Avalanche Effect In Semiconductors

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Content INFORMATION TO USERS This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1 .T h e sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. . This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced. 5. PLEASE NOTE: Some pages may have indistinct print. Filmed as received. " ' Xerox University Microfilms 300 North Zoeb Road Ann Arbor, Michigan 48106 K l 74-14,468 O K U T O , Wrji, 1939- A V A L A N C H E EFFECT IN S E M IC O N D U C T O R S . University of Southern C alifornia, Ph.D., 1974 M aterials Science University Microfilms, A X E R O X Company, Ann Arbor, Michigan L THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED AVALANCHE E F F E C T IN SEMICONDUCTORS by Yuji Okuto A D isse rta tio n P re s e n te d to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFO RNIA In P a r tia l F u lfillm en t of th e R eq u irem en ts fo r the D e g re e DOCTOR OF PHILOSOPHY (M aterials Science) Ja n u a ry 1974 UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFORNIA 8 0 0 0 7 This dissertation, written by under the direction of h.}.?.... Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillment of requirements of the degree of D O C T O R OF P H IL O S O P H Y Yuji Okuto DISSERTATION COMMITTEE Chairman ACKNOWLEDGEMENTS It is a p le a s u re to acknow ledge a s s is ta n c e re c e iv e d fro m s e v e ra l so u rc e s du rin g the p re p a ra tio n of th is th e s is . F i r s t , I w ant to sin c e re ly thank m y ad v iso r, P r o f e s s o r C. R . C row ell, fo r h is valuable counsel and guidance th ro u g h o u t the c o u rse of m y sc ien tific r e s e a r c h at the U n iv e rsity of S outhern C a lifo rn ia . I would like a lso to thank P r o f e s s o rs M . G ersh en zo n and S. P . P o rto fo r th e ir en co u rag em en t and for th e ir c ritic a l review ing of th is m a n u sc rip t. A ssista n c e in m any fo rm s a lso w as given b y m y c o l le a g u e s in the M a te ria ls Science D e p artm en t. The h elp of M . M. E . B eguw ala, S harad Jo sh i, Cheng H siuhg H uang, R onald Chwang, C. L , A nderson, G. I. R o b e rts , and V. L . R ideout w as a p p re c ia te d sin c e re ly . T hanks a re a lso due D r. G. A . B a raff, B ell T elephone L a b o ra to rie s , fo r u se fu l d is c u ssio n s. My g rad u a te stu d ies w ere fin an c ia lly su p p o rted by m any s o u rc e s . The su p p o rt given b y the N atio n al S cience F oundation, the A dvanced R e s e a rc h P r o je c t A gency, th e Jo in t S e rv ice s E le c tro n ic P ro g ra m , the U n iv e rsity of S outhern C a lifo rn ia and Nippon E le c tric Co. L td. is a p p re c ia te d . A m a jo rity of the equipm ent u se d in the ex p erim e n tal p o rtio n of th is w ork w as pro v id ed by the A rm ed F o rc e Office of S cien tific R e se a rc h u n der a Jo in t S e rv ic e s G ran t. I am tru ly g ra te fu l fo r th is su p p o rt. I would lik e sin c e re ly to thank to m y p a re n ts , M r. and M rs . S. Okuto, and a lso m y frie n d s , M r. and M rs . T . Y am agum a, for th e ir advice and encouragem ent. L a s tly , I w ant to e x p re ss m y d e ep e st g ra titu d e to m y w ife, E m iko, who m ade it po ssib le for m e to co m p lete th is w ork. TABLE OF CONTENTS ACKNOW LEDGEM ENTS LIST O F TABLES LIST O F FIGURES LIST O F SYMBOLS S ection I. INTRODUCTION 1-1 O bjectives 1-2 H isto ric a l B ackground I-3 O utline of the P r e s e n t W ork H .' AVERAGE DISTANCE FOR IONIZATION SCATTERING I I - 1 E n e rg y C o n serv atio n Condition II-2 A n aly tical A pproxim ation I I-3 C urve F ittin g to th e B araff R esu lt II-4 P h y s ic a l Significance of E m p iric a l P a r a m e te r s II-5 C urve F ittin g to E x p erim e n tal D ata III. NONLOCALIZED CONCEPT III— 1 T h resh o ld E n erg y E ffect III-2 F o rm u latio n of the N onlocalized C oncept (NLC) H I-3 P se u d o lo c al A pproxim ation H I-4 C o m p ariso n w ith E x p e rim e n ta l R e su lts IH -5 A pproxim ation L im it of th e P seu d o lo c al A pproxim ation HI- 6 E x ac t N onlocalized Solution III-7 S um m ary IV P a g e 1 1 2 14 17 17 20 22 24 26 35 35 37 45 52 59 65 73 Section Page IV. AVALANCHE BREAKDOWN VOLTAGE IN SEMICONDUCTOR JUNCTIONS IV -1 B reakdow n C ondition IV -2 S ty lized M odel IV -3 A n aly tical C urve F ittin g IV -4 B reakdow n Band Bending E stim a te and D isc u ssio n V.* AVALANCHE BREAKDOWN BAND BENDING M EASUREM ENTS V - l Significance of B reakdow n V oltage M e a su re m e n ts V -2 E x p e rim e n ta l A pproach V -3 D isc u ssio n and S um m ary VI.* SUMMARY AND CONCLUSIONS A PPENDIX A. DERIVATION OF THE M U LTIPLICATION EQUATION B.' E M PIR IC A L I-V CHARACTERISTICS OF A SCHOTTKY DIODE C . ‘ * ESTIM ATION OF IONIZATION COEFFICIEN TS FROM BREAKDOWN BAND BENDING R E FE R EN C ES 75 75 77 79 80 91 91 92 105 109 111 115 119 123 v LIST OF TABLES TABLE TABLE TABLE I Im pact Ionization P a r a m e te r s a t 300°K . II Im pact Ionization P a r a m e te r s a t 300°K obtained v ia the P s e u d o lo c a l A pproxim ation. III E m p iric a l C urve F ittin g P a r a m e te r s a t 300°K. vi P age 27 58 81 LIST OF FIGURES F ig u re 1-a 1-b 1-c 1-d P ag e E x p erim e n tally re p o rte d , io n izatio n co efficien t, 6 a, as a function of the e le c tr ic fie ld stre n g th , 8, fo r e lectro n s (e) and h o le s (h) in Ge a t 3 0 0 °K .r (A fter M ille r [1 2 ]) E x p erim e n tally re p o rte d io n izatio n c o efficien t, 7 a, as a function of the e le c tric fie ld s tre n g th , 8, for ele ctro n s (e) and h o le s (h) in Si a t 300°K . (A fter L ee e t a l. [ 1 3 ], M o ll and O v e rstra e te n [1 5 ] and O v e rs tra e te n and DeM an [16]) E x p erim e n tally re p o rte d io n iz atio n co efficien t, 8 a , a s a function of th e e le c tric fie ld stre n g th , < g , in GaAs at 300°K . (A fter L ogan and Sze [1 7 ]) E x p erim e n tally re p o rte d io n iz atio n co efficien t, 9 a , a s a function of th e e le c tric fie ld s tre n g th , 3, in GaP at 300° K.' (A fter L ogan and W hite [18]) R eplotted B araff r e s u lt (broken lin e s ) and our 19 an aly tic "one point fittin g " (solid lin e s ) w ith r a s a p a ra m e te r. The o rd in ate i s the ra tio of th e c a r r ie r en erg y lo s t via io n iz a tio n to the e n e rg y gained fro m th e e le c tric fie ld ,' The a b s c is s a is the n o rm a liz e d in v e rs e e le c tric fie ld (the ratio o f a c h a r a c te r is tic e le c tr ic fie ld , to the e le c tric field ). The s tra ig h t lin e for r -> ® is the Shockley a sy m p to te . V ll : Figure i 3 I 4 -a 4-b 4 -c 4-d 5 ! Page a v s. r fro m the one point fitting 23 Legend: J range o f a fro m a ll p o ssib le fittings to B araff r e s u l t . — fitting w hich to u ch es the B a ra ff r e s u lts a t a single point. o fitting a t z = 10. se le c tio n as m o st re p re s e n ta tiv e fitting. Io n izatio n co efficien ts, a, a s a function of 28 the e le c tric field stre n g th , $, for e le c tro n s (e) and h o le s (h) in Ge a t 300°K . Legend: --------- e x p e r im e n ta l.-------------- th e o re tic a l fit. • point u se d to d e te rm in e m ean fre e path fitting p a ra m e te r. (E ) --------- th e o re tic a l m ax im u m a 2 im posed by en erg y c o n serv a tio n , (3/2E ) ----------th e o re tic a l m axim um a for th re s h o ld io n izatio n e n erg y of 3 /2 Ionization co efficien t, a, a s a function of the 29 e le c tric fie ld stre n g th , $, fo r e le c tro n s (e) and holes (h) in Si a t 300°K. The leg en d is the sam e a s in F ig. 4 -a . Ionization co efficien t, a, a s a function of 30 the e le c tric field stre n g th , < 3 , in GaAs a t 300°K. T he legend is the sam e a s in F ig . 4 -a . Ionization co efficien t, a, a s a function of the 31 e le c tric fie ld stre n g th , < 3 , in GaP a t 300°K . The legend is th e sam e a s in F ig . 4 -a . Ionization coefficient as a function of the 33 in v e rse e le c tric field for e le c tro n s and h o les in GaAs a t se lec te d te m p e ra tu re . L egend: A77°K, A178°K, O 300°K , • 373°K [3 5 ], --------- p re d ic te d c u rv e s fro m p a ra m e te rs in T able I. • • • v m F ig u re 6 7 8- a 8-b Page S chem atic d e sc rip tio n of f(X), the p ro b ab ility 39 of finding a c a r r i e r w hich o rig in ated at X=0 in a co n stan t e le c tric field and h as not yet p roduced an e le c tro n -h o le p a ir, the d eriv ativ e of f(X) and the single c a r r ie r ionization co e ffic ie n t a: <X> denotes the a v erag e distance a t w hich io n izatio n sc a tte rin g o c c u rs and D is th e d a rk space d ista n c e . L egend: B roken lin e s : Schem atic r e p r e sen tatio n of an as yet unknown ex act r e latio n sh ip . T hin solid lin e s: A ssum ption im p lic it in applying p rev io u s th e o re tic a l calcu latio n s of <X>. Heavy so lid lin es: P r e s e n t n o n lo calized th eo ry . R elatio n sh ip s betw een ri(x), p(x), n e (x) and 43 p e (x). The u p p er h a lf shows conduction and v a le n ce bands v s . d ista n c e . Dn and Dp a re d a rk sp aces for e le c tro n s and h o les re s p e c tiv e ly . The low er h a lf shows n(x) and p(x) v s . x and the w ay in w hich e le c tro n and hole populations develop into ne (x) and Pe (x). Ionization c o efficien t, a, as a function of 53 the e le c tric field stre n g th , 5, for e le ctro n s (e) and h o le s (h) in Ge a t 300°K. L e g e n d : --------------e x p erim e n tal. --------- th e o re tic a l p re d ic tio n obtained by the p seu d o lo cal ap proxim ation. • point used to p re d ic t th e o p tical phonon sc a tte rin g m ean fre e p ath . Ionization co efficien t, a> as a function of the 54 e le c tric field stre n g th , #, for e le c tro n s (e) and h o les (h) in Si a t 300°K, The legend is th e sam e a s in F ig . 8-a . Ionization c o efficien t, a, as a function of 55 the e le c tric field stre n g th , < ? , in GaAs at 300°K . T he legend is the sam e as in F ig . 8- a and a.a a a a x (Eg) — — — th e o re tic a l m ax im u m a lim it im posed by E q. 73. ix Figure 8-d 9-a 9-b 9-c 9-d 10-a Page Io n iza tio n c o efficien t, a, a s a function of 56 th e e le c tr ic field stre n g th , < jf , in G aP at 300°K . The legend is the sam e as in F ig, 8 - a an d aa ^ a x (Eg) -------- th e o re tic a l m ax im u m a lim it im p o sed by E q. 73. C r itic a l m u ltip lic a tio n r a te , M c , for e le c - 61 tro n s (e) and h o le s (h) in Ge a t 300°K as a fu nction of the e le c tric field stre n g th . B eyond th e se lim its th e pseu d o lo cal ap p ro x im a tio n does not h o ld . C r itic a l m u ltip lic a tio n r a te , M , fo r e le c - 62 tro n s (e) and h o les (h) in Si a t 300°K as a fu n ctio n o f th e e le c tric field stre n g th . C r itic a l m u ltip lic a tio n r a te , M , in GaAs 63 a t 300°K a s a function of the e le c tric field s tre n g th . M . denotes the th e o re tic a l m in im u m v a lu e ^ lo r M_ a t infinite field . C C r itic a l m u ltip lic a tio n r a te , M c , in GaP 64 a t 3 0 0 °K a s a function of the e le c tric field s tre n g th , M c m in den o tes the th e o re tic a l m in im u m v alu e fo r M c a t infinite field . The so lu tio n of th e e x ac t n onlocalized fo rm - 68 u la tio n developed in III-2 for a Si p - i- n junction w ith 10(i i-re g io n a t 300°K. The m u ltip li c a tio n r a te , Mn , is below Mc . The upper c u rv e show s n(x) and p(x) v s. x . The m id d le c u rv e s show n 0 (x) and pg(x) v s . x fo r n(o) = 1. The m axim um num ber of n e and p e at ju n ctio n edges a re a lso shown. The lo w e st c u rv e s show the e x ac t io n izatio n c o e ffic ie n ts an e (x) and a (x) v s. x . F o r c o m p a ris o n (^^< 3) and 0 C a (< ?) obtained by the p se u d o lo c a l a p p ro x im atio n a re a lso shown by b ro k e n lin e s .’ B oundary d a rk sp aces Dn and D fo r e le c tro n s and h o les re s p e c tiv e ly a re a lso show n. x Figure 10-b 10-c 10-d 11 12 13 14 The e x a c t so lu tio n for a Si p - i- n junction with 0 . 5 |i i-r e g io n a t 300°K . Mn is above Mc . The leg en d is the sam e as th a t in F ig . 1 0 -a. The e x a c t so lu tio n for a GaAs p - i- n ju n ctio n w ith 10 [i i- r e g io n at 300°K. Mn is below M c. The leg en d is the sam e a s th a t in F ig .1 1 0 -a. The e x a c t so lu tio n for a GaAs p - i- n ju n ctio n w ith 0 .5 |i i-re g io n a t 300°K . Mn is above M c. The leg en d is th e sam e as th a t in F ig.' 1 0 -a. S tylized m o d el to ap p ro x im ate the e x ac t so lu tio n . T his fig u re is draw n fo r a p - i- n ju n ctio n . R atio of th e io n izatio n co efficien t obtained by E q . 85 to th a t obtained v ia the p seu d o lo cal ap p ro x im atio n fo r ele ctro n s, in Si a t 300°K a s a function of the e le c tric field stre n g th . P re d ic te d b reak d o w n band bending for a b ru p t ju n ctio n s in G e, Si, GaAs and G aP a t 300°K a s a function of th e background im p u rity d en sity . L e g e n d : ----------p re s e n t p r e d ic tio n s -------------- p re v io u s e s tim a te s [3 6 ]. P re d ic te d b reakdow n band bending fo r p - i- n ju n ctio n s in G e, S i, GaAs and G aP a t 300°K a s a function of the i-re g io n w idth. B roken lin e s c o rre sp o n d to V g m ^n obtained by E q s. b 96 and 97. Space c h arg e re g io n w idth and m axim um field a t b reak d o w n fo r a b ru p t jun ctio n s in Ge, Si GaAs and G aP a t 300°K as a function of the background doping. Page 69 70 71 78 82 85 86 88 xi Figure 16 17 18 19 20 21 22 23 24 P re d ic te d lo g a rith m ic te m p e ra tu re co efficien t of th e breakdow n band bending aro u n d room te m p e ra tu re for a b ru p t jun ctio n s as a function of the breakdow n band bending a t 300°K. F o rw ard I-V c h a r a c te r is tic s of P t-S i (A) and P tS i-S i (B) on n -ty n e Si a t 300°K w ith = 2 x 10*6 a to m s /c m ^ . R e v erse I-V c h a r a c te ris tic of a LA diode a t 300°K w ith N p = 2 x 10*° a to m s /c m ^ . Junction d ia m e te r is 1 m m . The (cap acitan ce)” ^, 1 /C ^ , v s. r e v e r s e b ia s , V, re la tio n for a 1 m m d ia m e te r LA diode with N p = 2 x 10*6 ato m s /c m ^ . MOS c ap a cita n c e, C, and equivalent p a ra lle l conductance, G (: n o t to sc a le ), a t 1 MHz and 30 Hz as fu n ctio n s of field p late b ia s . Wet th e rm a lly grow n SiC>2 a t 900°C. R e v e rse I-V c h a r a c te r is tic of a W -Si SA diode at 300°K w ith Nj-j = 3 x 10*7 a to m s/c m ^ Junction d ia m e te r is 0 . 4(im . P h o to c u rre n t v s . ap p lied r e v e r s e b ia s r e latio n sh ip a t 300°K fo r a A u-SA diode w ith Nj-j 3 x 10*7 a to m s /c m ^ . F o rw ard I-V c h a r a c te r is tic of a W -Si SA diode a t 300°K w ith Nj-j = 4 x 10*7 a to m s/c m ^ B reakdow n band bending (sum of breakdow n voltage and d iffusion p o ten tial) v s. im p u rity d en sity fo r Si a b ru p t ju n ctio n s a t 300°K. Solid cu rv e: a fte r N LC th e o ry . B roken line: a fte r Sze and G ibbons [3 6 ], " q m : a fte r M ille r [1 2 ]; "o ": LA diodes; "+ ": SA diodes; and a fte r L e p s e lte r and Sze [3 7 ]. Page 90 95 96 98 99 101 102 104 106 xii C o m p ariso n of the existing e x p e rim e n ta l d ata [4 8 ,6 0 ] and p red icted breakdow n band bending for abrupt ju n ctio n s in G aA s a t ro o m te m p e ra tu re . R e v e rse I-V c h a ra c te ris tic of a LA diode a t 300°K with ND = 2 x 101® a to m s /c m 3. Ju n ctio n d iam eter is 125 m ic ro m e te r s . B reakdow n band bending fo r Si a b ru p t ju n ctio n s at 300°K. Legend: O ex p erim e n tal r e s u lts , p red ictio n with o p tical phonon m ean free paths 48 A and 47 A for e le c tro n s and holes re s p e c tiv e ly at 3 0 0 ° K , p red ictio n with 45 A and 45 A, - - - - - p red ictio n w ith 42 A and 40 A , P re d ic te d ap p aren t ionization c o e ffic ie n t in Si a t 300°K a s a function of th e e le c tr ic field stre n g th . L egend: optical phonon m e a n fre e p ath 48 A and 47 A fo r e le c tro n s an d h o le s re sp e c tiv e ly at 3 0 0 ° K .--------- p re d ic tio n w ith 45 A and 45 A - - - - p red ic tio n w ith 42 A and 40 A. xiii LIST OF SYMBOLS io n izatio n coefficient sp a tia lly independent ap p aren t io n iz a tio n co efficien t obtained v ia the pseudolocal ap p ro x im atio n e x ac t io n iz atio n coefficient conventional th e o re tic a l io n izatio n co efficien t io n iz atio n co efficien t for e le c tro n s sp a tia lly independent e le c tro n a p p a re n t io n izatio n c o efficien t obtained v ia the p seu d o lo cal ap p ro x im atio n e x ac t io n iz a tio n coefficient for e le c tro n s conventional th e o re tic a l io n izatio n c o efficien t for e le c tro n s th e o re tic a l nonlocalized single c a r r i e r io n izatio n p ro b a b ility fo r e le c tro n s io n iz atio n co efficien t for holes sp a tia lly independent hole a p p a re n t io n izatio n c o efficien t obtained v ia the p seu d o lo cal ap p ro x im atio n e x ac t io n iz a tio n coefficient for h o les conventional th e o re tic a l io n izatio n co efficien t for h o le s th e o re tic a l nonlocalized single c a r r i e r io n izatio n p ro b a b ility fo r holes n o n lo calized single c a r r ie r io n izatio n p ro b ab ility lo g a rith m ic te m p e ra tu re co efficien t of c a r r i e r m ean fre e path fo r optical phonon sc a tte rin g c a r r i e r m e a n fre e path for o p tical phonon sc a tte rin g a t 0°K e le c tro n e n e rg y b a r r ie r betw een a m etal and a sem ico n d u cto r effectiv e b a r r i e r h eig h t fo r edge conductance c ap a cita n c e d a rk space d ista n c e d a rk space d ista n c e w here ab so rp tio n of optical phonons is ta k e n into account d a rk space d ista n c e for e le c tro n s d a rk space d ista n c e for h o le s e le c tric field stre n g th e le c tric field stre n g th a t th e point x m in im u m v alu e of the p ro d u c t of the e le c tric field and the sp ace c h arg e reg io n w idth at breakdow n e n e rg y a sso c ia te d w ith an in je cte d c a r r ie r band gap e n erg y th e e le c tric field stre n g th a t w hich the th re sh o ld e n e rg y is e x tra c te d fro m th e field in d istan c e X io n iz atio n th re s h o ld en erg y io n iz atio n th re s h o ld e n erg y for e le c tro n s io n iz atio n th re s h o ld e n erg y for h o les th e e le c tric field stre n g th a t w hich e n erg y equal to j th e o p tical phonon e n erg y is e x tra c te d fro m the fie ld in d ista n c e \ i E^ o p tic al phonon e n erg y i E ro o p tic a l phonon e n e rg y at 0°K ! f(X) p ro b a b ility a t X of finding a c a r r i e r w hich o rig in ated a t X = 0 an d h a s not yet pro d u ced an e le ctro n -h o le p a ir ! G conductance i I(V) ju n ctio n c u r r e n t at b ia s V l • Ij(V ) ju n ctio n c u r r e n t p re d ic te d by one dim ensional ju n ctio n th e o ry | IC(V) su rfa c e le a k a g e c u rre n t o ^data e x p e rim e n ta lly m e a s u re d sa tu ra tio n c u rre n t density : J , . .rec o m b in a tio n com ponent of J , , reco m b in atio n c d ata ! J sa tu ra tio n c u rr e n t d en sity 1 k B o ltzm an c o n sta n t ; M m u ltip lic a tio n fa c to r M c c ritic a l m u ltip lic a tio n fa c to r i M c ritic a l m u ltip lic a tio n fa c to r fo r e le c tro n s j ; M c ritic a l m u ltip lic a tio n fac to r fo r h o les j | Mn e le c tro n m u ltip lic a tio n fa c to r Mp h o le m u ltip lic a tio n fac to r ' n(x) e le c tro n d e n s ity at the point x | Njj io n ized im p u rity d en sity n (x) d en sity of e le c tro n s w hich a re e lig ib le to produce e e le c tro n -h o le p a irs a t x N su rfa ce sta te d en sity ss n, to ta l c a r r ie r d e n sity , i . e . the su m of the e le c tro n T d en sity and hole d e n sity N, to ta l num ber of activ e tra p s I p (x) hole d en sity at the point x p (x) d en sity of h o les w hich a re e lig ib le to produce e le c tro n -h o le p a irs a t x q e le c tro n c h arg e r ra tio of th e o p tical phonon e n e rg y to the ionization th re sh o ld e n erg y ! r ra d iu s of a ju n ctio n j J j R s e rie s re s is ta n c e ! s j T te m p e ra tu re I V applied b ia s | V avalanche breakdow n voltage j XX i V avalanche breakdow n band bending I B ( V _ . m in im u m breakdow n band bending Bm m z e ro b ia s diffusion p o ten tial I i W space c h arg e width j <X> a v erag e d ista n c e fo r io n izatio n sc a tte rin g y the lo g a rith m of the efficien cy w ith w hich e n erg y fro m j th e e le c tric field is u se d to p ro d u ce additional j c a r r ie r s ra tio of the ionization th re sh o ld e n e rg y to the e n e rg y gained fro m the e le c tric field in one m ean fre e path fo r o p tical phonon sc a tte rin g xviii SECTION I INTRODUCTION 1-1 O BJECTIV ES A valanche io n iz atio n in se m ic o n d u cto rs is known to be one of the two m e c h a n ism s w hich a re re sp o n sib le for junction b reakdow n. The o th e r is th e tunnel e ffect and is known to be re sp o n sib le only fo r v e ry low breakdow n voltage junctions [ 1], The avalanche e ffe c t lim its the m ax im u m voltage for the use of ju n ctio n s as r e c tif ie r s . The n o n lin e ar c u rre n t v s. voltage re la tio n sh ip a s s o c ia te d w ith av alan ch e io n izatio n i s , how ever, u tiliz e d in m an y p r a c tic a l d e v ic e s. Am ong th e m a re se m i conductor m icro w av e g e n e ra to rs [ 2 ,3 ] , such a s IM P ATT and I TRA P ATT d io d es, th e av alan ch e photo diode [ 4 ,5 ,6 ] w hich is a photo d e te c to r w ith a b u ilt-in gain and a re fe re n c e voltage (Z ener) diode [7 ] w hich m a in ta in s p ra c tic a lly a co n stan t v oltage independent of th e c u rr e n t th ro u g h th e junction. T hese ap p licatio n s have y ield ed m an y im p lic a tio n s fo r high field tr a n s p o rt.- ’ F o r optim um u se of ju n ctio n d e v ic e s, h o w ev er, a de- 1 ta ile d know ledge of av alan ch e io n izatio n is e s s e n tia l. The o b jectiv e of th is d is s e rta tio n is to provide an im p ro v ed p h y sic a l u n d e rstan d in g of the avalanche effect in se m i c o n d u c to rs. In p a r tic u la r , the p re v io u sly e x istin g understanding of th is e ffect is c ritic a lly exam ined in the lig h t of lim ita tio n s im p o sed b y e n erg y c o n se rv a tio n . A cco rd in g ly a sim ple an a ly tic a l e x p re s s io n to e stim a te the a v e ra g e d istan c e fo r ionization j s c a tte rin g in se m ic o n d u c to rs is obtained. The re la tio n sh ip ! 2 w hich e x is ts betw een conventional th e o re tic a l e s tim a te s of io n iz a tio n co efficien ts and e x p erim e n tal m e a s u re m e n ts is clarified.** T h is re la tio n sh ip le a d s us to m o dify th e d e fin itio n of th e io n iz a tio n co efficien t. To exam ine the im p ro v e m e n t m ad e by the p re s e n t a n a ly sis, the functional fo rm of th e o b se rv a b le a p p aren t io n izatio n co efficien ts as a function of e le c tric field stre n g th and la ttic e te m p e ra tu re is p re d ic te d in v a rio u s m a te r ia ls . The a g re e m e n t w ith e x p erim en t is e x ce llen t;' The breakdow n band bending fo r a b ru p t and p - i- n ju n ctio n s is a lso p re d ic te d . The re s u lts show sig n ifican t d iffe re n c e s fro m th e p re v io u s th e o re tic a l e s tim a te s for high field c o n fig u ra tio n s. In th e c o u rse of the w ork fo r th is d is s e rta tio n p re c is e breakdow n band bending m e a su re m e n ts w ere m ade w ith th e u se of sp e c ia lly developed m e ta l-n -s ilic o n p a ssiv a te d Schiottky b a r r i e r c o n fig u ra tio n s. The r e s u lts show e x c e lle n t a g re e m e n t w ith p r e s e n t th e o re tic a l pred ictio n s.^ 1-2 HISTORICAL BACKGROUND The avalanche effect h a s now b een know n fo r m o re th a n 20 y e a rs and h a s been d is c u s s e d in m an y te x t books: an ex ten siv e rev iew of both the e x p e rim e n ta l and th e o re tic a l a n a ly sis of th is effect p rio r to 1968 h a s b e en p ro v id ed by Chynoweth [ 8 ]. The f ir s t e x p e rim e n ta l r e p o r t of th is effect w as given by M cK ay and M cA fee in 1953 for Si and Ge p -n ju n ctio n s [9 ]. In 1954 M cK ay [1 0 ] ex p lain ed th is e ffe ct as being analogous to a gas d isc h a rg e [ l l ] , M cK ay a lso in tro duced the idea of "ionization c o e ffic ie n ts" a s "the n u m b er of th e io n izatio n sc a tte rin g s which tak e p lac e w hile an a v e ra g e c a r r i e r tra v e ls u n it path len g th along th e e le c tric field d ire c tio n . t ' The io n izatio n co efficien t w as f i r s t in tro d u c e d a s an ex p licit 3 ; function of th e e le c tric field stre n g th only. Once the io n izatio n | co efficien t, a , is defined as above the g e n e ra tio n eq u atio n fo r a i | ste a d y sta te can be w ritte n a s l(nW tpW ]. (1) | H ere dn(x)/dx and -d p (x )/d x a re g ra d ie n ts of e le c tro n and hole ; d e n sitie s n(x) and p(x) re sp e c tiv e ly a t the point x. T his eq u atio n can be in te g ra te d through the ju n ctio n as (2 ) : H ere M is th e m u ltip lic atio n fa c to r, defined as M = n(W )/n(0), j w h e re n(W) and n(0) a re d e n sitie s of c a r r i e r s e x tra c te d fro m ; the ju n ctio n (at x = W) and in jected into the ju n ctio n (at x = 0). 0 and W denote th e p o sitio n s w here the e le c tr ic field te r m in a te s . j In 1955 M ille r [1 2 ] pointed out th a t e le c tro n s and h o le s m ay ; have d iffe re n t io n iz atio n coefficients a and a re s p e c tiv e ly . n p | Once th is d iffe re n c e is introduced, E q s. 1 and 2 a re re p la c e d by th e follow ing equations: 1 ! and i' i i d n(x) _ d p(x) / % — \./ = _— _ a + a p(x) dx dx n pc (3) i rw rx -Mn=J 0 ^J0 (x) exp I (a ( x f) - O t (x ')) d x ' dx p n (4) i rw r w ~M ~ l a p W exP '(a (x') - - a _ ( x ') ) d x 'd x p Jo p Jx p (5) w a (x ) dx • 4 ■ H e re e le c tro n s a re a ssu m e d to be in je c te d fro m x = 0 and h o les ! fro m x = W . M and M a re e le c tro n and hole m u ltip licatio n I n p j fa c to rs and a re defined a s I I Mn S n(W )/n(0) when p(W) = 0 (6) > and I M = = p(0)/p(W ) when n(0) = 0 . (7) P ! Note th a t a s the ju n ctio n v o ltag e te n d s tow ards the breakdow n j v o lta g e , both and ten d to w a rd infinity. One c an a lso w rite E q. 3 in c u rre n t density fo rm and ! in tro d u c e d iffe re n t s a tu ra tio n v e lo c itie s fo r e le c tro n s and h o le s. : L ee and h is c o w o rk e rs [1 3 ] have exam ined th is situation and show n th a t th e b a sic equ ations ( i . e . , E q s. 4 and 5) a re e x actly ; the sa m e . W hen e le c tro n -h o le p a ir s a re g en erated inside the ; sp ace c h a rg e re g io n , n e ith e r e le c tro n n o r hole m u ltip licatio n fa c to rs o c c u r. R a th e r the o b se rv e d m u ltip licatio n fac to r b e - ; co m es a m ix tu re of M and M . T his situ atio n has been n p in v e stig a te d and fo rm u la te d by W oods et a l [1 4 ], E x p e ri- | m e n ta lly , th is situ a tio n o c c u rs freq u e n tly , as w ill be d isc u sse d | l a t e r , but r e p r e s e n ts an unde s ir able situ atio n for developing j an u n d e rsta n d in g of th e io n iz atio n e ffect in se m ic o n d u cto rs. i T h ese m u ltip lic a tio n eq u atio n s a re im p o rtan t, since the ; only e x p e rim e n ta lly o b se rv a b le q u a n titie s a re breakdow n v o lta g es I and m u ltip lic a tio n fa c to rs a s a function of applied voltage. I E ven though nobody h a s re p o rte d it yet, it is in p rin cip le i p o ssib le to deduce io n iz atio n c o efficien ts once a s e rie s of I breakdow n voltage m e a s u re m e n ts a re p e rfo rm ed on ju n ctio n s of ! d iffe re n t doping. In A ppendix C th is technique is d e m o n stra te d j for Si and the u se fu ln e ss of th is m ethod is d is c u s s e d . M ulti- ! | p licatio n fa c to r m e a s u re m e n ts have been p e rfo rm e d by in tro - j ducing e x ce ss c a r r i e r s by v a rio u s m ethods: v ia a lig h t I I p u lse [9 ], using an in je ctin g ju n ctio n [1 2 ] and by <X-particle j ! g en eratio n [1 0 ], T h ese tec h n iq u e s can be u sed to study the j I e le c tric field dependence of th e io n izatio n c o e ffic ie n ts. One can j i a lso m e a su re M and M in th e sam e junction by introducing n p ; e le c tro n s and h o le s in d ep en d en tly a t d iffe re n t ends of the junction. P ra c tic a lly a ll th e e x istin g e x p e rim e n ta l re s u lts w ere ! i I i obtained by th is m ethod. T y p ical e x p e rim e n ta l r e s u lts in j ; Ge [1 0 ], Si [1 3 ,1 5 ,1 6 ] , G aA s [1 7 ] and G aP [1 8 ] a re shown in i | F ig s . 1-a th ro u g h 1-d. j ! W here m o re th a n one e x p e rim e n ta l r e s u lt e x is ts (i.e . in j s Si) th e a g re e m e n t b etw een p re d ic tio n s obtained by d ifferen t ; a u th o rs is not alw ays good. T his d ifferen c e c o m e s fro m the j d ifficu lties a sso c ia te d w ith th e m u ltip lic atio n m e a su re m e n ts . The I ! m a jo r so u rc e s of e r r o r a re lis te d below . ] i 1. U n c e rta in ty of th e e le c tric field configuration: | The d iffe re n c e b etw een the p ra c tic a l e le c tric field j i co n fig u ratio n and th a t u sed to deduce the ionization j c o efficien ts c an in tro d u c e a p p re c ia b le e r r o r . W oods I : I and h is c o w o rk e rs [1 4 ] have u se d Schottky jun ctio n s i i I i w ith d iffused g u a rd rin g s to e n su re a sim ple and ! w ell defined field co n fig u ratio n . G ran t [1 9 ] has | obtained a d e ta ile d know ledge of the ju n ctio n con- j fig u ra tio n v ia th e ju n ctio n cap acitan ce v s . voltage j re la tio n sh ip and u se d th is in fo rm a tio n along with i ! ~ p h o to -m u ltip lic a tio n d ata to deduce the ionization j i - c o efficien ts in S i. Once diffused ju n ctio n s or alloyed ju n ctio n s a re u s e d , it is e s s e n tia lly im p o ssib le to obtain th e e x a c t im p u rity d istrib u tio n fro m the 6 i E o 6 e a 6 (10° V/cm) F ig . 1 -a. E x p e rim e n ta lly re p o rte d io n izatio n co efficien t, a, a s a function of the e le c tric field stre n g th , < ? , fo r e le c tro n s (e) and h o les (h) in Ge a t 300°K . (A fter M ille r [1 2 ]) i i j i V.."I I I i I i I I E o 1 0 £ ( lO 5 V/cm) F ig . 1-b. E x p e rim e n ta lly re p o rte d io n izatio n co efficien t, a , a s a function of the e le c tric field stre n g th , < ? , fo r e le c tro n s (e) and h o le s (h) in Si a t 300°K . (A fter L ee e t a l . , [1 3 ], M oll and O v e rs tra e te n [1 5 ] and O v e rstra e te n and DeM an [1 6 ]) 8 t i Ga As E a I 10 6 (l05 V/cm) F ig . 1 -c , E x p e rim e n ta lly re p o rte d io n izatio n c o e ffic ie n t, a , a s a function of the e le c tric fie ld stre n g th , #, in GaAs a t 300°K . (A fter L ogan and Sze [1 7 ]) i t 9 e o GaP 30 F ig . 1-d. E x p e rim e n ta lly re p o rte d io n izatio n co efficien t, a , a s a function of the e le c tric field stre n g th , £, in G aP a t 300°K . (A fter Logan and W hite [1 8 ]) 10 | cap a cita n c e v s, voltage re la tio n sh ip . In ste a d , one h a s to a ssu m e th e d istrib u tio n for one type of im - » p u rity . E ven when a Schottky b a r r ie r c o n fig u ra tio n j j is adopted to avoid th is d ifficu lty , once a d d itio n al deep im p u rity le v e ls e x is t, an u n c e rta in ty fo r the j d e te rm in a tio n of the im p u rity d istrib u tio n m a y e x is t i a s w as pointed out by R o b e rts and C ro w ell [2 0 ], j V oltage dependence of p h o to -in jectio n : j i E v en w hen the avalanche m u ltip lic atio n e ffe c t is allow ed fo r the p h o to cu rre n t m ay s till depend on th e j applied v o ltag e. T h ere e x is ts th re e p h y sic a l re a s o n s w hich c au se th is effect, i, e , , a) The w idth of the sp ace ch arg e reg io n in c re a s e s w ith th e ap p lied b ia s . j Thus fo r e le c tro n -h o le p a irs g e n erate d o u tsid e th e j j ju n ctio n th e d istan ce betw een the p o sitio n w h ere c a r r i e r g en eratio n o c c u rs and the ju n ctio n edge j i changes w ith applied b ia s . A cco rd in g ly e le c tro n o r j hole c u rre n t which w ill be c o llected by th e ju n c tio n j i changes w ith applied b ia s . T his e ffe ct h a s b e e n I I tak e n into account by m any a u th o rs [1 3 ,1 4 ] , W hen j th is e ffe ct is n eg lected , one o v e re s tim a te s th e io n - j iz a tio n co efficien ts in the low e le c tric field re g io n ' [1 5 ,1 6 ], b) Once the space ch arg e re g io n i s illu m in - i a ted by th e in jectin g lig h t, p h o to -in je c te d c u rre n t | changes a s a function of th e applied b ia s . T h is i s j b e ca u se th e w idth of the s p a c e 'c h a rg e c h a rg e re g io n j j is a function of the applied b ia s . H ow ard h a s fo rm u la te d th is situ atio n fo r u n ifo rm lig h t g e n e ra tio n j [ 2 1 ] , W oods e t a l [1 4 ] have fo rm u la te d fo r a m o re g e n e ra l case but have n o t c o n sid e re d how th is a ffe c ts the m e a su re m e n t of M and M • T h is ! ii | j c o rre c tio n is im p o rta n t since i t is v e ry d ifficult to i i '■ ! ' I ob tain e ith e r p u re e le c tro n o r hole in je ctio n s u n le ss j I S chottky c o n fig u ra tio n s a r e u se d , c) When a Schottky j b a r r i e r c o n fig u ra tio n is u se d , w hich e n s u re s a pure e le c tro n o r hole in je c tio n , the c u rr e n t in jected s till j depends on the ap p lied b ia s [ 7 ] . T h is e ffect w as I p a r tly ta k e n into acco u n t by W oods e t al [1 4 ], but a s w a s p o in ted out by A n d erso n [2 2 ] m o re detailed c o r re c tio n s fo r q u a n tu m -m e c h an ica l tunneling and phonon s c a tte rin g should be m ad e. 3. E ffe c t of th e th re s h o ld e n e rg y fo r io n izatio n s c a tte rin g : La th e h ig h e le c tr ic field re g io n the effects of the i th re s h o ld e n e rg y fo r io n izatio n sc a tte rin g [8 ] b eco m e important.*- T h is e ffe ct w as f ir s t fo rm ulated by O v e rs tra e te n and D eM an [1 6 ], The sam e kind of 1 a rg u m e n ts w ere p ro v id ed by W oods e t al [1 4 ] and ; G ra n t [1 9 ] in d ep en d en tly . La th e ir tre a tm e n ts th is c o rr e c tio n h as b e e n lim ite d to th a t fo r in je cte d c a r r i e r s only. H ow ever, as w ill be d isc u sse d in fu rth e r se c tio n s th e re a r e re a s o n s to co n sid er other a s p e c ts o f th re s h o ld e n erg y lim ita tio n s . La addition as i s show n in A ppendix B th e ir m a th e m a tic a l a n a ly s e s a r e n o t c o m p lete even for the c a se th ey ! c o n sid ered .^ S e v e ra l th e o r e tic a l a tte m p ts have b een m ade to estim ate ' io n izatio n co efficien ts.* The g e n e ra l a ssu m p tio n s th a t have been m ade a re a s follow s: I 1.' O p tical phonon sc a tte rin g [2 3 ] and io n izatio n j j s c a tte rin g a re th e only e n erg y lo s s m e c h a n ism s. j I j The e n e rg y lo s s e s E r and E ., a sso c ia te d with ! 12 I o p tic al phonon s c a tte rin g and io n iz a tio n sc a tte rin g I re s p e c tiv e ly , a r e a ssu m e d to be sin g le v alu ed , | j U su ally the th re s h o ld en erg y fo r th e io n izatio n sc a tte rin g h a s b e en u se d as an a d ju sta b le p a ra m e te r o r assu m e d to be 3 /2 of the band gap e n e rg y . T his m e a n s th a t the n e a rly fre e e le c tro n m odel is a ssu m e d [2 4 ], R ecen tly A n d erso n and C row ell [2 5 ] have e stim a te d th is quantity b a se d on d e ta ile d th e o re tic a l en erg y b and c a lc u la tio n s, 3, T he m ean fre e path fo r optical phonon s c a tte rin g , X, is a ssu m e d to be co n stan t [2 6 ,2 7 ] and u se d as an ad ju stab le p a ra m e te r . The f i r s t th e o re tic a l e stim a te w as given by W olff in 1954 ' [2 8 ], W olff solved the B o ltzm an tra n s p o rt equation fo r c a r r i e r s I in a high field w here the a v e ra g e e n erg y is m uch la r g e r than ! E ^ . M oll and O v e rstra e te n [1 5 ] have in v e stig a te d th is situ atio n in m o re d e ta il. T h eir r e s u lt is ra th e r e la b o ra te , b u t a s w as j show n by Chynoweth [8 ], th e ir high field lim it is c o n siste n t w ith ■W olff's re s u lt. F o r c a r r i e r s in a low field Shockley [2 9 ] p ro - ; p o sed a m odel in which e s s e n tia lly a b a llis tic so lu tio n w as a ssu m e d , M oll and M eyer [3 0 ] have solved a sim p lified th re e d im en sio n al b a llis tic p ro b le m and obtained a r e s u lt w hich is I e s s e n tia lly th e sam e [8 ] a s th a t obtained by Shockley. T h e ir ! r e s u lts a re a ~ exp I - 3EiE r \ (8) 2 2,2 q * in th e high field reg io n and in th e low field reg io n , w here q and $ a re the e le c tro n ic c h a rg e j and the e le c tric field stre n g th re s p e c tiv e ly . B ecause of th e ir sim p lic ity th e s e e x p re ssio n s a re u se fu l but a t the sam e tim e th e y cannot ex p lain the e x p e rim e n ta l data over a wide ran g e of j e le c tr ic field stre n g th [3 1 ]. In 1962 B araff [3 1 ] p ro p o sed a j m o re co m p lete th e o re tic a l tre a tm e n t. H is tre a tm e n t w as m o re j co m p licated th a n p re v io u s o n es. E s se n tia lly , B araff solved j th e B o ltzm an tr a n s p o rt equation for the a v erag e d istan ce for io n iz atio n s c a tte rin g . T hen the ionization coefficient w as defined I a s th e in v e rs e of th is d ista n c e . The final r e s u lt w as e stim a te d i i n u m e ric a lly . H is r e s u lts cover a w ider e le c tric field ran g e | th a n any p re v io u s single ap p ro ach and have been used by m any j a u th o rs to in te r p r e t th e ir experim ented r e s u lts . They find in j g e n e ra l, h o w e v er, th a t to fit th e ir re s u lts th ey had to choose j d iffe re n t v a lu e s of X fo r d iffe re n t junctions in the sam e se m i- i conductor a t th e sam e te m p e ra tu re [1 3 ,1 7 ,1 8 ,3 2 ], T his j i situ a tio n , h o w e v e r, is u n p h y sical since X is p resu m ab ly a j i la ttic e re la te d p a ra m e te r and m u st re m a in co n stan t in the sam e m a te ria l a t th e sam e te m p e ra tu re . T his situ atio n w as h a rd to j c la rify sin ce B a ra ff1 s r e s u lts w e re n u m e ric a l and difficult to | exam ine a n a ly tic a lly . By providing an aly tical e x p re ssio n s fo r I j the te m p e ra tu re dependences of E^ and X C row ell and Sze [3 3 ] j p re s e n te d an a n a ly tic a l e x p re ss io n w hich re p ro d u c e s B a ra ff's j r e s u lts . T h e ir re s u lts have a lso been u se d by m any a u th o rs . . i [ 5 ,7 ,3 4 ] to ex p lain th e ir e x p e rim e n ta l re s u lts but b e ca u se of j th e n a tu re of th e e x p re ss io n ( i . e . , a m ath em a tica l nine points fitting) the e x p re ss io n could not be evaluated p h y sically . 14 1-3 OUTLINE O F THE PR E S E N T WORK In Section II the B a ra ff r e s u lts w ill be exam ined p h y si c ally . An a n aly tic al e x p re s s io n w hich s a tis fie s p re v io u s p h y sical m odels (cf.** E q s. 8 and 9) and a lso an e n e rg y c o n se rv a tio n relatio n sh ip w ill be d ev elo p ed w ithin the fra m e w o rk of the B a ra ff m odel [3 1 ]. U sing re a so n a b le v a lu e s of E . [2 5 ], E^ [3 3 ] and a single value of X fo r e a c h m a te r ia l th is e x p re ssio n fits ex p erim en tal d ata fo r G aA s, Ge and Si o v er m o re th an th re e decades of th e io n iz atio n c o e ffic ie n ts, but does n o t fit ex istin g data for G aP and th e h ig h field re g io n for Si.' The e x p re ssio n also p re d ic ts th e te m p e ra tu re dependence of the io n iz a tio n coefficients with good a c c u ra c y in the c a se of GaAs [3 5 ], F o r the sim p lic ity of the e x p re s s io n th is su c c e ss is im p re s s iv e , but when th is w o rk w as d o n e, th e d isc re p a n c y fo r Si in the high field reg io n a p p e a re d to be a sy s te m a tic e r r o r since Si h a s b een studied the m o s t e x te n siv e ly of th e se m ic o n d u c to rs and the high field data h av e been r e p o rte d b y s e v e ra l in v e s tig a to rs . A ccordingly in S ection III i t is show n th a t th is d ifficu lty o rig i n a te s fro m the fa c t th at th e io n iz a tio n co efficien t obtained by e x p erim e n t and th e th e o re tic a l d efin itio n of the io n izatio n c o effi c ien t a re not c o n siste n t. T h is d iffic u lty is re so lv e d by in tr o ducing a nonlocalized co n cep t (N LC). The e x ac t re la tio n sh ip betw een th ese tw o q u a n titie s is fo rm u la te d .1 T h is fo rm u la tio n in d ic ate s th at th e e x p e rim e n ta lly obtainable io n izatio n co efficien ts a re not only e x p lic it functions of th e e le c tric field s tre n g th , as w as understood p re v io u s ly , b u t a ls o functions of the p o sitio n in sid e junctions o f in te r e s t. T he fe a s ib ility of s till re ta in in g th e concept of an io n izatio n c o efficien t w hich is only a function of th e e le c tric fie ld s tre n g th (cf. a P s e u d o lo c a l A pproxim ation) is examined;*1 T he a g re e m e n t b etw een the p re d ic tio n and ex istin g io n iz atio n co efficien t m e a su re m e n ts is e x ce llen t. V ia in te rn a l : in c o n s is te n c ie s th is a p p ro x im atio n show s its own ap p lica tio n i ; lim it. To exam ine the s e rio u sn e s s of the ap p lica tio n lim it an i e x a c t fo rm u la tio n is a lso solved for se le c te d p - i - n ju n ctio n s in * Si and G aA s, The r e s u lts in c o rp o ra te th e e x iste n c e of a ; b o u n d ary la y e r w ith a th ic k n e ss w hich c o rre sp o n d s to one io n iz atio n th re sh o ld p o te n tia l drop in w hich no in je c te d c a r r ie r c an o rig in a te io n izatio n sc a tte rin g . In th e r e s t of th e s tru c tu re , ; th e io n iz atio n c o efficien ts a re fa irly c o n stan t and th e ir values a re c lo se to th o se obtained v ia the p seu d o lo cal a p p ro x im atio n . | A cco rd in g ly in S ection IV a sty liz e d m o d el to ap p ro x im ate the | d istrib u te d io n izatio n e ffect in sid e ju n ctio n s w ith re a so n a b le i a c c u ra c y and a sim p le e x p re ss io n to r e p re s e n t a p seu d o lo cal io n iz a tio n co efficien t a re p re s e n te d ;' The breakdow n band I bending fo r G e, Si, GaAs and G aP a b ru p t and p - i - n jun ctio n s : a r e e stim a te d w ith th is m odel.'; The p re d ic tio n s show a p p re c i- ! able d ifferen c e fro m p rev io u s e s tim a te s [3 6 ], sp e c ia lly for high field c o n fig u ra tio n s.' To exam ine the ad eq u acy of our j p r e s e n t tr e a tm e n ts , p re c is e breakdow n band bending m e a s u r e m e n ts have b een m ade for Si Schottky ju n ctio n s (S ection V ). < In ste ad of em ploying a conventional guard rin g ty p e c o n fig u ra tio n [1 4 ,3 7 ], tw o kinds of sp e c ia lly developed c o n fig u ratio n s I w e re u se d : a la rg e a re a p o st ev ap o ratio n p a s s iv a te d Si Schottky diode (LA diode) and a sm a ll a re a point co n tact type ! p a ssiv a te d Si Schottky diode (SA d io d e),r W ith th e su c c e ssfu l I u se of th e se co n fig u ratio n s our th e o re tic a l p re d ic tio n s w ere co n firm ed for th e ran g e of im p u rity d e n sity of 1 x 1 0 ^ ' 3 18 3 I a to m s /c m to 3 x 10 a to m s /c m in n -ty p e Si. It is a lso j found e x p e rim e n ta lly th a t th e breakdow n m e c h a n ism in Si is j 18 j v ia avalanche fo r im p u rity d e n sitie s le s s th an 3 x 10 i 3 | a to m s /c m . 16 T h is w ork in d ic a te s th e adequacy of our p h y sic al a n a ly sis and a ls o in d ic a te s th a t one can u tiliz e th is tre a tm e n t for c h a ra c te riz a tio n of m o re co m p licated d e v ic e s. SECTION II AVERAGE DISTANCE FOR IONIZATION SCATTERING A sim ple a n a ly tic a l e x p re ss io n w hich re p re s e n ts the a v e ra g e d istan c e for the io n izatio n sc a tte rin g as a function of th e e le c tric field stre n g th and the la ttic e te m p e ra tu re is d e v e l oped in th is se ctio n [3 8 ].v T his e x p re ssio n is obtained by su p erim p o sin g an e n erg y c o n se rv a tio n rela tio n sh ip and two p r e v io u sly e x istin g p h y sic al m odels [2 8 ,2 9 ] on the B a ra ff r e s u lt [3 1 ] .' To exam ine the a p p lic a b ility of th e re su ltin g e x p re s s io n , th e e le c tric field dependence and th e te m p e ra tu re dependence of th e io n izatio n co efficien ts is p re d ic te d by adopting th e co n v en tio n al th e o re tic a l definition of th e ionization c o e ffic ie n ts. The p re d ic tio n show s e x c e lle n t a g re e m e n t with m o st of the e x istin g e x p e rim e n ta l r e s u lts but n o t w ith the h ig h fie ld d ata in S i/’ We co n sid er the im p lic a tio n s of th is in S ection III. I I - 1 ENERGY CONSERVATION CONSIDERATION As is m entioned in S ection I the object of the e x istin g th e o re tic a l tre a tm e n ts is to e s tim a te th e average d ista n c e , < X > , in a stead y sta te .' The io n izatio n coefficient is defined as (X = 1/<X>. T hen the ra tio of th e e n erg y lo st v ia th e io n iz a - S t tio n sc a tte rin g to the e n erg y gained fro m the e le c tric field c a n be e x p re s s e d in a sim ple m a n n e r, i.e .’ < i , a 4 E i^ q S S C (V > * * E i/q * 17 (10) | -18. ' ; P h y sic a lly th is q u an tity h a s to sa tis fy follow ing conditions: i | 1. W hen th e e le c tr ic field in c re a s e s , if the c ro s s se ctio n fo r io n iz a tio n sc a tte rin g is a stro n g function j 2 of the e n e rg y , (e.*g.'' (E - E .) above E . [2 9 ]), the chance of lo sin g e n e rg y to the la ttic e sy ste m should becom e s m a lle r . W hen the e le c tric field ten d s to w ard in fin ity E q, 10 should ten d tow ard unity: a • E ./q < ? -» l a s <?-»•« . (11) A 1 2, W hen E b e c o m e s sm a ll th is ra tio should becom e r u n ity fo r any & sin ce th e re e x is ts then no o th er e n e rg y lo s s m e c h a n ism except by ionization: a • E./qd?-* 1 a s E -» « , (12) A 1 r T hese e n erg y c o n se rv a tio n conditions m u st be sa tisfie d by W olff's [2 8 ] and B a ra ff's re s u lts [3 1 ], T his cannot be exam ined by W olff's r e s u lt sin ce the p re -e x p o n e n tia l fac to r is : not know n. On the o th e r hand th is can be exam ined w ith the B a raff re s u lt by re p lo ttin g h is u n iv e rs a l curve into o rd in ate tim e s a b s c is s a vs," a b s c is s a (cf. E q ;v 10)," The re s u lt is shown in F ig , 2 by b ro k en lin e s . T he r e s u lt does n o t sa tis fy e ith e r E q, 11 or Eq,'- 12." T h is is m a th e m a tic a lly u n d erstan d ab le since ; he had chosen the io n iz atio n m ea n fre e p ath fo r c a r r i e r s w ith e n erg y m o re th a n E . to be th e sam e a s th a t fo r the o p tical phonon sc a tte rin g and a ls o a ll e n erg y to be lo s t subsequent to ■ ionization,' In th is se c tio n we tak e the B a raff re s u lts in the I low field reg io n w h ere th e sp e cific m ean fre e p ath fo r e n e rg ie s j above E . is n o t im p o rta n t and look for an a p p ro p ria te e x p re ssio n f for a (or <X>) w hich s a tis fie s E q s, 8, 9> 11 and 12, B esides j A i th o se four a sy m p to tic re la tio n s h ip s we re q u ire th a t c c to be ! A an even, m o n o to n ic -in c re a sin g function of $ and (E j/E r ). I _____I— I ------1 — I ---1 I- _ 1 r * E r/E, =0.00: 0.01 - 10 - r % 0.02 \ 0 .0 3 \ 10- = 0.04 oo - 3 F ig . 2. R eplotted B a raff r e s u lt (broken lin e s) and our analytic "one point fittin g " (solid lin e s) w ith r a s a p a ra m e te r. T he o rd in ate is the ra tio of the c a r r i e r e n erg y lo s t v ia io n izatio n to the en erg y gained fro m th e e le c tric field . The a b s c is s a is th e n o rm a liz e d in v e rs e e le c tr ic field (the ra tio of a c h a ra c te ris tic e le c tric fie ld , S., to the e le c tr ic field ). The s tra ig h t lin e fo r r « is th e Shockley asy m p to te. I i I | 20 I I - 2 ANALYTICAL APPROXIM ATION In itia lly i t -will be helpful to define a few n o rm a liz e d p a r a m e te r s a s follow s: z = E ./q £ X (13) is th e ra tio of the io n izatio n th re sh o ld en erg y to the en erg y g ained fro m the e le c tric field in one m ean fre e path for o p tic al phonon s c a tte rin g . r = E /E . , (14) r i is the ra tio of the e n e rg y fo r o p tic al phonon to the io n izatio n s c a tte rin g 'e n e rg y , and y s Xn(a^E./q<j?) = y ( z ,r ) (15) is th e lo g a rith m of th e effic ie n cy w ith which en erg y fro m the e le c tr ic field is u se d to pro d u ce additional c a r r ie r s . C o n sid er a s e r ie s expansion w hich can sa tisfy th e above g e n e ra l re q u ire m e n ts , i .e .', ® , 2 2. n + 6 _ o » y = S Yn ( z + a ) + C , (16) n= -» w h ere a = a (r) is a m onotonic function of r and the c o e ffic ie n ts a r e independent of r and x .L In addition 0 < 6 < 1. T he Shockley m o d el (Eq. 9) re q u ire s that d y /d z 4 - 1 a s z -» oo. T h is condition y ield s the follow ing re s tric tio n s 6 = 2 t n < 0 , and Y o = -1.- (17) (18) (19) 21 If we w ish our r e s u lts to a p p ro a c h th o se of the W olff m odel in the high field lim it, we m u st p la c e the additional r e s tric tio n on. y th a t ,2 • • Y' -> -6 r a s z -¥ 0 • (20) dz H e re , the p re -e x p o n e n tia l te r m in E q . 9 h as been a ssu m e d to be q<?/E. to sa tis fy th e e n e rg y c o n se rv a tio n re la tio n in the high field lim it. T his re q u ire m e n t g iv es r is e to th e condition 2 (2n + 1) a 2n_1 = a “ 1- 6 r . (21) n= -oo E n e rg y c o n se rv a tio n a t h ig h fie ld (cf. E q . 11) re q u ire s th a t 2 Y a 2n+1 = -C . (22) II n= -co A ll th ese r e s tric tio n s can be s a tis fie d by E q. 21 and the follow ing equation: . 2 2.2’ . . 2 2.rt"2 2n+l.» /oo\ y = -(z + a )2 + a + 2 Y n {(z + a ) - a )} . (23) n= - o o We have not yet im p o sed th e condition y -> 0 a s r -* 0 w hich is a ls o re q u ire d by c o n se rv a tio n of e n erg y (cf. E q . 12). jlI-3 CURVE FITTIN G TO THE BA RA FF RESULT i In S ection II-2 , we p re s e n te d a g e n e ra l functional fo rm fo r a . F o rm a lly we could obtain an ex act e x p re ss io n if we u se d an infinite num ber of p o in ts fro m th e B a ra ff r e s u lts . ; Since th is is im p ra c tic a l and the high field asy m p to tic reg io n of th e B a raff r e s u lt is obviously in e r r o r , we have tr ie d to fit th e e x p re ss io n to the low er field p o rtio n s of th e B a raff r e s u lts . F ir s t, we trie d a two point fitting for e a c h value of r . ;We m u st re ta in the leading te r m in our e x p re ss io n (Eq. 22) as one of the two te r m s , b e ca u se th is te r m is the only one w hich ; su rv iv e s in the Shockley asy m p to tic reg io n . T he o th er te r m was i c h o sen fro m th e h ig h er o rd e r te r m s to su b seq u en tly sa tis fy the W olff re g im e via an a p p ro p ria te choice of o rd e r and w eighting. ; The e x p re ss io n for the two point fitting can th en be w ritte n as 2 a ( l - 6 a r ) y = a / -(z2 + a 2)^ + ( a - a ') £ (~ ) + 1} 2(a"a (23) a iw h ere a m u st alw ays exceed a '. N ote th a t w hen a ap p ro a ch e s ja* th e h ig h er o rd e r te r m d ro p s out quickly w hen z in c re a s e s beyond a. R e su lts fro m the two point fitting hav e su g g ested ith a t a ' w a . Then if we n e g le c t the h ig h er o rd e r te r m , we have a fa irly sim p le e x p re ssio n i . e . , y = a - (a2 + z2 )^ . (24) j H e re , a is a function of r and can be obtained fo r a single 'v a lu e of r fro m a single point fittin g . The re s u lta n t a is p lo tte d in F ig . 3 a s a function of r . The v e rtic a l lin e s fo r jeach r re p re s e n t the ran ge of a w hich c o rre sp o n d s to c u rv e j fittin g at d iffe re n t values of z. The open dots re p r e s e n t the | c a se w here o u r e x p re ssio n c ro s s th e B a raff r e s u lt a t z = 10. .01 r = Er/Ej 0 .1 F ig . 3. a v s. r fro m the one point fitting. L egend: g r a n g e of a fro m a ll p o ssib le fittin g s to B araff r e s u l t . -------------- fittin g w hich touches the B araff r e s u lts a t a single p o in t, o fitting a t z = 10. -i se le c tio n a s m o st re p re se n ta tiv e fitting. 24 In g e n e ra l, c u rv e s fro m E q. 24 c ro s s the B a r a ff re s u lts tw ice fo r a single v alu e of r . The d o tted lin e c o rre sp o n d s to the i j c a se w here our e x p re ss io n to u c h e s the B a ra ff re s u lt a t only a ' single point. The solid lin e c o rre s p o n d s to the v a lu e we have ; chosen as m o st re p re s e n ta tiv e of th e low field B a raff r e s u lts . ^ F o r th is line a (r) = 0 .2 1 7 r ‘ U 14 (25) : The re s u lts of th is fittin g p ro c e d u re can be co m p ared with the j B a raff re s u lts by an e x a m in a tio n of F ig . 2. The c u rv e fitting schem e re p re s e n ts the B a ra ff r e s u lt fa irly w ell fo r la rg e z and ; a p p e a rs reaso n ab le in th e h ig h field a sy m p to tic reg io n . The ; single point fitting of the c u rv e is not sig n ific an tly p o o rer than ; the two point fitting as fa r as the B a ra ff r e s u lts a re concerned. i E q s. 24 and 25 do not e x a c tly fit the W olff re s u lt (E q. 8) but provide a re a so n a b le a p p ro x im a tio n : in ste a d of an exponent 2 i of -3 rz , as z -* 0, our r e s u lt g iv es an asy m p to tic exponent \ of -2 . 38 r* ’ *4z^. As a m a tte r of fa c t, sin ce the B a ra ff re s u lts I do not appear to have s a tis f a c to r y a c c u ra c y at fields n e ar the : W olff reg io n , a two point fittin g of the B a ra ff r e s u lts is not ju stifie d . Such a fit m u st c o n ta in points fro m n e a r the W olff I reg io n to be sig n ifican tly b e tte r th an a one point fittin g . j I I - 4 PHYSICAL SIGNIFICANCE O F EM PIR IC A L PARAM ETERS Our -final e x p re s s io n , E q . 24, ex h ib its the Shockley asym ptotic reg io n and s a tis f ie s th e e n erg y c o n se rv a tio n r e - | latio n , but only q u a lita tiv e ly show s the W olff a sy m p to te. F ro m \ \ j E q s. 23 and 24 it is not im m e d ia te ly obvious how one can J deduce the e m p iric a l p a ra m e te r s E ., E^ and \ fro m e x p erim e n - J ta l d ata. We can get m o re in sig h t into th is p ro b lem by 25 | in tro d u c in g the follow ing two c h a ra c te r is tic fie ld s: j 3. s E ./q X > (25) | i i th e field a t w hich the th re sh o ld e n erg y is re a c h e d in one m ean fre e p ath , and A = E /qX , (26) r r the field a t w hich the phonon e n erg y is re a c h e d in one m ean fre e path. T hen fro m E q. 24 we have y = ln(OO E./q<?) = 0 .217(3./$ )1# 14 - {[0. 217(& /tf )1# U ] 2 i r * i r F ro m th is re s u lt we see th a t 3. d e te rm in e s th e slope of _ i 1 the y v s . 3 rela tio n sh ip a t low fie ld s , 3 d e te rm in e s the r _] field above w hich ap p reciab le c u rv a tu re o c c u rs in th e y v s. 3 re la tio n sh ip and E . d e te rm in e s the a sy m p to tic fo rm of a v s. 3 * w hen 3 » 3^» When a given se t of e x p e rim e n ta l d ata a re a n aly zed it b eco m es m o re d ifficu lt to obtain m ean in g fu l values of 3. , 3 and E. in th a t o rd e r. Note th at if C hynow eth's L aw i r i a p p e a rs to apply, effects of 3^ and E . cannot be se p a ra te d e m p iric a lly . A rea so n a b le value of can only be deduced fro m C hynow eth's Law if 3 « 3^ • T hen the io n iz atio n ra te m u st be e x tre m e ly sm a ll b e c a u se , on the a v e ra g e , the c h a rg e c a r r i e r m u st lo se a lm o st a ll its e n erg y via phonon g en eratio n . Note a ls o th a t then neithei th e ir r a tio is e sta b lish e d . Note a ls o th a t then n e ith e r E. n o r X a re d e te rm in e d , but only 26 j H -5 CURVE FITTIN G TO EXPERIM ENTAL DATA We have obtained the final e x p re ssio n fo r a s ! ( 0 . 3 1 7 i r r +< 5 -,2t*^ • < 28> Then if the definition of the ionization coefficient is c o rre c t ( i.e . a s 1/<X>), E q. 28 should d e scrib e ex istin g e x p erim e n tal a d a ta . E q. 28 h a s been used to study 10 se ts of e x p erim e n tal d ata for four d iffe re n t m a te ria ls (Ge [1 2 ], Si [1 3 ,1 5 ,1 6 ], GaAs [1 7 ] and G aP [1 8 ]). R ath er than attem pting a th re e point fittin g to d e te rm in e E ^, E . and \ we have chosen to show how the g e n e ra l fo rm of E q . 28 c o m p ares w ith the e x p e ri m e n ta l d ata for a re a so n a b le choice of p a ra m e te rs . We have u sed th e o re tic a l v alu es of E and E. and chosen \ to fit th e r i e x p erim e n tal d ata a t one value of a for each c a s e . T hese ; v a lu e s a re tab u lated in T able I . H ere E values have been r tak en fro m C row ell and Sze [3 3 ] and E. fro m A nderson and C row ell [2 7 ], In F ig s . 4 -a th rough 4-d the e x p erim e n tal ; d a ta a re plotted usin g so lid c u rv e s and values fro m E q. 28 are p lo tted by dashed c u rv e s . The points u sed to d e te rm in e th e \ v alu es a re in d icated by heavy d o ts. We have also plotted two s tra ig h t lin e s in the u p p e r p o rtio n of each d iag ra m . T hese . lin e s a re lim its im p o sed by co n serv atio n of en erg y when E . = E (solid lin e s) and E . = 3 /2 E (dotted lin e s). Thus in 1 g 1 g any case a„ can n e v er exceed the E. = E lin e s . Note th a t our & i g e x p re ss io n and choice of p a ra m e te rs re p re s e n ts the e x p erim e n tal d ata fa irly w ell fo r G e, GaAs and som e data fo r Si [1 3 ], but TABLE I IM PACT IONIZATION PA RAM ETERS AT 300°K M ATERIAL e le c tro n o r hole E i (eV) E r (meV) X (A) S. i (lO ^Y /cm ) *r (104V /c m ) Ge e .8 19 36 2 .2 5 .3 h .9 47 1 .9 4 .0 Si e 1.1 51 48 2 .3 10.6 h 0 0 • 44 4 .1 11.5 GaAs e /h 1.7 22 33 5 .2 6 .7 GaP e /h 2 .6 38 31 8 .4 12.3 28....| “ m ox(? E5)< ' / / / I E o Ge 10-z _ / 10 6 (I0 5 V/cnrO F ig . 4 -a . Ionization c o efficien ts, a , as a function of th e e le c tric field stre n g th , 5 , fo r e le c tro n s (e) and h o les (h) in Ge a t 300°K . L egend: e x p e rim e n ta l ----- — ; ----- — th e o re tic a l fit, • p oint u se d to d e te rm in e m ean fre e path fittin g p a ra m e te r . am a x (Eg) --------- th e o re tic a l m ax im u m a im p o sed b y e n erg y c o n serv a tio n , O jn a x ^ /Z E „) — th e o re tic a l m axim um a fo r th re s h o ld io n izatio n e n e rg y of 3 /2 E g . 4 r o- 29 max E o 10 - = 6 (10 V/cm) F ig , 4 -b . Ionization co efficien t, a, a s a function of the e le c tr ic field stre n g th , < £ , for e le c tro n s (e) and h o le s (h) in Si a t 300°K , The leg en d is th e sam e a s in F ig ,' 4 -a . 30 max I O - - E o GaAs 10 8 ( I05 V/cm) F ig . 4 -c . Io n izatio n c o e ffic ie n t, a, a s a function of the e le c tr ic fie ld stre n g th , 5, in GaAs a t 300°K . The leg en d is the sam e a s in F ig . 4 - a . 31 max £ o GaP 30 3 6 ( I05 V/cm ) F ig / 4 -d , Ionization co efficien t, a, a s a function of the e le c tric field stre n g th , 5, in G aP a t 300°K . The legend is th e sam e a s in F ig . 4 -a . i not for a ll d a ta on Si n o r for G aP. In Si som e data [15, 16] i ' se e m to sa tu ra te a t values of a , w hich would re q u ire E . « 10E . J L i g T his a p p e a rs to be m uch too la rg e since to a f ir s t ap p ro x im atio n E . « 3 /2 E • F o r G aP , on the o th e r hand, the ex p erim e n tal 1 g data shoots up m o re ste ep ly than o u r p red ic tio n and a lm o st c ro s s e s the e n e rg y c o n se rv a tio n asy m p to te. It is p ro b ab le th a t th e re is an a p p re c ia b le e r r o r in in te rp re ta tio n of the e x p e rim e n ta l r e s u lts in both the above Si and G aP r e s u lts . O ur e x p re ss io n can a lso be u sed to p re d ic t th e te m p e r a tu re dependence of a($) if the te m p e ra tu re dependences of X, < 3 and E . a re a ssu m e d . C row ell and Sze [33] assu m e d th a t r l the a p p ro p ria te E^ w as the av erag e en erg y lo s s per c o llisio n w ith an o p tic al phonon and th a t X w as the m ean fre e path for a c o llisio n independent of w h eth er a phonon w as g e n erated o r a b so rb e d . U nder th e se c irc u m s ta n c e s 3 is independent of te m p e ra tu re and ! X = X tan h (E /2kT ) (29) o ro w h ere X and E a re X and E a t 0°K. o ro r E x p e rim e n ta l d ata fro m m e a su re m e n ts on Si [3 9 ] and GaAs [3 5 ] w ere exam ined. The th re sh o ld energy w as a ssu m e d ; to be p ro p o rtio n a l to the band gap e n erg y [4 0 ,4 1 ]. The r e s u lts fo r the c a se of GaAs a re show n in F ig. 5. The d ata a re ! fitted w ith the p a ra m e te rs lis te d in T able I. Thus th e re a re no i a d ju sta b le p a ra m e te rs in th is fit. Our e x p re ssio n re p ro d u c e s j th e e x p e rim e n ta l d ata fa irly w ell ex cep t for the 300°K d a ta . The e x p e rim e n ta l d ata for 300°K do not appear to be c o n siste n t i w ith the d ata a t the o th er te m p e ra tu re s , but th e d ata a t th e | o th e r te m p e ra tu re s a re c o n siste n t w ith e a r lie r e x p erim e n tal [ d ata a t 300°K [1 7 ]. The c o rre la tio n is not as good in the case of Si a s in 'e o O 77 K- 3 00 373 5 4 3 1 /6 (ID6 V/ c m ) '1 F ig. 5. Io n izatio n c o e ffic ie n t as a function of the in v e rs e e le c tric field fo r e le c tro n s and h o le s in GaAs a t se le c te d te m p e ra tu re . L eg en d : A77°K, A l78°K , O 300°K, • 373°K, p re d ic te d c u rv e s fro m p a ra m e te rs in Table I. th a t of G aA s. The pu b lish ed d ata of a a s a function of te m p e ra tu re for Si, do h o w ev er, yield v a lu e s of a a t 300°K w hich f a re c o n sid era b ly h ig h er a t any given field th an the d ata p r e - j sented in F ig , 4 -b . SECTION in NONLOCALIZED CONCEPT In the lig h t of th e e n e rg y c o n se rv a tio n re la tio n sh ip w hich o rig in a te s fro m th e e x iste n ce of a th re sh o ld e n e rg y for th e io n izatio n sc a tte rin g , it is show n th a t th e re e x is ts a funda m e n ta l c o n tra d ic tio n in the conventional d efin itio n of the io n iz a tio n co efficien t [4 2 ], The re la tio n sh ip of th is conventional d e fin itio n to a new d efin itio n w hich is in tro d u c ed in th is se ctio n to in te r p r e t high field e x p e rim e n ta l r e s u lts is d isc u sse d . It is shown th a t th e p h y sic a l u n d e rstan d in g of the io n izatio n e ffect in se m ic o n d u cto rs is m o re co m p licated th an h a s b een a p p re c ia te d . The com plete fo rm u la tio n is p re s e n te d and re la tio n sh ip s b e tw een the p rev io u s th e o re tic a l r e s u lts and th e e x istin g e x p e ri m en tal r e s u lts a re m ade a p p a re n t, m -1 THRESHOLD ENERGY E F F E C T In the p re v io u s se c tio n a sim p le a n a ly tic a l e x p re ss io n for the io n izatio n co efficien t in se m ic o n d u c to rs h a s b een p r o posed ,0 The a g re e m e n t betw een the p re d ic tio n s and e x istin g e x p e rim e n ta l d ata is e x c e lle n t fo r G aA s, Ge and for Si in the low field re g io n . In th e high field re g io n fo r Si (F ig, 4-b) th e re e x is ts , h o w e v er, a la rg e d isa g re e m e n t th a t se e m s to dem and te n tim e s the band gap e n e rg y fo r E . to in te r p r e t hole data,* T his d isc re p a n c y a p p e a rs to be sy ste m a tic since Si h a s been studied the m o s t e x te n siv e ly of th e se m ic o n d u cto rs and th is e ffect h a s b een o b se rv ed s e v e ra l tim e s , 35 36 We show in th is se ctio n th a t th is d ifficu lty h a s o rig in a te d fro m the fac t th a t the n o n lo calized n a tu re of the io n izatio n p ro c e s s h a s not been c o n sid ere d in p re v io u s th e o re tic a l w ork. E x p e rim e n ta lly the io n izatio n c o efficien ts for e le c tro n s , , and h o le s, a , have p re v io u sly b e en in te rp re te d by E q . 3. On P the o th er hand th e o re tic a l re s u lts have b a s ic a lly been the e s tim a te s of the a v erag e d istan c e fo r io n izatio n sc a tte rin g s and th e io n izatio n coefficient w as defined a s the in v e rs e of th is q u an tity [3 1 ], T his e stim a te h a s g e n e ra lly been c o m p ared w ith e x p e rim e n ta l d ata. Thus we th en have a final solution in w hich i t is im p lic it th a t e v e ry c a r r i e r h a s a finite e n e rg y independent io n izatio n p ro b ab ility p e r unit p a th len g th in the d ire c tio n of th e e le c tric field. If, how ever, we rec o g n ize th a t to in itia te an io n izatio n sc a tte rin g a c a r r ie r h as to have enough k in etic e n e rg y (i. e . the ion izatio n th re sh o ld en erg y ), we have to a d m it the ex isten ce of a n onlocalized effect. T his co n cep t re q u ire s in tro d u ctio n of a d e sc rip tio n in which no c a r r i e r c a n in itia te an io n izatio n sc a tte rin g u n til it h as fa lle n th ro u g h a finite p o ten tial d ifferen c e . In high field jun ctio n s w ith low b reak d o w n v o ltag es th is lim ita tio n is v e ry im p o rta n t. Thus we in tro d u c e new q u an tities w hich we sh a ll c a ll the " th e o re tic a l n o n lo calized single c a r r i e r io n izatio n p ro b a b ilitie s " &nT(&) an<^ c t ^ (#) for e le c tro n s and h o le s re s p e c tiv e ly . F ro m th is we develop a re la tio n sh ip w hich c o rre la te s a (5) and a (g) w ith th e e x p e rim e n ta lly obtainable n r p r a p p a re n t io n izatio n co efficien ts and w hich s a tis fy E q . 3. We a lso show th a t the ot and a in a c tu a l ju n ctio n s a re n p g e n e ra lly not only functions of th e e le c tr ic field stre n g th but a lso functions of p osition in sid e a given sy ste m . T his type of d e sc rip tio n , how ever, m ak es the tre a tm e n t m o re co m p licated 37 i and th e r e s u lt sp ecific to each ju n ctio n co n fig u ra tio n . To j avoid th is difficulty, a t the expense of so m e p re c is io n in d e s c rip tio n , we develop h e re a p se u d o lo ca l ap p ro x im atio n in w hich a and c x ap p ear only a s functions of th e e le c tric field stre n g th , n p but o u r un d erstan d in g of the m ean in g of and a re q u ire s re d e fin itio n . W ith the aid of th e r e s u lt obtained in the p r e vious se ctio n , sp a tia lly independent a p p a re n t io n izatio n co e ffic ie n ts a (< ?) and a ($) a re e s tim a te d for a se le c tio n of na pa m a te r ia ls a t v a rio u s te m p e ra tu re s . T he p re d ic tio n s a re s a tis fa c to ry fo r m o st of the e x istin g io n iz a tio n c o efficien t m e a s u re m e n ts and reso lv e the p ro b lem of th e an o m alo u sly la rg e t h r e s hold e n erg y for Si. The p se u d o lo ca l a p p ro x im atio n is th u s a u sefu l m o d el. The r e s u lt, h o w e v er, c o n ta in s a s e rio u s p h y sic al flaw . F o r any e le c tric field th e re is a c r itic a l m u ltip lic atio n v alu e beyond w hich th is a p p ro x im atio n d o e s not h o ld p h y sically . T his is e sp e c ia lly se rio u s in th e high e le c tr ic field reg io n . To c la rify th is re s tric tio n ex act so lu tio n s a re p re s e n te d for se le c te d p - i- n ju n ctio n s. The r e s u lts show th e im p o rtan c e of boundary d a rk sp aces and a lso show th a t th e r e s t of the junction h a s an io n izatio n coefficient w hich is a lm o s t th e sam e a s th a t obtained by the pseudolocal a p p ro x im a tio n . : H I-2 FORMULATION OF THE NONLOCALIZED CONCEPT (NLC) In Section III-l th e p h ilo so p h ic al n eed fo r the in tro d u ctio n of a n onlocalized concept (NLC) for th e io n iz atio n p ro c e s s h a s been pointed out. L e t u s now define a n o n lo ca liz ed single c a r r i e r io n izatio n p ro b ab ility , ar (< S , X) in a c o n stan t e le c tric ; field as "the p ro b ab ility of in itia tin g an io n izatio n sc a tte rin g at • i point X for a c a r r ie r w hich o rig in a te d a t X = 0 w ith z e ro k in etic en erg y and w hich h a s not y et p ro d u ced an e le c tro n -h o le 38 p a ir . I1 ar (d?,X) h as to s a tis fy the follow ing tw o conditions: 1? a (<3,X) = 0 w hen 0 < X < D , (30) r e w here D = (E. - n E )/q<g . (31) e x r r H e re , De is the "d a rk sp a c e " d is ta n c e , n^ is the n e t n u m b er of o p tical phonons a b so rb e d b y th e c a r r i e r w hich o rig in a te s at X = 0 and obtains E . e n e rg y in the s h o rte s t d istan c e and q is th e e le c tro n ic c h a rg e . 2 / ct^(<?,X) -+ c o n stan t w hen X -> 0 0 • (32) T he f i r s t re q u ire m e n t (c f / E q s. 30 and 31) g u a ra n te e s th a t th e re is no io n izatio n p ro b a b ility fo r c a r r i e r s u n til th ey d rif t at le a s t one d a rk space d ista n c e a fte r th ey a re g e n e ra te d . In p rin c ip le , D could be a lm o st z e ro , sin c e th e r e is an 6 in fin ite s im a l p ro b a b ility of a c a r r i e r a b so rb in g a su c c e ssio n of o p tical phonons w ithout lo sin g any e n e rg y . H ow ever, g e n e ra lly E . is m uch g r e a te r th an E , (eg.' E . ~ 1 eV and E ~ 0. 05 eY l r i r fo r Si) and a b so rp tio n and e m iss io n p ro b a b ilitie s a re co m p arab le fo r high e n e rg y c a r r i e r s . T h u s, n^ in E q . 31 should be able to be a ssu m e d to be z e r o / The second r e s tr ic tio n is expected to hold sin ce beyond som e d ista n c e fro m th e o rig in , if a c a r r ie r h a s n o t y et p ro d u ced an e le c tro n -h o le p a ir , the in fo r m atio n co n cern in g its o rig in w ill be l o s t / It is d ifficu lt to te s t th is condition e x p e rim e n ta lly b e c a u se in p ra c tic e a sy s te m h a s a finite d im e n sio n and c a r r i e r s m a y le a v e th e sy s te m b efo re th e c a r r i e r d istrib u tio n re a c h e s a ste a d y sta te . The b ro k en lin e s in Fig.* 6 show the above situ atio n sc h e m a tic a lly .' H ere <X> is th e a v e ra g e d ista n c e a t w hich an io n iz atio n sc a tte rin g o c c u r s / f(X) is th e p ro b a b ility a t X of finding a c a r r i e r w hich o rig in a te d a t X = 0 and h a s not yet p ro d u ced an e le c tro n -h o le p a ir. T hus f(X) is re la te d to 39 ! X T3 X * 4 — X 0 D <X> M — <x> 0 D x jo •a it <X> : F ig . 6. S chem atic d e sc rip tio n of f(X ), th e p ro b a b ility of finding : a c a r r i e r w hich o rig in ated at X=0 in a c o n sta n t e le c tric field ! and h a s n o t yet produced an e le c tro n -h o le p a ir , th e d e riv a tiv e I of f(X) and th e single c a r r ie r io n izatio n c o e ffic ie n t a: <X> i d e n o te s th e a v erag e d istan c e a t w hich io n iz a tio n s c a tte rin g ; o c c u rs and D is the d a rk space d ista n c e . L egend: B roken : lin e s : S chem atic re p re s e n ta tio n of an as y et unknown e x ac t r e - | la tio n sh ip . T hin so lid lin e s : A ssu m p tio n im p lic it in applying j p re v io u s th e o re tic a l c alcu latio n s of <X >. H eavy so lid lin e s: i P r e s e n t n o n lo calized th e o ry . 40 a (< ?, X) a s follow s: r ar W,X) . - a . ^ p O) . (33) Two a p p ro x im atio n s to the "ex act" r e s u lt a re a ls o show n in F ig . 6. The situ a tio n im plied by p rev io u s th e o re tic a l r e s u lts (cf. E q . 4) is show n by th in solid lin e s . H e re f(X) is sim p ly an exponentially decaying function: f(X) = exp C-a^*) - X} , (34) f(<X>) = 1 /e (35) and a (<?,X) = 1/<X> = a,(5) when X > 0 . (36) r 1 — Thus f(X) in th is m odel s ta r ts to d e c re a s e a s soon a s a c a r r i e r o rig in a te s even though p h y sically th e re is no p o ss ib ility of m aking an io n izatio n sc a tte rin g fo r the ra n g e 0 < X < D . 1 G The b a s is fo r our nonlocalized ap p ro ach is show n by the h eav y solid lin e s in F ig. 6. H ere the d a rk sp ace d ista n c e is a ssu m e d to be D s E ./q tf , (37) ( i . e . , . n = 0 in E q . 31). We also assu m e f(X) to be an exponentially decaying function of X for X > D. T h is a p p ro x i m atio n y ie ld s, a (<S,X) s 0 when 0 < X < D (38) r — and a (5, X) = co nstant w hen X > D . (39) r 41 To keep the a v e ra g e d istan ce c o rr e c t the above eq uatio ns r e q u ire th a t 0 .( 5 , X) = (<X> - D(<?)}-1 when X > D . (40) T hus one can ev alu ate ot (5, X) fro m E q s. 37, 40 and a th e o re tic a l e stim a te of <X (#)>. A ccording to the p rev io u s se c tio n , in our p re s e n t notation E. i <X(4)> = D exp { [(£ -)2 + (0 .2 1 7 (^ i)1,14)2 ] 5 r r E - 0 . 2 1 7 ^ ) } . (41) r Once a (5 ,X) h as been defined a s above, re la tio n sh ip s r w hich c o rre sp o n d to E q s. 3 can be w ritte n for a co n stan t e le c tr ic field s tru c tu re as follow s, t s _ dp(x) = a { } n (x) + a {S) . y (42) dx dx n r e p r r e w h ere dn (x) dp (x) n r n = a (8) n (x) (43) and dx dx n r e dn (x) dp (x) —^ ----- = - — f = a (8) p (x) . (44) dx dx p r *e t W hen a n o n -z e ro e le c tric field g ra d ie n t e x is ts the analogue of th e se equations b ecom es m o re co m p licated and it b eco m es e x tre m e ly d ifficu lt to handle th e problem - rig o ro u sly . 42 H e re , the te r m s dn (x)/dx (and -dp (x)/dx) and dn (x )/d x (and n n P -dp (x)/dx) c o rre sp o n d to the g ra d ie n ts a t x in e le c tro n (and P h o le) d e n sity due to e le c tro n s and h o le s, re s p e c tiv e ly . an r ^ ) and a (<$) a re th e v a lu e s of the th e o re tic a l n o n lo calized single p r c a r r i e r io n izatio n p ro b a b ilitie s fo r e le c tro n s and h o le s a s defined by E q . 40. n (x) and p (x) a re defined a s the d e n sitie s of 6 G e le c tro n s and h o le s w hich a re eligible to p ro d u ce e le c tro n -h o le p a ir s a t x . n (x) and p (x) a re , w ithin the fra m e w o rk of E q s. G G 38 and 39, th o se c arrieT S which have been g e n e ra te d a t le a s t one d a rk sp ace d istan c e aw ay fro m x and have re a c h e d x w ithout p ro d u cin g e le c tro n -h o le p a ir s . Note th a t in E q s. 43 and 44 th e c o n stan t 00^(5, X) app ro x im atio n (cf. E q . 39) is playing an im p o rta n t ro le to sim p lify the re la tio n sh ip . O th e r w ise , E q s. 43 and 44 would becom e in te g ra l eq u atio n s. F ig u re 7 show s sc h e m a tic a lly how we develop th e r e la tio n sh ip s betw een n(x), p(x), n ^ x ^ ) and p e (xQ). The u p p er h a lf of F ig . 7 show s conduction and valence band edges v s . d ista n c e . The d a rk space d ista n c e s , D and D for e le c tro n s * * P and h o le s re s p e c tiv e ly , a re shown for a sp ecific p o sitio n x^. The low er h a lf of F ig . 7 shows n(x) and p(x) v s. x and th e way in w hich th e e le c tro n population a t (xq- D^) develops into n (x ). A s im ila r c o n stru ctio n for p (x) is a lso shown in F ig . e o e 7. The b o u n d a rie s of th e device a re not shown in th is fig u re . W ith th e a id e x p re ss e d as W ith th e a id of th is fig u re , we see th at n (x) and p (x) can be G G ----------- rx da. t o d x , pe(x) P(x) X-D X X+D F ig . 7. R e la tio n sh ip s betw een n(x), p(x), ne (x) and p e (x). The u p per h a lf show s conduction and valence bands v s . d ista n c e . Dn and Dp a re d a rk sp a ce s fo r e le c tro n s and h o les re s p e c tiv e ly . The low er h a lf show s n(x) and p(x) v s . .x and the way in w hich e le c tro n and hole populations develop into n e (x) and p e (x). 44 = n(x-D ) - f a (<3) n (x ') d x ' (45) n j n r e x-D n and x+D r p Pe (x ) = P(x + Dp ) - J ap r (tf) Pe ^ x / ) d x / • < 46) X The b o u n d ary conditions fo r a given device can a lso be deduced fro m th e NLC a s follow s n (x) = 0 w hen D > x > 0 , (47) e n — p (x) = 0 when w > x > W - D (48) e “ P n (x) = n(o) when x = D (49) e n and p (x) = p(W) w hen x = W - D . (50) c P H ere the c a r r i e r s (n(o) and p(W)) a re assu m ed to be in je cte d w ith z e ro k in etic e n erg y . W ith an in itia l e n erg y , E , 6 the in itia l b o u n d ary la y e r s w ill be re d u c e d to = (E.^ - E ^J/qS and D 7= (E. - E )/q 5 . (51) p ip e Thus if we know a (<$, X) and a (< ?, X) and the above n r Pr boundary co n d itio n s, E q s . 42 th ro u g h 50 a re coupled equations w hich c o m p lete ly d e s c rib e n(x) and p(x) w ithin a given sy ste m . T hese equations cannot be solved a n a ly tic a lly using any sta n d a rd function but can be so lv ed n u m e ric a lly . M o reover when the p ro b le m is so lv ed , the r e s u lt cannot be e x p re sse d in th e fo rm of E q, 3 u n le ss we allow a and a to becom e not only n p 45 functions of th e e le c tric field stre n g th but also functions of the p o sitio n in sid e the d e v ic e. T his a p p ro a ch is im p o rta n t for p ra c tic a l d ev ice c a lc u la tio n s but the r e s u lt is a c tu a lly specific fo r each device c o n fig u ra tio n as w ill be shown in S ection IH -6. In the follow ing se c tio n we a tte m p t to show the ex ten t to which it is p o ssib le to re c o n c ile the "ex act" approach w ith the con ven tio n al concept of th e io n izatio n co efficien t. I l l - 3 PSEUDOLOCAL APPROXIM ATION The conventional defin itio n of th e io n izatio n co efficien t h a s been found to be in su fficien t to d e sc rib e p h y sic al re a lity . The in fo rm a tio n th a t, in fa c t, one obtains fro m a m u ltip licatio n m e a su re m e n t is re la te d to d n (x)/dx = (-d p (x )/d x ). The ex isten ce of d iffe re n t v alu es for th e e le c tro n and the hole m u ltip licatio n in d ic a te s th a t d n (x )/d x is a function of the d e n sitie s of e le ctro n s and h o le s. M o re o v e r, th e fa c t th a t e x p e rim e n ta l m e a su re m e n ts can be e x p re s s e d in the fo rm of e le c tro n and h ole m u ltip licatio n fa c to rs in d ic a te s th a t d n (x)/dx is a lin e a r sum of n(x) and p(x) w ith a p p ro p ria te w eig h tin g s. P h y s ic a lly it is a lso obvious th at both e le c tro n s and h o les a re capable of m aking e le c tro n -h o le p a ir s . F ro m th e se o b se rv a tio n s d n (x )/d x can be expected to have the functional fo rm , J s i * L = . J e J * L . 0 n ( x ) + 0 p(x) (52) dx dx n a pa w here c t^ and a re e x p e rim e n ta lly obtainable "ap p a ren t io n izatio n c o efficien ts. I1 T his e x p re ss io n should be com pared w ith E q . 42. W hen the n o n lo calized concept is not c o n sid ered , it is im p lic itly a ssu m e d th a t n (x) = n(x) and p (x) = p(x). Then 6 6 = a. (8) and ct = & ,(# ). hi the NLC tre a tm e n t, although n a nl p a p i one can solve the p ro b le m g e n e ra lly , the solution is m o re 46 co m p licated . Then none of the te r m s in E q, 52 c o rre sp o n d to co efficien t how ever, is too co nvenient to d is c a rd lig h tly . In addition it is also the b a s is fo r in te rp re ta tio n of e x istin g e x p erim e n tal data. A cco rd in g ly th e r e s t of th is se ctio n is devoted to developing a p se u d o lo ca l ap p ro x im atio n w here the ap p a re n t ionization c o efficien ts a p p e a r only as functions of the e le c tric field stren g th and can be c o m p a re d w ith e x istin g ex - j p e rim e n ta l d ata. In th is a p p ro x im a tio n a ll the boundary t conditions w ill be n e g lected and E q s . 40, 42, 45 and 46 w ill ' be tak en as b a sic e q u atio n s. H ere the follow ing ap p roxim ations j w ill be used: i stre n g th only. H e re, a ssu m p tio n s 1 ., 2 . , and 3. a re se lf- j th o se in E q. 42. The co n cep t of th e conventional io n izatio n 1. A pparent io n iz a tio n c o e ffic ie n ts a re only functions of the e le c tric fie ld stre n g th ; (53) 2. E le c tro n s and h o le s have the sam e av erag e group velocity: n(x) + p(x) = = c o n sta n t (54) 3. E le c tric field is c o n stan t; <?(x) = $ = c o n stan t (55) 4. n (x) and p (x) can be e x p re s s e d a s lin e a r com bina- 6 6 tio n s of n(x) and p(x) w ith a p p ro p ria te w eighting; n (x) = A n(x) - B n e n n t (56) pe (x) = ApP(x) - Bpnt (57) A , B , A and B a re functions of the e le c tric field n n p p 47 e x p la n ato ry . The fo u rth a ssu m p tio n is n e c e s s a r y fo r E q. 3 to hold w ith a and a being p o sitio n independent q u a n titie s, n p Once th ese ap p ro x im atio n s a re m ade the m a th e m a tic a l tre a tm e n t is stra ig h tfo rw a rd . F ro m E q s. 3 and 54, dn(x) = (& (5) - a (5)) n(x) + a (< j? ) n . (58) dx na pa pa t F o r a co n stan t e le c tric field s tru c tu re th is equation h a s th e solution, n a (& ) n ( x > = a (i) -'a T « ) U ‘ e x p C ( “ n a W ) ' < x ' x o > V pa na + n(x ) exp {(a (< g ) - a (5)) (x - x )} . (59) O 1 1 2 1 p e l O A fter using th is r e s u lt to e lim in a te n(x - D^) in E q . 45, one obtains n g (x) = n(x) exp {(otp a (< ? ) - Dn } n .a (5) + - H r Z ---- r — r r r [1 - exp ((a (< S ) - a (5)) D }] a (5) - an a (< 2 ) Pa n * x - a (&) f n ( x * ) d x ; n r ' I e (60) e x-D n W hen E q. 56 is u se d to e lim in ate n ( x ') fro m the rig h t hand e side of E q . 60 and E q. 59 is used to e lim in a te n ( x ') fro m the in te g ra l, a fte r p e rfo rm in g the in te g ra tio n one obtains The above e x p re ss io n and E q. 55 m u st th en be tru e fo r a ll v alu es of n(x) and n . The co efficien ts of n(x) and n th u s U w yield two sim u ltan eo u s equations w hich can th en be solved for A and B . The r e s u lt is , n n 49 - exP < ap a (,S) " V ( S ,) Dn a (df) A + T i T Ce x P D r , ] - 1 c t \8) ~ o c \8l pa. na n pa na - (a (8) - a (d O ) D .} ] J . (63) pa na ~ J A and B can a lso be obtained in the sam e m an n er by in te r - P P changing the s u b s c rip ts n and p. Once A ^, B^, A^ and B^ a re fo rm a lly o b tain ed , E q s. 42, 52, 60 and 61 define the r e latio n sh ip s betw een a (5), a (8)f a (< S ) and a (5). Since na pa n r p r B^ and B have te r m s p ro p o rtio n a l to a and (X^ re sp e c tiv e ly , it is u se fu l to define B 7 = B a / a and B 7 s B O t /a . Then n n n r pa p p p r na A a + B 7 (A a - A a ) . a _ n n r n n n r p p r . (64) na 1 + B + B n p In terchanging n and p su b s c rip ts w ill yield a . The values of A and B 7 a re a ll p o sitiv e . T hus we note th a t a is a c na function of a and a , i. e . the a p p a re n t ionization coefficient n r p r c is a function of both single c a r r i e r io n izatio n co efficien ts. The coupling of the te r m s is such th a t when a and a a re r ° na pa d iffe re n t the e ffect is p ro p o rtio n a lly g re a te r on the sm a lle r a p p a re n t io n izatio n c o efficien t. A lso, since the A and B 7 e le m e n ts co n tain a (5), a (< jf), a (8) and a (8), the tr a n s - n a pa n r p r fo rm atio n b etw een (a (df), a (8 )) and (a (5), a (8)) is not na pa n r p r sim p le . When I (O C . - a ) D I and I (ot - O C ) D a re * I' n a pa n» I na pa p 1 c o n sid e ra b ly s m a lle r th an u n ity , h o w ev er, the A and B 7 te rm s do not e x p lic itly involve o t (5) and a (d 0« Then ' na pa 50 and A M 1/(1+ a (8) D ) (65) n nr n A « 1/(1 + a (5) D ) (66) P Pr P B ' « a (4) D (1 + i a («) D ) / ( l + o (4) D )2 (67) n n r n n r n n r n B 7 « a 16) D (1 + i a (6) D )/(l + a (8) D )2 . (68) p p r p p r p p r p H ere we see th a t in the low fie ld reg io n A^ and A^ ap p ro ach unity and and a p p ro a ch z e ro . T hen th e re is no fe e d back effect and the conventional lo c a liz e d m odel gives a fa irly good e stim a te . To be m o re p r e c is e , w hen E q s. 65 and 40 apply, A a is eq u al to a (cf. E q . 36). Thus a = a ” n n r n n f ^ na n£ when th e B 7 and B 7 te r m s a re n eg lig ib le . When the e le c tric n P field b eco m es la r g e r , A and A becom e s m a lle r and B and n p n B 7 s ta r t to co n trib u te to a (5) and a (5). T h ese effects p na pa a re due to the n o n lo calized n a tu re of the io n izatio n p ro c e s s . The situ a tio n is m uch s im p le r when a = a . T hen A n p and B a re the sam e in E q s. 65 and 67. Then a (< ? ) (1 + a ((J) D) a (8) = - £ ----------------£-------- _ (69) 1 + 4a (5) D + 2a (5) D r r and 2 ,„. _2 ~ (1 + 4a {& ) D) + (8a (< ?) D - 4a (< ?) D + 1) z ° r (< S ) = 2D (2a (6) D - 1) * (70) 51 The asy m p to tic lim its fo r low and h ig h fie ld c a s e s can e a s ily be deduced fo r th is c a se , i . e . 1, Low field lim it F ro m E q . 69 lim a (5) -> a f tf ) . (71) «*o a 1 T his r e s u lt is c o n siste n t w ith th a t d isc u sse d in connection w ith E q s .’ 65 th ro u g h 68. T h is follow s fro m th e fa c t th a t a t low field <X> is m uch g r e a te r th an D. A lte r n ativ ely b ecau se a^($) is s m a ll, th e re fo re th e d iffe re n c e betw een n(x) and n (x) is n e g lig ib le .' 6 2. High field lim it In the high field lim it <X> should ap p ro ach D. T hus fro m E q . 40 E . aa (4) D = <X> \ S - E •* " w hen 'S ‘> ” • (72) i T hen fro m E q . 68, in th e h ig h field re g io n “ .«*>■» s - ■ • (73) 1 T h is value is one h a lf of th a t obtained fo r th e asy m p to te given by E q. 41. If we w ere to a ssig n th e c a u se of an ex -. p e rim e n ta lly o b serv ed red u c ed a to th e th re s h o ld e n e rg y , we note th a t w ithout our p re s e n t d ev elopm ent th e high field ra n g e would a p p ear to be governed by tw ice th e th re s h o ld e n erg y needed to explain the low field ra n g e . F o r th e c a se w here a £ a c o rre sp o n d in g re la tio n sh ip s e x is t b u t a re n o t e a s y to n p obtain.^ 52 I II-4 COMPARISON WITH EX PERIM EN TA L RESULTS E quation 64 gives us a w ay to com pare the th e o re tic a l p re d ic tio n of the sin g le c a r r i e r ionization p ro b ab ility with the e x istin g e x p e rim e n ta lly obtainable a p p a re n t ionization coefficient w ithin th e fram e w o rk of our p seu d o lo cal approxim ation. The p ro c e d u re for th e c o m p a riso n is as follow s: 1. S e lec t ty p ic al e x p e rim e n ta l values of a and o t at any e le c tric field stre n g th and la ttic e te m p e ra tu re . 2 . C alcu late a and a a t th is p a rtic u la r point w ith n r p r the u se of E q . 64 and the th e o re tic a l values of E . xn and E . . *P 3. O btain X and X > the m ean free paths for o p tical n r p r phonon s c a tte rin g fo r e le c tro n s and holes re s p e c tiv e ly w ith th e u se of the e x p re ssio n obtained in S ection II (cf. E q . 41) and an ex p erim en tal value of E . r 4. C alcu late a (S) and a (8) at any o th er e le c tric n r p r field stre n g th and la ttic e te m p e ra tu re w ith the aid of th e p a ra m e te r s obtained above. 5. F in a lly c a lc u la te a (3) and a (#) w ith the use of na pa E q. 64 and co m p are w ith the r e s t of the existing e x p e rim e n ta l r e s u lts . Since E q . 64 in m o st c a s e s is a tra n sc e n d e n tia l ex p re s s io n fo r both a and a , ste p s 2 and 5 u su ally re q u ire c L r ite r a tiv e converging n u m e ric a l c a lc u la tio n s. A lso since E q . 64 and its eq u iv alen t fo r a.pa a r e coupled eq uations, to m ake th e so lu tio n m o re tr a c ta b le the e x p erim e n tal data should be tak en a t the sam e e le c tric field stre n g th and la ttic e te m p e ra tu re . T h is m ethod w as ap p lied to e x p e rim e n ta l data for Ge [1 2 ], Si [1 3 ,1 5 ,1 6 ] , GaAs [1 7 ] and G aP [1 8 ]. The re s u lts at 300°K a re shown in F ig s . 8 -a through 8 -d . H ere the 53 E o Ge 10 6 (10 5 v/cm) F ig . 8 -a . Io n izatio n co efficien t, a , as a function of th e e le c tric field stre n g th , ft, for e le c tro n s (e) and h o les (h) in Ge at 300°K . L e g e n d : -------------- e x p erim e n tal --------- th e o re tic a l p re d ic tio n obtained by the p se u d o lo c a l ap p ro x im atio n . * point used to p re d ic t the o p tical phonon sc a tte rin g m ean f re e path. 54 E o 3 6 (10 v/crn) F ig . 8 -b . Ionization c o efficien t, a, a s a function of the e le c tric field s tre n g th , 5, fo r e le c tro n s (e) and h o le s (h) in Si a t 300°K . The legend is th e sam e a s in F ig . 8 -a . I I j c t (cm”1 ) 55 a max GaAs 10 - r 6 ( I05 v/cm) F ig . 8 -c , Ionization co efficien t, a, a s a function of th e e le c tric field stre n g th , fi, in G aA s a t 300°K . The legend is the sam e a s in F ig . 8- a and a& mg_ x (Eg) th e o re tic a l m axim um a lim it im p o sed b y E q . 73. a (cm” 10 - r GaP 30 3 6 (io5 v/cm) F ig . 8-d . Io n izatio n co efficien t, a, a s a function of th e e le c tr ic field s tre n g th , < £ , in G aP a t 300°K , The legend is the sam e a s in F ig . 8 -a and m ax (Eg) th e o re tic a l m ax im u m a lim it im p o sed b y E q . 73. 57 v e rtic a l and h o riz o n ta l ax es a re a p p a re n t io n izatio n co efficien ts and e le c tric field stre n g th re s p e c tiv e ly . ,re " and "h" denote e le c tro n s and h o le s . E xisting e x p e rim e n ta l d a ta a re shown by b ro k en lin e s . The a p p aren t io n izatio n c o efficien ts p re d ic te d by th e p seu d o lo cal ap p ro x im atio n a re show n by so lid lin e s . The h eav y dots on e ac h g rap h a re the p o in ts u se d to e stim a te the m ea n fre e p ath s fo r o p tical phonon s c a tte rin g . A sym ptotic v a lu e s of the m ax im u m a p p a re n t io n izatio n c o efficien t d isc u sse d in connection w ith E q . 73 a re a lso show n by so lid stra ig h t lin e s fo r the c a s e s of GaAs and G aP . T he p h y sic al p a ra m e te rs u se d for th e se p re d ic tio n s a re lis te d in T able I I . A s can be se e n , the p re s e n t a n a ly sis re p ro d u c e s the e x p e rim e n ta l d ata fa ir ly w ell. E sp e c ia lly in the c a se of Si a g re a t im p ro v e m e n t h a s been m ade in the high field re g io n due to the u se of the N EC . The lig h tly dotted line show s the p re v io u s conventional r e s u lt for a „ and a F o r the c a s e s of GaAs and G aP , the n f p* e x p e rim e n ta l r e s u lts a re la r g e r th an the p re s e n t e stim a te s for th e high field re g io n . The follow ing re a s o n s a r e su g g ested fo r th is tendency. 1. GaAs and GaP a re p o lar c ry s ta ls and th e dom inant sc a tte rin g m ec h an ism is p o la r r a th e r th an n o n -p o lar o p tical phonon sc a tte rin g . T hus the m ean fre e path m ay be expected to be la r g e r for h ig h e r en erg y c a r r i e r s and the c o n sta n t m ea n fre e path a p p ro x im a tio n m a y be in ad eq u ate. T h is would r e s u lt in a h ig h e r io n izatio n co efficien t a t in te rm e d ia te e le c tric fie ld s, b u t should not s e rio u s ly a ffe ct the high field asy m p to tes w hich should be independent of the m ean fre e path for phonon s c a tte rin g . 2. As w ill be d isc u sse d in th e follow ing se c tio n , th e re e x is ts a m u ltip licatio n fa c to r lim it on ap p licatio n of TABLE II IM PACT IONIZATION PA RAM ETERS AT 300°K OBTAINED VIA THE PSEUDOLOCAL APPROXIM ATION M ATERIAL e le c tro n o r hole E . 1 (eV) E r (meV) X o ‘ (A) Ge e • 8 19 39 h .9 51 Si e . 1.1 51 48 h 1.8 47 GaAs e /h 1.7 22 33 G aP e /h 2.6 38 31 59 the co n v en tio n al d e fin itio n of the ion izatio n co e ffic ie n t. T h u s, i f the e x p e rim e n t w as p e rfo rm e d beyond th is lim ita tio n (i."e.', w ith a high m u ltip lic a tion fa c to r in the h ig h e le c tr ic field reg io n ), the r e sult m ig h t n o t be in te rp re ta b le b y th e conventional method." E ven though the above c o n sid e ra tio n s m a y explain som e of the deviation b etw een e x p e rim e n t and th e o ry , a m o d ificatio n of the th e o ry cannot ju s tify th e e x p e rim e n ta l high field for G aP . T he e x p erim en tal d a ta c r o s s e s over the m ax im u m io n izatio n co efficien t obtained by E q . 73 a s sinning a m in im u m th re sh o ld e n erg y equal to th e band g a p / T his lim ita tio n is deduced b a sic a lly fro m the e n e rg y c o n se rv a tio n re la tio n sh ip u sin g a i stra ig h tfo rw a rd a rg u m e n t. T hus th e re does not a p p ea r to be a ju stific a tio n of the GaP e x p e rim e n ta l d ata in th is field region. . . V .. : .• ' V . •: ° 1 . ■ :v . . ' m - 5 A PPLICA TIO N L IM IT O F THE PSEUDOLOCAL APPROXIM ATION In Section IH -3 A , B , A and B w e re defined a s being n n p p independent of n(x) and p (x). T h is se em s to im p ly th a t th is m ethod can be u s e d at a n y p o in t in sid e a ju n ctio n so long as i th e boundary re g io n s a r e n e g le c te d . T his i s , in fa c t, not c o rr e c t for s tru c tu re s w ith a su fficien tly high m u ltip lic a tio n , sin ce fo r som e r a tio s o f n(x): n^, p(x): n^ o r n(x): p(x), n (x) o r p (x) can be p re d ic te d to b e n e g a tiv e .’ Such a re s u lt e e is unphysical. T h e c o rre sp o n d in g c r itic a l r a tio s , M cn and M can be obtained by se ttin g n (x) = 0 and p (x) = 0 in Cp G G ; E q s. 56 and 57 re s p e c tiv e ly . Then M and M a re not e a sy to calcu late for the case a = £ a , cn cp n p b u t fo r a = a , M can be w ritte n as n p c 2(1 + ar (8) D) M c = a (<$) D (2 + a (8) D) (76) a r T h is e x p re ss io n h a s two asym ptotic lim its : M -* 1 /a (8) D when < S -* 0 (77) c a and M -> 4 when < 3 -> « , (78) c N u m e ric a lly c alcu late d M and M a s functions of cn cp e le c tr ic field for d iffe re n t m a te ria ls a t 300°K a re shown in F ig s . 9 -a th ro u g h 9 -d . As expected fro m th e above re a so n in g , M c s ta r ts off fro m a la rg e value at low field and s a tu ra te s a t the high field end. The im p lica tio n of th is re s u lt is th a t if th e r e is any point in sid e a device w h ere the ra tio of n^/n or n /p is h ig h er th an M o r M re s p e c tiv e ly , the p seu d o lo cal i C H Cp ap p ro x im atio n does n o t h old. F o r s tru c tu re s in which the m u ltip lic a tio n ex ceed s M , the a p p a re n t io n izatio n co efficien ts c should be tre a te d a s sp a tia lly dependent p ro p e rtie s . E ven fo r m u ltip lic a tio n le s s th an M the boundary conditions a re im p o r ta n t. In th e follow ing se ctio n we show th a t if is a ssu m e d to be z e ro fo r one e le c tro n d a rk space fro m the point of 1 0 - = 10 £( 10 5 v/cm) F ig . 9 -a . C ritic a l m u ltip lic atio n r a te , M c , for e le c tro n s (e) and h o le s (h) in Ge a t 300°K as a function of the e le c tr ic field stre n g th . Beyond th e se lim its th e p seu d o lo cal ap p ro x im atio n does not h old. 1 0 - = £ (105 v/cm) F ig . 9 -b . C ritic a l m u ltip licatio n r a te , M c , for e le c tro n s (e) and h o les (h) in Si a t 300°K a s a function of the e le c tric field stre n g th . GaAs 1 0 - = IV L min £ (I05 v/cm) F ig . 9 -c . C ritic a l m u ltip lic a tio n r a te , Mc , in GaAs a t 300°K a s a function of the e le c tric field s tre n g th . M c m jn d en o tes the th e o re tic a l m in i m u m value for M c at infinite field . 64 GaP 30 £(I05 v/cm) F ig . 9-d. C ritic a l m u ltip lic a tio n r a te , M c , in G aP at 300° K as a function of the e le c tric field stre n g th . Mc m jn denotes th e th e o re tic a l m in i m um value for M a t in fin ite fie ld . c . i | j j 65 e le c tro n in je c tio n and if a is sim ila rly assum ed to be z ero fo r one h o le d a rk space fro m the other end of the s tru c tu re , th e p re d ic tio n s b ased on th is m odified pseudolocal co n cep t com e v e ry c lo se to the "e x a c t" solution which follow s fro m th e fo rm u la tio n in S ection III-2. I ll- 6 EXACT NONLOCALIZED SOLUTION In th e p rev io u s se c tio n s we saw th at the conventional d efin itio n of the io n izatio n coefficient, which re q u ir e s th a t a = a ($) and a = a (<?), is not e n tire ly a p p ro p ria te to d e s- n n p p c rib e the p h y sic al re a lity of the ionization p ro c e s s in sid e a sy s te m . T h is is p a rtly b ecau se the boundary conditions have ta c itly been n eg lected in the pseudolocal ap p ro x im atio n . F ro m th e p seu d o lo cal ap p ro x im atio n its e lf, how ever, we can not obtain any e stim a te of th e e r r o r introduced by its u s e . ' A cco rd in g ly we p re s e n t h e re som e exact nonlocalized so lu tio n s to co m p are w ith p seu d o lo cal re s u lts , p -i-n s tru c tu re s w ere c h o sen fo r ex am in atio n b e ca u se the single c a r r ie r io n izatio n p ro b a b ility w as defined fo r the case of a constant e le c tric field . The i-re g io n th ic k n e s se s and applied field w ere c h o sen so th a t e x ac t solutions could be obtained for e le c tro n and hole m u ltip lic atio n s g re a te r th an and le s s than the v a lu e s in the p re v io u s se ctio n . The fo rm u latio n for the ex act solution h a s b een p re s e n te d in S ection III-2. E xact so lu tio n s w ere obtained u sin g an ite ra tiv e approach to the stead y sta te so lu tio n . The fundam ental equations w ere f i r s t m an ip u lated a s follow s. 66 | F ro m E q s , 42, 45, 46, 47 and 48, for x > D j i i x -D x-D J r n 11 r x | 1 < X r , , . P « ^ x / ) d X / + I n J X ' ) d X 7 - f a « r n /»^x , ) d x ' o p r e J D n r e J x _D n r e n n + n (0) (79) | . I and for x ^ W -D - p W W -D x+D p (x) = f a n ( x ,) d x / + f ap p (x 7 )dx 7 - f a P p (x ;)dx 7 *e J n r e J p r r e J p r r e x+D x+D x p p + p(W) . (80) j T h ese coupled in te g ra l equations have boundary conditions se t by eq u atio n s 47 and 50. T he p ro c e d u re fo r solving the c a se of e le c tro n in jectio n a t x = 0 w as a s follow s: 1. In itia lly n (x) and p (x) w ere a ssu m e d to be z ero 6 G th ro u g h o u t the sy s te m ex cep t at the point x = D^. j T h e re n (D ) w as se t equal to n(0) as is c o n siste n t e n w ith th e N LC , 2. E q . 79 w as th en u se d to g e n erate t r i a l values of n^(x) up to x = W. 3. T h is ap p ro x im atio n to ^ ( x ) w as th en u sed in E q . 80 to g e n e ra te ap p ro x im ate v a lu e s of p (x) w ith the b o u n d a ry condition p (W -D ) = p(W) = 0. e p 4 . A b e tte r a p p ro x im atio n to ne (x) v s. x w as th en o b tain ed usin g E q . 79 w ith the know ledge of the p re v io u s ly obtained p (x) v s . x. • G 5. S teps 3 and 4 w e re th en re p e a te d u n til the r e s u lt 67 converged. E q . 42 w as th en used to c alcu late n(x) and p(x). F o r hole in jectio n n and p w ere in te rc h a n g e d . If both types of c a r r i e r a re in jected , both n e 0^n ) an<* Pe ^ " ^ p ) no* z e ro « If an e le c tro n -h o le p a ir is g e n e ra te d a t som e point x /, n (x) 6 and p (x) w ill exhibit eq u al d isc o n tin u itie s a t x '+ D and r e n x ' - Dp re sp e c tiv e ly . P re d ic tio n s for se le c te d Si and GaAs p - i- n ju n ctio n s a re shown in F ig s. 10-a th ro u g h 10-d. In each fig u re the upper cu rv e show s n(x) and p(x) m e a s u re d dow nw ard fro m n^ as functions of x. The c u rv e s in the m id dle p o rtio n show n (x) e and p (x) v s. x . The c u rv e s in the lo w e st p o rtio n show "ex act" G io n izatio n coefficients a (< £ , x) and a (< $ , x) v s . x w h e re, ne pe a (<J, x) and a (< ?, x) a re defined a s ne pe 1 6x1 a (<S,x) = —j - r - — (81) ne n(x) dx and dn (x) V w ’x)E i k r - t ~ < 8 2 ) F o r co m p ariso n the a n a (^) an<^ a a (^) obtained fro m the p se u d o lo c a l approxim ation a re a lso shown in the lo w e st p o rtio n of each fig u re . The p red ic tio n s show th e follow ing c h a ra c te r is tic s : 1. n (x) and a (5 ,x) a re z e ro fo r x < D . p (x) e ne n e and a ($,x) a re z e ro fo r x > W -D . T his is due pe P to the NLC re q u ire m e n ts , 2. T hroughout th e r e s t of the re g io n , a (< jf, x) and 116 a (<S!,x) a r e n e a rly c o n stan t and v a lu e s a re clo se pe to those obtained fro m th e p seu d o lo cal, ap p ro x im atio n . 6=264 KV/cm Mn =266 255 225 ne P 0 5 10 D ISTA N CE (fj.) F ig . 10--a. The solution of the ex act nonlocalized fo rm u latio n d e - j veloped in III-2 for a Si p - i- n ju n ctio n w ith 10(a i-re g io n a t 300?K. T he m u ltip lic a tio n r a te , Mn , is below M c. The u p per curve show s I n(x) and p(x) v s. x. The m iddle c u rv e s sh o w n e (x) and pe (x) v s. x ; fo r n (o )= l. The m ax im u m n u m b er of ne and pe a t junction edges | a re a lso show n. The lo w e st c u rv e s show the e x ac t ionization co e ffic ie n ts c c ^ x ) and < X p e (x) v s . x. F o r co m p ariso n < x na(( j) andOtpa (< £ ) obtained b y th e p seu d o lo cal ap p ro x im atio n are a lso shown by b ro k e n lin e s . B oundary d a rk sp a c e s Dn and Dp fo r e le c tro n s and j h o le s re s p e c tiv e ly a re a ls o show n. j 6= 418 KV/cm 664 Si p-i-n 588 491 0-0 0*5 DISTANCE {jjl) F ig . 10-b. The ex act so lu tio n fo r a Si p - i- n junction w ith 0 .5 (i i-r e g io n a t 300°K . Mn is above Mc • The legend is th e sam e a s th at in F ig . 10-a. £ = 2 8 0 KV/cm GaAs p- 1 - n 25 24 D DISTANCE ( /ex) F ig . 1 0 -c. The e x ac t solution fo r a GaAs p - i- n ju n ctio n w ith 10 |a i-re g io n at 300°K . Mn is below M c . The legend is the sam e as th a t in F ig . 1 0 -a. 71 6 =426 K V/cm P(X) n (x) GaAs 29 2 8 ne 0 0 D n DISTANCE (/i. ) F ig . 1 0 -d . The e x ac t solu tio n fo r a GaAs p - i- n ju n ctio n w ith 0 .5 |J i-re g io n a t 300°K . M is above M . The leg en d is the sam e a s th a t in F ig . 10- a . 72 T his effect is m o re a p p a re n t in ju n ctio n s w ith wide i-re g io n s th an in n a rro w ju n ctio n s b e ca u se the effect of the b o u n d ary is th e n le s s s e rio u s . 3. At th e boundary w here a c a r r i e r sp e c ie s is e x tra c te d the c o rre sp o n d in g a (5 , x) alw ays re a c h e s 6 a m axim um . T his ten d en cy i s m o re a p p a re n t the n a rro w e r the ju n ctio n . T his h ap p en s b e c a u se the ra tio of n to n s ta r ts o ff a s z e ro at th e point of e in jectio n and grow s s te a d ily a c r o s s the s tru c tu re . 4. F o r the case of Si, n e a r th e p oint of e le c tro n ex tra c tio n a (A, x ) is s m a lle r th a n a (5) and n e a r ••ne ■ na the point of hole e x tra c tio n a (<?, x) ex ceed s p e a , (<?). B ecause (X > a , d n (x )/d x is la r g e s t n e a r pa n p the i- n junction. T h e re fo re n e a r x = 0 a g re a te r fra c tio n of the h o le s a re e lig ib le to m ake io n izatio n sc a tte rin g s than is the c a se fo r e le c tro n s at x = W. T his m a k e s a (x= W) and a (x= 0) c lo s e r ne pe to each other th an the v a lu e s of a and a . n a pa 5. The definition we have c h o se n in E q s. 81 and 82 w as for convenience only and th e r e is no p h y sic a l need to define io n izatio n c o e ffic ie n ts th is w ay in o rd e r to analyze m u ltip licatio n in r e a l s tr u c tu r e s . The r e su lts indicate s e rio u s d iffic u ltie s w ith th e conven tio n a l concept of th e io n iz a tio n c o efficien t fo r high field stru c tu re s if one i s c o n c e rn e d w ith the sp a tia l d istrib u tio n of ionizing e v e n ts . 6. The re s u lts su g g est th a t if one defines a to be z e ro for the d istan ce f r o m th e in je ctio n and if one s im ila rly m ak e s a z e ro for the d ista n c e D 7 p a p fro m the o th er end of th e ju n c tio n , the p seu d o lo cal "............................................................. ' 73 v a lu e s of a and a a re rea so n a b le ap p ro x im atio n s na pa ov er the r e s t of th e junction, H I-7 SUMMARY In th e lig h t of e n e rg y c o n serv atio n c o n sid e ra tio n s we have show n th e p h ilo so p h ical need for the in tro d u c tio n of the non lo c a liz e d concept to u n d e rsta n d the avalanche io n izatio n p ro c e s s in se m ic o n d u c to rs. W ith the u se of a sty lized sp a tia l depen den ce of th e io n izatio n p ro b a b ility of a single type of c a r r i e r , a n o n lo ca liz ed concept h a s b een fo rm u la te d . T his fo rm u latio n in d ic a te s the inadequacy of the conventional d efin itio n of the io n iz atio n co efficien t. T his definition a ssu m e s in c o rre c tly th a t the lo c a l r a te of p ro d u ctio n of e le c tro n s by e le c tro n s is p ro p o rtio n a l to the lo c a l e le c tro n population. Since the conven tio n a l concept of the io n izatio n co efficien t is too convenient to d is c a rd lig h tly , a p seu d o lo cal approxim ation w here the io n iz a tio n c o efficien t a p p e a rs only a s a function of e le c tric field stre n g th h a s been developed. It w as found th a t th e pseudolocal ap p ro x im atio n can p re d ic t the existing high field e x p e rim e n ta l io n iz atio n co efficien t d a ta w ith good a c c u ra c y . T he anom alously la rg e io n iz atio n th re sh o ld e n e rg ie s p re v io u sly re q u ire d to in te r p r e t the high e le c tric field data for Si h as b een explained. E v en though the p seu d o lo cal app ro x im atio n is a u se fu l a p p ro a ch , th is p ic tu re h a s an a p p lica tio n s lim it b ecau se boundary co n d itio n s a re n eg le cted . T h ese boundary conditions a re m o re im p o rta n t th e lo w er th e breakdow n voltage of the ju n ctio n . To ex am in e th is lim it the "ex a c t" nonlocalized fo rm u latio n w as so lv ed for se le c te d p - i- n ju n ctio n s. The re s u lts in d ic a te th a t th e io n iz atio n c o efficien t its e lf cannot in p rin cip le be defined so le ly as a field dependent quantity. E x act so lu tio n s re q u ire z e ro io n iz atio n co efficien t in sid e boundary d a rk sp a c e s at p -i and i-n ju n ctio n s fo r e le c tro n s and h o les re s p e c tiv e ly . T hrough out the r e s t of the re g io n io n izatio n c o efficien ts a re sp a tia lly dependent but a re fa irly co n stan t and c lo se to the v alu es ob tain ed v ia the p seu d o lo cal a p p ro x im atio n . SECTION IY AVALANCHE BREAKDOWN VOLTAGE IN SEMICONDUCTOR JUNCTIONS ; To d e m o n stra te th e significance of the p re s e n t th e o re tic a l tr e a tm e n t, th e breakdow n band bending is p re d ic te d for a b ru p t and p - i- n ju n ctio n s in G e, Si, GaAs and GaP a t v a rio u s te m - : p e r a tu r e s [4 3 ], The p re d ic tio n is m ade p o ssib le by in tro d u cin g a sty liz ed m odel to d e s c rib e the d istrib u te d io n izatio n p ro c e s s ■ in sid e a ju n ctio n and a lso a sim ple e m p iric a l e x p re ssio n w hich re p ro d u c e s th e a p p a re n t io n izatio n coefficient obtained v ia th e p se u d o lo c a l a p p ro x im atio n . The p re d ic tio n show s a p p re c ia b le d iffe re n c e fro m the p re v io u s th e o re tic a l e s tim a te s fo r high e le c tr ic field configurations,^ T his is due to the th re sh o ld e n e rg y e ffe ct w hich is in tro d u c ed by the NLC tre a tm e n t. W here e x p e rim e n ta l d a ta e x ists the a g re e m e n t is e x cellen t, IV -1 BREAKDOWN CONDITION P h y s ic a lly the avalanche breakdow n v o ltag e, V , is A defined as the applied voltage w here e le c tro n and hole m u ltip li c a tio n fa c to rs re a c h infinity,'’ E x p erim e n tally , since the in fin ite ; c u rr e n t condition cannot be obtained, one m u st m ake som e type o f e x tra p o la tio n to define th e breakdow n voltage [4 4 ], The th e o re tic a l e s tim a te , h o w e v er, can be defined som ew hat m o re ! p r e c is e ly a s th e p o ten tial drop a c ro s s the junction a t w hich th e i follow ing condition is sa tis fie d . 75 76 lim. M -»» n W f x = J an ^ x ) exP j (a “ an (x 0) d x 7 dx . (83) 0 0 Note th a t w hen re a c h e s in fin ity a ls o re a c h e s infinity sim ultaneously," 0 and W denote c o o rd in a te s w h ere the e le c tric field , 5, te rm in a te s . The p o te n tia l d ifferen c e betw een x = 0 and x = W, V _ , is th e to ta l band bending a t breakdow n. The breakdow n v o ltag e is le s s th an by the z e ro b ia s diffusion p o te n tia l, Y ^ , i;T e . , W | <S(x) d x = V B = Ya + Vd . (84) 0 P re v io u sly Y h a s been e s tim a te d [3 6 ] w ith the u se of E q, 83 B by assu m in g a and a a r e only functions of th e e le c tric field n p stren g th ,'' E x istin g p re d ic tio n s u n d e r e stim a te fo r high e le c tric field co n fig u ratio n s [4 5 ,4 6 ], T his is p a rtly b ecau se th e effect of th e th re s h o ld e n e rg y fo r the io n izatio n sc a tte rin g is not c o n sid e re d . The im p o rta n c e of th is e ffe ct h as been pointed out since 1957 [4 7 ], M any a u th o rs [ 1 4 ,1 5 ,1 6 ,1 9 ] tr ie d to in c o rp o ra te th is e ffe c t b u t th e ir c o rre c tio n h a s been lim ite d to in je c te d c a r r i e r s only and th e ir m a th e m a tic a l tre a tm e n ts a r e n o t co m p lete (cf. A ppendix A), As is shown in the p re v io u s se c tio n (cf," S ection III-6), once th is effect is co m p le te ly tak en into acco u n t, th e p ic tu re of th e io n izatio n p ro c e s s in sid e a ju n ctio n h a s to be m o d ified d ra s tic a lly ( i,e , in boundary d a rk sp a ce s and a lso in sid e the bulk sem ico n d u cto r). In th is se ctio n we show how th e r e s u lts obtained in the p rev io u s M n = 1 77 se ctio n can be u sed to p re d ic t V r,. To sim p lify the calcu latio n J D we in tro d u ce a sty lized m odel for io n izatio n in ju n ctio n s (in S ection IV -2) and an a n a ly tic a l e x p re ss io n to re p re s e n t a p p a re n t io n iz atio n co efficien ts v ia a p se u d o lo ca l a p p ro x im atio n (in S ection IV -3). In S ection IV -4, p re d ic te d v a lu e s a re p re s e n te d fo r a b ru p t and p - i- n junctions in G e, Si, GaAs and G aP a t ro o m te m p e ra tu re . The r e s u lts show e x c e lle n t a g re e m e n t w ith e x istin g e x p e rim e n ta l d ata fo r Si and GaAs a b ru p t ju n ctio n s (cf. F ig s . 24 and 25). IV -2 STYLIZED MODEL The ex act solutions for v a rio u s p - i- n ju n ctio n s and the c h a r a c te r is tic s of th ese r e s u lts a re p re s e n te d in S ection III-5 . T h ese r e s u lts show why e a r lie r p re d ic tio n s voider e stim a te d V , sp e c ia lly fo r high field co n fig u ratio n s: the boundary d a rk b sp a c e s in tro d u ce a p o ten tial dro p w hich does n o t co n trib u te d ire c tly to io n izatio n . The th re s h o ld e n e rg y re q u ire m e n t a lso ten d s to su p p re ss the p o sitiv e feedback e ffe c t w hich is e s s e n tia l fo r breakdow n. In p rin c ip le , the b e s t e stim a te of would be obtained by solving th e ex act fo rm u la tio n fo r th e c a se of in fin ite m u ltip lic a tio n . In re a lity , h o w ev er, th is ap p ro ach d e m an d s a p p re c ia b le com puter tim e and the fo rm u la tio n is v e ry co m p licated when an e le c tric field g ra d ie n t e x is ts . T hus a sty liz ed m o d el to rep ro d u c e th e e x ac t solu tio n is p ro p o se d h e r e . In th is m o d el e le c tro n and hole a p p a re n t io n iz atio n c o efficien ts a re a ssu m e d to be zero w ithin th e ir c o rre sp o n d in g boundary d a rk sp a c e s and a re a ssu m e d to be an a (< 2 (x)) and ap a (< ? (x)) th ro u g h th e r e s t of the re g io n . F ig , 11 show s th is situ atio n sc h e m a tic a lly . T h is sty liz ed m odel h a s the follow ing re la tiv e ly 78 i a n o ( £ > a.„( 6) D , W-D, W F ig . 11.' S tylized m odel to ap p ro x im a te th e e x a c t solution. T his fig u re is d raw n for a p - i- n junction. 79 I m in o r d iffe re n c e s fro m th e e x a c t so lu tio n s: 1. In th e e x ac t so lu tio n the io n izatio n co efficien ts grow slow er th a n the step functions assu m e d h e r e . T his tendency is s tro n g e r for h ig h field co n fig u ratio n s and p ro d u ces an u n d e re stim a tio n of V.^. 2. The a p p a re n t io n izatio n co efficien ts a re la r g e r for the sp e c ie s w ith the la r g e r io n izatio n co efficien t (eg. e le c tro n in Si) and s m a lle r fo r the sp e c ie s w ith the s m a lle r io n iz a tio n co efficien t (eg. h o le s in Si). T his is due to b o u n d ary effe cts on the feedback w hich w ere not c o m p le te ly tak en into account when a (<g) and C X (<g) w e re .deduced, n a pa IV-3 ANALYTICAL CURVE FITTIN G The p re c is e e x p re ss io n s for th e p seu d o lo cal io n izatio n co efficien ts a re too co m p lica te d a lg e b ra ic a lly for e a sy u se in p ra c tic a l device c h a ra c te riz a tio n . T hus we have a ttem p ted to re p ro d u c e the r e s u lts w ith a sim p le a n a ly tic a l e x p re ssio n . The t r ia l function had th e fo rm a (c?) = a < g n exp {-(b/tf)1 1 1 } (85) H ere a and b a re c o n sta n ts and n and m a re in te g e rs . A conventional le a s t sq u a re fittin g p ro c e d u re w as u se d to d e te r m ine th e se c o n sta n ts. T h is m eth od w as exam ined in v a rio u s m a te ria ls at v a rio u s te m p e r a tu re s . The choice of n = 1 and m = 2 w as found to be the b e s t fo r a ll c a s e s , a and b w ere found to be lin e a r functions of the la ttic e te m p e ra tu re around 300°K , i .e . 80 a = a 300 (1 + c(T-300)} (86) and b = b 300 f l + d (T “ 300> } • (87) The v a lu e s of the c o n sta n ts, range of th e e le c tric field o v er w hich the fitting is v a lid , and the a v erag e e r r o r s a re lis te d in T able III along w ith the th re sh o ld e n erg y [2 5 ], o p tical phonon e n erg y [3 3 ] and a lso m ean free path fo r o p tical phonon sc a tte rin g [4 2 ] a t 300°K , F ig. 12 show s th e ra tio of th e io n iz atio n co efficien t obtained by the p re s e n t e x p re ss io n to th a t obtained v ia th e p seu d o lo cal appro x im atio n for e le c tro n s in Si a t 300°K as a function of the e le c tric field stre n g th . The a g re e m e n t of the sim p le e x p re ss io n with the p seu d o lo cal io n iz a tio n c o efficien ts is e x c e lle n t over a wide enough ra n g e of the e le c tr ic field to be ap p licab le to m o st p ra c tic a l d ev ices. IV -4 BREAKDOWN BAND BENDING ESTIM ATE AND DISCUSSION The V _ e stim a te h a s been p e rfo rm e d w ith the u se of B th e b a s ic equation W n J o X x exp I [a ( ^ ( x ') / x ') - a (g (x ')»* ')] dx ' dx I p n (88) o TABLE m A PPA R EN T IONIZATION PA RA M ETERS (300°K) Ge Si GaAs G aP e /h e h e h e /h e /h E . i eY 0.8 0 .9 1. 1 1.8 i .7 2.6 E r m eV 22 53 22 37 X o A 39 51 4 8 47 33 31 a 300 v ”1 0 .569 0.559 0.426 0.243 0 .2 9 4 0.191 h.-. x lO"4 ® ^-1 6 .3 3 7 .8 7 3.05 5 .35 8 .5 0 8 .3 8 b300 x I05 V /c m 3 .3 2 2 .7 2 4 .8 1 6 .5 3 5 .8 6 9.91 d x lO"4 ® ^ 1 9 .3 4 8 .8 2 6.86 5 .6 7 7 .1 7 5 .9 5 a x 10^ V /c m 0 . 7 - 7 1 - 1 0 1 - 20 3 - 4 0 E r r o r % 3 4 4 4 5 4 E x p re ss io n a(fi) = a300 {1 + c(T - 300)} 3 exp -D>300 {1 + d(T - 300)}/ 2 0 Si ( e le c tro n ) 3 0 0 K 0 00 0 5 10 £ ( x I05 v /c m ) F ig . 12. R atio of the io n izatio n co efficien t obtained by E q. 85 to th a t obtained v ia the p seu d o lo cal app ro x im atio n for e le c tro n s in Si a t 300°K as a function of th e e le c tric field stre n g th . 83 H e re a(< g (x ),x ) =0 fo r 0 <x <D (89) n n a (<?(x),x) h 0 fo r W -D <x <W (90) P P and in the r e s t of the re g io n the io n izatio n co efficien ts w ere a ssu m e d to be e x p re ss e d as an (S (x ),x ) = an < 2 (x) exp J - ( W # (x))2 J . (91) The in te rc h a n g e of s u b s c rip ts n and p w ill give the c o rre sp o n d in g r e s u lt for h o le s . E q s. 88 th ro u g h 90 can be com bined into a single equation (cf. A ppendix A for d e ta ils), N ote th a t when D -» 0 and D -» 0, E q. 92 b e co m es the n P conventional re la tio n sh ip . The p r e s e n t re la tio n sh ip is the one w hich should be u se d to deduce a p p a re n t io n iz a tio n co efficien ts fro m m u ltip lic a tio n m e a su re m e n ts . If one h as to u se a c o m p u te r, h o w e v er, E q s. 89 th ro u g h 91 a re ju s t a s good a s E q. 92. F ig s . 13 and 14 show re s u lts fo r G e, Si, GaAs and G aP a b ru p t and p - i- n ju n ctio n re s p e c tiv e ly . H e re the o rd in a te s a re the breakdow n band bending in u n its of v o lts. In F ig . 13 the a b s c is s a is th e background im p u rity d en sity and in F ig . 14 the a b s c is s a is th e w idth of th e i-re g io n . In both fig u re s one n o tic e s th e sa tu ra tio n of V _ . T his tendency is p re d ic te d to be m o re a p p a re n t in J 3 p - i - n ju n ctio n s th a n in step ju n c tio n s. T h is is due to the th re s h o ld e n erg y . T his lim it can e a s ily be d e riv e d a s follow s fo r p - i- n ju n ctio n s when . The conventional breakdow n condition p ro v id e s u s w ith a m in im u m e s tim a te of the b re a k down band bending. T his condition is a . W ~ 1 (93) a A ccording to E q . 69 a t high e le c tric field th e follow ing r e la tio n sh ip h o ld s, i. e. Abrupt Junction Okuto a Crowell Sze a Gibbons 3 00°K 10“ ' I 0 ,w 1 0 " 10’ BACKGROUND DOPING ( a to m s /c c ) F ig . 13. P re d ic te d breakdow n band bending fo r a b ru p t ju n ctio n s in G e, Si, GaAs and G aP a t 300°K as a function of the background im p u rity d e n sity . L egend: ------ p re s e n t p r e d ic tio n s ---------------------p re v io u s e s tim a te s [3 6 ], 0 0 cn V B (volts) P-i-n diode Theoretical Limit for Avalanche Breakdown GaP G aA s" 300° K Ge -3 F ig . 14. P re d ic te d breakdow n band bending fo r p - i- n ju n ctio n s in G e, Si, GaAs and G aP a t 300°K a s a function of th e i-re g io n w idth. B roken lin e s c o rre sp o n d to V g obtained by E q s. 96 and 97. a » D -+ 1/2 a 87 (94) F ro m E q s. 93 and 94 one obtains E. q(<3» W) . ~ 2 m in Thus V_. . = (<S*W) . « 2 E ./q . (96) B m in m m i F o r the c a se a a the situ atio n is m o re co m p lica te d but n p one can a ls o w rite as the analogy to E q. 96 th e follow ing: . « (E. + E . ) / q . (97) B m in m ip E q s. 96 and 97 can be understood p h y sic a lly a s fo llo w s. When the e le c tric field in c r e a s e s , the a p p a re n t io n izatio n co efficien ts in c re a s e . At an in fin itely la rg e field th ey a ls o b ecom e in fin itely la rg e . N e v e rth e le ss, both boundary d a rk sp a c e s r e q u ire d efin ite th re sh o ld p o ten tial d ro p s w h atev er the value of the field . Thus the m in im u m band bending cannot b ecom e le s s th an the su m of th ese two p o ten tials (cf. E q s. 96 and 97). T his V can be used a s a m e a su re to id en tify the breakdow n J d m ech an ism ; i . e . once a junction show s le s s th an V „ . , the " ■ a m in breakdow n m ech an ism is due to tunneling. N ote, th is a rg u m en t cannot in g e n e ra l be u se d the o th er w ay aro u n d . The m axim um e le c tric field stre n g th , & , and w idth of the ® m ax space c h arg e reg io n a t breakdow n, W, fo r a b ru p t ju n ctio n s a re shown in F ig . 15 as a function of th e background im p u rity d e n sity . The p re d ic te d lo g arith m ic te m p e ra tu re c o e ffic ie n ts, 0, Abrupt Junction 300°K GaAs ■ « ___I __ L io ,w 1 0 " 10' BACKGROUND DOPING (a to m s /c c ) F ig . 15. Space ch arg e re g io n w idth and m ax im u m field a t breakdow n fo r a b ru p t jun ctio n s in Ge, Si, GaAs and GaP a t 300°K as a function of the background doping. M AXIM UM FIELD ( V / c m ) 89 of the breakdow n band bending, around ro o m te m p e ra tu re for a b ru p t ju n ctio n s a r e show n in Fig# 16. H ere d is defined as d 4 n V (T) t - ■ d T B (98) The tendency of th e re s u lts a g re e s w ith e x istin g e x p e rim e n ta l re s u lts [4 8 ], Note th a t 0 is p re d ic te d to becom e negative fo r high field c o n fig u ra tio n s. o 'O h ■o > m "O III <3. •5 Abrupt Junction 1-0 Ge GaAs 0-5 GaP 0 0 0-5 BREAKDOWN BANDBENDING (v o lts ) F ig . 16. P re d ic te d lo g a rith m ic te m p e ra tu re co efficien t of the breakdow n band bending around ro o m te m p e ra tu re for a b ru p t ju n ctio n s a s a function of the breakdow n band bending a t 300°K . SECTION Y i l AVALANCHE BREAKDOWN BAND ! j BENDING M EASUREM ENTS The w ork p re s e n te d in th is se ctio n w as p e rfo rm e d to ! d e m o n stra te c o n siste n c y betw een e x p e rim e n ta l breakdow n voltage : m e a su re m e n ts [4 9 ] and the th e o re tic a l p re d ic tio n s d isc u sse d in i i th e p re v io u s se c tio n s. The e x p e rim e n ta l m e a su re m e n ts w ere 16 | m ade on n -ty p e Si w ith im p u rity d e n sity , N , betw een 1 x1 0 1 18 3 1 and 3 x 10 a to m s /c m , F o r the m e a su re m e n ts two new ; kinds of p a ssiv a te d S chottky d io d es w e re developed: a L arg e A rea P o s t-E v a p o ra tio n -P a s s iv a te d Schottky D iode and a Sm all 1 - i A rea P o in t C ontact P a s s iv a te d Schottky D iode. The diodes have i no conventional g u a rd rin g s tru c tu re but have n e a rly id ea l i c u rre n t v s. voltage re la tio n s h ip s . The NLC tre a tm e n t w as found to be e s s e n tia l to u n d e rsta n d the breakdow n p ro c e s s for 16 3 N ^ > 1 x 10 a to m s /c m in Si. In Si the breakdow n m echanism ; w as found to be due to avalanche p ro c e s s e s a t dopings a t le a s t a s la rg e a s 3 x 10*^ a to m s /c m ^ , i V - l SIGNIFICANCE OF BREAKDOWN VOLTAGE M EASUREM ENTS I I In p re v io u s se c tio n s it w as shown th a t the n o n lo calized I n a tu re of the io n iz a tio n phenom enon o rig in a te d fro m a th re sh o ld e n erg y e ffe ct is e s s e n tia l to d e sc rib e the p ro c e s s in se m i- i c o n d u cto rs.' W ith th e u se of th e n o n lo calized concept (NLC) i ; j and a sim ple a n a ly tic a l e x p re s s io n fo r the a v e ra g e d istan c e for ; io n izatio n s c a tte rin g , e x istin g e x p e rim e n ta l d ata fo r the io n iz a- 92 tio n co efficien t have been in te rp re te d p h y sic ally w ith good a c c u ra c y . W ith the aid of a p p ro p ria te boundary co n d itio n s, w hich a lso o rig in ated fro m the NLC th e o ry , the breakdow n band bending h a s been p re d ic te d in s e v e ra l m a te ria ls fo r se le c te d e le c tric field co n fig u ratio n s. B reakdow n band bending is the sum of th e e x p erim e n tal breakdow n voltage and the diffusion v o lta g e. The NLC p re d ic ts a lim itin g v o ltag e in the re v e rs e breakdow n of n a rro w high field p - i- n ju n ctio n s. F o r ab ru p t ju n ctio n s the breakdow n v o lta g es a re h ig h er th an p re v io u s e s tim a te s and a re le s s dependent on ju n ctio n doping fo r high im p u rity d e n sitie s . Thus a breakdow n voltage m e a su re m e n t is one of th e m o st c ru c ia l and stra ig h tfo rw a rd e x p e rim e n ta l te s ts of the NLC th e o ry . F ro m an e x p e rim e n ta l point o f view th is is a lso tru e b ecau se as long as the ju n ctio n p a r a m e te r s a re w ell defined th e breakdow n voltage m e a su re m e n t is m o re sim p le and a c c u ra te th an m u ltip lic atio n m e a su re m e n ts (som e e x p e rim e n ta l d iffic u ltie s in th e se m e a su re m e n ts w e re d isc u sse d in S ection 1-2 and fu rth e r d e ta ils a re given in th e follow ing sectio n ). V -2 EX PERIM EN TA L APPROACH B reakdow n voltage m e a su re m e n ts of u n ifo rm ly doped m e ta l-n -ty p e silico n Schottky b a r r i e r diodes w ere u n d e rta k en in ste a d of p -n o r p - i- n diode stu d ie s for the follow ing re a so n s: 1. A Schottky diode p ro v id es a m o re a b ru p t e le c tric field te rm in a tio n a t th e m e ta l-se m ic o n d u c to r in te rfa c e J. th an p ' -n or p - i- n ju n c tio n s. 2. It is an e m p iric a l fact th a t th e te rm in a tio n of the e le c tric field at the sem ico n d u cto r end of th e space c h arg e reg io n in u n ifo rm ly doped ju n ctio n s is b e tte r defined th an in p - i- n ju n ctio n s. C onventional ungu arded p la n a r m e ta l-se m ic o n d u c to r Schottky | ju n c tio n s, h o w e v er, n o rm a lly show a t le a s t the follow ing th re e phenom ena w hich ten d to m ake th e c u rre n t v s . v o lta g e, I-V , ; ; c h a r a c te r is tic s n o n -id e a l and th u s to p a rtia lly o b scu re a b re a k - j I down v o ltag e m e a su re m e n t. | 1. An in su latin g in te rfa c e la y e r betw een the m e ta l and sem ico n d u cto r in c re a s e s the diode n -v alu e and the s e r ie s re s is ta n c e [5 0 ], 2. S urface leak ag e c u rre n t due to su rfa c e s ta te s n e a r th e ; edge of the m e ta l c o n tact in c re a s e s the n -v a lu e , m a y becom e a dom inant com ponent of the c u rre n t and m a y • be one of the re a s o n s fo r soft low voltage breakdow n. 3. The m e ta l p late co n fig u ratio n ten d s to c o n ce n tra te th e e le c tric field a t th e junction edges and yield an j edge breakdow n w hich is le s s th a n the bulk b r e a k down v o ltag e [ 5 1 ] , P re v io u s ly th e se d iffic u ltie s have been avoided by u sin g I d iffused g u ard rin g m e ta l-s ilic id e Schottky s tru c tu re s [3 7 ] o r a i : i ; g ate c o n tro lle d s tru c tu re [5 2 ], The above m ethods, h o w e v er, j 1 ' I | involve m an y co m p licated fa b ric a tio n p ro c e s s e s . j We have developed two new kinds of p a ssiv a te d Si ; Schottky d io d e s. W ith th e se diodes we have m ade breakdow n voltage m e a s u re m e n ts on n -ty p e Si w ith im p u rity d e n s itie s , N , i 16 18 , 3 betw een 1.5 x 10 and 3 x 10 a to m s /c m . i a . L a rg e A re a P o s t-E v a p o ra tio n -P a s s iv a te d P tS i-S i Schottky Diode - (LA diode) We ach iev ed one solution fo r the p ro b lem s d isc u sse d i I I above b y fa b ric a tin g la rg e a re a p o st-e v a p o ra tio n -p a s siv a te d i i j P tS i-S i Schottky diodes (LA diode). P t w as ev ap o rated th ro u g h a j | m e ta l m a s k on the d e s ire d p o rtio n of a Si w afer. P tS i w as th en | ! . . . . . . . . . . . . . . . . . . . . . . ' _ s ; fo rm ed in an oxygen am b ien t on th e ev ap o rate d a re a and sim u l- i ta n e o u sly a SiO_ la y e r w as grow n on the exposed Si su rface. | The SiO_ w as th e n re m o v e d w h ere ohm ic co n tacts w ere d e sire d j L i t j and A u-Sb ohm ic c o n ta c ts ap p lied . T his p ro c e s s p ro v id es a / p o sitiv e SiO_ s e a l aro u n d th e P tS i la y e r and red u c es surface ! C i \ leak ag e c u rr e n t. T he s tru c tu re h a s p ra c tic a lly no in te rfa c e ; i | la y e r p ro b lem sin c e it h a s a re g ro w n P tS i-S i in te rfa c e . In a : I i p re lim in a ry s tr u c tu r e the above two fe a tu re s w ere d em o n strated se p a ra te ly . T he e x p e rim e n ta l d ata for the fo rw ard I-V c h a ra c - ! te r is tic s b e fo re and a fte r th e grow th of th e P tS i c o n firm the j I ab sen ce of an in te rfa c e la y e r (cf. F ig . 17). The P tS i was obtained by a 10 m in u te h e a t tre a tm e n t a t 320° C in a 15% H„ ; 2 i 1 and 85% Ng a m b ie n t. N ote th a t th is te m p e ra tu re is m uch ; lo w er th an th a t ex p ected fro m a p hase d ia g ra m [5 3 ], The n - value im p ro v ed fro m 1 .1 2 to 1.03 due to the fo rm atio n of the <PtSi-Si ju n ctio n . T he h e a t tre a tm e n t w as p e rfo rm e d in a r e - j ducing am b ien t to avoid any im p ro v e m e n t in su rface leakage j d u rin g th is p r o c e s s . In fa c t, b e fo re the h e a t tre a tm e n t the 1 P t-S i ju n ctio n could m a in ta in -5 v o lts at 0. 1 m A but the P tS i-S i : i : ju n ctio n could only m a in ta in -2 v o lts a t th e sam e c u rre n t. The j ; leak ag e c u rre n t w as im p ro v e d by th e grow th of the SiO_ la y e r « j ! w ithout any d e g ra d a tio n in fo rw a rd I-V c h a ra c te ris tic s . The ! I I ; breakdow n c h a r a c te r is tic b ecam e th a t of bulk breakdow n. A ty p ic a l r e v e r s e I-V c h a r a c te r is tic of I m m d ia m e te r LA diodes 16 3 w ith = 2 x 10 a to m s /c m is shown in F ig . 18. T his j diode h a s a b reak d o w n voltage of 41 v o lts . T his value is con- i s is te n t with th a t o b tain ed by L e p s e lte r and Sze on carefu lly p re p a re d diffused g u a rd rin g P tS i-S i Schottky diodes [ 3 7 ] . The j I re v e rs e I-V c h a r a c te r is tic s of the LA diodes b efore the b re a k - i j -2 2 ! jd o w n a re a ls o n e a r ly id e a l. The (capacitance) , 1/C , v s. j i r e v e r s e v o lta g e , V , re la tio n sh ip show s th e absence of significant j FORWARD CURRENT (Amp/Junction) I0"2-t = - r : i *T - A • •* * •* • • • . .* 10 • • • 4 10 • • ' • • 9 9 9 10s A 9 • 9 9 9 9 # 9 9 # 9 B 9 ,0 6 -|= . • 6 B 2 S 4 - 1-8 20 mils <f> • • A Pt - Si n = 1.12 < / > =0.71 eV I0 7 • n = U 2 9 9 B PtSi - Si -a ' / n = l-03 1 0 -^ . < £ = 0 .8 6 eV 9 9 9 io9 l / . 1 . L — -----1 —------1 ---------1 ------- , u 0 .1 .2 .3 .4 .5 .6 FORWARD BIAS (Volts) F ig . 17. F o rw a rd I-V c h a r a c te r is tic s of P t-S i (A) and P tS i-S i (B) on n -ty p e Si at 300°K w ith N p = 2 x 10*6 a to m s /c m 3. REVERSE CURRENT (Amp/Junction) 96 -2 6CIS7-3-34 40 mil < £ -6 -7 -8 - 9 HO r2 io'' io° io 1 REVERSE BIAS (Volt) F ig . 18. R e v e rse I-V c h a ra c te ris tic of a LA diode a t 300°K w ith ND = 2 x 1016 a to m s /c m 3 Ju n ctio n d ia m e te r is 1 m m . ................................................................... “ . 9 7 ......... edge e ffe c ts [5 0 ] (cf. F ig . 19). The freq u en cy dependence of th e ju n ctio n cap a cita n c e is a lso exam ined betw een 20 Hz and 1 MHz to a s s u re the ab sen ce of any ap p re cia b le am ount of deep i j le v e l im p u ritie s [ 2 0 ] , ; F o r th is co n fig u ratio n the m e ta l should not oxidize du rin g j th e h e a t tre a tm e n t and it is convenient fo r the m e ta l to have a hig h alloying te m p e ra tu re . G row th conditions for the SiOg a re j a lso im p o rta n t. The optim um oxide w as a w et th e rm a l SiO , 16 3 grow n fo r 30 m in u tes a t 6 00°C for N ^ ~ 10 a to m s /c m . i H eat tre a tm e n t in a red u cin g am b ien t m ak es the diode lea k y w ith i a low voltage soft breakdow n [5 4 ], T his fu rth e r c o n firm s th a t \ th e S iO , p lay s an im p o rta n t ro le in th is configuration. T he SiO u se d in th is w ork w as obtained in the sy ste m j d is c u s s e d by A n d erso n in h is d is s e rta tio n [2 2 ], The in trin s ic ! j c h a r a c te r is tic s of th e SiO ^-Si sy ste m w ere exam ined by m aking \ MOS d io d es. T y p ical C -V re la tio n sh ip s a t v a rio u s fre q u e n c ie s a re show n in F ig . 20 for w et th e rm a l SiO^ grow n a t 900°C. The r e s u lt in d ic a te s th e e x iste n ce of an in v e rsio n la y e r [55] 11 2 w ith su rfa c e sta te d e n sity , Ngg, of 5 x 10 s ta te s /c m . N ote I th a t th e re e x is t som e fa s t sta te s w hich can resp o n d at le a s t up i to 1 M H z. S iO , grow n in d ry O , shows p ra c tic a lly the sam e 12 2 c h a r a c te r is tic s w ith a slig h tly h ig h er N (~ 2 x 10 s ta te s /c m ). s s The above co n fig u ratio n was u sed su c c e ssfu lly fo r im - 16 17 3 p u rity d e n sitie s betw een 1.5 x 10 and 4 x 10 a to m s /c m . The im p u rity d e n sitie s w ere d e te rm in e d to w ithin 10% fro m the 2 1/C v s . V re la tio n s h ip s . b. S m all A re a P o in t C ontact P a ss iv a te d Schottky Diode (SA diode) A sm a ll a re a point co n tact p a ssiv a te d Schottky b a r r i e r co n fig u ratio n h a s a ls o b een u sed su c c e ssfu lly . The p h y sical a p p e a ra n c e of th is diode is a lm o st like th at of a conventional I/C 2 0 5x10' Ll_ 6 C IS 7 -3 -3 4 40 milscjb Z0 30 REVERSE BIAS (Volt) 40 F ig . 19. The (cap acitan ce)- , 1 /C , v s . re v e r s e b ia s , V , re la tio n fo r a 1 m m d ia m e te r LA diode w ith = 2 x 10*^ a to m s /c m . 99 - ■ 70 l MHz 30 Hz -■ 60 - - 50 - 4 0 - 1 0 5 0 volts F ig . 20. MOS c ap a cita n c e, C, and equiv alent p a ra lle l conductance, G (: n o t to s c a le ), a t 1 MHz and 30 Hz as functions of field p late b ia s . W et th e rm a lly grow n SiC>2 a t 900°C . ' ' ' 100 point co n ta ct diode except fo r the ex isten ce of a SiO^ la y e r w hich m ak e s an in tim ate co n tact with the b a r r i e r m e ta l. T his s tru c tu re w as obtained by f ir s t grow ing the SiO la y e r on a ! C * i p re p a re d Si w a fer. Ohmic contacts w ere th en m a d e . F in a lly a j re c tify in g m e ta l-se m ic o n d u c to r junction w as fo rm ed by b rea k in g J th ro u g h the SiOg w ith a m e ta l point using e ith e r m e c h a n ic a l j fo rce o r an e le c tric a l p u lse . A l, Au, Cu, W and p h o sp h o r- I b ro n ze w ire s w ere u sed su c ce ssfu lly . The th ic k n e ss of the © o SiO- w as v a rie d fro m about 50A to 1500A w ith th e u s e of both | £ t ! w et and d ry oxidation. If the SiO^ is too th ic k i t is h a rd to j b re a k th ro u g h and if it is too th in the diode b e co m es u n sta b le and lea k y . The optim um condition is not c r itic a l, b u t w et SiO grow n a t 800°C for 10 m in u tes w as m o st s u c c e s s fu l. The j L t \ ro le of the SiO- is sim ila r to th a t in the LA dio d e. A ty p ic a l j C t r e v e r s e I-V c h a ra c te ris tic is shown in F ig . 21 fo r a W -Si j s tru c tu re . The breakdow n is sh a rp . The p reb re ak d o w n c u rre n t j is p ro p o rtio n a l to the applied voltage a s ex p ected th e o re tic a lly j (cf. A ppendix B). P h o to m u ltip licatio n in the junction w as d e m o n s tra te d w ith | a H e-N e la s e r . Since the junction a re a w as m u ch s m a lle r th an j i the lig h t sp o t, se p a ra te m e a su re m e n ts of th e e le c tro n and h o le ! m u ltip lic a tio n s could not be obtained. T y p ically a p h o to m u lti p lic a tio n fa c to r of the o rd e r of 100 could be o b tain ed n e a r b re a k - ; down (cf. F ig . 22). T his fu rth e r c o n firm s the e x is te n c e of | avalanche breakdow n. The ex isten ce of the m ax im u m in the j i m u ltip lic a tio n is due to the in te ra c tio n of th re e p ro p e rtie s of ; the sy ste m : (1) the voltage drop a c ro s s the s e r ie s r e s is ta n c e ! of the s tr u c tu r e , (2) the n o n lin ear I-V c h a r a c te r is tic s around th e breakdow n and (3) the ex isten ce of an a p p re c ia b le m u lti- j p lied d a rk c u rre n t n e a r breakdow n. The v o ltag e fo r m ax im u m g ain is v e ry clo se to the breakdow n voltage [ 5 ,6 ]. The -4 c o 7B2SI-4-P2 • « — o c 1 3 r 5 Q . e < .~ 6 LJ G O CC £ i o 9 LU cr , -io -ii REVERSE BIAS (Volt) F ig . 21. R e v e rse I-V c h a ra c te ris tic of a W -S i SA diode a t 300°K w ith ND = 3 x 1017 a to m s /c m ^ . Ju n ctio n d ia m e te r is 0 .4 (Jim. PHOTO CURRENT (Amp./Junc.) 102 n -6 r 8 7C IS9-2 - I REVERSE BIAS (Volt) F ig . 22. P h o to c u rre n t v s. ap p lied r e v e r s e b ia s r e latio n sh ip a t 300°K fo r a A u-SA diode w ith ~ 3 x 101? a to m s/c m ^ , 103 ju n ctio n cap a cita n c e w as too sm a ll in c o m p a riso n w ith s tra y cap a cita n c e fo r a conventional cap acitan ce m e a su re m e n t. The ju n ctio n a re a w as e stim a te d fro m th e sp read in g re s is ta n c e a t breakdow n [5 6 ], F o r e x am p le , the W -Si SA diode shown in F ig . 21 h a s a sp read in g re s is ta n c e of about 1000 0 and the Si re s is tiv ity is 0. 04 0 cm . T hus the d ia m e te r of the junction is about 0 .4 |im . The ju n ctio n a re a w as found to be co n tro llab le by changing the SiO^ th ic k n e s s. The fo rw ard I-V c h a ra c te ris tic is s im ila r to th a t of a S chottky diode. An n -v alu e of a p p ro x i- 17 3 m a te ly 1 .0 w as obtained fo r ~ 3 x 10 a to m s /c m . F ig . 23 show s the I-V c h a ra c te r is tic of a slig h tly p o o re r th an a v erag e W -Si SA diode. T his show s th re e c u rre n t reg io n s - - low , m ed iu m and h ig h c u rre n t re g io n s w here su rfa c e reco m b in atio n , diffusion and s e r ie s r e s is ta n c e re s p e c tiv e ly dom inate th e c h a ra c te r is tic [5 7 ], F ro m the ju n ctio n a re a and the c o rre c te d s a tu ra - tio n c u rre n t d e n sity , J g , ( i . e . , J g = J d a ta - J reo o m b in atio n > « « b a r r i e r h eig h t w as e stim a te d to be 0.61 eV . A fter a c o rre c tio n of 0 .0 5 eV fo r im age fo rc e lo w erin g [5 8 ], th is b a r r ie r height is c o n siste n t w ith the re p o rte d value of 0 .6 6 eV fo r W -Si Schottky b a r r i e r s [5 9 ]. We obtained 0.76 eV for the A u-Si z e ro field b a r r i e r h eig h t on o u r SA diodes which is v e ry close to th e re p o rte d value of 0 .7 8 eV [6 0 ], T hese m e a su re m e n ts c o n firm th a t th e ju n ctio n s a re Schottky b a r r ie r s . The im p u rity d e n sitie s w e re d e te rm in e d fro m the c ap acitan ce-v o ltag e re la tio n sh ip s of Schottky diodes d e p o site d n e a r the SA d io d es. As o b se rv e d w ith the LA d io d es, if the sam p les w ere h e a t tre a te d in a red u cin g a m b ien t, th e SA diode leakage c u rre n ts in c re a s e d . T h is c o n firm s, ag ain , the im p o rtan ce of the SiOg la y e r. T his co n fig u ratio n w as u se d for the full im p u rity range 1A 18 3 of N ^ : 2 x 10 to 3 x 10 a to m s /c m • FORWARD CURRENT (Amp/Juhc.) 10 10 i-7 10' 10 rll . . . 1 • . - .1 . . 1 ! / R C DC SRC / / . * / / O i / o / / ° / / ° / ■ / - p / - W - SA Diode f > / / - / / A / / / / / / n o f / — °— Jdata 12 9 V/ // + / ----------------drecom 2 * 0 J ? / / sP / k- v J d a t a - . a / . n d° / / drecom / > / / ' ° • / / ♦ ° / 7 2l . / ....\ L - . ____ 1 1 i 0.1 0.2 0.3 0.4 FORWARD BIAS ( v o lt) 0.5 0.6 F ig , 23. F o rw a rd I-V c h a ra c te r is tic of a W -Si SA diode at 300°K w ith = 4 x 10*^ a to m s /c m ^ . 1 ....................... 105.... V -3 DISCUSSION AND SUMMARY ! B reakdow n voltage m e a su re m e n ts w ere m ade a s a function 16 I of im p u rity d e n sity fo r the ran g e of N = 1 .5 x 1 0 to 3 x j 18 3 ■ D 10 a to m s /c m . T his ran g e w as chosen b ecau se a t h ig h im p u rity d e n sitie s th e re e x is ts an a p p re c ia b le d ifferen ce b etw een th e NLC p re d ic tio n s and p rev io u s e s tim a te s . The I-V c h a ra c t e r i s t i c s w e re m e a su re d w ith d c. The breakdow n v o ltag e w as ' d e te rm in e d by lin e a r e x tra p o la tio n of the I-V re la tio n sh ip under breakdow n conditions to z e ro c u rre n t d en sity to avoid s e r ie s re s is ta n c e and th e rm a l effe cts [4 4 ]. B reakdow n band bending I ^ 1 w as d e te rm in e d by adding the d iffusion p o ten tial to th e e x p e ri- i m e n ta l breakdow n v o ltag e. T his c o rre c tio n is sig n ific an t for ; | high im p u rity d e n s itie s . The diffusion p o ten tials have been 2 i ; e stim a te d fro m 1/C v s . V re la tio n s h ip s . The m ax im u m e r r o r | ! fo r th e breakdow n band bending d e te rm in a tio n is e s tim a te d to be j th e la r g e r of 0 ,2 v o lts o r two p e rc e n t. The e r r o r in the N ^ j d e te rm in a tio n is expected to be le s s than 10%, The re la tio n sh ip ! i b etw een breakdow n band bending and im p u rity c o n c e n tra tio n is show n in F ig .’ 24. H e re th e solid cu rv e is the NLC p re d ic tio n , th e b ro k en lin e is the e a r lie r p re d ic tio n b y Sze and G ibbons [3 6 ], The "o" and re p re s e n t d a ta fro m th e LA and SA 1 d io d es re s p e c tiv e ly . denotes th e data obtained on a g u ard ; rin g Schottky diode by L e p s e lte r and Sze [ 3 7 ] , The e x p e ri m e n ta l r e s u lts m atch the NLC p re d ic tio n w ith good a c c u ra c y , ’ M ille r 's d ata [1 2 ] show n by " g " a re a lso v e ry clo se to the NLC p re d ic tio n . M ille r 's r e s u lts w e re obtained fro m d iffused ju n c tio n s. T hus a c o rre c tio n could not be p e rfo rm e d . In | addition M ille r did not p re s e n t su fficien t d ata th a t th e e le c tric | field co n fig u ratio n could be re d e te rm in e d for h is d io d es. j E ven though we have not p e rfo rm e d any e x p e rim e n t in J Si Abrupt Junction 3 0 0 K — - Okutoa Crowell SzeaGibbons o [O' BACKGROUND DOPING ( a to m s /c c ) F ig . 24. Breakdown, band bending (sum of breakdow n voltage and diffusion potential) v s. im p u rity d e n sity fo r Si a b ru p t ju n ctio n s a t 300°K . Solid cu rv e: a fte r NLC th e o ry . B roken lin e : a fte r Sze and Gibbons [3 6 ], " O ": a fte r M ille r [1 2 ]; "o ": LA diodes; SA diodes; and a fte r L e p s e lte r and Sze [ 3 7 ] , .............. .......... .............."................' ' ........ 107" G aA s, it is in te re s tin g to exam ine the a g re em e n t b etw een our p re d ic tio n and p re v io u sly ex istin g e x p erim e n tal re s u lts [4 8 ,6 1 ] , A s is show n in F ig , 25, e x p erim e n tal re s u lts show a re a so n a b le a g re e m e n t w ith the p re s e n t p re d ic tio n s. The a g re e m e n t i s not good a s th a t for the c a se of Si, T his m ay be b e ca u se th e e x p e rim e n ta l r e s u lts w ere obtained on diffused ju n ctio n s an d do n o t include a c o rre c tio n and also becau se the m a te r ia l D c h a r a c te r is tic s for GaAs a re not as w ell e sta b lish e d as th o se fo r Si, T h ese c o m p a riso n s (cf. F ig s, 24 and 25) su g g e st the follow ing c o n clu sio n s: 1, F o r p ra c tic a l d evice breakdow n voltage c a lc u la tio n s 16 3 fo r KTp > 1 x 10 a to m s/c m in Si and G aA s, the NLC th e o ry p ro v id es a sig n ifican tly b e tte r e s tim a te th an p rev io u s calcu latio n s, 2, F o r Si, th e breakdow n m ec h an ism a p p e a rs to b e the 18 a v alan ch e p ro c e s s a t le a s t up to N = 3 x 10 a to m s /c m ^ and according to M ille r's d ata m a y also 19 3 extend up to N ^ = 1 x 10 a to m s /c m . F o r G aA s the tra n s itio n betw een the avalanche b reak d o w n to th e tunnel breakdow n seem s to lo ca te a t aro u n d 18 3 « 4 x 10 a to m s/c m . 10' GaAs Abrupt Junction 3 0 0 K LU < 10 o 1 0 ° BACKGROUND DOPING ( a t o m s / c c ) F ig . 25. C o m p ariso n of th e ex istin g e x p e rim e n ta l d ata [4 8 ,6 0 ] and p re d ic te d breakdo-wn band bending for a b ru p t jun ctio n s in GaAs a t ro o m te m p e ra tu re . SECTION VI SUMMARY AND CONCLUSIONS In th e lig h t of e n erg y co n serv a tio n c o n sid e ra tio n s the ex istin g u n d e r standing of the avalanche io n iz atio n e ffe ct in sem ic o n d u cto rs w as c ritic a lly exam ined. A cco rd in g ly a sim ple a n a ly tic a l e x p re ss io n fo r the m ean d ista n c e fo r io n iz atio n sc a tte rin g in sem ico n d u cto rs h a s been obtained w ithin the fram e w o rk of the B a ra ff m odel and s a tis fie s tw o p re v io u sly e x istin g p h y sic a l m o d e ls. The com plete re la tio n sh ip s b e tw een conventional th e o re tic a l p re d ic tio n s and e x p e rim e n ta lly o b se rv ed io n izatio n co efficien ts w ere fo rm u la te d . T h is r e latio n sh ip le d u s to m odify the d efin itio n of th e io n izatio n co e ffic ie n t (eg, io n izatio n coefficients a re n o t only functions of th e e le c tric field stre n g th but a lso functions of th e p o sitio n in sid e a ju n ctio n ). Once p rev io u s know ledge of th e io n izatio n th re s h o ld e n erg y and the o p tical phonon e n erg y is a cc ep ted , th e p re s e n t tre a tm e n t re q u ire s only know ledge of th e m ea n fre e p a th for o p tic al phonon sc a tte rin g as a ad ju sta b le p a ra m e te r . P o s itio n independent a p p a re n t io n izatio n co efficien ts w ere p r e d icted a s functions of th e e le c tric field stre n g th and the la ttic e te m p e ra tu re v ia a pseu d o lo cal ap p ro x im atio n in v a rio u s m a te ria ls by a ssig n in g a single value for the o p tic al phonon m ean fre e p ath for e ach sp ecies of c a r r i e r s fo r e ach m a te ria l. The p re d ic tio n s show e x cellen t a g re e m e n t w ith the e x istin g e x p er im e n ta l r e s u lts . The com plete n o n lo calized fo rm u la tio n w as a lso solved and ex act solutions for p - i- n ju n ctio n s in Si 109 ................................................. ■............................................ .....................................l i o ! and GaAs w ere o b tain ed . By exam ining e x ac t solutions the r e - , la tio n sh ip s betw een th e m u ltip lic atio n fa c to r, breakdow n band bending and the a p p a re n t io n izatio n co efficien ts w ere c la rifie d . I The p re d ic te d breakdow n band bending show s ap p re cia b le d ifferen c e fro m th e p re v io u s p re d ic tio n s fo r high field con fig u ratio n s and is in re a so n a b le a g re e m e n t w ith th e existing e x p e rim e n ta l d ata.' To c ritic a lly exam ine the p re s e n t t r e a t m e n ts p re c is e b reak d o w n band bending m e a su re m e n ts w ere p e rfo rm e d in n -S i w ith the u se of two kinds of sp e cia lly developed p a ssiv a te d m e ta l-S i Schottky d io d es. The re s u lts fu rth e r c o n firm th e ad eq u acy of th e p re s e n t tre a tm e n ts . It is e x p e rim e n ta lly o b se rv e d th a t the breakdow n m ec h an ism in Si 18 3 is v ia the avalanche a t le a s t up to « 3 x 10 a to m s /c m • T hroughout th e p re s e n t w ork we d e m o n stra te d th at the i e n erg y c o n se rv a tio n re la tio n sh ip is a u se fu l and p rev io u sly unused c rite rio n to u n d e rsta n d the avalanche e ffe c t in s e m i c o n d u cto rs. A PPEN D IX A DERIVATION O F THE M U LTIPLICA TIO N EQUATION In th is tre a tm e n t we a ssu m e th a t n e le c tro n s a re in - o je c te d fro m x = 0 and a lso th e sam e sa tu ra tio n v e lo c ity for e le c tro n s and h o les; thus n = n(x) + p(x) = c o n stan t . (A -l) H ere n, is the to tal d e n sity of c a r r i e r s . One can divide a ju n ctio n in to th re e re g io n s , i . e. D n R egion I 0 < x < w here E ^ = J (g(x)dx, o a (x) = 0 n and ap (x) = R egion II D ^ x ^ W-D n p an (x) = and a (x) s a , (<g(x)). P pa 111 a (5 (x )). pa W w here E . = f <g(x)dx, 1P V -D P a (<?(x)) na 112 R egion HI W -Dp < x < W w here a (x) = a (8 (x )) n na and a (x) = 0. P T hen th e m a th e m a tic a l tre a tm e n t is s tra ig h tfo rw a rd . R egio n I = ap ^ P W » D D n n n(Dn ) = n^. J a (x) exp - J a (x ') dx ' dx n o p x P D n + n exp - f a (x)dx . (A -3) ° J o P R egion II = a (x) n(x) + a (x) p(x) . (A-4) ax n p Thus W -D W-D P /• P n(W -D ) = n f a (x) exp f { a j * ') ~ a ( x ') } d x 'd x P t J _ p J n p P W -D 'D r x n r P + n(D ) exp f a (x) - a (x)}dx . (A-5) n ^ n p n 113 Region III = an (x) n(x) , (A-6) w n^ - n(W -D p) exp J an (x)dx . (A-7) W-D P By elim in a tin g 1 1 ( D ) and n(W -D ) fro m E q s. A -3 , A-5 and A -7 one o btains the re la tio n sh ip W W "Dp 1 - M~ exp I L “n (x ,d x ~I “p(x)dxj n ' D o n W -D W -D P P = f a (x) exp J { a ^ x 7 ) - a ( x 'J j d x 'd x T) ^ x ^ n W • exp a (x)dx W-D n P D D n _ n + j a (x) exp J - a ( x ') d x 7 J p J p o x W • exp ( Ji n n dx W-D / W P ) j j an (x)dx -J ap (x )d x | . (A-8) It is obvious th a t one can obtain a c o rresp o n d in g rela tio n sh ip fo r M in th e sam e m a n n e r. E q . A -8 is a g e n eral e x p re ssio n P w hich can be u se d fo r any e le c tric field configuration. It m ig h t be w o rth pointing out th a t w hen W becom es la rg e , the th re s h o ld e n erg y c o rre c tio n b e co m es le s s im p o rtan t since the band bending in c re a s e s but th e p o ten tial d ro p s a c ro s s both boundary d a rk sp a c e s s ta y c o n sta n t. T his c o rre c tio n is also le s s im p o rtan t for g rad e d ju n ctio n s since the e le c tric field in sid e the boundary d a rk s p a c e s is sm a ll and th u s th e c o n tri bution to the io n izatio n in te g ra l is sm a ll. APPENDIX B EM PIRICA L I-V CHARACTERISTICS O F SCHOTTKY DIODES E ven w hen no ap p re cia b le in te rfa c e la y e r e x is ts betw een a m e ta l and a se m ic o n d u cto r, th e actu al I-V of a Schottky diode is not th a t w hich one expects fro m th e sim p le one d im en - : sio n al junction -theory. R a th e r the actu al c u rr e n t, I(V), can be w ritte n as I(V) = Ij(V ) + IS (V) . (B -l) H ere 1^ (V) is th e c u rre n t com ponent d e riv e d by the one d im en sio n al ju n ctio n th e o ry and is p ro p o rtio n a l to the ju n ctio n a re a . I_ (V) is the su rfa c e leakage c u rre n t due to tra p p in g le v e ls , o F o r the fo rw a rd c h a ra c te ris tic s tr a p s a ct a s re c o m b in a tio n c e n te rs and I (V) h a s an ex p ected n -v alu e of tw o. T hen o when -V » k T / q , the fo rw a rd I-V can be w ritte n a s I(V) = a . exp (B -2) w here a and b a re c o n sta n ts, $ is the b a r r i e r h eig h t betw een th e m etal and th e sem ico n d u cto r and i s an effectiv e b u ilt-in p o ten tial betw een th e m e ta l and the sem ico n d u cto r s u rfa c e . In ; E q . B -2 , the second te r m is due to the su rfa c e re c o m b in a tio n j c u rre n t and the f i r s t te r m is due to the one d im e n sio n a l diode i c u rre n t, i . e . diffusion c u rre n t plus tu n n el c u rre n t. 115 116 M o re o v e r, once an a d d itio n al s e r ie s re s is ta n c e , R , s is included i t is obvious th a t E q . B -2 can be w ritte n as /q < V -R I) - I \ / q(V -R I) - } ' \ X(V) = a exp { -------- — --------- ) + b exp ( ------- ^ --------- j (B-3) T his eq u atio n should be ab le to re p ro d u c e p ra c tic a l e x p e ri m e n ta l r e s u lts (eg. F ig s . 18 and 24). U nder re v e rs e b ias tra p s a c t a s g e n e ra tio n c e n te rs and Ig(V) is p ro p o rtio n a l to the n u m b er of tr a p s w hich a re exposed u n d er the e le c tric field . The n u m b er of tr a p s involved, N^, is p ro p o rtio n a l to the a r e a w hich c o v e rs the su rfa c e space ch arg e reg io n and to N gg, th e su rfa c e sta te d e n sity . Then Nt = tTT(rJ + W)2 " n r J* ’ N s s (B"4) h e re r is th e ra d iu s of the ju n ctio n m e ta l p late and W is the J w idth o f th e sp a ce c h arg e reg io n ,' Since W is p ro p o rtio n a l to (V + i-g)2 f ° r a b ru p t ju n c tio n s, w hen N gg is co n stan t, Ig(V) can be w ritte n as Ig(V) = c . ' t j . {V+«B }2 + d . {V+«B } (B-5) h e re c and d a re b ia s independent c o n sta n ts. In E q. B-5 when V e x ce ed s a few v o lts $ can be n e g le c te d .' T hen E q. B -l B can be w ritte n a s 2 — I(V) = n r j o J X (V) + c • r j • V 2 + d .V (B-6) H ere Jj(V ) is th e c u rre n t d en sity due to th e one d im en sio n al ju n ctio n th e o ry and can be assu m e d to be c o n sta n t. E q . B -6 in d ic a te s th a t th e re e x is t th re e ty p ic a l I-V re la tio n sh ip s depending on the ju n ctio n s iz e , i. e . I(V) = co n stan t w h ere r : la rg e JL ^ I(V) « V 2 w h ere rj.r m ed iu m I(V) « V w h ere r : sm a ll . J It is obvious th a t F ig. 19, F ig . B - l and F ig . 22 c o rre sp o n d c a s e s 1, 2 and 3 re s p e c tiv e ly . and 1. 2. 3. REVERSE CURRENT (Amp/Junction) 118 6CIS7 - 3 - 5 5mils < $ > -6 -10 r Z REVERSE BIAS (Volt) Fig." B - l . R e v e rse I-V c h a r a c te ris tic of a LA diode a t 300°K w ith = 2 x 10*° a to m s /c m ^ . Ju n ctio n d ia m e te r is 125 m ic ro m e te rs . APPENDIX C ESTIMATION OF THE IONIZATION C O EFFICIEN TS FROM BREAKDOWN BAND BENDING P re v io u sly the a p p a re n t io n izatio n co efficien ts have been p re d ic te d fro m m u ltip lic a tio n m e a s u re m e n ts . Once th e io n iz a tio n th re sh o ld e n e rg y e ffe c t is taken into acco u n t, the re la tio n ship betw een m e a su re d m u ltip lic a tio n fa c to rs and the io n izatio n co efficien ts b e co m es co m p lica te d (cf. A ppendix A). B e sid e s, th e m u ltip lic atio n m e a s u re m e n t is a co m p licated e x p e rim e n t : and involves m any u n c e rta in tie s a s w ere d isc u sse d in S ection I. In Section V, th e re la tio n sh ip betw een the breakdow n band bending and th e a p p a re n t ~ioniza,tion c o efficien ts w as fully fo rm u la te d . E x p e rim e n ta lly , the b reakdow n band bending m e a s u r e m e n ts a re re la tiv e ly re p ro d u c ib le and e a sy to p e rfo rm (cf. ; F ig . 24). A cco rd in g ly , a p p a re n t io n iz atio n c o efficien ts w ere e stim a te d fro m th e breakdow n band bending in Si a t 300°K . T his e stim a te w as p e rfo rm e d by scanning o p tical phonon m ean ' fre e paths fo r e le c tro n s and h o le s and p re d ic te d band bendings w e re c o m p ared w ith the e x p e rim e n ta l r e s u lts . X^ and X^ w ere i scanned independently b etw een 42 A and 49 A for e le c tro n s and i 40 A and 47 A for h o le s a t 300°K . T y p ical re s u lts a re show n in ; F ig . C - l along w ith the e x p e rim e n ta l data (O). It is found th a t X = X s 4 5 A is a re a so n a b le ch o ice. T he co rre sp o n d in g n p I io n izatio n c o efficien ts a re show n in F ig . C -2 . It is obvious I th a t the d iffe re n c e s betw een th re e c a s e s a re e x p e rim e n ta lly ' d istin g u ish ab le (cf. F ig . C - l) . The re s u ltin g d iffe re n c e s in 119 Abrupt Junction cn O > 00 > BACKGROUND DOPING (atoms/cc) F ig . C - l. B reakdow n band bending fo r Si a b ru p t ju n ctio n s a t 300°K . L egend: O e x p e ri m e n ta l r e s u lts , p re d ic tio n w ith o p tical phonon m ean fre e p ath s 48 A and 47 A fo r e le c tro n s and h o le s re s p e c tiv e ly a t 3 0 0 ° K ,--------------p re d ic tio n w ith 45 A and 45 A , ----- - - p re d ic tio n w ith 42 A and 40 A. c m 121 i 6 10 300 I04 ^ 'O3 I0 2 10 1 8 ( x 10® V / cm) 10 F ig , C -2 . P re d ic te d a p p a re n t io n iz atio n c o efficien t in Si at 3009K as a function of the e le c tr ic field stre n g th . L egend: o p tical phonon m e a n fre e path 48 A and 47 k for e le c tro n s and h o le s re s p e c tiv e ly a t 300°K , ----- -------- p re d ic tio n w ith 45 k and 45 k - - - - - p re d ic tio n w ith 42 A and 40 A, 122 the a p p a re n t io n iz a tio n coefficients e stim a tio n is rea so n a b ly ;sm a ll. It is n u m e ric a lly found th a t to d e te rm in e the o p tical jphonon m ean fre e p a th w ithin 10% one h as to have knowledge of the im p u rity d e n sity and the b reak d o w n band bending w ithin a few p e rc e n t. In p rin c ip le if a = a is a ssu m e d one V data is r e - n p B iquired to p re d ic t the p seu d o lo cal io n iz a tio n coefficient a s a function of th e e le c tr ic field s tre n g th (assum ing E . and E^ a re know n). W hen O t = £ a th is m ethod re q u ire s two V w ell n p B se p a ra te d d ata to p r e d ic t and (X ^ (again assu m in g E .^ , E .^ an d E a re known). In th is case, if one w ishes to se p a ra te a and r n ;a w ith re a so n a b le a c c u ra c y th o se V d ata have to be a c c u ra te p B w ith in a few p e rc e n t. E ven though th is m ethod h a s a d isa d v an tag e b e c a u se th e re is no sim ple a c c e s s to se p a ra te the e le c tro n and th e hole c o n trib u tio n s d ire c tly , h o w ev er, because of its sim p lic ity , th is m eth o d can be u s e d as an approxim ate check. . R E FE R EN C ES 1. C. Z en e r, P ro c . Roy. Soc. A 145, 523 (1934). 2. M icrow ave Sem iconductor D ev ices and T h e ir C irc u it A pplications (H. A. W atson e d . ) M cG raw -H ill Book C o ., New Yorjc (1969). 3. S. M . Sze and R . M. R y d e r, P ro c . IE E E 59, 1140 (1971). 4. K. M . Johnson, IEEE T ra n s . E le c tro n . D ev ices ED 12, 55 (1965). 5. K atsuhiko N ishida, Jap an J . A ppl. P h y s . 9 _ , 481 (1970). 6. H, M elchior and W. T . L ynch, IE E E T ra n s , E le c tro n . D evices E D -U , 829 (1967). 7. S. M . Sze, P h y sic s of S em ico n d u cto r D e v ic e s, John W iley and Sons, In c ., New Y ork (1969). 8. A. G. Chynoweth, "C harge M u ltip lic a tio n P h e n o m e n a ," Sem iconductors and S e m im e ta ls , (R. K. W illardson and A. C. B e e r, e d s .) V ol. 4, A cadem ic P r e s s , New Y ork (1968). 9. K. G. M cKay and K. B. M cA fee, P h y s . R ev. 91. 1079 (1953). 10. K. G. M cKay, P h y s. R ev. 94, 877 (1954). 11. L . B. L oeb, F undam ental P r o c e s s o f E le c tric a l D isch arg e in G a se s, Jo h n W iley and Sons, I n c ., New Y ork (1939). 12. S. L . M ille r, P h y s. R ev. 105, 1246 (1957). 123 124 13. C. A. L ee , R . A. L ogan, R . L . B a td o rf, J. J . K leim ack and W. W iegm ann, P h y s. Rev. 134, . A761 (1964). 14. M . H, W oods, W. G. Johnson and M . A. L a m p e rt, S o lid -S t. E le c tro n . 16, 381 (1973). 15. J . L , M oll and R . V a n O v e rstra e te n , S olid-S t. E le c tro n . (> , 147 (1963). 16. R . V anO ver s tra e te n and H. DeM an, S olid-S t. E le c tro n . 13, 583 (1970). 17. R . A. L ogan and S. M . S ze, J . P h y s. Soc. Jap an Suppl. 21, 434 (1966). 18. R . A, Logan and H. G. W hite, J . A ppl. P h y s. 36, 3945 (1965). 19. W. N . G ra n t, S olid-S t. E le c tro n . 16, 1189 (1973). 20. G. 1. R o b e rts and C. R . C ro w ell, J . A ppl. P h y s. 41, 1767 (1970). 21. N. R . H ow ard, J . E le c tro n . C ontrol 13_, 537 (1962). 22. C. L . A n d erso n , P h .D . T h e s is , U n iv e rsity of Southern C alifo rn ia (1973). 23. E . J . R yder and W. Shockley, P h y s . R ev. 81, 139 (1951). 24. J . L . M oll, P h y s ic s of S em ico n d u cto rs, M cG raw -H ill Book C o ., New Y ork (1964). 25. C. L , A nderson and C. R . C row ell, P h y s . R ev. B5, 2267 (1972). 26. W. Shockley, B ell S y stem T ech. 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Creator Okuto, Yuji (author)

Core Title Avalanche Effect In Semiconductors

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Degree Doctor of Philosophy

Degree Program Materials Science

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Tag engineering, materials science,OAI-PMH Harvest

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Advisor Crowell, Clarence R. (committee chair), Gershenzon, Murray (committee member), Porto, Sergio P.S. (committee member)

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