DSpace at VNU: Reference levels signal forms and determination of emisstion factor in DLTS

JOURNAL OF SCIENCE.

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 (t2) - 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

S (T )= C( t j ) - C( t 2) for v a r io u s tj a n d t2

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

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

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 being th e signal form s is firs 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:

w h e r e C0 is C(t=oo), AC=ZAC; = C(t=O )-C0 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 t2=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:

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 ang's sig n al an d set th e believe th a t i t rea 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)]/2d 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 )]l2d ~ 111, 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

( 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 2d 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:

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

3 2 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

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)]/2d = 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 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

= 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

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

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

ĨA

3

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%

o

t/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%

(A

'5

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

n &

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

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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C o m p u tin g, 1 2 ( 1 9 9 1 ) , 3 6 2 - 3 8 2

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12 z S u , J w F a r m e r , J A p p ly P h y s6 8 (1 9 9 0 ), p 4 0 6 8