l l e n s , C. L. A n d e r s o n , and H. L. Dunlap, "P lanar, i o n - i m p l a n t e d G a A s b ipo la r t r an s i s to r s , " Tech. Dig. IEEE IEDM, p. 826, 1980. 83 84 [14] , "P lanar, i o n - i m p l a n t e d b ipo la r de v i c e s in G a A s , " Sol id-State Electron., v o l . 26, no. 8, pp. 7 1 7 - 7 2 1 , 1983. [15] H. T. Yuan, F. H. Doe rbeck , and W. V. M c l e v i g e , "Ion imp lanted G a A s b ipo la r t r an s i s t o r s , " Electron. Lett., v o l . 16, no. 16, pp. 637 -638 , 1980. [16] H. T. Yuan, W. V. M c l e v i g e , F. H. Doerbeck , and E. G. D ie r schke , " G a A s b ipo la r integrated c i rcu i t t e chno l ogy , " Tech. Dig. IEEE /EDM, pp. 398 -400 , 1980. [17] F. H. Doerbeck , W. M . Duncan, W. V. M c l e v i g e , and H. T. Yuan, " Fab r i ca t i on and h i gh - tempera tu re cha rac te r i s t i c s o f i o n - i m p l a n t e d G a A s b ipo la r t r an s i s t o r s and r ing o s c i l l a t o r s , " IEEE Trans. Ind. Electron., v o l . IE-29, no. 2, pp. 136-139, 1982. [18] H. Kraut le, P. Na rozny , and H. Benek ing , "Latera l pnp G a A s b ipo la r t r an s i s to r prepared by ion imp lan ta t i on , " Electron. Lett., v o l . 18, pp. 259 -260 , 1982. [19] , "Latera l PNP G a A s b ipo la r t rans i s to r w i t h m i n i m i z e d subst rate current, " IEEE Electron Devices Lett., v o l . E D L - 3 , no. 10, pp. 315 -317 , 1982. [20] K. V. Va idyanathan , C. L. A n d e r s o n , H. L. Dunlap, and R. A . J u l l e n s , "Ion imp lan ta t i on of w i d e bandgap s e m i c o n d u c t o r s , " F inal Report , Cont rac t No. N 0 0 1 - 7 3 - 7 9 - C - 0 0 2 8 , Nava l E l e c t r on i c s S y s t e m s C o m m a n d , Wa sh i n g t on , D.C. 20375. [21] M a s s i m o Vanz i , Semiconductor Device Analysis P r o g r am, in tegrated C i r cu i t s Labo ra to r y , S t a n f o r d U n i v e r s i t y , January 1980 V e r s i o n . [22] M . Kurata, Numerical Analysis for Semiconductor Devices, p. 3, D. C. Heath and C o m p a n y , Lex i ng ton , Ma s s a chu se t t s , 1982. [23] S. C. Choo , "Carr ier g e n e r a t i o n - r e c o m b i n a t i o n in the s pa ce - cha r ge reg ion of an a s y m m e t r i c a l p -n j unc t i on , " Solid-State Electron., v o l . 11, pp. 1069-1077, 1968. [24] C. T. Sah, R. N. N o y c e , and W. S h o c k l e y , "Carr ier generat ion and r e c o m b i n a t i o n in P - N j unc t i on s and P - N j unc t i on cha rac te r i s t i c s , " Proc. IRE, v o l . 45, pp. 1228-1243, 1957. [25] A . M . S e k e l a , D. L. Feucht, and A . G. M i l n e s , " D i f f u s i o n length s tud ie s in n ga l l i um a r sen ide , " Gallium Arsenide and Related Compounds [ Conf. Ser.-lnst. Phys. ] , no. 24, pp. 245 -253 , 1975. [26] R. D. Ryan and J . E. Eberhardt, "Ho le d i f f u s i o n length in high pur i ty n - G a A s , " Solid-State Electron., v o l . 15, pp. 865 - 868 , 1972. [27] H. C. C a s e y , Jr., B. I. M i l l e r , and E. P i nka s , "Va r i a t i on of m i n o r i t y - c a r r i e r length w i t h carr ier c oncen t r a t i on in G a A s l i qu id -pha se • ep i tax i a l l aye r s , " J. Appl. Phys., v o l . 44, no. 3, pp. 1281-1286, 1973. 85 [28] H. C. Ca sey , J r . and F. S te rn , " C o n c e n t r a t i o n - d e p e n d e n t ab so rp t i on and spontaneous e m i s s i o n o f heav i l y doped G a A s , " J. Appl. Phys., v o l . 47, no. 2, pp. 631 -643 , 1976. [29] A . G. M i l n e s , " Impur i ty l eve l s in ga l l i um ar sen ide, " Adv. in Electronics and Electron Physics, v o l . 61, pp. 63 -160 , A c a d e m i c P re s s , N e w York , 1983. [30] H. K re s se l and J . K. But ler , Semicondunctor Lasers and Heterojunction LEDs, pp. 4 6 - 4 7 , A c a d e m i c P re s s , N e w York , 1977. [31] P. Van Halen and D. L. Pu l f r e y , " A c c u r a t e , short se r ie s app rox imat i on s to F e r m i - D i r a c integra l s o f order - 1/2 , 1/2, 1, 3/2, 2, 5/2, 3, and 7/2," J. Appl. Phys.,, v o l . 57, no. 12, pp. 5271 -5274 , 1985. [32] J . S. B l a k e m o r e , " S e m i c o n d u c t i n g and other major p rope r t i e s of ga l l i um ar sen ide, " J. Appl. Phys., v o l . 53, no. 10, pp. 123-181, 1982. [33] R. P. Me r ten s , R. J . van Ove r s t r ae ten , and H. J . de Man , "Heavy dop ing e f f e c t s in s i l i c o n , " Adv. in Electronics and Electron Physics, v o l . 55, pp. 77 -118 , A c a d e m i c P re s s , N e w York , 1981. [34] J . S. B l a k e m o r e , " Intr ins ic dens i t y n.(T) in G a A s : Deduced f r o m band gap and e f f e c t i v e mas s parameters and de r i ved i ndependent l y f r o m Cr accep to r capture and e m i s s i o n c o e f f i c i e n t s , " J. Appl. Phys., v o l . 53, no. 1, pp. 520 -531 , 1982. [35] J . P. Ba i l be , A . Ma r t y , and G. Rey , " In f luence of degeneracy behav iour o f h o m o j u n c t i o n G a A s b i po l a r t r an s i s to r , " Electron. Lett., v o l . 20, no. 6, pp. 258 -259 , 1984. [36] In Re fe rence 30, p. 19. [37] V. F. D v o r y a n k i n , O. V. E m e l ' y a n e n k o , D. N. Na s l edov , D. D. N e d e o g l o , and A . A . Te l e g i n , " E l e c t r i c a l p rope r t i e s of ep i tax ia l f i l m s o f n - t y p e G a A s , " Sov. Phys.-Semi cond., v o l . 5, no. 10, pp. 1636-1640, 1972. [38] D. E. H i l l , " A c t i v a t i o n energy o f ho le s in Z n - d o p e d G a A s , " J. Appl. Phys., v o l . 41 , pp. 1815-1818, 1970. [39] S. M . S ze , Physics of Semiconductor Devices, pp. 2 2 - 27 , 2nd ed., John W i l e y & son s , N e w York , 1981. [40] D. E. H i l l , " Infrared t r a n s m i s s i o n and f l u o r e s c e n c e o f doped Ga l l i um A r s e n i d e , " Phys. Rev., v o l . 133, no. 3 A , pp. 866 -872 , 1964. [41] 0 . V. E m e l ' y a n e n k o , T. S. Lagunova , and D. N. N a s l e d o v , " S ca t te r i ng o f current den s i t i e s in ga l l i um ar sen ide in the p re sence of s t rong degeneracy , " Sov. Phys.-Solid State, v o l . 2, pp. 176-180, 1960. [42] T. Katoda and T. Sugano, "Hal l e f f e c t , S c h o t t k y barr ier capac i t ance , and p h o t o l u m i n e s c e n c e spec t ra mea su remen t s f o r G a A s ep i tax ia l layer and their c o r r e l a t i o n , " J. Electrochem. Soc, v o l . 121, no. 8, pp. 1066-1073, 1974. 86 [43] W. W a l u k i e w i c z , L. L a g o w s k i , L. J a s t r z e b s k i , M . L i ch tens te i ge r , and H. C. G a t o s , " E l ec t ron m o b i l i t y and f r e e - c a r r i e r ab so rp t i on in G a A s : De te rm ina t i on o f the c o m p e n s a t i o n ra t i o , " J. Appl. Phys., v o l . 50, no. 2, pp. 899 -908 , 1979. [44] H. M . Cox and J . V. D i Lo renzo , "Cha rac te r i s t i c s of an A sC I 3 /Ga/H 2 t w o - b u b b l e r G a A s CVD s y s t e m f o r M E S F E T app l i c a t i on s , " Gallium Arsenide and Related Compounds [ Conf. Ser.-lnst. Phys. ], no. 33b, pp. 11-22, 1977. [45] D. J . A s h e n , P. J . Dean, D. T. J . Hurle, J . B. M u l l i n , A . R o y l e , and A . M . Wh i t e , "The i nco rpo ra t i on o f res idua l impur i t i e s in vapour g rown G a A s , " Gallium Arsenide and Related Compounds [ Conf. Ser.-lnst. Phys. ] , no. 24, pp. 229 -244 , 1975. [46] F. D. R o s i , D. M e y e r h o f e r , and R. V. J e n s e n , " P rope r t i e s o f p - t y p e G a A s prepared by copper d i f f u s i o n , " J. Appl. Phys., v o l . 31, no. 6, pp. 1105-1108, 1960. [47] Sh. M . Ga s an l i , 0 . V. Eme l ' y anenko , V. K. E rgakov , F. P. Ke saman l y , T. S. Lagunova , and D. N. N a s l e d o v , "De te rm ina t i on o f the impur i ty c oncen t r a t i on f r o m the Hall e f f e c t and the hole m o b i l i t y in Z n - d o p e d Ga l l i um A r s e n i d e c r y s t a l s , " Sov. Phys.-Semi cond., v o l . 5, no. 10, pp. 1641-1644, 1972. [48] J . V i l m s and J . P. Garrett , "The g rowth and p rope r t i e s o f LPE G a A s , " Solid-State Electron., v o l . 15, pp. 443 - 455 , 1972. [49] J . V i l m s and W. E. Sp i ce r , "Quantum e f f i c i e n c y and rad ia t i ve l i f e t i m e in p - t y p e Ga l l i um A r s e n i d e , " J. Appl. Phys., v o l . 36, no. 9, pp. 2 815 - 2821 , 1965. [50] W. W a l u k i e w i c z , L. L a g o w s k i , L. J a s t r z e b s k i , and H. C. G a t o s , " M i n o r i t y - c a r r i e r m o b i l i t y in p - t y p e G a A s , " J. Appl. Phys., v o l . 50, no. 7, pp. 5040 -5042 , 1979. [51] C. H i l s um and B. H o l e m a n , "Carr ier l i f e t i m e in G a A s , " Proc. Int'l Conf. on Semicond. Phys., pp. 962 -966 , 1960. [52] R. J . N e l s o n and R. G. S obe r s , " M i n o r i t y - c a r r i e r l i f e t i m e and internal quantum e f f i c i e n c y o f s u r f a c e - f r e e G a A s , " J. Appl. Phys., v o l . 49, no. 12, pp. 6103 -6108 , 1978. [53] M . E t tenberg , H. K r e s s e l , and S. L. G i l be r t , " M i n o r i t y carr ier d i f f u s i o n length and r e c o m b i n a t i o n l i f e t i m e in G a A s ; G e prepared by l i qu i d - pha se ep i t axy , " J. Appl. Phys., v o l . 44, no. 2, pp. 8 27 - 831 , 1973. [54] G. B. S c o t t , G. Duggan, and P. D a w s o n , " A p h o t o l u m i n e s c e n c e s tudy o f b e r y l l i u m - d o p e d G a A s g rown by mo lecu l a r beam ep i taxy , " J. Appl. Phys., v o l . 52, no. 11, pp. 6888 -6894 , 1981. [55] K. L. A s h l e y and J . R. B ia rd , "Op t i c a l m i c r o p r o b e r e spon se o f G a A s d i ode s , " IEEE Trans. Electron Devices, v o l . ED -14 , no. 8, pp. 429 -432 , 1967. 87 [56] C. J . Hwang , "Dop ing dependence o f hole l i f e t i m e in n - t y p e G a A s , " J. Appl. Phys., v o l . 42, no. 11, pp. 4408 -4413 , 1971. [57] L. L a g o w s k i , L. J a s t r z e b s k i , and H. C. Ga to s , " A p p l i c a t i o n of scann ing e l ec t r on m i c r o s c o p y to de te rm ina t i on o f su r face r e c o m b i n a t i o n v e l o c i t y : G a A s , " Appl. Phys. Lett., v o l . 27, no. 10, pp. 537 -539 , 1975. [58] G. A . A c k e t and J , J . Scheer , "Re laxat ion o s c i l l a t i o n s and r e c o m b i n a t i o n in ep i tax ia l n - t y p e Ga l l i um A r s e n i d e , " Solid-State Electron., v o l . 14, pp. 167-174, 1971. [59] L. F. Shamp ine and R. C. A l l e n Jr., Numerical Computing : An Introduction, pp. 87 -108 , Saunders , Ph i l ade lph i a , 1973. [60] A . Neug ro s che l , M . A r i e n z o , Y. K o m e n , and R. D. Isaac, "Exper imenta l s tudy o f the m i n o r i t y - c a r r i e r t ranspor t at the p o l y s i l i c o n - m o n o s i l i c o n i n te r face , " IEEE Trans. Electron Devices, v o l . ED - 32 , pp. 807 -816 , 1985. [61] H. K. G u m m e l , " A s e l f - c o n s i s t e n t i te rat i ve s cheme for o n e - d i m e n s i o n a l s teady s tate t r an s i s to r ca l cu l a t i on s , " IEEE Trans. Electron Devices, v o l . E D - 1 1 , no. 10, pp. 455 -465 , 1964. [62] A . D e M a r i , " A n • accurate numer i ca l s teady s tate o n e - d i m e n s i o n a l s o l u t i o n of the P - N junc t i on , " Solid-State Electron., v o l . 11, pp. 33 -58 , 1968. [63] - , " A n accurate numer i ca l o n e - d i m e n s i o n a l s o l u t i o n o f the P - N junc t i on under arb i t rary t rans ient c ond i t i on s , " Solid-State Electron., v o l . 11, pp. 1021-1053, 1968. [64] D. L. S cha r fe t te r and H. K. G u m m e l , " L a r ge - s i g na l ana l y s i s o f a s i l i c o n Read d iode o s c i l l a t o r , " IEEE Trans. Electron Devices, v o l . ED - 16 , no. 1, pp. 64 -77 , 1969. [65] J . W. S l o t b o o m , " I terat ive s cheme f o r 1- and 2 - d i m e n s i o n a l D.C.-t ran s i s to r s imu l a t i on , " Electron. Lett., v o l . 5, pp. 677 -678 , 1969. [66] H. W . Loeb , R. A n d r e w , and W. L o v e , " A p p l i c a t i o n o f 2 - d i m e n s i o n a l s o l u t i on s of the S h o c k l e y - P o i s s o n equat ion to i n v e r s i o n - l a y e r M.O.S.T. d e v i c e s , " Electron. Lett., v o l . 4, pp. 352 -354 , 1968. [67] J . E. Sch roeder and R. S. Mu l l e r , " IGFET ana l y s i s through numer ica l s o l u t i o n o f P o i s s o n ' s equat ion , " IEEE Trans. Electron Devices, v o l . E D - 1 5 , no. 12, pp. 9 54 - 961 , 1968. [68] E. M . Butur la, P. E. C o t t r e l l , B. M . G r o s s m a n , K. A . Sa l sbu rg , M . B. L a w l o r , and 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 te e lement s i m u l a t i o n of s e m i c o n d u c t o r d e v i c e s , " Proc. Int. Solid-State Circuits Conf., pp. 7 6 - 7 7 , 1980. [69] N. S h i g y o , M . Konaka , and R. L. M . Dang, " T h r e e - d i m e n s i o n a l s i m u l a t i o n of inver se n a r r o w - c h a n n e l e f f e c t , " Electron. Lett., v o l . 18, no. 6, pp. 274 -275 , 1982. 88 [70] F. N. T r o f i m e n k o f f , " F i e l d - dependen t m o b i l i t y ana l y s i s o f the f i e l d - e f f e c t t rans i s to r , " Proc. IEEE, v o l . 53, no. 11, pp. 1765-1766, 1965. [71] V. L. Da la i , A . B. Dreeben, and A . T r i ano , "Temperature dependence of ho le v e l o c i t y in p - G a A s , " J. Appl. Phys., v o l . 42, no. 7, pp. 2864 -2867 , 1971. [72] W. V. M c l e v i g e , M . J . Hel ix , W. V. Va idyanathan, and B. G. S t r ee tman , " E l e c t r i c a l p r o f i l i n g and o p t i c a l a c t i v a t i o n s tud ies o f B e - i m p l a n t e d G a A s , " J. Appl. Phys., v o l . 48, no. 8, pp. 3342-3346, 1977. [73] J . L indhard, M . S cha r f f , and H. E. Sch i t f t t , "Range concep t s and heavy ions ranges, " Mat. Fys. Medd. Dan. Vid. Se/sk., v o l . 33, pp. 1-42, 1963. [74] J . F. G i bbon s , W. S. J o h n s o n , and S. W. M y l r o i e , Projected Range Statistics, 2nd ed. D o w d e n , Hutch in son , and Ro s s , S t roudburg , Penn s y l v an i a , 1975. [75] H. R y s s e l and I. Ruge, lonenimplantation, Teubner, S tuttgart , 1978. [76] S. Se lberher r and E. Guer re ro , " S i m p l e and accurate rep re sen ta t i on of imp lan ta t i on parameter s by l o w order p o l y n o m i a l , " Sol id-State Electron., v o l . 24, pp. 591 -593 , 1981. [77] D. V. Mo r gan , F. H. E i s en , and A . Ez i s , " P r o spec t s f o r ion bomba rdmen t and ion imp lan ta t i on in G a A s and InP dev i c e f a b r i c a t i o n , " Proc. /EE, v o l . 128, no . 4, pp. 109-130, 1981. [78] S. K. Ghandh i , VLSI Fabrication Principles, pp. 345 -346 , J ohn W i l e y & Son s , 1983. [79] C. A . S t o l t e , " Ion imp lan ta t i on and mate r i a l s f o r G a A s integrated c i r cu i t s , " in Semiconductors and Semi metals, v o l . 20, pp. 89 -158 , A c a d e m i c P re s s , 1984. [80] T. H. Chen and M . S. Shur, " A n a l y t i c a l m o d e l s o f i o n - i m p l a n t e d G a A s FET ' s , " IEEE Trans. Electron Devices, v o l . ED - 32 , no. 1, pp. 18 -27, 1985. [81] S. G. L iu , E. C. Doug la s , C. P. W u , C. W. Magee , S. Y. Narayan, . S. T. J o l l y , F. Ko l ond ra , and S. J a i n , "Ion imp lan ta t i on of sulfur and s i l i c o n in G a A s , " RCA Review, V o l . 41 , pp. 227 -262 , 1980. [82] C. L. A n d e r s o n and H. L. Dunlap, " L o w - t e m p e r a t u r e anneal ing behav io r o f G a A s imp lan ted w i t h Be, " Appl. Phys. Lett., v o l . 35, no. 2, pp. 178-180, 1979. [83] In Ref. 78, pp. 643 -644 . APPENDIX A C • — C C This Program i s written i n structured WATFIV f o r implementing C C the a n a l y t i c a l model. The program can handle 2 donor l e v e l s C C and 3 acceptor l e v e l s . C C C C Variable Names C C C C Level Name Ion Cone. Act. Energy Degen. C C C C DONOR 1 NAMEl DNl ENINl DGl C C DONOR 2 NAME2 DN2 ENIN2 DG2 C C ACCEPTOR 1 NAME3 DN3 ENIN3 DG3 C C ACCEPTOR 2 NAME4 DN4 ENIN4 DG4 C C ACCEPTOR 3 NAME5 DN5 ENIN5 DG5 C C C C LAYER(I) = The name of the Ith semiconductor layer C C WIDTH(I) = The width of the Ith semiconductor region, i n C C Microns C C T = The temperature i n K e l v i n C C DIEL =Permittivity of GaAs C C MO = Ele c t r o n r e s t mass i n Kg C C NI = D i l u t e i n t r i n s i c c a r r i e r concentration i n cm-3 C C CAYT = Boltzmann's constant * temperature C C Q = E l e c t o n i c charge i n Coulombs C C C C C CHARACTER*10 NAMEl,NAME2,NAME3,NAME4,NAME5,LAYER(3) CHARACTER*3 VAR,BIAS EXTERNAL FDHF,FD3HF,EXPN REAL NCONC,MOB,NETDOP,NETION,JEP,JBNO,JBNW,JCBOP,JGEN,JBREC REAL JREC,JB,JC,ME,MH,SUMD(3),NI,NIEFF(3) INTEGER COUNT DIMENSION ME(3),MH(3),FC(3),FV(3),ETTA(3),FL(3),BGAP(3) DIMENSION WIDTH(3),TAU(3),DCONST(3),DIFFL(3),PCONC(3) DIMENSION NCONC(3),EMOB(3),PMOB(3),NETDOP(3),CION(3),XXI(3) COMMON NAMEl,NAME2,NAME3,NAME4,NAME5 COMMON DNl,DN2,DN3,DN4,DN5 COMMON DI1,DI2,DI3,DI4,DI5 COMMON EN1,EN2,EN3,EN4,EN5 COMMON ENINl,ENIN2,ENIN3,ENIN4,ENIN5 COMMON DGl,DG2,DG3,DG4,DG5 COMMON T,CN,CP,EG,EFME,EFMH,FNC,FNV,CAYT,Q,ETA,XI,MO COMMON SUMTD,SUMTA,SUMID,SUMIA,NI DIEL = 13.1 * 8.854E-14 T=300 NI=2.0E06 MO=9.1095E-31 CAYT=8.6173E-5 * T Q = 1.6022E-19 C0UNT=1 C C C Sta r t i n i t i a l i z a t i o n f o r each region of t r a n s i s t o r . C C ITER = The number of i n i t i a l i z a t i o n s . C C C READ,ITER DO 100 KK=1,ITER DO 200 1=1,3 READ 1,LAYER(I),WIDTH(I) WIDTH(I)=WIDTH(I) * 1.0E-4 READ 2,NAME1,DN1,ENIN1,DG1 READ 2,NAME2,DN2,ENIN2,DG2 READ 2,NAME3,DN3,ENIN3,DG3 READ 2,NAME4,DN4,ENIN4,DG4 READ 2,NAME5,DN5,ENIN5,DG5 1 FORMAT(Tl,All,T15,E10.4) 2 FORMAT(Tl,All,T15,E8.2,T25,E6.1,T35,E6.1) IF ( I .EQ. 1) PRINT 3 3 FORMAT ('1') PRINT 4, LAYER(I),'PARAMETERS' 4 FORMAT(T28,All,A10,/) PRINT 5,'DOPANT *,'DENSITY','ACTIVATION ENERGY','DEGENERACY * 5 FORMAT(A10,9X,A10,7X,A17,4X,A10) C C C Calcu l a t e reduction i n a c t i v a t i o n energy C C SUMTD = Tot a l donor concentration C C SUMTA = Tot a l acceptor concentration C C EN's = A c t i v a t i o n energy of each impurity state C C ENIN's = d i l u t e a c t i v a t i o n energy of each impurity state C C C SUMTD=DN1+DN2 SUMTA=DN3+DN4+DN5 C IF( ENIN1 .GT. 0.0 ) EN1= ENIN1 - (1.90E-•8 * SUMTD**0.3333) IF( ENIN2 .GT. 0.0 ) EN2= ENIN2 - (1.90E-•8 * SUMTD**0.3333) IF( ENIN3 • GT. 0.0 ) EN3= ENIN3 - (2.34E-•8 * SUMTA**0.3333) IF( ENIN4 • GT. 0.0 ) EN4= ENIN4 - (2.34E-•8 * SUMTA**0.3333) IF( ENIN5 • GT. 0.0 ) EN5= ENIN5 - (2.34E-•8 * SUMTA**0.3333) IF( ENIN1 .EQ. 0.0 ) EN1= 0.0 IF( ENIN2 .EQ. 0.0 ) EN2= 0.0 IF( ENIN3 .EQ. 0.0 ) EN3= 0.0 IF( ENIN4 .EQ. 0.0 ) EN4= 0.0 IF( EN IN 5 .EQ. 0.0 ) EN5= 0.0 IF( EN1 .LE . 0 . 0 ) EN1= 0 . 0 IF( EN2 .LE . 0 . 0 ) EN2= 0 . 0 IF( EN3 .LE . 0 . 0 ) EN3= 0 . 0 IF( EN4 .LE . 0 . 0 ) EN4= 0 . 0 IF( EN5 .LE . 0 . 0 ) EN5= 0 . 0 91 7 C-C C C C C C C C-) IF ( NAME1 .NE. 'DUMMY IF ( NAME2 .NE. 'DUMMY IF ( NAME3 .NE. 'DUMMY IF ( NAME4 .NE. 'DUMMY IF ( NAME5 .NE. 'DUMMY FORMAT(A12,8X,E10.4,9X,E10.4,8X,E10.4) PRINT 7 FORMAT('/') PRINT 6,NAME1,DN1,EN1,DG1 ) PRINT 6,NAME2,DN2,EN2,DG2 ) PRINT 6,NAME3,DN3,EN3, DG3 ) PRINT 6,NAME4,DN4,EN4,DG4 ) PRINT 6,NAME5,DN5,EN5,DG5 -C C c c c c c c -c c-c c c c c c c c-c-c c c c c c re-c a l c u l a t e the energy bandgap of GaAs by r e l a t i n g the bandgap narrowing to the e f f e c t i v e i n t r i n s i c c a r r i e r concentration. EGO = I n t r i n s i c bandgap of GaAs at room temperature EG = Actual bandgap of GaAs NIEFF= E f f e c t i v e i n t r i n s i c c a r r i e r concentration, using the empirical formulae of Bailbe et a l . IF ( I .EQ. 2) THEN DO NIEFF(I) = 9.0E05 + 3.38E-3 * - 6.72E05 3.38E-3 * - 3.47E05 NIEFF(I) = NIEFF(I) ELSE DO NIEFF(I) = 9.0E05 + NIEFF(I) = NIEFF(I) END IF EG0=1.519-(5.405E-4*T**2)/(T+204) EG1=CAYT * ALOG ( NIEFF(I)*NIEFF(I)/NI/NI ) EG= EG0-EG1 SQRT(SUMTA) * ALOG(SUMTA*l.E-17) SQRT(SUMTD) * ALOG(SUMTD*l.E-17) Compute the electron and hole concentration by solvi n g the equation of d e t a i l e d balance. An i n i t i a l i s chosen for the Fermi energy c a l c u l a t i o n . EF = Lower bound of the i n t e r v a l EFG = Upper bound of the i n t e r v a l RE = I n i t i a l r e l a t i v e error AE = Absolute error EF=0.0 EFG=1.65 RE=l.E-6 AE=l.E-6 CALL ZERO(EF,EFG,RE,AE,IFLAG) FL(I) =EF IF ( IFLAG .GE. 2 ) THEN DO PRINT, 'CONVERGENCE FAILED' ELSE DO END IF A=ALOG10(CN) B=ALOG10(CP) -C C C C C C C C -C -C C C C C C C -C Cal c u l a t e the ele c t r o n m o b i l i t y : If p-type material, assume number of electrons = number of holes f o r m o b i l i t y c a l c u l a t i o n . EHFAC = (weak f i l e d H a l l f a c t o r ) * * - l RATIO = Electron m o b i l i t y i n p-GaAs/ Electron m o b i l i t y i n n-GaAs 92 • Y = A IF( I .EQ. 2) Y = B IF ( Y .LT. 14 ) Y = 14.0 EHFAC = 0.851 EMOB(I)=-.05355 - 69783.16*Y + 17348.328*Y**2 - 1579.599*Y**3 EMOB(I)= EMOB(I) +63.03560142*Y**4 - .9353191689*Y**5 EMOB(I)= EHFAC * EMOB(I) RATIO =684.326936722 - 168.174778461*Y + 15.4966483265*Y**2 S - 0.633308960591*Y**3 + .00967786832189*Y**4 IF ( I .EQ. 2 ) EMOB(I)= RATIO *EMOB(I) C C C Calculate the hole m o b i l i t y : C C If n-type material, assume hole concentration = electron C C concentration f o r mo b i l i t y c a l c u l a t i o n . C C PHFAC = (weak f i e l d H a l l f a c t o r ) * * - l C C C Z = B PHFAC =0.80 IF ( I .EQ. 1 .OR. I .EQ. 3) Z = A PMOB(I)=502511.268 -156107*Z + 19205.6*Z**2 - 1168.926*Z**3 PMOB(I)= PMOB(I) + 35.20885*Z**4 - .42017908*Z**5 PMOB(I)= PHFAC*PMOB(I) C C C Computing the minority c a r r i e r parameters. C C TAU's = Lifetimes C C DCONST's = D i f f u s i o n constants C C DIFFL's = d i f f u s i o n lengths C C C IF ( I .EQ. 1 .OR. I .EQ. 3) THEN DO IF( CP/FNV .GE. 1.00E-22 ) THEN DO C=1+.35355*CP/FNV - 9.9E-3*(CP/FNV)**2 § + 4.45E-4*(CP/FNV)**3 ELSE DO C=l + .35355*CP/FNV - 9.9E-3*(CP/FNV)**2 END IF DCONST(I)=PMOB(I)*CAYT*C E= 545.075507209 - 99.5742908474*A + 5.96752406063*A**2 $ -0.11903779718*A**3 TAU(I)=10**E DIFFL(I)=SQRT(ABS(TAU(I)*DCONST(I))) ELSE DO C IF( CN/FNC .GE. 1.00E-22 ) THEN DO D=l + .35355*CN/FNC - (9.9E-3)*(CN/FNC)**2 $ + 4.45E-4*(CN/FNC)**3 ELSE DO D = 1 + .35355*CN/FNC-(9.9E-3)*(CN/FNC)**2 END IF DCONST(I)=EMOB(I)*CAYT*D F= -7.63238159E-6 - 678.5724587 * B + 157.93508506*B**2 $ -13.7579663247 * B**3 + 0.531235178933 * B**4 $ - 0.0076716569056 * B**5 TAU(I)=10**F DIFFL(I)=SQRT(ABS(TAU(I)*DCONST(I))) END IF c c c Compute the ph y s i c a l parameters •C C •c SUMID = DI1 + DI2 SUMIA = DI3 + DI4 + DI5 SUMD(I) = SUMID CION(I) = ABS( SUMID -SUMIA ) PCONC(I)=CP NCONC(I)=CN ETTA(I) =ETA XXI(I) = XI ME(I) =EFME MH(I) = EFMH FC(I) = FNC FV(I) = FNV BGAP(I)=EG NETDOP(I)=ABS(SUMTD-SUMTA) 200 CONTINUE C C C P r i n t out the device parameters C C C IF( COUNT .EQ. 1) THEN DO PRINT 8,LAYER(1),LAYER(2),LAYER(3) 8 FORMAT(T43,A10,T58,A4,T70,A7) PRINT 9,'SEMICONDUCTOR LAYER WIDTH =',WIDTH(1),WIDTH(2), $ WIDTH(3) PRINT 9,'ENERGY BAND GAP = ',BGAP(1),BGAP(2),BGAP(3) PRINT 9,'FERMI LEVEL =',FL(1),FL(2),FL(3) PRINT 9,'EF - EG / KT =',ETTA(1),ETTA(2),ETTA(3) PRINT 9,'-EF /KT =',XXI(1),XXI(2),XXI(3) PRINT 9,'EFFECTIVE INTRINSIC CONC =',NIEFF(1),NIEFF(2), $ NIEFF(3) PRINT 9,'EFFECTIVE MASS OF ELECTRON = ',ME(1),ME(2) ,ME(3) PRINT 9,'EFFECTIVE MASS OF HOLE =',MH(1),MH(2),MH(3) PRINT 9,'EFFECTIVE DENSITY OF ELECTRON STATES =',FC(1),FC(2), $ FC(3) PRINT 9,'EFFECTIVE DENSITY OF HOLE STATES =',FV(1),FV(2), S FV(3) PRINT 9,'HOLE CONCENTRATION IN VAL. BAND =',PCONC(l), $ PCONC(2),PCONC(3) PRINT 9,'ELECTRON CONCENTRATION IN COND. BAND =',NCONC(l), $ NCONC(2),NCONC(3) PRINT 9,'ELECTRON MOBILITY =',EMOB(l),EMOB(2),EMOB(3) PRINT 9,'HOLE MOBILITY =',PMOB(l),PMOB(2),PMOB(3) PRINT 9,'MINORITY CARRIER DIFFUSION CONSTANT =',DCONST(l), $ DCONST(2),DCONST(3) PRINT 9,'MINORITY CARRIER DIFFUSION LENGHT =',DIFFL(1), $ DIFFL(2),DIFFL(3) PRINT 9,'MINORITY CARRIER LIFETIME =',TAU(1),TAU(2),TAU(3) PRINT 9,'NET DOPANT DENSITY =*,NETDOP(l),NETDOP(2),NETDOP(3) PRINT 9,'DENSITY OF IMMOBILE IONS =',CION(l),CION(2),CION(3) 9 FORMAT C0',A38,3E13.5) ELSE DO END IF c c C Calculate the current components and gains of the transistor C C i n the analytical model C C S = Emitter surface recombination velocity C C VCE = emitter-collector bias voltage C C C S = 2.0E06 VCE =5.0 BIAS= 'VBE' C • C C Read i n the parameter for sensitivity studies C C VAR = The name of the parameter C C N = The number of parameter values for evaluation C C PARM = The value of the parameter C C C READ 10,VAR 10 FORMAT (A3) READ,N DO 100 J=1,N READ,PARM PRINT 11,'TRANSISTOR BIAS VOLTAGES & CURRENT COMPONENTS' 11 F0RMAT(*1',T15,A50,/) IF( VAR .EQ. 'VCE') THEN DO VCE=PARM PRINT 9,'EMITTER-COLLECTOR VOLTAGE =',VCE ELSE DO END IF IF( VAR .EQ. 'NE ') PRINT 9,'EMITTER DOPANT DENSITY =',SUMD(3) IF( VAR .EQ. 'NC ') PRINT 9,,'COLLECTOR DOP. DENSITY =',SUMD(1) IF( VAR .EQ. 'SV ') THEN DO S=PARM PRINT 9,'SURFACE RECOMB. VEL =',S ELSE DO ENDIF IF( VAR .EQ. *WB ' ) THEN DO WIDTH(2)=PARM*1.0E-4 PRINT 9,'BASE WIDTH =',WIDTH(2) ELSE DO ENDIF IF( VAR .EQ. 'WE ' ) THEN DO WIDTH(3)=PARM*1.0E-4 PRINT 9,'EMITTER WIDTH =',WIDTH(3) ELSE DO END IF IF( VAR .EQ. 'TE ' ) THEN DO TAU(3)=PARM DIFFL(3)=SQRT(ABS(TAU(3)*DCONST(3))) PRINT 9,'EMITTER LIFETIME =',TAU(3) ELSE DO ENDIF IF( VAR .EQ. 'TB ' ) THEN DO TAU(2)=PARM DIFFL(2)=SQRT(ABS(TAU(2)*DCONST(2))) PRINT 9,'BASE LIFETIME =',TAU(2) ELSE DO ENDIF c c C C a l c u l a t i o n of various current components i n the a n a l y t i c a l C C model C C C DO 300 K=l,41 C IF (VAR .EQ. 'VCE') BIAS=VAR IF( K.EQ.l ) PRINT 12,BIAS,'JP(-XE)',' JN(0) ',' JN(XB)', $ ' JREC ', ' JGEN ,,'JP(XC) ', ,JN(0)-JN(XB)', $ ' GE ', 'INJ EFF','B T FAC,' JB ', $ ' JC ', ' BETA ' 12 FORMAT(5X, A3, 8X, 6 ( 3X, A7, 6X) , 3X, A12, /, 6 (3X, A7, 6X)) C C C Set the bias co n d i t i o n s . C C VCB = Base-collector voltage C C VEB = Base-emitter voltage C C C IF( VAR .NE. 'VBE' ) THEN DO VEB = 0.02*(K-1) + 0.50 ELSE DO IF( K .LE. 20) VCE = K*0.1/20. IF( K .GT. 20 .AND. K .LE. 30) VCE= (K-20)*0.01+0.1 IF( K .GT. 30 .AND. K .LE. 41) VCE= (K-30)*4.8/11.+0.2 VEB=PARM ENDIF IF( K.EQ.l .AND. VAR .EQ. 'VBE1 ) PRINT 9, ' VBE=',VEB VCB = VEB-VCE C- C C COMPUTATION OF THE BUILT-IN VOLTAGES C C BEVBI = Base-emitter j u n c t i o n voltage C C BCVBI = Base-collector junction voltage C C C BEVBI = CAYT * ALOG( PCONC(2) / PCONC(3) ) BCVBI = CAYT * ALOG( PCONC(2) / PCONC(l) ) C XE1 = 2*DIEL*(BEVBI-VEB)*CION(2) XE2 = XE1/(1.6022E-19*CION(3)*(CION(2)+CION(3))) XE = SQRT( ABS(XE2) ) XC - 2*DIEL*(BCVBI-VCB)*CION(2)/(Q*CION(l)*(CION(2)+CION(l))) XC - SQRT( ABS(XC) ) XBO = 2*DIEL*(BEVBI-VEB)*CION(3)/(Q*CION(2)*(CION(2)+CION(3))) XBO = SQRT( ABS(XBO) ) XBW = 2*DIEL*(BCVBI-VCB)*CION(l)/(Q*CION(2)*(CION(2)+CION(l))) XBW = SQRT( ABS(XBW) ) C C C WE = Neutral emitter width C C WB = Neutral basewidth C C C WE = WIDTH(3) - XE WB = WIDTH(2) - XBO -XBW DVEB = EXPN( VEB/CAYT) - 1.0 DVCB = EXPN( VCB/CAYT) - 1.0 EE = EXPN( 2*WE/DIFFL(3) ) BB = EXPN( 2*WB/DIFFL(2) ) c c C Computation of the emitter back-injected hole current, JEP, C C as given i n the model. C C C PXE = PC0NC(3) * DVEB IF ( S .EQ. 0.0 ) THEN DO PY = 2 * PXE * EXPN( WE/DIFFL(3) ) /( EE + 1 ) ELSE DO IF ( S .GT. 1.E10 ) THEN DO PY = 0.0 ELSE DO PY = 2 * PXE * EXPN( WE/DIFFL(3) ) PY = PY / ( S*DIFFL(3)/DCONST(3)*(EE - 1) + (EE + 1) ) END IF END IF JEP = Q*DC0NST(3)*( PXE*(EE + 1) - 2*PY*EXPN(WE/DIFFL(3)) ) JEP = JEP /( DIFFL(3) * ( EE - 1 ) ) C C C Computation of the el e c t r o n d i f f u s i o n currents at the C C base depletion edges, JBNO and JBNW C C ' C JBNO = DVEB*( BB + 1 ) - DVCB*2*EXPN( WB/DIFFL(2) ) JBNO = Q*DC0NST(2)*NC0NC(2)*JBN0 /( DIFFL(2) * ( BB - 1 ) ) JBNW = DVEB*2*EXPN( WB/DIFFL(2) ) - DVCB*( BB + 1 ) JBNW = Q*DCONST(2)*NCONC(2)*JBNW /( DIFFL(2) * ( BB - 1 ) ) C C C Computation of the generation current i n the reversed-biased C C base-collector junction, JGEN, and the c o l l e c t o r leakage C C current JCBOP. C C C JCBOP = Q * DCONST(l) * PCONC(l) / DIFFL(l) JGEN = -Q / 2.0 * (XBW+XC) * SQRT( 1. / (TAU(1)*TAU(2)) ) $ / COSH( 0.5*ALOG(TAU(1)/TAU(2)) ) C • C C Computation of the recombination current i n the forward C C biased emitter-base junction, as given by Choo's equations. C C C B = EXP(-0.5*VEB/CAYT ) * COSH( 0.5*ALOG( TAU(3)/TAU(2) ) ) ALPHA = 2 * SINH( 0.5 * ( BEVBI - VEB ) / CAYT ) AAA = TAU(2) * PCONC(2) /( TAU(3) * NC0NC(3)) GAMMA = SQRT(AAA)+SQRT(1./AAA) + 2*B*C0SH( (BEVBI-VEB)/2/CAYT ) IF ( B .NE. 1.0 ) THEN DO IF ( B .LT. 1.0 ) THEN DO FCNB = 1/SQRT( 1-B*B ) * ATAN( ALPHA/GAMMA*SQRT( 1-B*B ) ) ELSE DO BBB = ALPHA * SQRT( B*B -1 ) / GAMMA FCNB = 1 / SQRT( B*B - 1) * 0.5 * AL0G( (1+BBB)/(1-BBB) ) END IF ELSE DO FCNB = ALPHA / GAMMA ENDIF JREC =Q*SQRT( PCONC(l)*NCONC(l) )*(XBO+XE)*2*SINH(VEB/2/CAYT ) JREC = JREC*FCNB / ( SQRT( TAU(2)*TAU(3) )*( BEVBI-VEB )/CAYT ) 97 C C C Computation of the base current density, JB, and the C C c o l l e c t o r current density, JC. C C C JBREC = JBNO - JBNW JB = JEP + JBNO - JBNW - JCBOP - JGEN + JREC JC = JBNW + JCBOP + JGEN C C C Computation of other t r a n s i s t o r parameters. C C BETA = DC t r a n s i s t o r gain C C EMIEFF = Emitter i n j e c t i o n e f f i c i e n c y C C BTFAC = Base transport factor C C GE = Emitter Gummel number C c c BETA = JC / JB EMIEFF=JBNO/(JEP+JBNO+JREC) BTFAC=JBNW/JBNO PAR1=DIFFL(3)*S/DCONST(3) PAR2=WE/DIFFL(3) GE=NC0NC(3) * DIFFL(3) * NIEFF(3) * NIEFF(3) / DCONST(3)/4.0E12 $ *(PAR1*SINH(PAR2)+C0SH(PAR2)) / (PAR1*C0SH(PAR2)+SINH(PAR1)) GE=Q*4.0E12 *DVEB/(JREC+JEP) IF( VAR .EQ.'VCE') THEN DO PRINT 13, VCE, JEP, JBNO, JBNW, JREC,JGEN,JCBOP,JBREC, $ GE, EMIEFF, BTFAC, JB, JC, BETA ELSE DO PRINT 13, VEB, JEP, JBNO, JBNW, JREC,JGEN,JCBOP,JBREC, $ GE, EMIEFF, BTFAC, JB, JC, BETA END IF 13 FORMAT(7(E13.6,3X),E13.6,/,6(E13.6,3X),/) 300 CONTINUE 100 CONTINUE STOP END C : C C This function c a l c u l a t e s the exponential of the v a r i a b l e x. C C The overflow error i s avoided by s e t t i n g the minimun f(x) to C C be 0 f o r x < -100, and the maximum f(x) to be exp(lOO) f or C C x > 100. C C C FUNCTION EXPN(X) IF ( X.LT.-100.0) THEN DO EXPN=0.0 ELSE DO IF ( X.LE.100.0 .AND. X.GE.-100.0) THEN DO EXPN=EXP(X) ELSE DO IF ( X.GT.100 ) THEN DO EXPN=EXP(100) ELSE DO END IF END IF END IF RETURN END c c C This function evaluates the Fermi-Dirac Integral of order C C 3/2 using short series expressions by Van Halen and Pulfrey. C C C FUNCTION FD3HF(X,EXPN) IF ( X .LE. 0.0 ) THEN DO FD3HF = EXPN(X) - 0.176826*EXPN(2*X) + 0.064772*EXPN(3*X) $ - 0.033677*EXPN(4*X) + 0.021353*EXPN(5*X) $ - 0.011451*EXPN(6*X) + 0.003032*EXPN(7*X) ELSE DO IF ( X .GT. 0.0 .AND. X .LE. 4.0) THEN DO FD3HF = 0.867200 + 0.765101*X + 0.302693*X**2 $ + 0.062718*X**3 + 0.005793*X**4 $ + 0.001342*X**5 + 0.953657*X**6 ELSE DO IF ( X .GT. 4.0 ) THEN DO FD3HF = 0.300901*X**2.5 + 1.85581*X**0.5 $ + 0.466432*X**(-1.5) + 7.71648*X**(-3.5) $ + 120.535*X**(-5.5) + 800.702*X**(-7.5) $ + 2189.84*X**(-9.5) ELSE DO END IF END IF END IF RETURN END C C C This function evaluates the Fermi-Dirac Integral of order C C 1/2 using short series expressions by Van Halen and Pulfrey. C C — C FUNCTION FDHF(X,EXPN) IF ( X .LE. 0.0 ) THEN DO FDHF = EXPN(X) - 0.353568*EXPN(2*X) + 0.192439*EXPN(3*X) $ - 0.122973*EXPN(4*X) + 0.077134*EXPN(5*X) $ - 0.036228*EXPN(6*X) + 0.008246*EXPN(7*X) ELSE DO IF ( X .GT. 0.0 .AND. X .LE. 2.0 ) THEN DO FDHF = 0.765147 + 0.604911*X + 0.189885*X**2 $ + 0.020307*X**3 + 0.004380*X**4 $ + 0.000366*X**5 + 0.000133*X**6 ELSE DO IF ( X .GT. 2.0 .AND. X .LE. 4.0 ) THEN DO FDHF = 0.777115 + 0.581307*X + 0.206132*X**2 S + 0.017680*X**3 + 0.006549*X**4 $ + 0.000784*X**5 + 0.000036*X**6 ELSE DO IF ( X .GT. 4.0 ) THEN DO FDHF - 0.752253*X**(1.5) + 0.928195*X**(-0.5) $ + 0.680839*X**(-2.5) + 25.7829*X**(-4.5) $ + 553.636*X**(-6.5) + 3531.43*X**(-8.5) $ + 3254.65*X**(-10.5) ELSE DO END IF END IF END IF ENDIF RETURN END c-c c c c c c c c This subroutine c a l c u l a t e the c a r r i e r concentration f o r each semiconductor region by solv i n g the equation f o r charge n e u t r a l i t y . It searches f o r a zero of the t h i s non-linear equation between the given values B and C u n t i l the width of the i n t e r v a l (B,C) has collapsed to within a tolerance s p e c i f i e d by the stopping c r i t e r i o n , ABS(B-C) .LE. 2*(RW*ABS(B)+AE). •C C C C C C C C •C SUBROUTINE ZERO(B,C,RE,AE,IFLAG) CHARACTER*10 NAME1,NAME2,NAME3,NAME4,NAME5,LAYER REAL B,C,RE,AE,NIEFF INTEGER IFLAG COMMON NAME1,NAME2,NAME3,NAME4,NAME5 COMMON DN1,DN2,DN3,DN4,DN5 COMMON DI1,DI2,DI3,DI4,DI5 COMMON EN1,EN2,EN3,EN4,EN5 COMMON ENIN1,ENIN2,ENIN3,ENIN4,ENIN5 COMMON DG1,DG2,DG3,DG4,DG5 COMMON T,CN,CP,EG,EFME,EFMH,FNC,FNV,CAYT,Q,ETA,XI,MO COMMON SUMTD,SUMTA,SUMID,SUMIA,NI EXTERNAL EXPN,FDHF,FD3HF DATA ER/6.0E-8/ RW=AMAX1(RE,ER) IC=0 ACBS=ABS(B-C) A=C INDEX = 1 EF=A 20 ETA=(EF-EG)/CAYT XI=-EF/CAYT C C C Compute e f f e c t i v e mass of electrons using F-Dirac s t a t i s t i c s C C C AA=(1.38134E-47) * 3.09*CAYT/EG*FD3HF(ETA,EXPN) AA=AA + 1.38134E-47*FDHF(ETA,EXPN) AA=AA + (6.8135E-46) * EXPN(ETA-.476/CAYT) AA=AA + (3.54638E-46) * EXPN(ETA-.284/CAYT) X = FDHF(ETA,EXPN) EFME=(AA / X )**(2./3.) C C C Compute e f f e c t i v e mass of holes using F-Dirac s t a t i s t i c s C C C BB=(1.82164E-47) * 13.3875*CAYT/EG*FD3HF(XI,EXPN) BB= BB + (1.82164E-47+3.07395E-46)*FDHF(XI,EXPN) Y = FDHF(XI,EXPN) EFMH=(BB / Y )**(2./3.) C C C FNC= Density of states i n the conduction band C C FNV= Density of states i n the valence band C C CN = Ele c t r o n concentration i n the conduction band C C CP = Hole concentration i n the valence band C C C FNC=(5.55446E+60) * (EFME*T)**1.5 FNV=(5.55446E+60)* (EFMH*T)**1.5 CN=FNC * FDHF(ETA,EXPN) CP=FNV * FDHF(XI,EXPN) 100 c c C Compute the d e n s i t i e s of the io n i s e d impurities C C DNs = Density of dopants C C DIs = Ionized dopant concentration C c c Dl 1=0.0 IF ( NAMEl .NE. 'DUMMY ' ) THEN DO P0WER=( EF-EG+EN1)/CAYT DI1=DN1/(1. + ( DG1*EXPN(POWER) )) ELSE DO END IF TF( EN1 .EQ. 0.0 ) DI1=DN1 DI2=0.0 IF ( NAME2 .NE. 'DUMMY ' ) THEN DO POWER=( EF-EG+EN2)/CAYT DI2=DN2/(1. + ( DG2*EXPN(POWER) )) ELSE DO END IF IF( EN2 .EQ. 0.0 ) DI2=DN2 DI3=0.0 IF ( NAME3 .NE. 'DUMMY ' ) THEN DO POWER=( EN3 -EF )/CAYT DI3=DN3/(1. + ( DG3*EXPN(POWER) )) ELSE DO END IF IF( EN3 .EQ. 0.0 ) DI3=DN3 DI4=0.0 IF ( NAME4 .NE. 'DUMMY ' ) THEN DO POWER=( EN4 -EF )/CAYT DI4=DN4/(1. + ( DG4*EXPN(POWER) )) ELSE DO ENDIF IF( EN4 .EQ. 0.0 ) DI4=DN4 DI5=0.0 IF ( NAME5 .NE. 'DUMMY ' ) THEN DO POWER=( EN5 -EF )/CAYT DI5=DN5/(1. + ( DG5*EXPN(POWER) )) ELSE DO ENDIF IF( EN5 .EQ. 0.0 ) DI5=DN5 C- C C TEST i s the charge n e u t r a l i t y equation. In t h i s subroutine, C C The Fermi energy i s computed by i t e r a t i o n s u n t i l TEST has C C become very small. C C C TEST=(CN+DI3+DI4+DI5)-(CP+DI1+DI2) C C C The following perform the i t e r a t i o n s to f i n d the value C C of the Fermi Energy u n t i l the stopping c r i t e r i o n i s C C reach. C C C IF ( INDEX .EQ. 1 ) GO TO 30 IF ( INDEX .EQ. 2 ) GO TO 40 IF ( INDEX .EQ. 3 ) GO TO 8 30 FA = TEST INDEX = 2 101 EF = B GO TO 20 40 FB=TEST FC=FA KOUNT=2 FX=AMAX1(ABS(FB),ABS(FC)) 1 IF ( ABS(FC) .GE. ABS(FB) ) GO TO 2 A=B FA=FB B=C FB=FC C=A FC=FA 2 CMB=0.5*(C-B) ACMB=ABS(CMB) TOL=RW*ABS(B)+AE IF ( ACMB .LE. TOL ) GO TO 10 P=(B-A)*FB Q=FA-FB IF ( P .GE. 0. ) GO TO 3 P=-P Q="Q 3 A=B FA=FB IC=IC+1 IF ( IC .LT. 4 ) GO TO 4 IF ( 8.*ACMB .GE. ACBS ) GO TO 6 IC=0 ACBS=ACMB 4 IF ( P .GT. ABS(Q)*TOL ) GO TO 5 B=B+SIGN(TOL,CMB) GO TO 7 5 IF ( P .GE. CMB*Q ) GO TO 6 B=B+P/Q GO TO 7 6 B=0.5*(C+B) 7 INDEX = 3 EF = B GO TO 20 8 FB=TEST IF ( FB .EQ. 0. ) GO TO 11 KOUNT=KOUNT+l IF ( KOUNT .GT. 200) GO TO 15 IF ( SIGN(1.0,FB) .NE. SIGN(1.0,FC ) ) GO TO 1 C=A FC=FA GO TO 1 10 IF ( FB*FC .GT. 0. ) GO TO 13 IF ( ABS(FB) .GT. FX ) GO TO 12 IFLAG=1 RETURN 11 IFLAG=2 RETURN 12 IFLAG=3 RETURN 13 IFLAG=4 15 RETURN IFLAG=5 RETURN END c C The following i s a typical input f i l e containing data needed C in the program. The doping levels and widths in the emitter, C base, and collector are the typiacal values used i n the C analytical model. In this case, a sensitivity study of the C effect of surface recombination velocity of 10**4 and 10**6 C cm/sec on the transistor gain i s intended. /DATA i j . COLLECTOR 2.0 N-DOPANT 1.00E+16 0.0058 2.0 DUMMY 0.0 0.0 0.0 DUMMY 0.0 0.0 0.0 DUMMY 0.0 0.0 0.0 DUMMY 0.0 0.0 0.0 BASE 0.40 DUMMY 0.0 0.0 0.0 DUMMY 0.0 0.0 0.0 BERYLLIUM 1.00E+17 0.0195 4.0 DUMMY 0.0 0.0 0.0 DUMMY 0.0 0.0 0.0 EMITTER 0.25 DUMMY 0.0 0.0 0.0 SELENIUM 1.00E+17 0.0058 2.0 DUMMY 0.0 0.0 0.0 DUMMY 0.0 0.0 0.0 DUMMY 0.0 0.0 0.0 SV 2 1.0E4 1.0E6 

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