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

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 .

1. 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 ra 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 ra 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 1 0 0 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 4 ] ( 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 5 1 2 or 1 0 2 4 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 2 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 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 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 ( 2 )
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 1 0 0 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 2 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 2 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 1 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.
2.

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 4 Q r « f

( 1)

w h e r e C 0 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 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 ra 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 ra 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 ra tu re ran g e . W ith th e in crease of te m p e ra 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, ra 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)]/ 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 fa c t t h a t ln [(t+ d )l(t-d )]l 2d ~ 1 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 .2.

o f C n(t) a n d c „ = e 1 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 = 0 .3 6 8 .

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 2 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 *,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 be 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 tr a l p o s it io n a t t b u t p r o d u c e s th e m a x im a l o u t p u t a lo n g th e
r e fe 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 la s s ic a l L a n g 's s ig n a l S(T ) is o f


H o a n g N a m N h a ty P h a m Q u o c T r ie u

32

o r d e r 1 : S (T )[l1. W ith a ll p o s s ib le a , th e s y s t e m S (T ) io1 fo rm s a c la s s o f s ig n a ls - t h e

L a n g 's s ig n a l c la ss . T h e fa 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 to a l t

w h en

0 c a n a ls o b e o b s e r v e d b y d if f e r e n t ia t in g S (T )íaỉ a c c o r d in g to c, ( le a v in g a ll o th er ejxi
= 0 ) a n d s e t i t t o 0. T h e r e s u lt is: e max(S (T )ial)= a ln[(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 it c a tc 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 ig n a l c la s s a s s o c ia t e s e a c h p o in t X in t h e p la n e \y=Cn(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 li n e y= e~a a n d t h e v e r t ic a l lin e x=t, s o t h a t X lie s in t h e
in t e r s e c t b e t w e e n t h e s e t w o lin 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
fa c to r e , = a / i . It is n a t u r a lly to u n ify X w ith e t a n d w r ite e ,= e /a ,t ) . F rom t h e a n a ly s is
a b o v e it 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
p r o c e s s in g

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 fe r e n c e le v e ls

sy ste m

u s in g

th e

d o u b le

b o x ca r.

The

fo llo w in g

in

th e s ig n a l

r e la t io n s

co m e s

s tr a ig h t fo r w a r d .
( 6)

A ,[ei(a ,t)+ ei(b ,t)]= x,ei(a ,t)+ x ,ei(b%
,t)= = 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 fo llo w a lin e a r a lg e b r a on 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 ty o f th e L a n g 's s ig n a l form : it s h o w s c e r ta in
s e p a r a b ilit 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 tic e

s h o u ld

be

b o th

s p e c tr o s c o p ic

and

r e s o lu b le .

Up

to

now ,

th e

on ly

s p e c tr o s c o p ic s i g n a ls t h a t b r o u g h t b e tte r r e s o lu tio n w e r e from t h e t r a n s fo r m a tio n o f
t h e w h o le t r a n s ie n t . T h e s e s ig n a ls , 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 and
t h e ir 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 iffe r e n t.
T h is s e c t io 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 la s s (to t h e la t e r o n e th e L a n g 's c la s s S (T ){al r e d u c e s a s a
s p e c ia l case)', p o s s e s s in g t h e s a m e a lg e b r a ic s tr u c tu r e o f t h e r e fe r e n c e le v e ls a s th e
L a n g ’s s ig n a l fo rm

a n d a ls o

f u lf illin 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 and


R e fe r e n c e le v e ls , s ig n a l fo r m s a n d d e te r m in a tio n o f .

s p e c tr o s c o p ic .

The

fa ct t h a t

th ere

33

m ay

e x i s t o th e r s p e c tr o sc o p ic s ig n a ls th a n th 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
fro m th e te m p e r a tu 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 th e
p e a k -s h a p e fu n c tio n from t h e C(t)=f( T) is

to e ith 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
(or d o n e b y L a n g , to s u b s t r a c t C(t*2) from

Cdz )

- w h ic h

d if f e r e n t ia t io n

e v id e n tly
w hen

reduces

C(t ) ’s

th e

to

th e

b eco m e

in f in it e s im a lly c lo se ). T h e s e c la s s e s

a re

s u m m a r iz e d in th e T a b le 1 , w h e r e th e la s t

Tempe rature [KJ

c o lu m n s h o w s th 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
m a x im a l

n o is e le v e l (in % o f th e

s ig n a l)

th a t

d oes

not

F ig .3. T h e d e v e lo p m e n t o f c a p a c ita n c e a t

d istu r b

t h r e e s u c c e s s iv e t im e s fo r th e L a n g 's n-

t h e ir e mftX m o r e t h a n 5% fro m th e c o rr ect

G a A s e x a m p le w it h tw o tr a p s E = 0 .4 4 e V

v a lu e .

a n d 0 .7 5 e V .
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 la s s if ie d in to tw o d iffe r e n t g ro u p s.

1 st is

The

c o n s is t in g

th e

fin it

o f th e

e le m e n t

c la s s e s

w it h

group,
s ig n a ls

fo rm ed from t h e f in it n u m b e r o f C(t). T h e

2 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 e r o f C(t).
T h is c la s s if ic a t io n ca n

b e e x te n d e d to

co v er a ls o t h e 3 rd c la s s o f s ig n a l fo rm s,
w h ic h

deal

a lg o r ith m s ,

w ith

th a t

is

th e

n o n -a n a ly tic

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),

in p u t s a n d

o u t p u t s th e

c o n s id e r e d

as

th e

Te m pe rature [K]

t,

T

as

th e

p eak s can be

s ig n a l

form

if

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 ic a l L a n g ’s
S (T ) fo rm fo r a s a m p le w it h

o n e tr a p

E = 0 .4 4 e V .

it

s a t is f ie s t h e c o n d itio n s for t h e s ig n a l fo rm s, 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
b e p r e s e n t e d in a n o th e r p a p e r . T h is w ork se t fo c u s o n t h e l 8t g r o u p o f s ig n a l fo rm s.


H o a n g N a m N h a ty P h a m Q u o c T r ie u

34

Table 1. T h e

fin it e le m e n t s ig n a l c la sse s: s ig n a l fo rm s, th e ir e max a n d r e fe r e n c e le v e ls
.mes ----------------------------------

( £ { < ) - C { t)a

1
ĨA
3
C3
o

a=

for a ----2,
e„,„= (//íy«í2AC/(l+p-2Co)l
em«= (//i)MAC/(|i-C0)l

e~°,

u su a lly p = 5-10

2

Ì
3

3

/*“s

4

C { i)e p c « y c « r

1.0-1.5%
a=

u su a lly ỊẤ-1 ,
0.2
k o n ly sca les th e gra p h

a = ln [a A C /(c

3-5%

eraax=a l n ự / t 2y ( t,- t ĩ} ~ a /t

e~°

1-1.5%

for t 2= 2tj:

e~°,

0.5%

< w = (i/o in (i+1/i+/C 7C 0)

a = ln ạ + J ì + 4 C / C ồ)

a A C -ịC ụ ^ V C ậ ý )

for ì ị —ỉ ị —Ị (u n ita r signal):

e~*

ca n n o t be u se d with the

emax= - 0 / 0 ỉn(-l + ^ l+ l i c i )

ứ = - ln ( - l+ ^ l + l/e g )

C ( r / A _aC(0

5

fo r X >1, u su a lly Ẳ = 2
C ' Q j " - C M

1

&
.5

' 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 - 1 o r 2

8

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 ( 0 ]

1.5-2%

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

f o r 0 < 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 ! 7 ơ z

o
t/i
(A

1.5-2%

1-1.3%

n o rm a liz e d C n(tJ

9

1-1.3%

estimation for fj-A / :
«W* 1.21188215//

ç ( / 2) i n ç a 1) - q ,( /1)in q,(iz)
n e e d n o rm a liz e d C n(t)

a ~ 1 .2 1 1 8 8 2 1 5

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 2. 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 .5. 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

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

0 .7 5 e V .

tw o

tra p s

E = 0 .4 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 .4 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 .5 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.

4.

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
1.

D .v. Lang, J. Appl. Phys. 45 (1974) p .3023.

2.

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 2 , (1 7 9 5 ), 2 4 -7 6 .

3.

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 , 1 2 ( 1 9 9 1 ) , 3 6 2 -3 8 2 .


H oang N a m Nhaty P h a m Quoc Trieu

36

4.

S. Weiss, R. Kassing, S o lid S tate E le c t r o n ic s , Vol. 31, 12 (1988) p. 1733

5.

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.

6.

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 5 2 0 0 2 , p .1 1 5 -1 1 9 .

7.

c . Hurtes, M. Boulou, A. Mitonneau, D. Bois, A p p l. P h y s. L e tt , 32 (1978) p.821-

823.
8.

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.

9.

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 ra , 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 5 (1 9 8 4 ) p .3 4 5 3 .
12. z . Su, J .w .F a r m e r , J . A p p ly . P h y s 6 8 (1 9 9 0 ), p .4 0 6 8 .

13. I. Thurzo, D. Pogany, K. Gmucova, S o lid - S t a t e E le c t ., 35 (1992) p.1737-1743.
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