VNU JOURNAL OF SCIENCE Mathematics - Physics T.XVIII, N04, 2002 R E F E R E N C E L E V E L S , S IG N A L F O R M S A N D D E T E R M IN A T IO N O F E M IS S IO N F A C T O R IN D L T S Hoang Nam Nhat and Pham Quoc Trieu D e p a rtm e n t o f P h ysics , College o f Sciences, V N U A b s tr a c t The existence of referen ce le v els of sig n a ls w hich d eterm in e directly th e tem p eratu re d ep en d en ce of em issio n factor in deep level transient p h en om en a is d iscu ssed T h e b asic algebraic stru ctu re of reference le v e ls in the cla ssic a l DLTS is stu d ied and various sig n a l forms w ith derived reference levels are given W e then d em o n stra te th e u se o f th ese sign al forms and com pare them w ith th e c la ssic a l DLTS double boxcar sign al K e y w o rd s S ig n a l fo r m s , r e fe r e n c e le v e ls , D L T S , d e e p tr a p Introduction T h e e x i s t e n c e o f t h e d e e p le v e ls is a n im p o r ta n t p h e n o m e n o n in s e m ic o n d u c to r p h y s ic s It is w e ll-k n o w n th a t th e y cau se m any c o n s id e r a b le b e h a v io u r s of m a t e r ia ls T h e c h a r a c t e r iz a t io n o f t h e d e e p tr a p s fa c e d m a n y d if f ic u lt ie s u n t il 1974 w h e n L a n g h a s in t r o d u c e d a s p e c tr o s c o p ic m e th o d c a lle d t h e D e e p L e v e l T r a n s ie n t S p e c tr o s c o p y (D L T S ) [ 1] T h is a llo w s to d educe fro m c a p a c it a n c e d e c a y s b a s ic p h y s ic a l th e e x p o n e n t ia l C(() = ACe '*'1 t h e p a m eters of th e tr a p s s u c h a s t h e a c t iv a t io n e n e r g y , ca p tu r e c r o s s - s e c t io n c o n c e n t r a t io n has b een The and L a n g 's m e th o d w id e ly a c c e p te d to d a y a s th e sta n d a rd to ol, a lt h o u g h it h a s s e v e r a l li m it a t io n s s u c h a s t h e s lo w r u n a n d r e l a t iv e l y lo w r e s o lu t io n T o e x t r a c t t h e tr a p p a r a m e t e r s fro m t h e e x p o n e n t ia l d ecays, in tr o d u c e d th e Lang s ig n a l has form S (T )= C (ti)-C (t 2) - t e c h n ic a lly r e a liz e d F ig A L a n g 's S (T )= C ( t j ) - C( t 2) s e t t in g s and dependence d e t e r m in e for v a r io u s draw s th e of th e m e th o d S (T ) scan s tj and t2 te m p e r a tu r e The te m p e r a tu r e s m a x im u m T o f th e e m is s io n fa c to r Cnax s e t fo r th b y t h e te w in d o w s u s in g a d o u b le b o x c a r c ir c u it , w h ic h m o n ito r s t h e c a p a c it a n c e t r a n s i e n t s a t tw o d if f e r e n t t im e s T h is f u n c t io n S ( T ) h a s a d e s ir a b le p r o p e r ty t h a t i t s h o w s m a x im a l g a in a t c e r ta in te m p e r a tu re r e la t e d to t h e d o u b le b o x c a r r a te w in d o w s s e t t in g S o b y s c a n n in g t h e S ( T ) o v e r t e m p e r a t u r e s e v e r a l t im e s o n e o b t a in s th e 28 R e fe r e n c e le v e ls , s ig n a l f o r m s a n d d e te r m i n a t io n o f 29 fu n c t io n a l d e p e n d e n c e o f e m is s io n fa cto r on t e m p e r a t u r e e= f(T) a n d c a n c o n s tr u c t t h e A r h e n iu s p lo t In ( e l T 2) v e r s u s 0 /T for t h e d e t e r m in a t io n o f tr a p p a r a m e te r s ( F ig l) T h e k ey e le m e n t in t h is te c h n iq u e is th u s th e d e t e r m in a t io n o f th e t e m p e r a t u r e d e p e n d e n c e e -f(T ) U p to now , m a n y a t t e m p t s h a v e b e e n m a d e in t h i s fie ld to im p r o v e t h e D L T S m e th o d A m o n g t h e t e c h n iq u e s t h a t h a v e b e e n r e p o r te d [2 -1 ] ( th e li s t is c e r t a in ly n o t c o m p le te ), t h e r e a re tw o t h a t a t t r a c t e d g e n e r a l a t t e n t io n : th e F o u rier a n d the L a p la c e te c h n iq u e T h e s e a r e b o th t r a n s fo r m a tio n m e t h o d s m a n ip u la t in g w ith th e w h o le r a n g e o f m e a s u r e d d a ta , u s u a lly d ig it a lly re c o r d e d or p o in ts R e c a ll t h a t t h e c la s s ic a l S (T ) u s e s o n ly p o in ts a n d t h r o w s t h e r e s t a w a y In g e n e r a l th e F o u r ie r a n d t h e L a p la c e s ig n a l fo r m s s h o w m o r e s e n s i t iv e p e a k s t r u c t u r e o f th e g a in , b u t s in c e t h e y n o t in v o lv e a n y r a t e w in d o w t h e e x a c t e m is s io n fa c to r a t th e m a x im a l g a in c a n n o t b e c a lc u la te d in a d v a n c e T h u s t h e c o r r e s p o n d e n c e o f t h e p e a k s a n d th e d e e p c e n t e r s a p p e a r s in t h e s e c a s e s s o m e h o w s u b t le a n d a r b itr a r y A c o m m o n f e a t u r e o f a ll s p e c tr o s c o p ic m e t h o d s is t h e p r e s e n t a t io n o f th e a n a ly t ic a lg o r ith m c o n v e r t in g t h e s e t o f th e c a p a c it a n c e t r a n s i e n t s C(t)y e a c h o f th e m h a s b e e n re c o rd e d a t s o m e p r e s e t t e m p e r a t u r e T , in to t h e s p e c ific v a lu e s of c e r t a in a n a ly t ic f u n c t io n s f n( T ), s h o w in g t h e p e a k s t r u c t u r e s a c c o r d in g to T T h e f n(T) h a v e tw o im p o r ta n t p r o p e r tie s: ( 1) th e y a r e sp ectro sco p ic in t h e c o n t e x t t h a t e a c h o f t h e p e a k s in /„ (T) c a n b e a s s o c ia t e d w ith o n e s p e c if ic d e e p c e n t e r a n d ( ) th e y a r e lin e a r y i.e t h e A r h e n iu s p lo t [ln (e /T 2) v e r s u s 0 /T ] t r a n s f o r m a t io n o f th e m a x im a o f a r b itr a r ily c h o s e n p ea k is lin e a r T h e f u n c t io n s f n(T ) r e p r e s e n t th e a lg o r ith m a n d u s u a lly t h e m e th o d is n a m e d a f t e r f n(T ) H e r e in a f t e r t h e f n( T) a r e r e fe r e e d to a s t h e s ig n a l fo r m For s h o r t w e m a y r e m o v e t h e in d e x n d e n o t in g th e t im e - s e t t in g s a n d u s e f{ T) in s t e a d o f f n( T ) T h e d if f e r e n t s ig n a l fo r m s in v o lv e th e d iffe r e n t n u m b e r o f m e a s u r e d d a t a a n d h a v e t h e d if f e r e n t a b ilit y in s e p a r a tio n o f th e o v e r la p p in g d e e p c e n t e r s T h e c la s s ic a l L a n g 's s ig n a l fo rm , for e x a m p le , in v o lv e s o n ly p o in t s in t h e w h o le t r a n s ie n t , w h e r e a s t h e F o u r ie r a n d th e L a p la c e s ig n a l fo rm s a r e c o m p o s e d p r in c ip a lly o f t h e w h o le t r a n s ie n t T h e r e is n o t k n o w n u n t il to d a y a n y o th e r s p e c tr o s c o p ic s ig n a l form t h a n t h e a b o v e t h r e e In t h is w o rk w e p r e s e n t t h e s tu d y o f t h e a lg e b r a ic s t r u c t u r e o f th e L a n g 's c la s s ic a l s ig n a l form S(T ) s h o w in g t h a t t h is fo rm p o s s e s s e s a d e s ir a b le p r o p e r ty o f h a v in g a s o -c a lle d reference level of s ig n a l w h ic h d ir e c t ly d e t e r m in e s th e r e la t io n s h ip e=fÇT) T h is p r o p e r ty o f D L T S w a s n o t r e p o r te d a n y w h e r e b efo re W e t h e n in tr o d u c e t h e c l a s s e s o f m a n y o th e r s ig n a l fo r m s h a v in g t h e s a m e a lg e b r a ic s t r u c tu r e o f t h e r e fe r e n c e le v e ls a n d r e d u c in g t h e L a n g 's fo r m a s a s p e c ia l c a s e In c o n tr a s t to t h e L a n g ’s fo rm t h a t in v o lv e s v a lu e s o f C(t)y t h e r e is a c la s s o f fo r m s w h ic h in v o lv e o n ly s in g le v a lu e C(t) T h is is a s u r p r is in g f a c t t h e s e fo r m s a ls o p r o v id e t h e p e a k s t r u c t u r e o f g a in a c c o r d in g to T T h e L a n g 's s ig n a l fo rm is e x t e n d e d in to t h e c la s s o f s ig n a l fo r m s w h ic h c o n t a in s m a n y o t h e r fo r m s p r o v id in g H o a n g N a m N h a ty P h a m Q u o c T r i e u 30 th e sam e re s u lts as th e L an g ’s form T he fact th a t th e re e x ist m any a n aly tic functions f( T) fulfilled th e re q u ire m e n t of being th e signal form s is firs t described in th is paper The reference levels in Lang's signal form S(T) and their algebraic structure T he dependence of th e cap acitance tra n s ie n t C(t) on tim e t is considered in g en eral case as: C (0 = Co + l Q r « f ( 1) w h e r e C is C(t=oo), AC=ZAC; = C(t=O)-C a n d i d e n o t e s t h e n u m b e r o f p r e s e n t d e e p tra p s W ith re sp e c t to th e n o rm alized cap acitance given as C n(t)=(C(t)-Co) / AC, a n d d e n o te t j = t - d f t 2=t+di w e r e d e f in e t h e L a n g 's s ig n a l for t h is g e n e r a l c a se : S { T ) = C n ( t - d ) - C n (t + d) = ỵ ,{ A C i / A C )[e-‘‘('~d) - e ' ,ti,+d)] Suppose t h a t th e tr a p s a re in d ep e n d en t an d not o v erlap p in g each o ther (they a re fa r each from o th er in th e te m p e tu re scale), one m ay d iffe re n tia te th is sig n al according to som e em ission factor e„ leaving th e o th er ones zeroed, to determ in e th e signal m axim al g ain in th e given te m p e tu re range We modify th e re s u lt from [1] w ith r e s p e c t to t h e v a r ia b le s t a n d d m e n tio n e d above: e m«x=Zn [(*+d )/(t -d )]/2 d (3) T his re la tio n show s th a t by fixing th e r a te w indows (by t a n d d) one also selects th e em ission factor to w hich th e L an g ’s sig n al re a c ts m ostly when it scans th ro u g h th e s e t te m p e tu re ran g e W ith th e in crease of te m p e tu re th e tra p begins to re le a se electrons an d it re le a se s m ostly w hen th e em ission factor is high enough, is in g th e L ang's sig n al to m axim um B ut w hen th e tr a p becom es b lank, th e em ission process slow s down re s u ltin g in th e drop of L an g 's signal T his in t u it iv e u n d e r s ta n d in g o f t h e e m is s io n p r o c e s s - a lth o u g h n o t c o r r e c t, o ffers c e r ta in physical m ean in g to the L ang's sig n al an d set th e believe t h a t i t re a lly depicts th e physical tra p s O n e t h in g t h a t s e e m s e it h e r u n o b s e r v e d or a t t r a c t e d n o c o n s id e r a b le a t t e n t io n from t h e L a n g 's t im e is t h a t t h e r e la t io n (3) u s e d to o b ta in t h e e mBX a lm o s t e q u a ls 1It n u m e r ic a lly U s in g th e E u le r n u m b e r d e f in it io n fo r m u la lim(l + l / « ) ” =e o n e c a n « —►«5 w ith o u t d iffic u lty p ro v e t h a t ln[(t+d)/(t-d)]/ d r e a lly c o n v e r g e s t o / t w h e n d-> G iv in g t h e fa c t t h a t ln [(t+ d )l(t-d )]l 2d ~ 1 1, t h e e max a lw a y s c o r r e s p o n d s to C n(t)=e~i R e f e r e n c e l e v e l s , s i g n a l f o r m s a n d d e t e r m i n a t i o n o f 31 (e is E u le r n u m b e r ) T h is s p e c ia l fe a tu r e o f t h e c la s s ic a l d o u b le b o x ca r t e c h n iq u e is illu s t r a t e d in F ig 2, w h e r e o n e c a n s e e t h a t t h e e max o c c u r s e x a c t ly w h e n C n(t) p a s s e s t h r o u g h t h e c r o s s -p o in t o f t h e g a te c e n t r a l p o s it io n t a n d t h e lin e c „ = e T h is m e a n s t h a t d e s p it e o f t h e v a r ia tio n in t h e r a t e w in d o w p o s itio n s , th e o n ly area o f im p o r ta n c e w a s C n(t)=e~l T h e e v id e n t c o n s e q u e n c e fo llo w s im m e d ia te ly t[ms] t h a t to d e t e c t t h e fu n c tio n a l d e p e n d e n c e o f t h e e m is s io n fa cto r on th e t e m p e r a tu r e eị -f(T) o n e s im p ly ch eck t h e c r o s s -p o in ts F ig o f C n(t) a n d c „ = e to o b ta in d ir e c tly th e d o u b le v a lu e of e m is s io n fa c to r (e^l/t) The b o x ca r a c c o r d in g w e c a ll C n= e “1 t h e d ecrea ses F or t h is reason reference level o f t h e s ig n a l form S (T ) It fe a t u r e te c h n iq u e : [ t - df t+ d ] s h o w s w in d o w c o r r e s p o n d in g to th e g iv e n t e m p e r a tu r e T s p e c ia l to T w hen th r o u g h of th e th e r a te m a x im u m C n(t) th e th e area c n( t h l/e = is a g r e a t a d v a n t a g e for t h e s ig n a l form to p o s s e s s t h e r e fe r e n c e le v e l s in c e t h is m e a n s t h a t e- f ( T ) c a n b e d e r iv e d d ir e c tly from it s r e fe r e n c e le v e l A lth o u g h t h e L a n g 's s ig n a l o n ly a p p r o a c h e s t h is r e fe r e n c e le v e l in t h e lim it c a s e w h e n t h e g a t e w id th d is in f in it e s im a lly s m a ll, th e r e is a lo t o f o th e r s ig n a l fo rm s a s d is c u s s e d in th e n e x t s e c tio n , w h ic h h a v e e x a c t r e fe r e n c e le v e l T h e im p o r ta n c e o f r e fe r e n c e le v e ls fo llo w s from th e fa c t th a t th e y le a d to th e u n d e r s ta n d in g o f t h e a lg e b r a ic s tr u c tu r e o f th e e x p o n e n t ia l d e c a y s in g e n e r a l a n d o f th e c a p a c ita n c e t r a n s ie n t p a r tic u la r ly W e n o w in tr o d u c e t h e s o -c a lle d L a n g 's s ig n a l c la ss a n d d e r iv e th e a lg e b r a ic s t r u c t u r e fo r t h is c la s s C o n s id e r th e m o v in g o f g a t e from t to t'= a ty for a is a p o s it iv e r e a l n u m b e r S in c e emax d e p e n d s in v e r s e ly o n t it fo llo w s t h a t th e e m is s io n fa c to r eị(t) d e te c te d on th e b a s is o f emax(t) c h a n g e s as: ei( t t) = e t(at) = H a t = (lla )C i(t) T h e t r a n s ie n t a s s o c ia te d w ith t h is e f t ' ) w ill h a v e a t t im e t t h e v a lu e e q u a l to t h e v a lu e o f th e t r a n s ie n t a s s o c ia t e d w ith Ci(t) a t t im e t l a : e *,1, u su a lly Ẳ = C ' Q j " - C M & ' 1* ụ / t ) i n ị ầ C b ứ J ( \ - C 0lnk)} n ee d n o rm a liz e d C n(t) b u t n o t f o r a~l c ( f , ) " / c ( / 2r u su a lly n - o r 5-a C 0)] e~° y a - ln [ầ C ỉn )J ( 1-C (JnX )] w c to /«[AC/fa-Co)] 3-5% emax= (i/iyn[aAC/(e"'-aC0)] - C ( /) ln [a C ( ] 1.5-2% /w[2AC/(l+p-2C0)| f o r < a < J, u su a lly a= 0.2 '5 £ Lang /;/[2AC/(P-2C0)] W=(//i)/,/[2AC/(p-2Co)] e [ p c { ty c ụ f) k e - ( c ụ ) - n Ý ! z o t/i (A 1.5-2% 1-1.3% n o rm a liz e d C n(tJ 1-1.3% estimation for fj-A / : ôW* 1.21188215// ỗ ( / 2) i n ỗ a 1) - q ,( /1)in q,(iz) n e e d n o rm a liz e d C n(t) a ~ 1 8 T h e f i n i t elem en t s ig n a l cla sses T h e s ig n a l fo rm s a r e c o m p o s in g from o n e s in g le C(t) or fro m num ber o f a fin it T h e L a n g 's c la s s is a s p e c ia l c a s e w h e r e t h e n u m b e r o f C(tt) is It is w o r th to a d o p t t h e fo llo w in g n o ta tio n A c c o r d in g to t h e n u m b e r o f C(tj) t h e y c o n s is t o f th e s ig n a l form is c a lle d th e u n it a r y or b in a r y s ig n a l form A m o n g th e u n ita r y s ig n a l fo rm s, th e P o is s o n o n e s - d e r iv e d from th e 1000/T P o is s o n d is t r ib u t io n f u n c tio n , d e s e r v e m o st a t t e n t io n s in c e th e y p r o v id e sh a r p p ea k a n d t h e ir r e s is t ib ilit y to n o is e is h ig h T h e G a u s s ia n fo r m s a ls o p o s s e s s good p eak stru ctu re but th e y seem m ore s e n s i t iv e to n o is e B o th t h e s e tw o F ig T h e A r h e n iu s p lo t c o n s tr u c te d u s in g t h e G a u s s ia n s ig n a l fo rm N o 1 ) for t h e L a n g 's e x a m p le n - (T a b le G aA s w it h e V tw o tra p s E = 4 e V and R e fe re n c e l e v e l s y s i g n a l f o r m s a n d d e t e r m i n a t i o n o f 35 c l a s s e s a r e o f e~a r e fe r e n c e le v e l c la s s w ith e max= a /£ F ig c o m p a r e s s o m e o f th e m w it h th e c la s s ic L a n g 's form w h ic h b e lo n g s to t h e m id d le q u a lity s ig n a ls T h e L a n g 's s ig n a l form , w o r k a b le in th e in t e r fe r e n c e o f 1-1.5% n o is e , is th e b e s t form a m o n g t h e b in a r y o n e s b u t is c o m p a r a b le to t h e G a u s s ia n fo r m s (1.5% ) a n d is w o rse th a n t h e P o is s o n fo r m s (3-0% ) A c o m m o n f e a t u r e o f th e f in it e le m e n t fo rm s is t h a t th e y a ll h a v e e~a refe r e n c e le v e l w ith a p r e s e t T h e e max d e p e n d s o n ly on t a n d is a lw a y s a l t T h is e n a b le s th e s tr a ig h tfo r w a r d c o n s tr u c tio n o f t h e fu n c tio n a l d e p e n d e n c e e=f( T): a t e a c h T w h e n th e C(t) is r e c o r d e d , th e t im e t w h e r e C(t) c r o s s e s t h e h o r iz o n ta l lin e c= e ° d e t e r m in e s e (T ) = a / t S o th e r e p e a te d s c a n n in g o f C(t) o v er t h e w h o le te m p e r a tu r e r a n g e a s for t h e c la s s ic a l D L T S is n o t n e e d e d T h e u s e o f t h e u n it a r y s ig n a l form s e v e n m a k e s th e m e a s u r e m e n t p r o c e s s m o re f a s t e r in o n e a s p e c t t h a t w e d o n ’t n e e d to s c a n t h e w h o le t im e t a n d c a n s e t fo c u s o n to t h e s p e c ific a r e a T h is to p ic is h o w e v e r t h e s u b je c t o f th e fu r th e r s tu d y T h e e x is t e n c e o f t h e u n ita r y s ig n a l fo rm s i t s e l f is a s u r p r is in g fa ct F ig il lu s t r a t e s t h e u s e o f t h e G a u s s ia n s ig n a l form to d e t e r m in a t e t h e t r a p s in t h e L a n g ’s e x a m p le n - G a A s C o n c lu s io n T h e e x is t e n c e o f r e fe r e n c e le v e ls o f s ig n a ls a n d m a n y s ig n a l fo r m s in D L T S is d is c u s s e d h e r e for t h e fir s t tim e W e s h o w e d t h a t t h e s e t o f t h e r e fe r e n c e le v e ls fo rm s a lin e a r a lg e b r a w h ic h h o ld s v a lid for t h e p r e s e n t e d c l a s s e s o f s ig n a l form s T h e r e fe r e n c e le v e ls a llo w th e d ir e c t d e t e r m in a t io n o f e = /iT ) in a g e o m e tr ic a l w a y B e s id e s t h e L a n g ’s s ig n a l c la s s , o b t a in in g fro m t h e m o d ific a tio n o f th e L a n g 's c la s s ic a l form S (T ), th e tw o o th e r s ig n a l c l a s s e s - t h e G a u s s ia n a n d t h e P o iss o n c la s s e s , a re d is c u s s e d T h e e x is t e n c e o f a u n it a r y c la s s o f s ig n a ls is p ro b a b ly th e m o s t in t e r e s t in g r e s u lt o f t h is w ork T h e u n it a r y s ig n a l fo r m s a r e , in o n e h a n d , m o re p e r s is t e n t to n o is e , in th e o th e r , red u ce th e n eed of r e p e a t in g th e m e a s u r e m e n t T h e y p r o v id e v e r y good r e s u lt s c o m p a r e d to t h e c la s s ic a l D L T S References D v Lang, J Appl Phys 45 (1974) p 3023 D e P r o n y , B a r o n G a sp a r d R ic h e , E s s a i é x p e r im e n t a l e t a n a ly tiq u e : su r le s lo is d e la d ila t a b ilit é d e flu id e s ô la s t iq u e e t s u r c e lle s d e la fo rce e x p a n s iv e de la v a p e u r d e l'a lk o o l, d if f é r e n t e s t e m p é r a t u r e s , J o u r n a l de l'école P olytech n iq u e y V o l.l, c a h ie r 2 , (1 ), -7 M R O s b o r n e , G K S m y th , A m o d ifie d P r o n y a lg o r ith m fo r f it t in g fu n c tio n s d e f in e d by d iffe r e n c e e q u a tio n s , S I A M J o u r n a l o f S c ie n tific a n d S ta tis tic a l C o m p u tin g , ( 9 ) , -3 H oang N a m Nhaty P h a m Quoc Trieu 36 S Weiss, R Kassing, S o lid S tate E le c t r o n ic s , Vol 31, 12 (1988) p 1733 L Dobaczewski, p Kaczor, I.D Haw kins, A.R.Peaker, M a t S c i.a ĩid T e ch 11 (1994) p 194-198 H oan g N a m N h at, P h a m Quoc T rieu , P ro ce e d in g s o f the V G S 0 , p 1 -1 c Hurtes, M Boulou, A Mitonneau, D Bois, A p p l P h y s L e tt , 32 (1978) p.821- 823 J Morimoto, M Fudamoto, K Tahira, T Kida, s Kato, T M iyakawa, J a p J A p p l P h y s 26 (10) (1987) p.1634-1640 M Pawlowski, R ev S c i I n s t r u m , 70 (1999) p 3425-3428 10 M O kuyam a, H T a k a k u , Y H am ak aw a, S o lid - S ta te E le c t., 26 (1983) p.689- 694 11 F R S h a p iro , S D S e n tu r ia , D A d ler, J A p p ly P h y s , 5 (1 ) p 12 z Su, J w F a r m e r , J A p p ly P h y s (1 9 ), p 13 I Thurzo, D Pogany, K Gmucova, S o lid - S t a t e E le c t , 35 (1992) p.1737-1743 14 K Ikeda, H Takaoka, J a p J A p p l.P h y s ,21 (1982) p.462-466  ... re q u ire m e n t of being th e signal form s is firs t described in th is paper The reference levels in Lang's signal form S(T) and their algebraic structure T he dependence of th e cap acitance... re s u ltin g in th e drop of L an g 's signal T his in t u it iv e u n d e r s ta n d in g o f t h e e m is s io n p r o c e s s - a lth o u g h n o t c o r r e c t, o ffers c e r ta in physical... ed from t h e f in it n u m b e r o f C(t) T h e nd is in fin ite th e c o n s is t in g o f th e e le m e n t c la s s e s w it h g ro u p s ig n a ls fo rm ed from t h e in f in it e n u m b