ScanGate document - VNU Journal of Science

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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 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 E M I S S I O N F A C T O R I N D L T S

Hoang Nam Nhat and Pham Quoc Trieu

D e p a r tm e n t o f P h y sics

,

College o f Sciences, V N U

A b s tr a c t. The existence o f referen ce le v e ls of sig n a ls w hich d eterm in e directly th e tem p era tu re d ep en d en ce o f em issio n factor in deep lev el tran sien t p h en o m en a is d iscu sse d . T h e b a sic algebraic stru ctu re o f reference le v e ls in th e c la ssic a l DLTS is stu d ie d and various sig n a l forms w ith derived referen ce levels are g iv en . W e th en d e m o n str a te th e u se o f th e s e sig n a l forms and com pare them w ith th e c la ssic a l D LTS dou ble boxcar sig n a l.

K e y w o rd s.

S ig n a l fo r m s , r e f e r e n c e le v e l s , D L T S , d e e p tr a p .

1. Introduction

T h e e x i s t e n c e o f t h e d e e p l e v e l s is a n im p o r t a n t p h e n o m e n o n i n s e m ic o n d u c to r p h y s ic s . It i s w e ll- k n o w n t h a t t h e y c a u s e m a n y c o n s id e r a b le b e h a v io u r s o f m a t e r i a ls . T h e c h a r a c t e r iz a t i o n o f t h e d e e p tr a p s f a c e d m a n y d if f ic u lt ie s u n t il 1 9 7 4 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 t r 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 t r o s c o p y (D L T S ) [1]. T h is a llo w s

to d e d u c e fr o m t h e e x p o n e n t ia l c a p a c i t a n c e d e c a y s

C(() = ACe

'*'1^{t h e} b a s ic p h y s i c a l p a r a m e t e r s o f t h e t r a p s s u c h a s t h e a c t iv a t i o n e n e r g y , c a p t u r e c r o s s - s e c t io n a n d c o n c e n t r a t io n . T h e L a n g 's m e t h o d h a s b e e n w id e ly a c c e p t e d t o d a y a s t h e s t a n d a r d

to o l,

a lt h o u g h i t h a s s e v e r a l l i m i t a t i o 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 i v e l y lo w r e s o lu t i o n . T o e x t r a c t t h e t r a p p a r a m e t e r s fr o m t h e e x p o n e n t ia l d e c a y s , L a n g h a s in t r o d u c e d t h e

s ig n a l fo rm S ( T )= C (ti) -C (t

2

)

- t e c h n i c a l l y r e a l iz e d u s in g a d o u b le b o x c a r c i r c u it , w h ic h m o n it o r s t h e c a p a c i t a n c e t r a n s i e n t s

a t t w o d if f e r e n t t i m 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 i n a t c e r ta in te m p e r a tu r e

r e la t e d to t h e d o u b le b o x c a r r a t e w in d o w s s e t t i n 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

F ig A .

L a n g 's m e th o d s c a n s S (T )= C

( t j ) - C( t 2)

for v a r io u s

tj

a n d

t

2 s e t t i n g s a n d d r a w s t h e t e m p e r a t u r e d e p e n d e n c e o f S (T ). T h e m a x im u m d e t e r m in e t h e t e m p e r a t u r e s T o f th e e m is s io n f a c t o r

Cnax

s e t fo r th b y t h e r a te w in d o w s .

2 8

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f u 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 c to r o n t e m p e r a t u r e

e= f(T)

a n d c a n c o n s t r 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 1 0 0 0 /T for t h e d e t e r m i n a t i o n o f t r a p p a r a m e t e r s ( F i g . l ) . T h e k e y e l e m e n t in t h i s t e c h n iq u e is t h u s t h e d e t e r m i n a t i o n o f t h 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 n o w , 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 t h 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 t e d [2 -1 4 ] ( t h e l i s t is c e r t a in ly n o t c o m p le t e ) , 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 rie r a n d th e 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 f o r m a t io n m e t h o d s m a n ip u la t in g w it h t h e w h o le r a n g e o f m e a s u r e d d a t a , u s u a l ly d ig it a lly r e c o r d e d 5 1 2 o r 1 0 2 4 p o in t s . R e c a ll t h a t t h e c l a s s ic a l S (T ) u s e s o n ly 2 p o in t s 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 t h 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 i v e p e a k s t r u c t u r e o f t h e g a in , b u t s in c e t h e y do 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 i s s i o n fa c to r a t t h e m a x im a l g a in c a n n o t b e c a l c u l a t e 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 t h 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 l e 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 t h e a n a ly t ic a lg o r it h m c o n v e r t in g t h e s e t o f t h e c a p a c i t a n c e t r a n s i e n t s

C(t)y

e a c h o f t h e m h a s b e e n r e c o r d 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 , i n t o t h e s p e c ific v a l u e s o f 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 t a n t p r o p e r tie s : (1) t h e y a r e

sp ectro sco p ic

i n 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 it h o n e s p e c if ic d e e p c e n t e r a n d (2) t h 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 1 0 0 0 /T ] t r a n s f o r m a t io n o f t h e m a x im a o f a r b it r a r ily c h o s e n p e a 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 t h e a lg o r it h m a n d u s u a lly t h e m e t h 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 f e r e e d to a s t h e

s ig n a l fo r m .

F or 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 t h e t i m e - s e t t i n 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 t h e d if f e 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 ili t y in s e p a r a t io n o f t h 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 l a s s ic a l L a n g 's s i g n a l fo r m , fo r e x a m p le , in v o lv e s o n ly 2 p o in t s i n 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 t h e L a p la c e s ig n a l fo r m 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 i e 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 t h e r s p e c t r 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 i s w o r k w e p r e s e n t t h e s t u 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 t h e L a n g 's c l a s s ic a l s ig n a l form S (T ) s h o w i n g t h a t t h i s 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

referen ce level

o f s ig n a l w h ic h d ir e c t ly d e t e r m in e s t h e r e la t io n s h ip

e=fÇT).

T h is p r o p e r t y o f D L T S w a s n o t r e p o r t e d a n y w h e r e b e fo r e . W e t h e n in t r o d u c e t h e c l a s s e s o f m a n y o t h e r s ig n a l fo r m s h a v i n g t h e s a m e a lg e b r a ic s t r u c t u r e o f t h e r e f e r e n c e l e v e l s 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 t r a s t to t h e L a n g ’s fo rm t h a t in v o lv e s 2 v a l u e s o f

C(t)y

t h e r e is a c l a s s o f fo r m s w h ic h in v o lv e o n ly 1 s i n g l e v a lu e

C(t).

T h is is a s u r p r i s i n 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 t o T . T h e L a n g 's s ig n a l fo r m is e x t e n d e d in t o t h e c l a s s o f s ig n a l fo r m s w h ic h c o n t a i n s m a n y o t h e r fo r m s p r o v id in g

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3 0 H o a n g N a m N h a t y P h a m Q u o c T r i e u

th e sam e r e s u lts as th e L an g ’s form . T he fact t h a t th e re e x is t m any a n a ly tic functions f( T) fulfilled th e re q u ire m e n t of b ein g th e sig n al form s is f ir s t described in th is p ap er.

2. The reference levels in Lang's signal form S(T) and their algebraic structure

T he d ep en d en ce of th e c ap a citan ce tr a n s ie n t C(t) on tim e t is considered in g e n eral case as:

C(0 = Co + l 4 Q r « f (1)

w h e r e C0 is

C(t=oo),

AC=ZAC; =

C(t=O )-C

0 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 s p e c t to th e n o rm aliz e d c ap a citan ce given as C n(t)=(C(t)-Co) / AC, a n d

d e n o t e

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 fo r t h is g e n e r a l c a s e : 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 e p e n d e n t a n d n o t o v e rla p p in g each o th er (th ey a re fa r each from o th e r in th e te m p e ra tu re scale), one m ay d iffe re n tia te th is sig n a l according to som e em ission facto r e„ leav in g th e o th e r ones zeroed, to d e te rm in e th e sig n al m ax im al g a in in th e given te m p e ra tu re ran g e . W e modify th e re s u lt from [1]

w it h r e s p e c t t o t h e v a r ia b le s

t

a n d

d

m e n t io n e d ab ove:

e m«x=Zn [( *+d

)/(t -d )]/2 d

(3 )

T h is re la tio n show s t h a t by fixing th e r a te w indow s (by t a n d d) one also selects th e em issio n facto r to w hich th e L an g ’s sig n a l re a c ts m ostly w hen it scan s th ro u g h th e s e t te m p e ra tu re ra n g e . W ith th e in cre ase of te m p e ra tu re th e tra p begins to re le a s e electro n s a n d it re le a s e s m ostly w hen th e em issio n factor is high enough, ra is in g th e L an g 's sig n a l to m axim um . B u t w h en th e tr a p becom es b lan k , th e em ission process slow s dow n re s u ltin g in th e drop of L a n g 's signal. T his

in t u it iv e u n d e r s t a n d in g o f t h e e m is s io n p r o c e s s - a lt h o u g h n o t c o r r e c t, o ffe rs c e r t a in

p h y sical m e a n in g to the L an g 's sig n a l a n d se t 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 i t 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 fro m t h e L a n g 's t im e is t h a t t h e r e l a t io n (3 ) u s e d to o b t a in t h e e mBX a lm o s t e q u a ls 1

It

n u m e r ic a lly . U s i n g t h e E u le r n u m b e r d e f in it io n f o r m u la lim(l + l / « ) ”

=e

o n e c a n

« —►«5

w it h o u t d if f ic u lt y p r o v e t h a t

l n[(t +d)/ (t -d)]/

2

d

r e a lly c o n v e r g e s t o 1

/ t

w h e n

d->

0. G iv in g t h e f a c t t h a t

ln [(t+ d )l(t-d )]l

2

d ~

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

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( e is E u le r n u m b e r ). T h is s p e c ia l f e a t u r e o f t h e c l a s s i c a l d o u b le b o x c a r t e c h n iq u e i s il lu 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 t e c e n t r a l p o s it io n

t

a n d t h e li n 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 t io n in t h e r a t e w in d o w p o s it io n s , t h e o n ly a r e a o f im p o r t a n c e w a s

C n(t)=e~l.

T h e e v i d e n t c o n s e q u e n c e f o llo w s im m e d ia t e ly 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 c to r o n t h e t e m p e r a t u r e

eị - f ( T)

o n e s im p ly c h e c k t h e c r o s s - p o in t s o f

C n(t)

a n d c „ = e 1 to o b t a in d ir e c tly t h e v a lu e o f e m is s io n fa c to r

( e ^ l / t )

c o r r e s p o n d in g to t h e g iv e n t e m p e r a t u r e T. F o r t h i s r e a s o n w e c a l l C n= e“1 t h e

reference leve l

o f t h e s ig n a l fo rm S (T ). It 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 f e 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 fro m it s r e f e r e n c e le v e l.

A lt h 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 f e 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 2

d

is in f in i t e s i m a lly s m a ll, t h e r e is a lo t o f o th e r s ig n a l fo r m s a s d is c u s s e d in t h e n e x t s e c t io n , w h ic h h a v e e x a c t r e f e r e n c e le v e l. T h e im p o r t a n c e o f r e f e r e n c e l e v e ls fo llo w s from t h e fa c t t h a t t h e y le a d to t h e u n d e r s t a n d in g o f t h e a lg e b r a ic s t r u c t u r e o f t h 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 t h e c a p a c it a n c e t r a n s ie n t p a r t ic u la r ly . W e n o w in t r 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 t h 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 t h e m o v in g o f g a t e fro m

t

to

t'= a ty

fo r 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 f o llo w s t h a t t h e e m is s io n fa c to r

eị(t)

d e t e c t e d o n t h e b a s is o f

emax(t)

c h a n g e s as:

ei( t t)

=

e t(a t)

=

H a t

=

(lla )C i(t).

T h e t r a n s ie n t a s s o c ia t e d w it h 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

*,</•)/

=e -e,<t)uo =CnỢ/a) =\CnỢ)\]/a

S o w e c a n c o n s t r u c t a

m o d ifie d L a n g 's s ig n a l

, to b e c a lle d o f t h e

order a

as:

S ( T ) ia] = c n( t - d ) 1/a- c n( t + d ) 1/a (4)

w h ic h s t i l l h a s a c e n t r a l p o s it io n a t

t

b u t p r o d u c e s t h e m a x im a l o u t p u t a lo n g t h e r e f e r e n c e le v e l c „ = e ~ ° ( e = 2 .7 1 8 2 8 2 ) . O f c o u r s e , t h e c l a s s ic a l L a n g 's s ig n a l S (T ) i s o f

t[ms]

F ig .2.

T h e s p e c ia l f e a t u r e o f th e d o u b le b o x c a r t e c h n iq u e : t h e r a te w in d o w

[ t - d f t+ d

] s h o w s m a x im u m a c c o r d in g to T w h e n t h e

C n(t)

d e c r e a s e s th r o u g h th e a r e a

c n( t h

l / e = 0 .3 6 8 .

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o r d e r 1: S ( T ) [l1. W it h a ll p o s s i b l e a , t h e s y s t e m S (T) io1 fo r m s a c l a s s o f s ig n a ls - t h e

L a n g 's s ig n a l c la s s .

T h e f a c t t h a t t h e

e

^ o f S (T )ta] r e a lly c o n v e r t s t o

a l t

w h e n

0

c a n a ls o b e o b s e r v e d b y d i f f e r e n t i a t in g S (T )íaỉ a c c o r d in g to c, ( le a v i n g a ll o th e r

ejxi

=0) a n d s e t i t t o 0. T h e r e s u l t is: e max(S (T )ial)=

a l n[(t +d)/ (t -d)]/

2

d

= a e ^ C S O lV 1*) =

a l t .

W h e n a < l , t h e S (T) io1 c a t c h s C n=e"° a t lo w e r T a n d w h e n a > l i t c a t c h e s

Cn=e"a

a t h ig h e r T c o m p a r e d to S (T ).

T h is s i g n a l c l a s s a s s o c i a t e s e a c h p o in t

X

in t h e p la n e

\y= C n(t

) 9

x=t]

w ith s o m e h o r iz o n t a l r e f e r e n c e le v e l l i n e y = e~a a n d t h e v e r t ic a l li n e

x=t,

s o t h a t

X

lie s in t h e i n t e r s e c t b e t w e e n t h e s e t w o l i n e s . E a c h p o in t

X

t h u s d e t e r m in e s a u n iq u e e m is s io n f a c to r e , = a / i . I t is n a t u r a lly to u n if y

X

w it h

e t

a n d w r ite e , = e / a ,t ) . F ro m t h e a n a ly s is a b o v e i t is o b v io u s th a t:

e ,( a ,t ) = a e ị ( l ,t ) = e ị ( l ,ư a ) (5)

e i(û ,t)n= a ne , ( l , t ) n= a ne i( l , t n)= e i(a n,t n) = e i( l , ( t / a ) n)

T h is t e l l s u s a b o u t t h e e q u iv a le n c e o f a ll r e f e r e n c e l e v e ls in th e s ig n a l p r o c e s s in g s y s t e m u s in g t h e d o u b le b o x c a r . T h e f o llo w in g r e la t io n s c o m e s s t r a ig h t f o r w a r d .

A ,[ei(a ,t)+ ei(b ,t)]= x,ei(a ,t)+ x ,e i(b%,t)= - (6)

= X.aei(l,t)4*X,bei( l,t ) = ^ ( a + b ) e i( l , t ) = ei(X (a+ b ),t)

[ej(a,tn)x e,(b,t,n)]>‘ = ei(a,tn)xx ei(b,tm)x =

= a \ > , ( l , t ) nXx b xe i( l , t ) mx= (a b )xe i( l , t ) x(n+m) =

= e ,((a b )x,t x<n+m>)

O n e m a y n o t ic e t h a t t h e y f o llo w a lin e a r a lg e b r a o n w 2.

3. The signal classes and forms

T h e r e is a n im p o r t a n t p r o p e r t y o f t h e L a n g 's s ig n a l form : i t s h o w s c e r ta in s e p a r a b ili t y w h e n t h e d if f e r e n t t r a p s o v e r la p . T h e s ig n a l t h a t is w o r th th e u s e in p r a c t ic e s h o u ld b e b o t h s p e c t r o s c o p ic a n d r e s o lu b le . U p t o n o w , t h e o n ly s p e c t r o s c o p ic s i g n a l s t h a t b r o u g h t b e t t e r r e s o lu t io n w e r e fro m t h e t r a n s f o r m a t io n o f t h e w h o le t r a n s i e n t . T h e s e s i g n a l s , h o w e v e r , do n o t p o s s e s s t h e r e f e r e n c e le v e ls an d t h e i r a lg e b r a ic s t r u c t u r e s a r e q u it e d if f e r e n t .

T h is s e c t i o n d e s c r ib e s tw o c l a s s e s o f t h e s ig n a l fo r m s , w h ic h w e c a ll h e r e th e G a u s s ia n a n d t h e P o is s o n c l a s s (to t h e la t e r o n e t h e L a n g 's c l a s s S ( T ) {al r e d u c e s a s a s p e c ia l ca se)', p o s s e s s i n g t h e s a m e a lg e b r a ic s t r u c t u r e o f t h e r e f e r e n c e le v e ls a s th e L a n g ’s s ig n a l fo r m a n d a ls o f u lf il lin g t h e r e q u ir e m e n t o f b e in g r e s o lu b le an d

(6)

s p e c t r o s c o p ic . T h e fa c t t h a t t h e r e m a y e x i s t o t h e r s p e c tr o s c o p ic s i g n a l s t h a n t h e L a n g 's o n e c a n b e in t u it iv e ly r e c o g n iz e d fr o m t h e t e m p e r a t u r e d e p e n d e n c e o f

C(t)

( F ig .3 ) . T h e s im p le s t w ay h o w to c r e a t t h e p e a k - s h a p e f u n c t io n from t h e

C(t)= f(

T) i s t o e it h e r d if f e r e n t ia t e

C(t)

a c c o r d in g to T (o r d o n e b y L a n g , to s u b s t r a c t

C(t*

2

)

from

Cd z )

- w h ic h e v id e n t ly r e d u c e s to t h e d if f e r e n t i a t io n w h e n t h e

C(t

) ’s b e c o m e i n f i n i t e s i m a l l y c lo s e ). T h e s e c l a s s e s a r e s u m m a r iz e d in t h e T a b le 1, w h e r e t h e la s t c o lu m n s h o w s t h e e s t im a t io n for m a x im a l p s e u d o - r a n d o m n o is e le v e l (in % o f t h e m a x im a l s ig n a l) t h a t d o e s n o t d is tu r b t h e ir e mftX m o r e t h a n 5% fro m t h e c o r r e c t v a lu e .

In g e n e r a l, t h e s ig n a l c l a s s e s c a n b e c l a s s if i e d in to tw o d if f e r e n t g r o u p s . T h e 1st is t h e

f i n i t e le m e n t g r o u p

, c o n s is t in g o f t h e c l a s s e s w it h s ig n a ls fo r m e d fro m t h e f in it n u m b e r o f

C (t).

T h e 2nd is t h e

in fin ite e le m e n t g ro u p

c o n s is t in g o f t h e c l a s s e s w it h s ig n a ls fo r m e d fro m t h e in f in i t e n u m b e r o f

C(t).

T h is c l a s s if i c a t io n c a n b e e x t e n d e d to c o v e r a ls o t h e 3 rd c l a s s o f s ig n a l fo r m s, w h ic h d e a l w it h t h e n o n - a n a ly t ic a lg o r it h m s , t h a t i s th e

fr a c ta l g r o u p

. P r in c ip a lly , a n y n o n - a n a ly t ic a lg o r ith m

F (t,T ,C (t,T ))

t a k in g

C(t), t,

T a s t h e in p u t s a n d o u t p u t s t h e p e a k s c a n b e c o n s id e r e d a s t h e s ig n a l fo rm i f it s a t i s f i e s t h e c o n d it io n s for t h e s ig n a l fo r m s, b e p r e s e n t e d in a n o t h e r p a p e r . T h is w o rk se

Tem pe rature [KJ

F ig .3.

T h e d e v e l o p m e n t o f c a p a c it a n c e a t t h r e e s u c c e s s i v e t i m e s fo r t h e L a n g 's

n-

G a A s e x a m p le w it h t w o t r a p s E = 0 .4 4 e V a n d 0 .7 5 e V .

Te m pe rature [K]

F ig A .

C o m p a r is o n o f s o m e s e le c t e d s ig n a l fo r m s to t h e c l a s s i c a l L a n g ’s S (T ) fo rm fo r a s a m p l e w it h o n e t r a p E = 0 .4 4 e V .

T h e s t u d y o n t h e 2 nd a n d 3 rd g r o u p s w ill t f o c u s o n t h e l 8t g r o u p o f s ig n a l fo r m s .

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Table 1.

T h e fin it e le m e n t s ig n a l c la s s e s : s ig n a l fo rm s, t h e ir e max a n d r e fe r e n c e le v e ls

1

( £ { < ) - C { t ) a e [ p c { t y c ụ f )

.mes---

W=(//i)/,/[2AC/(p-2Co)] a = /;/[2AC/(P-2C0)]

1.5-2%

u su a lly p = 5-10 ĨA3

C3

o Ì

3

2

C { i ) e p c « y c « r for a ----2,

e„,„= (//íy«í2AC/(l+p-2Co)l

e~°,

/w[2AC/(l+p-2C0)|

1.5-2%

3

k e - ( .c ụ ) - n Ý ! 7 ơ z u su a lly ỊẤ-1, 0.2

k o n ly sc a le s th e g ra p h

em«= (//i)MAC/(|i-C0)l

a = /«[AC/fa-Co)]

1.0-1.5%

ot/i

/*“s 4 - C ( /) l n [ a C ( 0 ]

f o r 0 < a < J, u su a lly a= 0.2

emax= (i/iyn[aAC/(e"'-aC0)]

a = ln [ a A C /(c 5 -a C 0)]

3-5%

'5(A

£ c

5

C ( r / A _aC(0

f o r X >1, u su a lly Ẳ = 2

w ụ / t ) i n ị ầ C b ứ J ( \ - C 0ln k)} e~° y

a - ln [ ầ C ỉn )J ( 1 -C (JnX )]

3-5%

C ' Q j " - C M ' 1* eraax=a l n ự / t 2y ( t ,- t ĩ } ~ a / t e~° 1-1.5%

to n e e d n o r m a liz e d C n(t) b u t n o t f o r a ~ l

c ( f , ) " / c ( / 2 r

u su a lly n - 1 o r 2

for t2=2tj: e~°, 0.5%

Lang

&

1 < w = (i/o in (i+1/i+ /C 7C 0) a = ln ạ + J ì + 4 C / C ồ)

.5

a A C - ị C ụ ^ V C ậ ý ) for ì ị—ỉ ị—Ị (u n ita r sig n a l): e~* 1-1.3%

8

ca n n o t b e u s e d w ith th e n o r m a liz e d C n(tJ

emax= -0 /0 ỉn(-l + ^ l+ l i c i ) ứ = - ln ( - l+ ^ l + l/e g )

9 ç ( / 2) i n ç a 1) - q ,( /1)in q ,(iz)

n e e d n o r m a liz e d C n(t)

estimation for fj-A / :

«W* 1.21188215// a ~ 1 .2 1 1 8 8 2 1 5

1-1.3%

T h e f i n i t e le m e n t s ig n a l cla sses

T h e s ig n a l fo r m 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 a fin it n u m b e r o f T h e L a n g 's c l a 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( t t)

is 2. It is w o r th t o a d o p t t h e f o llo w in g n o t a t io n . A c c o r d in g t o 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 t h e s ig n a l form is c a lle d t h e u n it a r y o r b in a r y s ig n a l form .

A m o n g t h e u n it a r y s ig n a l fo r m s, t h e P o is s o n o n e s - d e r iv e d fro m t h e P o is s o n d is t r ib u t io n f u n c t io n , d e s e r v e m o s t a t t e n t io n s in c e t h e y p r o v id e s h a r p p e a k a n d t h e ir r e s i s t i b i l i t y t o 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 g o o d p e a k s t r u c t u r e b u t t h e y s e e m m o re s e n s i t i v e to n o is e . B o t h t h e s e tw o

1000/T

F ig .5.

T h e A r h e n iu s p lo t c o n s t r u c t e d u s i n g t h e G a u s s ia n s i g n a l fo rm N o . 1 (T a b le 1) fo r t h e L a n g 's e x a m p le

n

- G a A s w it h tw o t r a p s E = 0 .4 4 e V a n d 0 .7 5 e V .

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c l a s s e s a r e o f

e~a

r e f e r e n c e le v e l c l a s s w it h e max= a / £ . F ig .4 c o m p a r e s s o m e o f th e m w it h t h e c l a s s ic L a n g 's fo rm w h ic h b e lo n g s to t h e m id d le q u a lit y s ig n a ls . T h e L a n g 's s i g n a l fo rm , w o r k a b le in t h e in t e r f e r e n c e o f 1-1.5% n o is e , is t h 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 t o t h e G a u s s ia n fo r m s (1.5% ) a n d i s w o r se t h 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 t h e f in it e l e m e n t fo r m s is t h a t t h e y a ll h a v e

e~a

r e fe r e n c e le v e l w it h

a

p r e s e t . T h e e max d e p e n d s o n ly o n

t

a n d is a lw a y s

a l t .

T h is e n a b le s t h e s t r a ig h t f o r w a r d c o n s t r u c t io n o f t h e f u n c t io 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 t h e

C(t)

i s r e c o r d e d , t h 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 t a l lin e

c= e

° d e t e r m in e s e (T

) = a / t .

S o t h e r e p e a t e d s c a n n in g o f

C(t)

o v e r t h e w h o le t e m p e r a t u r e r a n g e a s for t h e c l a 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 fo rm s e v e n m a k e s t h e m e a s u r e m e n t p r o c e s s m o r e 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 t o 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 if ic a r e a . T h is to p ic is h o w e v e r t h e s u b j e c t o f t h e fu r th e r s t u d y . T h e e x i s t e n c e o f t h e u n it a r y s ig n a l fo r m s i t s e l f is a s u r p r is in g fa c t. F i g .5 i l 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 i n t h e L a n g ’s e x a m p le

n -

G a A s.

4 . C o n c l u s i o n

T h e e x i s t e n c e o f r e f e r e n c e l e 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 fo r t h e f ir s t t im 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 r m 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 fo rm s.

T h e r e f e r e n c e l e v e l s a llo w t h 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 t r 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 l a s s , o b t a in in g fro m t h e m o d if ic a tio n o f th e L a n g 's c l a s s ic a l fo rm S (T ), t h e tw o o t h 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 is s o n c l a s s e s , a r e d is c u s s e d . T h e e x i s t e n c e o f a u n it a r y c l a s s o f s i g n a ls is p r o b a b ly t h e m o s t i n t e r e s t i n g r e s u l t o f t h i s w o rk . 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 r e p e r s i s t e n t t o n o is e , in t h e o t h e r , r e d u c e t h e n e e d o f r e p e a t in g t h 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 g o o d r e s u lt s c o m p a r e d t o t h e c l a s s i c a l D L T S .

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36