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THE USE OF MICROWAVES TO CHARACTERIZE OPTICALLY STIMULATED SEMICONDUCTORS

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300 N. ZEEB ROAD, ANN ARB O R, Ml 48106
18 BE DFO RD ROW, LO NDO N WC1R 4EJ, E N G L A N D
8107305
Br o w n , H a r o ld K e n t
THE USE OF MICROWAVES TO CHARACTERIZE OPTICALLY
STIMULATED SEMICONDUCTORS
The Ohio State University
University
Microfilms
International
P h .D .
300 N. Zeeb Road, Ann Arbor, MI 48106
Copyright 1980
by
Brown, Harold Kent
All Rights Reserved
1980
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T H E U S E O F M IC R O W A V E S TO C H A R A C T E R IZ E
O P T IC A L L Y S T IM U L A T E D SE M IC O N D U C T O R S
D IS S E R T A T IO N
P r e s e n t e d in P a r tia l F u lfillm e n t o f th e R e q u ir e m e n ts f o r
th e D e g r e e o f D o c to r o f P h ilo s o p h y in th e G ra d u a te S c h o o l o f
T h e O hio S t a t e U n iv e r s it y
by
H a ro ld K ent B r o w n , B . S . ,
M .S.
$ % $ afc *
T h e O hio S t a t e U n iv e r s it y
19 8 0
R ea d in g C o m m itte e :
A p p ro v ed B y
D r . R obert J . G arbacz
D r . R o b e r t G . K ou you m jia n
D r . M a r lin O . T h u r s to n
D r . M a r lin O . T h u r sto n
D e p a r tm e n t o f E l e c t r ic a l
E n g in e e r in g
(c )
C o p y r ig h t by
H a ro ld K ent B ro w n
1980
ii
T o M y W ife and O ur P a r e n ts
m
ACKNOW LEDGEM ENTS
I w is h to g r a t e f u lly a c k n o w le d g e th e e n c o u r a g e m e n t and
g u id a n c e g iv e n to m e b y m y a d v is o r ,
P r o f e s s o r M a r lin O . T h u r s to n .
A l s o , th e a s s i s t a n c e and c o n tin u a l su p p o r t th ro u g h o u t m u ch o f th e
e x p e r im e n ta l and t h e o r e t ic a l d e v e lo p m e n t by P r o f e s s o r R o b e r t J .
G a r b a c z , R o b e r t G . K o u y o u m jia n , John M e a d o r s , and B e n e d ik t
M unk h a s b e e n g r e a t ly a p p r e c ia te d .
And f in a lly ,
I w is h to th ank
m y c o lle a g u e s B am d ad B a sta n i and G ian G e r o s a f o r t h e ir s im u la t н
in g c o n v e r s a t io n s and s u p p o r t.
iv
V IT A
June 12, 1 9 5 5 . . . .
B o r n , W in s t o n - S a le m , N o rth C a r o lin a
1975 ............................
B .S .E .E .,
F lo r id a In s titu te o f T e c h n o lo g y ,
M e lb o u r n e , F lo r id a
1 9 7 6 -1 9 7 7 ..................
T e a c h in g A s s o c i a t e , D e p a r tm e n t o f E le c t r ic a l
E n g in e e r in g , T h e O hio S t a t e U n iv e r s it y ,
C o lu m b u s, O hio
1 9 7 7 ..............................
M .Sc.,
T h e O hio S t a t e U n iv e r s it y ,
C o lu m b u s , O hio
1 9 7 7 -1 9 8 0 ...............
R e s e a r c h A s s i s t a n t , S o lid S t a t e L a b o r a to r y ,
T h e O hio S t a t e U n iv e r s it y
P U B L IC A T IO N S
" P r o p a g a tio n o f E le c tr o m a g n e t ic W a v e s T h ro u g h S i l ic o n w ith an
A r b itr a r y C o n c e n tr a tio n D is tr ib u tio n " .
F IE L D S
M a jo r F ie ld :
OF
STUDY
E le c t r ic a l E n g in e e r in g
S t u d ie s in S e m ic o n d u c to r D e v ic e s and S o lid S t a t e P h y s i c s .
D r . M . O . T h u r s to n and D r . J . M .
S w a r tz
л?
F IE L D S
OF
S t u d ie s in E le c t r o m a g n e t is m .
D r. R .J .
ST U D Y
D r. R.G.
CO N T.
K o u y o u m jia n and
G arbacz .
S t u d ie s in D ig ita l and C o m p u te r A r c h it e c t u r e . D r . K . J . B r e e d in g .
S t u d ie s in Q uantum M e c h a n ic s .
S t u d ie s in A p p lied S t a t i s t i c s .
D r . P . E . W ig e n .
D r. R .C .
S r iv a s t a v a .
TABLE
OF
C O N T EN TS
Page
D EDICATIO N
iii
ACKNOW LEDGM ENTS
iv
V IT A
v
L IS T O F T A B L E S
ix
L IS T O F F IG U R E S
x
INTRODUCTION
1
C h a p ter
I
T H E O R E T IC A L IN V E ST IG A T IO N
4
A.
In tr o d u cto r y R e m a r k s
4
B.
O p tic a l A b s o r p tio n
7
C.
C a r r ie r R e d is tr ib u tio n
D.
M ic r o w a v e
T r a n s m is s io n and R e fle c tio n
C o e f f ic ie n t s
E.
II
17
R e v ie w
26
35
E X P E R IM E N T A L R E S U L T S
38
A.
D e s c r ip tio n o f A p p a r a tu s
38
B.
A p p a ra tu s C a lib r a tio n and D a ta C o lle c tio n
44
C.
P r e s e n t a tio n o f C o lle c t e d D a ta and
C o r r e la tio n to T h e o r e t ic a l R e s u lt s
46
T A B L E OF C O N T EN TS CONT.
Page
III
D E V IC E A P P L IC A T IO N
A.
O p tic a l D e te c to r
B.
C o n tr o l o f M ic r o w a v e T r a n s m is s io n
T h ro u g h a P e r io d ic S u r f a c e
IV
SUM M ARY
63
63
67
71
A.
T heory
71
B.
E x p e r im e n ta l
73
C.
F in a l R e m a r k s
75
A P P E N D IX E S
A
C o m p u te r P r o g r a m s
76
B
S a m p le D a ta
94
B IB L IO G R A P H Y
98
v iii
L IST O F T A B L E S
P age
TABLE
C o e ffic ie n ts f o r E q s . 1 -5 7 th rou gh
1 - 6 7 d e s c r ib in g th e c o n d u c tiv ity o f
s ilic o n .
26
2.1
R e s u lts o f S t e p 1 .
47
2.2
R e s u lts o f S t e p 2 .
49
2 .3
R e s u lts o f S t e p 3 and a l i s t o f
c o n s ta n ts u s e d th ro u g h o u t the c a lc u la н
tio n s .
50
R e s u lts o f S t e p s 4 and 5 .
50
1 .1
2 .4
ix
L IS T O F FIGURES
P age
FIG U R E
1 .1
1 .2
1 .3
2.1
2.2
2 .3
2 .4
2 .5
2.6
2 .7
R e fle c tio n and t r a n s m is s io n o f p la n e
w a v e s by a s e m ic o n d u c to r a t n o r m a l
in c id e n c e .
8
E x a g g e r a te d v ie w o f th e e le c t r o n s
d iffu sin g at a f a s t e r r a te than th e h o l e s .
20
I llu s tr a tio n d e p ic tin g a w a fe r (sh a d ed
a r e a ) in a w a v e g u id e w ith a p r e s c r ib e d
c o o r d in a te s y s t e m .
28
B lo c k d ia g r a m illu s t r a t in g th e e x p e r iн
m e n ta l a p p a r a tu s .
39
O p tic a l p o r tio n o f th e e x p e r im e n ta l
a p p a ra tu s.
41
I llu s tr a tio n d e p ic tin g th e m ic r o w a v e
a p p a ra tu s.
43
B lo c k d ia g r a m d e s c r ib in g th e c o m p u ta н
tio n a l s t e p s u s e d in th e n u m e r ic a l a n a l y s is .
48
E le c t r o n - h o le p a ir c o n c e n tr a tio n th ro u g h th e
s i l i c o n w a fe r a ft e r th e lig h t p u ls e p a s s e d .
52
P lo t o f th e s p a c e - t im e m a tr ix w h ic h i s
th e p lo t o f th e r e s u lt s o f s te p 7 .
54
C o m p a r is o n o f th e e x p e r im e n ta l and
t h e o r e t ic a l r e s u l t s .
56
x
P lo t o f n o r m a liz e d tr a n s m it te d m ic r o w a v e
e n e r g y v e r s u s tim e w ith in fin ite s u r f a c e
r e c o m b in a tio n v e lo c it y on o n e s id e o f a
s i l i c o n w a fe r and z e r o s u r f a c e r e c o m b in a н
tio n v e lo c it y on th e o t h e r .
57
S lo p e o f th e c u r v e s illu s t r a t e d in F ig u r e
2 . 8 a t th e .5 e n e r g y m a r k a s a fu n c tio n
o f ta u .
58
P lo t o f n o r m a liz e d tr a n s m itte d m ic r o w a v e
e n e r g y v e r s u s tim e w ith in fin ite s u r f a c e
r e c o m b in a tio n v e lo c it y on both s u r f a c e s o f
a s ilic o n w a fe r .
59
S lo p e o f th e c u r v e s illu s t r a t e d in F ig u r e
2 . 1 0 a t th e . 5 e n e r g y m a r k a s a fu n c tio n
of ta u .
60
P lo t o f n o r m a liz e d tr a n s m itte d m ic r o w a v e
e n e r g y v e r s u s tim e w ith z e r o s u r f a c e
r e c o m b in a tio n v e lo c it y on both s u r f a c e s
o f a s ilic o n w a fe r .
61
S lo p e o f th e c u r v e s illu s t r a t e d in F ig u r e
2 . 1 2 a t th e . 5 e n e r g y m a r k a s a fu n c tio n
o f ta u .
62
T h e r e s p o n s e o f an e x is t e n t o p tic a l d e t e c t o r .
65
T h e R F r e s p o n s e o f th e s ili c o n on s a p p h ir e
w a fe r .
66
I llu s tr a tio n sh o w in g th e p e r io d ic a r r a y and
w a f e r m ou n ted in w a v e g u id e .
69
P lo t s illu s t r a t in g th e fr e q u e n c y r e s p o n s e o f
no s a m p le , p e r io d ic a r r a y o n ly , and p e r io d ic
a r r a y w ith s i l i c o n w a f e r .
70
xi
P age
FIGURE
B .1
B .2
B .3
P lo t o f th e in c id e n t in te n s ity o f th e l a s e r
p u ls e illu m in a tin g th e s i lic o n w a f e r .
95
P lo t o f th e in t e n s it y o f th e l a s e r p u ls e
e x itin g th e s ilic o n w a f e r .
96
P lo t o f th e tr a n s m itte d m ic r o w a v e e n e r g y
in r e s p o n s e to th e in c id e n t lig h t p u ls e .
97
x ii
INTRO DUCTIO N
A tte m p ts h a v e b e e n m a d e in th e p a s t to c h a r a c t e r iz e s e m i н
c o n d u c to r p r o p e r t ie s u s in g m ic r o w a v e r a d ia tio n .
g a to r s [ 1]
E a r ly i n v e s t iн
s u s p e n d e d a b lo c k o f s e m ic o n d u c to r m a t e r ia l in f r e e
s p a c e , m e a s u r e d th e s c a t t e r e d r a d ia tio n , and c a lc u la te d th e
e f f e c t iv e m o b ility and lif e t im e o f th e s a m p le fr o m th e m e a s u r e н
m e n ts.
P r o b le m s a r o s e w ith t h is a p p ro a c h b e c a u s e th e s c a t t e r e d
r a d ia tio n w a s s e n s i t i v e to th e a n g le o f in c id e n t r a d ia tio n .
O th er
e x p e r im e n t e r s [ 2 ] fa b r ic a te d d io d e s and p la c e d th em in to a w a v e н
g u id e w ith th e le a d s p a r a lle l to th e e l e c t r i c fie ld ;
how ever,
r e a s o n a b le c o r r e la t io n b e tw e e n e x p e r im e n ta l and t h e o r e t ic a l r e s u lt s
w a s n ot s h o w n .
M o r e r e c e n tly [ 3 ] ,
th in d e v i c e s , p r im a r ily
s o l a r c e l l s , h a v e b e e n p la c e d in s id e a w a v e g u id e in s u c h a w a y
th at th e e n tir e g u id e c r o s s s e c t io n w a s c o v e r e d .
In th is la t t e r
w o r k , lif e t im e o f th e m a t e r ia l w a s c la im e d to h a v e b e e n m e a s u r н
ed a c c u r a t e ly , but no c o n c lu s iv e r e s u lt s w e r e s h o w n .
It i s th e in te n t o f t h is w o r k to c o r r e la t e th e d y n a m ic
r e s p o n s e o f th e m ic r o w a v e r e f le c t io n and t r a n s m is s i o n c o e f f ic ie n t s
1
2
o f a s e m ic o n d u c to r w a f e r w ith an in c id e n t lig h t p u ls e a s a s t im u н
lu s .
T h e s e m ic o n d u c to r i s m ou n ted in a w a v e g u id e , c la m p e d
b e tw e e n tw o f l a n g e s , c o v e r in g th e e n t ir e c r o s s - s e c t i o n a l a r e a .
A
p u ls e o f lig h t fr o m a l a s e r i s c o u p le d in to th e w a v e g u id e illu m in н
a tin g th e s e m ic o n d u c to r w a f e r .
A s th e lig h t c r e a t e s e l e c t r o n -
h o le p a ir s in th e s e m ic o n d u c t o r , th e d y n a m ic r e s p o n s e o f th e
m ic r o w a v e t r a n s m is s io n and r e f le c tio n c o e f f ic ie n t s i s c h a r a c t e r н
iz e d .
B oth t h e o r e t ic a l and e x p e r im e n ta l a n a ly s e s a r e p r e s e n t e d .
T h e s o lu tio n i s d iv id e d in to th r e e s e q u e n tia l m o d u le s ,
o p tic a l a b s o r p tio n , c a r r i e r r e d is tr ib u tio n and m ic r o w a v e c h a r a c t e r н
iz a t io n .
In e a c h o f th e m o d u le s , th e fu n d a m en ta l e q u a tio n s a r e
u s e d , s u c h a s M a x w e ll's e q u a tio n s f o r th e o p tic a l a b s o r p tio n and
m ic r o w a v e c h a r a c t e r iz a t io n m o d u le s and th e c o n tin u ity e q u a tio n s
f o r th e c a r r i e r r e d is tr ib u tio n m o d u le .
T h e fu n d a m en ta l e q u a tio n s
a r e r e d u c e d to a d i s c r e t e fo r m and a r e s o lv e d on a d ig ita l c o m н
p u te r .
T h e s o lu t io n s a r e th en c o r r e la t e d w ith e x p e r im e n ta l
r e s u lts .
In a d d itio n to th e th e o r e tic a l c a lc u la t io n s and e x p e r im e n ta l
v e r if ic a t io n , tw o p o s s ib le a p p lic a tio n s a r e p r e s e n te d u sin g s e m i н
c o n d u c to r s in w a v e g u id e .
O ne d e v ic e i l l u s t r a t e s th e f e a s ib il it y o f
u s in g s ilic o n in c l o s e p r o x im ity to a p e r io d ic a r r a y to a llo w
s w itc h in g a c tio n e v e n in th e band p a s s .
T h e o th e r d e v ic e i s a
3
la r g e - a r e a o p tic a l d e te c to r w h ic h i s r e s is t a n t to d a m a g e fr o m
e x c e s s i v e o p tic a l e n e r g i e s .
o f t h e s e d e v ic e s
E x p e r im e n ta l c h a r a c t e r i s t i c s o f both
a re p resen ted .
T h is w o r k i s p r e s e n te d in fo u r c h a p t e r s .
In C h a p ter I, th e
t h e o r e t ic a l a n a ly s is i s d e v e lo p e d le a d in g to th e e q u a tio n s th a t a r e
to b e s o lv e d on a d ig ita l c o m p u te r .
D e s c r ib e d in C h a p te r II a r e
th e e x p e r im e n ta l a p p a ra tu s and th e m eth o d o f d a ta c o lle c t i o n . A ls o
p r e s e n te d i s a c o m p a r is o n b e tw e e n th e t h e o r e t ic a l and th e e x p e r iн
m e n ta l r e s u lt s w ith s a m p le c a lc u la t io n s .
In C h a p te r III, tw o
d e v ic e a p p lic a tio n s and th e c o r r e s p o n d in g e x p e r im e n ta l r e s u lt s a r e
d is c u s s e d .
T h e c o n c lu s io n s in C h a p te r IV h ig h lig h t im p o r ta n t
a s p e c t s o f th e w o r k and p r o p o s e a r e a s f o r fu r th e r s tu d y .
CHAPTER
T H E O R E T IC A L
A.
I
IN V E ST IG A T IO N
In tro d u cto ry R e m a r k s
E le c tr o m a g n e t ic w a v e s m a y b e u s e d to in t e r r o g a te c e r ta in
in te r n a l p r o c e s s e s in s e m ic o n d u c t o r s .
B y s tim u la tin g a s e m i н
c o n d u c to r s a m p le w ith an in c id e n t lig h t p u ls e , e le c t r o n - h o le p a ir s
a r e g e n e r a te d p r o v id e d th a t th e a s s o c ia t e d photon e n e r g y o f th e
lig h t p u ls e i s la r g e r th an th e b a n d -g a p o f th e m a t e r ia l.
The
p r e s e n c e o f an in c r e a s e d n u m b e r o f e le c t r o n s and h o le s in c r e a s e s
th e c o n d u c tiv ity o f th e s e m ic o n d u c to r and r e d u c e s th e t r a n s m is s io n
o f m ic r o w a v e s .
A s th e e le c t r o n s and h o le s r e tu r n to e q u ilib r iu m
by r e c o m b in in g , th e m ic r o w a v e r e f le c t io n and t r a n s m is s io n
c o e f f ic ie n t s r e la x to t h e ir r e s p e c t iv e s t e a d y - s t a t e v a lu e s .
The
p u r p o se o f t h is c h a p te r i s to d e v e lo p th e o r y w h ic h r e la t e s th e
in t e n s it y o f th e in c id e n t lig h t p u ls e to th e m ic r o w a v e r e f le c t io n
and t r a n s m is s io n c o e f f ic ie n t s v ia th e e l e c t r o n - h o le p a ir r e c o m н
b in a tio n m e c h a n is m w ith in th e s e m ic o n d u c t o r .
In S e c t io n B , th e e le c t r o n and h o le c o n c e n tr a tio n s a r e
co m p u te d a ft e r th e lig h t p u ls e h a s p a s s e d th ro u g h th e s a m p le .
4
5
T w o m e th o d s a r e p r e s e n t e d , o f w h ic h o n e i s u s e d i f th e b a n d -g a p
i s a p p r o x im a te ly e q u a l to o r la r g e r than th e in c id e n t p h oton
e n e r g y , and th e o th e r i s u s e d if th e b a n d -g a p i s m u ch s m a lle r
than th e in c id e n t photon e n e r g y .
F o r th e c a s e w h en th e b a n d -g a p i s la r g e r than th e in c id e n t
photon e n e r g y , th e o p tic a l a b s o r p tio n c o e f f ic ie n t and th e P o y n tin g
v e c t o r in s id e th e s e m ic o n d u c to r a r e r e la te d to th e in c id e n t and
tr a n s m itte d e n e r g i e s o f the lig h t p u ls e .
T h e c a r r ie r c o n c e n tr a н
tio n c a n b e c o m p u te d by a s s u m in g th a t f o r e v e r y photon o r e q u iн
v a le n t e n e r g y a b s o r b e d , o n e e le c t r o n - h o le p a ir i s g e n e r a t e d , and
th a t th e d iv e r g e n c e o f th e P o y n tin g v e c t o r g iv e s th e photon e n e r g y
a b so r b e d p e r u n it v o lu m e p e r s e c o n d in th e s e m ic o n d u c to r .
W hen th e band gap i s m u ch l e s s than th e in c id e n t photon
e n e r g y , a m o r e c o m p lic a te d m o d e l i s n e e d e d .
It i s a s s u m e d th a t
th e e x c e s s c a r r i e r c o n c e n tr a tio n in th e s a m p le is s a tu r a te d a t a
c o n c e n tr a tio n C D e x te n d in g a d is ta n c e
and th at b eyon d
in to th e in c id e n t s u r f a c e ,
th e r e i s an e x p o n e n tia l d e c a y o f c a r r i e r s .
K now ing th e o p t ic a l a b s o r p tio n c o e f f ic ie n t fo r th e s e m ic o n d u c to r ,
th e in d ic e n t and tr a n s m it te d e n e r g ie s o f th e lig h t p u ls e a r e r e la te d
to C Q and z^ , th e r e b y g iv in g th e c o n c e n tr a tio n d is tr ib u tio n .
6
U sin g e it h e r o f th e c a r r i e r d is tr ib u tio n s o f th e e l e c t r o n h o le p a ir s co m p u ted in S e c t io n B a s an in it ia l c o n d itio n , d e r iv a н
tio n s in S e c t io n C d e s c r ib e th e tim e d e c a y p r o c e s s o f th e e x c it e d
e le c t r o n - h o le p a i r s .
P r o c e e d in g w ith th e c o n tin u ity e q u a tio n s fo r
e le c t r o n s and h o le s w h ile a s s u m in g th a t th e e x c it e d e le c t r o n and
h o le c o n c e n tr a tio n s and t h e ir r e s p e c t iv e f i r s t and s e c o n d d e r iv a н
t i v e s a r e a p p r o x im a te ly e q u a l, a s im p le d iffu s io n eq u a tio n is
d e v e lo p e d w ith a s s o c ia t e d a m b ip o la r lif e t i m e and d iffu s io n c o e f f ic н
ie n t s .
T h e fin a l eq u a tio n i s th en fo r m u la te d s o th a t it m a y b e
s o lv e d n u m e r ic a lly u sin g a d ig ita l c o m p u te r .
A l s o , th e e q u a tio n s
r e la tin g th e c o n d u c tiv ity to th e c o n c e n tr a tio n a r e p r e s e n te d in
S e c t io n C .
A s th e e x c it e d e le c t r o n s and h o le s d e c a y , th e t r a n s m is s io n
and r e f le c t io n c o e f f ic ie n t s o f th e m e d iu m a t m ic r o w a v e fr e q u e n c ie s
w ill v a r y .
In S e c t io n D , a r e v ie w o f th e c o m p u ta tio n o f t h e s e
m ic r o w a v e r e f le c t io n and t r a n s m is s io n c o e f f ic ie n t s i s p r e s e n t e d .
A c o m p le te tr e a tm e n t w a s d e v e lo p e d p r e v io u s ly [ 4 ] . P r e s e n t e d
h ere is
a
d e r iv a tio n w h ic h s t a r t s w ith M a x w e ll's e q u a tio n s and
a s s u m e s an a r b it r a r ily v a r y in g c o n d u c tiv ity in th e z d ir e c t io n .
S i n c e th e c o n d u c tiv ity a p p e a r s in n u m e r ic a l f o r m , th e fin a l
e q u a tio n s a r e tr a n s fo r m e d s o th a t th e y m a y b e s o lv e d b y a d ig ita l
c o m p u te r .
In S e c t io n E , a r e v ie w o f th e e q u a tio n s n e e d e d f o r th e
n u m e r ic a l a n a ly s is i s p r e s e n t e d .
T h e c o m p u te r p r o g r a m s s o l v н
in g t h e s e e q u a tio n s a r e l is t e d in A p p en d ix A , w h ile th e s o lu tio n s
a r e c o r r e la t e d w ith e x p e r im e n ta l d a ta in C h a p te r II, S e c t io n C .
B.
O p tic a l A b s o r p tio n
A s a s h o r t p u ls e o f lig h t p r o p a g a te s th ro u g h a s e m ic o n d u c н
to r, th e in te n s ity o f th e lig h t p u ls e d e c a y s w h ile e le c t r o n s a r e
e x c ite d to a h ig h e r e n e r g y l e v e l , th e r e b y c r e a tin g e le c t r o n - h o le
p a ir s .
T h e n u m b er o f e le c t r o n - h o le p a ir s p r o d u ce d c a n b e c o m н
puted by d e te r m in in g th e in t e n s it y o f th e lig h t p u ls e and r e la tin g
it to th e n u m b e r o f p h o to n s a b s o r b e d .
C a se I
If th e b a n d -g a p o f a s e m ic o n d u c to r m a t e r ia l i s a p p r o x im a te н
ly eq u a l to o r la r g e r th an th e in c id e n t photon e n e r g y , th e r e i s
lit t le a b so r p tio n by d ir e c t in te r b a n d t r a n s it io n s , and th e a b s o r p н
tio n c o e f f ic ie n t d u e to o t h e r p r o c e s s e s ca n b e a s s u m e d to b e
s m a ll and c o n s ta n t th ro u g h o u t th e m a t e r ia l.
T h ree h om ogen eou s
m e d ia , c h a r a c t e r iz e d by th e p r o p a g a tio n f a c t o r s k 1 , k , and k_ ,
1
2
o
and s e p a r a te d b y p la n e b o u n d a r ie s , a r e c o n s id e r e d a s sh o w n in
F ig .
1 .1
th r e e a r e:
[ 5] .
T h e e le c t r o m a g n e t ic f ie l d s in m e d ia o n e th ro u g h
z
F ig u r e 1 .1
:
Region 3
Region 2
Region 1
0
z
d
R e fle c tio n and t r a n s m is s io n o f p la n e
w a v e s by a s e m ic o n d u c to r a t n o r m a l
in c id e n c e .
M ed iu m 1
_╗
Ei
f*
=
-?
Ht
ik ^ z-ico t
x|_E░e
+ E1S
[* ?<1
=
-ik .. z -ic o t "I
J
ik^ z -ic o t
y f e r
Eoe
-ik-j z -ic o t
- Eie
M ed iu m 2
*
Es
_
=
. [ 1 + ik2 z
x [ E2 e
H?
s
=
JK 2
y|
\ _ W2
-icot
, F - - lk2 z l
e
+
2e
J
+ ik g 2
_ - i k g z l -ico t
E? e
- E? e
Ie
M ed iu m 3
E
t
-?
H.
=
x E e
3
=
A ^3
-y
<^3
ikSo.
0z - iw t
3
i ^ z - icot
Eoо
3
T h e p r o p a g a tio n f a c t o r s fo r e a c h o f th e m e d ia a r e :
10
( 1- 8 )
(1 -9 )
In E q s . 1 - 3 and 1 - 4 ,
E g,
Eg , and kg m u s t b e d e te r m in e d
b e fo r e th e fie ld d e s c r ip t io n i s c o m p le t e .
T h e p r o p a g a tio n fa c to r
kg ca n b e co m p u ted b y u s in g e x p e r im e n ta l d a ta f o r th e m a g n itu d e
o f th e o p tic a l t r a n s m is s io n c o e f f ic ie n t T .
F r o m S t r a t t o n 's w o r k
[ 5 ] , th e t r a n s m is s io n c o e f f ic ie n t i s found to b e
4e
T
(1 + Z 1 2 )(1 + Z 2 3 )
i (k 2 _ kg)d
1 + r -i2r 2 3 e
2 ik g d
Re [ Z 13]
( 1- 1 0 )
w here
2 jZ
Z, j
=
1,2,3
( 1- 1 1 )
z, j
=
1, 2,3
( 1- 1 2 )
Z =
1,2,3
V i k.
1 - Z
j*
1 + z
, and
=
S in c e
Cg
$Z
+
ia
(1 -1 3 )
i s th e o n ly unknow n p a r a m e t e r , it m a y b e v a r ie d u n til
th e c o r r e s p o n d in g c o m p u te d v a lu e o f T Q a g r e e s w ith th e e x p e r i -
11
+
m e n ta l v a lu e .
?
T o d e te r m in e Eg and E g , th e e x p e r im e n ta lly
m e a s u r e d in c id e n t and tr a n s m it te d lig h t in t e n s it ie s w ill b e b e u s e d .
T h e m e a n in t e n s it y o f th e lig h t p u ls e i s d e s c r ib e d b y th e
r e a l p a rt o f th e c o m p le x P o y n tin g v e c t o r
Sr
=
ReC^EXH*]
(1 - 1 4 )
B y in te g r a tin g o v e r th e e n tir e s u r f a c e o f th e s e m ic o n d u c t o r , th e
to ta l e n e r g y a b s o r b e d p e r u n it t im e i s
S r ? nda
?T
?
(1 - 1 5 )
I
T h u s,
г
T
= AB
2
(1 -1 6 )
w h e r e A B i s th e illu m in a te d a r e a o f th e w a fe r sh o w n in F ig u r e
1.1.
4-p c a n a 1so b e s ta te d in t e r m s o f th e in c id e n t in te n s ity
and th e o p tic a l r e f le c t io n and t r a n s m is s i o n c o e f f ic i e n t s , Rq and
T , r e s p e c t iv e ly ;
- R e [ S 0 ] [ 1 - R 0 - T 0 ]A B
w here
R e (S 0 ) i s th e in c id e n t in t e n s it y and th e r e f le c t io n
c o e f f ic ie n t i s [ 5 ]
(1 - 1 7 )
12
2ikгcl
1 + r 12 r 2 3 e
(1 -1 8 )
2 ik 2 d
F u r t h e r , th e e n e r g y c o u p le d in to th e s e m ic o n d u c to r p e r u n it tim e
i s g iv e n by
fit
R e [ S Q] [ 1 - R q ]A B
_+i2
=
-
E
AB
(1 -1 9 )
2 CO
B y c o m b in in g E q s .
1-16,
1-17,
and 1 - 1 9 ,
b e c o m p u te d , le a d in g to
-+|2
( 1 - R 0 )e
2 a Qd
2 -
Tc
( 1- 2 0 )
R e [S c ]
2and
fir
(e
-2 a 0d
-e
2 )
and
(1 -R 0 ) e
e
; i2
( 1- 2 1 )
20! 2 d
(e
Eg and Eg
Tc
R e [S c]
fir
K now ing
?2ttod
2 -
| E* | 2
and
-2 a 2d
-e
)
| Eg | 2 d o e s not le a d to a s o lu tio n o f
s in c e p h a se r e la t io n s h ip s a r e not k n ow n .
H ow ever,
ta k in g th e r e a l p a r t o f the d iv e r g e n c e o f th e P o y n tin g v e c t o r in
th e s e m ic o n d u c to r g iv e s th e e n e r g y a b s o r b e d p e r u n it v o lu m e p e r
u n it t im e a s a fu n c tio n o f z ;
T h e r e f o r e , know ing o n ly
v a lu e s o f a 2 and jSg
i + i2
| E g|
and
i - 12
| Eg |
and th e co m p u te d
i s s u f f ic ie n t f o r th e d e s ir e d s o lu t io n .
S in c e the e n e r g y is a b s o r b e d in qu antu m in c r e m e n t s and
f o r e v e r y photon a b s o r b e d o n e e le c t r o n - h o le p a ir i s g e n e r a te d
E q . 1 -2 2 c a n b e s e t e q u a l to -C (z )h to ,
C (z )
=
&2a 2
2
+ . 2 ?о? 2 Z
Eg | e
+
th u s
-
,2
aa2 z
(1 -2 3 )
to M2 h
w h e r e C (z ) i s th e n u m b e r o f e le c t r o n - h o le p a ir s g e n e r a te d p e r
u n it v o lu m e p e r l a s e r p u ls e , and
h i s P la n c k 's c o n s ta n t d iv id e d
by 2 n .
C a s e II
If the in c id e n t photon e n e r g y i s s u b s ta n tia lly h ig h e r than
th e b a n d -g a p e n e r g y , th e in c id e n t lig h t w i l l b e v e r y e f f e c t iv e in
g e n e r a tin g e le c t r o n - h o le p a ir s and th e a b s o r p tio n c o e f f ic ie n t w i l l
b e h ig h .
In a d d itio n , if th e in c id e n t lig h t p u ls e i s o f s u f f ic ie n t
in t e n s it y to e x c it e m o s t o f th e a v a ila b le e le c t r o n s to a h ig h e r
e n e r g y l e v e l , th e a b s o r p tio n c o e f f ic ie n t i s n ot c o n s ta n t th ro u g h o u t
th e s e m ic o n d u c to r .
U n d er s u c h c o n d it io n s , th e a b s o r b in g
m e d iu m i s d iv id ed in to tw o r e g io n s .
In th e f i r s t r e g io n , c l o s e
14
to th e in c id e n t s u r f a c e , it i s a s s u m e d th a t m o s t o f th e a v a ila b le
e le c t r o n s a r e e x c it e d , g iv in g a c o n c e n tr a tio n C Q p e n e tr a tin g to a
depth z 1 in to th e in c id e n t s u r f a c e .
B y th e tim e th e lig h t p u lse
h a s p e n e tr a te d in to th e s e c o n d r e g io n ,
it s in t e n s it y w ill h a v e
d e c r e a s e d s u c h th a t th e a b s o r p tio n c o e f f ic ie n t ca n b e a s s u m e d to
b e c o n s ta n t.
In t h is r e g io n , th e c o n c e n tr a tio n d is tr ib u tio n w ill b e
o f th e e x p o n e n tia l f o r m .
T h u s , th e c o n c e n tr a tio n d is tr ib u tio n
- 2 0 ! 2 ( z - z i)
C (z )
=
C D [*x(z) -
) + M (z -z 1 )e
]
(1 -2 4 )
w ill b e c o n s id e r e d , w h e r e :
/i(z )
i s th e u n it s t e p fu n c tio n ,
C0
i s th e s a tu r a tio n le v e l f o r th e c o n c e n tr a tio n
c l o s e to th e s u r f a c e ,
i s th e a b s o r p tio n c o e f f ic ie n t in th e r e g io n
b e tw e e n z 1 and d ,
z1
i s th e depth o f s a tu r a tio n .
In th e r e g io n o f z^ < z <
c o n d u c to r is
s
and
d , th e EM w a v e in s id e the s e m i н
15
+ ik ^ z - z .,)
Hs
=
y
E2 e
_
-
-ico t
-ik g C z-z-j)
E2 e
e
(1 -2 6 )
C0/i2
w here E
2
i s a s s u m e d to be n e g lig ib le s in c e ol0 and d a r e a s s u m e d
^
to be la r g e en ou gh to e lim in a te any in te r n a l r e f l e c t io n s .
The
m e a n in t e n s it y o f th e lig h t p u ls e in th is r e g io n is
R e [S s ]
=
z
0
+ ,2 - 2 a 2 ( z - z 1>
E2 | e
z. < z < d
2a; n .
(1 - 2 7 )
w here S
s
i s th e P o y n tin g v e c t o r .
C om p u tin g th e e n e r g y a b s o r b e d
in th e r e g io n z^ < z < d ,
' ^ ? 2 , + .2
| e2 I e
Re [ V * S s ] =
CO
(1 - 2 8 )
^2
and r e la tin g R e ( V * S s ) to quantum a b s o r p tio n le a d s to
-2 a 2(z -z )
C Qe
HS a 2
h co =
2 - 2 a 2 ( z - z 1)
| Eg | e
(1 -2 9 )
? Mo
+ i2
A l s o , r e la tin g th e tr a n s m itte d in t e n s it y , R e [ S fc] , to | Eg |
Re [ s ]
=
2
2
^2 +Oto
|E
+ , 2 ~2 a 2 ( d ~z l )
| e
T
(1 -3 0 )
and c o m b in in g E q s . 1 - 2 9 and 1 - 3 0 g i v e s a s o lu tio n f o r C Q in t e r m s
o f z 1;
16
Re [ S T ]
^2 a 2
T 2 3 h 00
2 a 2 ( d - z 1)
(1 -3 1 )
^ 2 +Ct2
w h e r e T 0 ? is th e t r a n s m is s i o n c o e f f ic ie n t b e tw e e n m e d ia tw o and
2o
th r e e ;
T.
23
=
R e [ Z 2*3 ]
(1 - 3 2 )
1 + Z,2 3
S o f a r , o n ly o n e o f th e n e e d e d e q u a tio n s r e la tin g C Q and z^ to
e a c h o th e r h a s b e e n d e te r m in e d .
1 -2 4
T o c o m p le te th e s o lu t io n ,
z = 0 to z = d
i s in te g r a te d fr o m
Eq.
to g iv e th e to ta l n u m b e r
o f e le c t r o n - h o le p a ir s g e n e r a te d in th e s e m ic o n d u c to r p e r lig h t
p u lse ; g iv in g th e s e c o n d n e e d e d eq u a tio n
-2 a 2 (d -z 1
'T
z
1
+ ----- [ 1 - e
2 a,
(1 - 3 3 )
w h e r e C-p i s r e la te d to th e e x p e r im e n ta lly m e a s u r e d in c id e n t and
tr a n s m itte d in t e n s it ie s by
(1 - | r 1 2 | ) R e [ S 0 ] - R e [ S T ]
AB
'T
C o m b in in g E q s .
(1 -3 4 )
h u)
1 -3 3 and 1-31 g iv e s a tr a n s c e n d e n ta l e q u a tio n fo r
w h ic h m u st b e s o lv e d by s u c c e s s i v e a p p r o x im a tio n .
O n ce z^
i s known CQ m a y b e found u s in g E q . 1-31 th u s g iv in g th e n eed ed
c o e f f ic ie n t s f o r E q . 1 - 2 4 .
17
C.
C a r r ie r R e d is tr ib u tio n
In th e p r e v io u s s e c t i o n , th e c o n c e n tr a tio n d is tr ib u tio n C ( z )
o f th e e le c t r o n - h o le p a ir s g e n e r a te d b y an in c id e n t lig h t p u ls e w a s
d e te r m in e d .
A fte r th e lig h t p u ls e p a s s e s , th e e le c t r o n s and h o le s
tend to d iffu s e to l e s s p o p u la ted a r e a s o f th e s e m ic o n d u c t o r .
A ls o ,
a r e c o m b in a tio n p r o c e s s ta k e s p la c e th a t a llo w s th e e le c t r o n s and
h o le s to a n n ih ila te e a c h o th e r s o th e to ta l n u m b e r o f p a ir s d e c r e a s e s .
T h e c o n tin u ity e q u a tio n s th a t d e s c r ib e th is p r o c e s s , o n e f o r e l e c н
tr o n s ,
an
?
?*
?R n + Hn V ?(n E ) + D n V
=
and a n o th e r fo r
-гPat
=
P
n
(1 - 3 5 )
h o le s ,
?R
-
n V ? (p E )+ D V 2 p
p
p
(1 - 3 6 )
p
r e la t e th e to ta l c h a n g e o f e le c t r o n s and h o le s p e r u n it v o lu m e
p e r u n it t im e
^ ? and
at
, r e s p e c t iv e ly , w ith th e ir r e s p e c t iv e
at
r e c o m b in a tio n , d r if t , and d iffu s io n r a t e s p e r
u n it v o lu m e p e r u n it
t im e .
1 -3 6 a r e d e fin ed a s:
T h e v a r ia b le s u se d in E q s .
n
n'
+
1 -3 5 and
no
n
-
to ta l e le c t r o n c o n c e n tr a tio n p e r u n it v o lu m e
n'
-
e x c it e d e le c t r o n c o n c e n tr a tio n
nQ -
e le c t r o n b a ck g ro u n d c o n c e n tr a tio n
(1 - 3 7 )
18
p
=
P*
+
PQ
(1 -3 8 )
p
-
to ta l h o le c o n c e n tr a tio n p e r u n it v o lu m e
p1
-
e x c it e d h o le c o n c e n tr a tio n
p^
-
h o le b ack g ro u n d c o n c e n tr a tio n
n - nQ
R
R
=
n
n
-
T
n
r
(1 -3 9 )
n
r e c o m b in a tio n r a te fo r e le c t r o n s
- lif e t im e f o r th e e le c t r o n s
P " P0
R
-=
P
R
P
-
r e c o m b in a tio n r a te fo r h o le s
T
P
-
lif e t i m e fo r th e h o l e s .
D
-
d iffu s io n c o e f f ic ie n t f o r
e le c t r o n s
-
d iffu s io n c o e f f ic ie n t f o r
h o le s
D
P
n
P
Hn -
m o b ility f o r th e e le c t r o n s
/Li P
m o b ility f o r th e h o le s
T h e a s s u m p tio n w ill be m a d e th ro u g h o u t th e fo llo w in g
d e r iv a tio n th a t th e p a r tia l d e r iv a t iv e s w ith r e s p e c t to tim e f o r n
and p a r e a p p r o x im a te ly e q u a l.
m a d e f o r s p a c e d e r iv a t i v e s .
A s i m i l a r a s s u m p tio n w ill b e
In r e a li t y , h o w e v e r , a s th e e le c t r o n s
19
and h o le s r e d is t r ib u t e , th e e le c t r o n s ten d to d if fu s e f a s t e r than
th e h o l e s , a s s u m in g th a t th e e le c t r o n s h a v e a l a r g e r d iffu s io n
c o e f f ic ie n t than do th e h o l e s , a s illu s t r a t e d in F ig u r e 1 . 2 . W h ile
th e in b a la n c e o f e le c t r o n s and h o le s d e v e lo p s , an e l e c t r i c fie ld
i s fo r m e d b e tw e e n th e e le c t r o n s and h o l e s .
A s a r e s u lt , th e
d iffu s io n o f th e h o le s i s a id e d b y th e f i e l d .
T h e b a s ic a s s u m p н
tio n s t a t e s th at a lth o u g h th e e le c t r o n and h o le d iffu s io n c o e f f ic ie n t s
a r e d if f e r e n t , th e e l e c t r i c f ie ld fr o m c h a r g e u n b a la n c e p r e v e n ts
th e e x c ite d e le c t r o n and h o le c o n c e n tr a tio n s and t h e ir r e s p e c t iv e
d e r iv a t iv e s fr o m d iffe r in g s ig n if ic a n t ly fr o m e a c h o th e r th ro u g h o u t
th e s e m ic o n d u c t o r .
T h is , in tu r n , a llo w s th e u s e o f a s im p le
d iffu s io n e q u a tio n p r o v id e d th a t e f f e c t iv e o r " a m b ip o la r" m o b ility ,
li f e t i m e , and d iffu s io n c o e f f ic ie n t s a r e u s e d .
M u ltip ly in g E q . 1 -3 5 by fx^p and E q .
1 -3 6 by
ju^n , ad d in g
th e tw o , and le ttin g
bp
bt
bn
bt
(1 - 4 1 )
Vn
(1 - 4 2 )
and
Vp
le a d s to
=
T
-------
1
I-------------- 1
1-------------- 1
"
T
I
"
T
Electrons
Holes
c
o
4->
03
г4->
?r*
C
CD
U
c
o
o
j______i______ i
?______i______ i---------1--------- 1--------- 1
Displacement
F ig u r e 1 .2 :
E x a g g e r a te d v ie w o f th e e le c t r o n s d iffu sin g a t a f a s t e r r a te
than th e h o le s .
21
bn
-r?? ( U p + jLtn)
bt ^ p
n '
- R/Ltp+/xnR
+ n_fJL (p -n )E ? V n +
n p
'n
p
'p'n
'
=
/UppDnV 2 n + ^ nnDpV 2 p
(1 -4 3 )
C o m b in in g E in s t e in 's r e la t io n s h ip s fo r h o l e s ,
D
kT
=
P
q
U
(1 -4 4 )
P
and f o r e le c t r o n s
Dn
kT
? Mn
=
(1 -4 5 )
q
g iv e s
DpM ?
=
On ,xp
(1 -4 6 )
p
O
A s s u m in g V n e q u a ls V P and c o m b in in g E q s . 1 - 4 3 and
1 -4 6 g iv e s
bn
"Rn ^ pP + ^ nnRp
( P - n >E ?
Vn
+
bt
MpP + Mnn
W
+P)
y 2n
+
MpP + Mnn
(1 -4 7 )
u p +a n
P
n
N oting that (n-p)V n
i s s m a ll and that n' is a p p ro x im a tely equal
to p' g iv e s the fin a l resu ltj
w here T
3
and D
a r e th e a m b ip o la r lif e t i m e and d iffu s io n
cl
c o e f f ic ie n t s , r e s p e c t iv e ly ;
Ta
=
D
=
n ?m + u ) + ( u p
^ p
n'
d o
? ------------ ;--------------
+ u n )
^ n o'
(1 - 4 9 )
and
M Dn(2n* + nQ + pQ)
a
(1 -5 0 )
n ?^ D + ^n) + ^ pPo + ^ n no
U sin g th e r e s u lt s o f S e c t io n B a s th e in itia l c o n d itio n
(le ttin g n ( z ,t ) a t t im e e q u a ls z e r o b e C (z ) ) and E q . 1 - 4 8 ,
th e
e le c t r o n - h o le p a ir c o n c e n tr a tio n c a n b e co m p u te d th ro u g h o u t th e
s e m ic o n d u c to r a s a fu n c tio n o f t im e by u s in g n u m e r ic a l m e th o d s .
A p p ro x im a tin g E q . 1 -4 8 by th e d if fe r e n c e eq u a tio n
k+1
n'
m
k-1
k
k
n 1 . ? - 2n '
m+1
m
- n1
m
Dam
26
k
+ n
m -1
k
n'
m
2
h
w here
"Tam
(1 - 5 1 )
m
i s th e s p a t ia l in d ex
k
i s th e te m p o r a l in d ex
h
i s th e in c r e m e n ta l s p a tia l d is p la c e m e n t
6
i s th e in c r e m e n ta l te m p o r a l d is p la c e m e n t
23
and s o lv in g f o r n ^
g iv e s ,
1
*
,k +1
n*
m
c
k
ni
m
7
E
Am +1
, +
am
L
_
,k -1
n'
m
26
h2
(1 - 5 2 )
1
J.
2 D am
h2
Tam
A p p ly in g th e s u c c e s s i v e o v e r r e la x a tio n m eth o d (S O R ), an e r r o r
f a c t o r i s d e v e lo p e d ,
n
D am
.k
+ n'^
m -1
m +1
n .k+1 . n .k-1
m______ m
2 6
h
E rror
? n'
m
=
2 D am
h
(1 -5 3 )
W * E rror
(1 -5 4 )
am
and a new n'
m
n'
m
i s co m p u ted by
n'
m
w h e r e W i s th e r e la x a tio n f a c t o r .
by
T h e d e s ir e d s o lu tio n i s o b ta in e d
r e p e a te d ly u s in g E q s . 1 -5 3 and 1 - 5 4 o v e r a ll r a n g e s o f m
?
and k u n til th e e r r o r i s r ed u c ed to to le r a b le l i m i t s . *
*
O n ce n f i s
T h e v a lu e s o f n o v e r th e r a n g e s o f m and k a r e r e f e r r e d to
a s th e s p a c e - t im e m a t r ix .
24
kn ow n , th e e le c t r o n and h o le c o n c e n tr a tio n s b e c o m e known r e s p e c t н
iv e ly a s ,
n
=
n'
+
n
(1 - 5 5 )
p
=
n'
+
pQ
(1 - 5 6 )
and
T h e c o n d u c tiv ity o f th e s e m ic o n d u c to r i s r e la t e d to n and p
by
=
q (" ^ n +
PMp)
(1 -5 7 )
w h e r e q i s th e e l e c t r i c c h a r g e o f an e l e c t r o n .
B e r z [ 6 ] d e te r н
m in ed th e m o b ilit ie s , u n and *Up to b e
jU
n
=
1
r
I" 1
F
L ^ n
+
+
?
^ henl
" in ^ h e n
(1 - 5 8 )
J
and
1
At
P
=
p---------------- ;-------? ------- ?
r
?\ I"
1
^1 n + ^ h en 1
F
w here
,
L /^
7
jUhe , and
+
M1 n ^ h e n
(1 - 5 9 )
J
a r e th e c o m p o n e n ts r e la te d to l a t t ic e ,
e le c t r o n - h o le , and im p u r ity s c a t t e r in g ,
r e s p e c tiv e ly .
The
s t a t i s t i c a l fa c to r F i s
F
=
.0 9 5 4
+
.4 7 3 S -
-3 8 3 S 2
.0 9 5 6
+
.8 6 6 S -
.7 7 6 S 2
(1 - 6 0 )
25
' i v
v = n ,p
(1 -6 1 )
^1 v ^ h ev
v + ^ hev
T h e in d iv id u a l m o b ility c o m p o n e n ts a r e r e la te d to n ,p , th e t e m н
p e r a tu r e (T ^)
and to ta l im p u r ity c o n c e n tr a tio n (N ) b y ,
A
V 300
/
(1 -6 2 )
"In
N 1n
1 +
(n + p )
(1 -6 3 )
1n
1p
- 2 .4
T,
C
in
(1 -6 4 )
300
T
?i p
\ ?2 .5
i X
(1 -6 5 )
300
V~2~
hen
hep
J
00/
\330 C
2
=
M
hen
N
p
n
1n
( 1- 6 6 )
(1 -6 7 )
26
T h e c o e f f ic ie n t s A , B , C ^ , and Cg a r e li s t e d in T a b le 1 .1 f o r
s ilic o n .
T a b le 1 .1
C o e ffic ie n ts fo r E q s . 1 - 5 7 th ro u g h 1 - 6 7
d e s c r ib in g th e c o n d u c tiv ity o f s i lic o n
C o e ffic ie n ts
A
V a lu e s
2 . 4 5 x 10 21
D.
O
1350
c ,
C
X
<7
B
480
2
M ic r o w a v e T r a n s m is s io n and R e fle c tio n P r o p e r t ie s
In th is s e c t io n , a th e o r e tic a l e x p r e s s io n d e s c r ib in g the
p r o p a g a tio n o f an e le c t r o m a g n e t ic w a v e th ro u g h b u lk s e m ic o n d u c to r
m a t e r ia l in a w a v e g u id e i s d e r iv e d .
T r a n s m is s io n and r e f le c t io n
c o e f f ic ie n t s o f th e m a t e r ia ls a r e th en c o m p u te d .
T h e r e s u lt s
of
S e c t io n C a r e u s e d to g e n e r a te th e c o n d u c tiv ity p r o file in the
d ir e c t io n o f p r o p a g a tio n ( z - d ir e c t io n ) .
T h e v a r ia b le s u s e d th ro u g h н
ou t th e d e r iv a tio n a r e d e fin ed a s fo llo w s :
27
<7( z )
-
c o n d u c tiv ity a s a fu n c tio n o f z in th e
s e m ic o n d u c to r
-
p e r m itt iv ity o f th e s e m ic o n d u c to r
~
p e r m e a b ilit y o f th e s e m ic o n d u c to r
jUg
? ., e 1 3
, Hq -
p e r m itt iv ity o f s p a c e in th e w a v e g u id e
p e r m e a b ilit y o f s p a c e in th e w a v e g u id e
60
- a n g u la r fr e q u e n c y in r a d ia n s p e r s e c o n d
x
- u n it
v e c t o r in
th e x
d ir e c t io n
y
- u n it
v e c t o r in
th e y
d ir e c t io n
z
- u n it
v e c t o r in
th e z
d ir e c t io n
b
-
w id th o f th e w a v e g u id e a lo n g th e y d ir e c t io n
N ote th at th e c o o r d in a te s y s t e m to b e u s e d h e r e i s r o ta te d
and s h ifte d fr o m th at c h o s e n in S e c t io n s B and C .
in F ig u r e 1 . 3 .
It i s d e p ic te d
T h is c h a n g e in th e c o o r d in a te s y s t e m
th e n u m e r ic a l s o lu t io n .
s im p l i f i e s
T h e w a v e i s a s s u m e d to p r o p a g a te in th e
m in u s z d ir e c tio n and
a s s u m e d to b e r e a l c o n s t a n t s .
,
?3 ,
/Li1 , /Hg,
M3 ancl w
arо
A ls o an e jWt tim e d e p e n d e n c e
is
a s s u m e d th ro u g h o u t and s u p p r e s s e d f o r c o n v e n ie n c e .
T o a c h ie v e th e d e s ir e d s o lu t io n , a r e c u r s io n r e la tio n
d e s c r ib in g th e e l e c t r i c f ie ld th ro u g h o u t th e s e m ic o n d u c to r i s
d e v e lo p e d . S t a r t in g w ith M a x w e ll's e q u a tio n s in th e s e m ic o n d u c to r ,
28
X
Wafer
Incident
Electric Field
Transmitted
E lectric Field
-2
F ig u r e 1 . 3 :
I llu s tr a tio n d e p ic tin g a w a fe r (sh a d e d a r e a )
in a w a v e g u id e w ith a p r e s c r ib e d c o o r d in a te
sy ste m .
29
VXH
=
[< r(z )
-?
VXES
=
?+
-JWM2 HS
+
^ * 2 ] Es
(1 -6 8 )
and
(1 -6 9 )
and tak in g th e c u r l o f E q . 1 - 6 9 , and c o m b in in g it w ith E q . 1 -6 8
g iv e s th e w a v e e q u a tio n ,
VXVXEs
=
-jWjLi2 ( <T(z) + j w e 2 ) E s
(1 -7 0 )
S in c e th e tim e -in d e p e n d e n t r e p r e s e n ta tio n o f th e T E ^ q tr a n s m itte d
w a v e in th e w a v e g u id e i s
Et
=
[ 7 ] o f th e fo r m
x Eg s in |
| exp
(1 -7 1 )
z
<
0
th e e l e c t r i c f ie ld in th e s e m ic o n d u c to r is p r o p o se d to b e o f a
s i m i l a r fo r m
Es
=
X Eg s in
?
M
l - ^ l f(z ) ,
d < z < 0
Lb J
w h e r e f ( z ) i s a fu n c tio n to b e d e te r m in e d .
in to E q .
(1 -7 2 )
S u b s titu tin g E q . 1 -7 2
1 - 7 0 g i v e s th e fin a l fo r m o f th e w a v e e q u a tio n w h ic h i s
to b e s o lv e d ;
= [ ( 7 )2 +
L'
'
^ ?2
(CTл
+ ^ f 2) ]
J
d < z < 0
(1 -7 3 )
30
A r e c u r s io n r e la t io n i s fo r m e d by f i r s t a p p r o x im a tin g th e
s e c o n d d e r iv a tiv e o f f ( z ) a s
.2 . , N
d f(z )
dz
f
~
,? - 2 f m
m+1
2
h
+ f
.
m -1
(1 - 7 4 )
2
w here
h
i s an in c r e m e n ta l le n g th
m
is
z
i s eq u a l to m h
nh
i s th e s e m ic o n d u c to r w a fe r th ic k n e s s
an in te g e r ;
N e x t, c o m b in in g E q s .
m = 1, 2 ,
.. . ,
n
1 -7 3 and 1 - 7 4 and s o lv in g fo r
g iv e s
th e in ten d e d r e c u r s io n r e la tio n s h ip ;
2
f m+1
=
\ f
(f-)
T
+
+ j ? e s ) + 2 ] f m - f m -1
(1 - 7 5 )
T o s t a r t th e r e c u r s io n , f^ and f m u st
K now ing th a t
the ta n g e n tia l c o m p o n en t o f
c o n tin u o u s a c r o s s an in t e r f a c e , f
b e d e te r m in e d
th e e l e c t r i c
.
f ie ld i s
i s co m p u ted by s e t t in g E q s .
1 -71 and 1 -7 2 eq u a l a t z = 0;
5E3 s<"( f - ) eXp[(^0->( f - f - ?V
3
)
^=
(1 -7 6 )
x E g s in
f(░)
31
w h ic h r e d u c e s to
(1 -7 7 )
0
T o s o lv e fo r f , f ( z ) i s exp an d ed arou n d z = 0 by u s in g th e
M a cL a u r in s e r i e s and th en e v a lu a tin g it f o r z = h s o th a t
00
f
1
+ y
o
'
i=1
1
j
J 2 _ Q f(z )
i !
6zx
(1 - 7 8 )
z=0
T h e f i r s t te r m h a s a lr e a d y b e e n d e te r m in e d to be u n ity .
s e c o n d te r m i s d e te r m in e d f i r s t by sh o w in g th a t
?+
The
A
5 ( E ^ .* x ) / 6 z
is
c o n tin u o u s a c r o s s th e b o u n d a ry a t z = 0 , and th en by a p p ly in g
E q s . 1-71 and 1 - 7 2 .
A p p ly in g o n e o f M a x w e ll's e q u a tio n s ,
(1 - 7 9 )
VXEt
to E q . 1 - 7 1 , th e ta n g e n tia l c o m p o n en t a t
1
&(Et * x )
yj
y (Ht -y)
WjLtg
becom es
(1 - 8 0 )
6z
A l s o , in a s i m i l a r f a s h io n , th e ta n g e n tia l c o m p o n e n t o f H f o r th e
s e m ic o n d u c to r i s
5 (E s ? x)
a
y
j
y (Hs - y )
(1 -8 1 )
S in c e th e ta n g e n tia l c o m p o n en t o f H i s c o n tin u o u s a c r o s s th e
32
in t e r f a c e a t z = O (a s s u m in g no f r e e s u r f a c e c u r r e n t d e n s it ie s ) ,
E q . 1 - 8 0 and 1-81
c a n b e s e t eq u a l i f e v a lu a te d a t z = 0 , g iv in g
6 (E t - x )
& (E ? x )
v s
b z
H i
z = 0
S u b s titu tin g E q s . 1 -71
&z
^3
z = 0
and 1 -7 2 in to Eq 1 -8 2 le a d s to
& f(z)
bz
(1 - 8 2 )
M2
w
(1 -8 3 )
^ 3 ?3
M3
z = 0
w h ic h i s th e s e c o n d te r m o f E q . 1 - 7 8 .
T h e r e m a in in g t e r m s o f E q .
th e k
th
1 -7 8 a r e d e te r m in e d by ta k in g
d e r iv a tiv e o f E q . 1 - 7 3 , g iv in g a g e n e r a l fo r m
k
2) f(z)
5z
e>k~2 f(z )
(f)B
л
JU2 ? 2
'
]
bz
k -2
(1 - 8 4 )
k -2
E
(k -2 )!
?
p !(k -2 -p )!
5P a ( z )
?
6zp
f 2 ? Pf ( z )
---------
6z
?P
p=0
Now th a t fQ and f^ h a v e b e e n c a lc u la t e d , th e e l e c t r i c f ie ld th ro u g h н
o u t th e s e m ic o n d u c to r i s d e fin ed by E q s .
1 -7 2 and 1 - 7 5 .
To
33
c o m p u te th e t r a n s m is s io n and r e f le c t io n c o e f f i c i e n t s , th e fie ld
a c r o s s th e in t e r fa c e a t z = d m u s t b e m a tc h e d .
T h e r e f le c t io n and t r a n s m is s io n c o e f f ic ie n t s
R
and T ,
r e s p e c t iv e ly , a r e d e fin e d by
R
S
E sd
'
E ld
E td
(1 -8 5 )
and
Eto
( 1- 86)
E id
w here
E. .
id
i s th e in c id e n t e l e c t r i c f ie ld a t z = d w ith th e
w a fe r a b s e n t
E
sd
,
i s th e to ta l e l e c t r i c fie ld in s id e th e s e m ic o n d u c to r
at z = d
E
i s th e tr a n s m itte d e l e c t r i c f ie ld in th e w a v e g u id e
at z = 0 .
It s h o u ld b e n oted th a t in t h is d e r iv a tio n a ll th e e l e c t r i c f i e l d s
a r e a s s u m e d to r e m a in p a r a lle l to x .
M a tc h in g th e H f ie ld s a t
z = d and u s in g M a x w e ll's e q u a tio n s , E q s . 1 - 7 2 , 1 - 8 5 ,
1 - 8 6 , th e
t r a n s m is s io n and r e lf e c t io n c o e f f ic ie n t s c a n b e c o m p u te d to b e
34
T
Mo
5f(z)
2 ( f ) - ? 2V .
&z
f(d )
+
z = d
(1 -8 7 )
and
R
//n \*
2
f^ ((-b -) - ? * V i)
/ / 7T \
f(d) ( y
2
\ \
^1
- M^
SfQO
5z
M-,
Ml
z = d
^ f(z )
5z
z = d
(1 - 8 8 )
S in c e f(z ) a s e x p r e s s e d in E q . 1 -7 5 i s n o t in c lo s e d f o r m ,
E q s.
1 -8 7 and 1 -8 8 m u s t b e a p p r o x im a te d , s u c h th at
1
T
(1 - 8 9 )
M1 (f (nh) - f (n -1 )h j
f(n h )
and
^
Mi
M2
R
f(n h )^
2 . w2M iC i^ +
f(nh) - f ( ( n - 1 ) h )
h
fjjn h l -
(1 - 9 0 )
35
w h e r e th e
f(nh) - f ( ( n - 1 ) h )
h
E.
(1 - 9 1 )
R ev iew
P r e s e n t e d in th is s e c t io n i s a r e v ie w (th ro u g h r e f e r e n c e )
o f th e m a jo r e q u a tio n s u se d th ro u g h o u t t h is c h a p t e r .
F o r c o m p le te
d e t a ils p le a s e r e f e r to th e a p p r o p r ia te s e c t io n , f o r th is i s in ten d н
ed o n ly a s a s u m m a r y .
T o b e g in th e c a lc u la t io n s , T Q, th e o p t ic a l t r a n s m is s io n
c o e f f ic ie n t and R e (S Q) , th e in c id e n t o p tic a l in t e n s it y a r e n e e d e d .
T h e s e q u a n titie s a r e u s u a lly o b ta in e d e x p e r im e n t a lly . T o c h a r a c t e r н
i z e th e m a t e r ia l, e "
of Eq.
E q s . 1 - 7 th rou gh 1 - 1 3
1 - 8 m u s t b e d e t e r m in e d .
U sin g
?" i s co m p u te d b y t r ia l and e r r o r .
T h ro u g h o u t th e r e s t o f th e c a lc u la tio n s ? " i s u s e d in d ir e c t ly
th rou gh E q s . 1 - 8 and 1 -1 3 le a d in g to u s e fu l v a lu e s f o r
and yS^*
T h e e le c t r o n - h o le p a ir c o n c e n tr a tio n th ro u g h o u t th e s e m i н
c o n d u c to r , a ft e r th e lig h t p u ls e , i s g iv e n by E q . 1 - 2 3 .
n e e d e d c o e f f ic ie n t s ,
E q s. 1 -2 0 ,
1 -2 1 ,
| E^ | 2 ,
and
| E^ | 2 , and
R Q a r e g iv e n
The
by
1 -1 8 , r e s p e c t iv e l y .
C om pu tin g th e c a r r i e r c o n c e n tr a tio n a s a fu n c tio n o f s p a c e
and tim e i s p e r fo r m e d in tw o s t e p s .
F i r s t , an in itia l g u e s s o f
36
a s p a c e - t im e m a t r ix i s n e e d e d .
m a k in g a g u e s s .
T h e r e i s no b e s t m eth o d o f
U s u a lly a s im p le t h e o r e t ic a l m o d e l i s s u f f ic ie n t ,'
h o w e v e r , a p o o r g u e s s m a y s u b s ta n tia lly in c r e a s e co m p u ta tio n
t im e .
T h e in itia l g u e s s u s e d in th is w o r k i s d e s c r ib e d in th e
c a lc u la t io n s p r e s e n t e d in C h a p ter 2 - C .
th e im p le m e n ta tio n o f E q s .
1 -5 4 ,
1 -5 3 ,
T h e s e c o n d s t e p in v o lv e s
1 -4 9 ,
and
1 -5 0 .
T h e s e e q u a tio n s a r e u s e d to it e r a t e th ro u g h th e s p a c e - t im e m a tr ix
u n til a s o lu tio n i s o b ta in ed fo r th e e l e c t r o n - h o le p a ir c o n c e n tr a н
tio n w ith a t o le r a b le e r r o r .
N e x t, th e c o n d u c tiv ity o f th e s e m ic o n d u c to r is co m p u te d a s
a fu n c tio n o f s p a c e and t im e .
U tiliz in g E q s . 1 -5 7 th rou gh 1 -6 7
e a c h e le m e n t o f th e s p a c e - t im e m a t r ix i s c o n v e r te d fr o m c a r r i e r
c o n c e n tr a tio n to c o n d u c tiv ity .
C o n d u c tiv ity i s th e d om in an t
v a r ia b le in th e c o m p u ta tio n o f th e m ic r o w a v e t r a n s m is s io n and
r e f le c t io n c o e f f i c i e n t s .
C o m p u ta tio n o f th e m ic r o w a v e r e f le c t io n and t r a n s m is s io n
c o e f f ic ie n t s i s a c h ie v e d w ith th e im p le m e n ta tio n o f E q s . 1 -7 5 ,
1 -7 7 ,
1 -7 8 ,
1 -8 4 ,
and
1 -8 9 th ro u g h 1?9 1 .
E q u ation 1 -7 5 i s
th e e q u a tio n th at d e s c r ib e s th e n o r m a liz e d e l e c t r i c fie ld in th e
s e m ic o n d u c t o r .
T o s t a r t th e ite r a tio n p r o c e s s E q s .
1 -7 7 ,
1 -7 8 ,
and 1 - 8 4 a r e u s e d in a c o m b in e d fa s h io n w h e r e th e p a r tia l d e r i v it iv e in E q . 1 - 8 4 i s co m p u te d n u m e r ic a lly fr o m th e s p a c e - t im e
37
m a t r ix .
A fte r E q . 1 -7 5 i s ite r a te d th ro u g h th e s a m p le , E q s .
1 - 8 9 th rou gh 1-91 a r e u s e d to m a tc h th e b o u n d a ry c o n d itio n s and
c o m p u te th e m ic r o w a v e t r a n s m is s io n and r e f le c t io n c o e f f i c i e n t s .
S a m p le c a lc u la tio n o f th e a b o v e e q u a tio n s a r e illu s t r a t e d
in C h a p ter 2 , S e c t io n C .
A l s o , c o m p a r is o n o f e x p e r im e n ta l and
th e o r e t ic a l r e s u lt s i s p r e s e n t e d .
CHAPTER
E X P E R IM E N T A L
A.
II
R ESU LTS
D e s c r ip tio n o f A p p a ra tu s
T h e a p p a r a tu s ca n b e d iv id e d in to th r e e s e c t io n s ; o p t ic a l,
m ic r o w a v e , and d e te c tio n .
T h e in itia l s t im u lu s i s a lig h t p u ls e
w ith a w a v e le n g th o f 1 .0 6 m i c r o m e t e r s , g e n e r a te d b y a Q s w itc h e d n e o d im iu m VAG l a s e r .
A n o p tic a l s y s t e m
is u sed
to
c o u p le th e lig h t in to a m ic r o w a v e w a v e g u id e , th u s illu m in a tin g
th e s e m ic o n d u c to r s a m p le a s illu s t r a t e d in F i g . 2 . 1 .
A run o f
x -b a n d w a v e g u id e i s u s e d to su p p o r t p r o p a g a tio n o f a 10 g ig a н
h e r tz c o n tin u o u s w a v e (CW ) m ic r o w a v e w h ic h i s g e n e r a te d by a
k ly s tr o n o s c i l l a t o r , and gu id ed to th e s e m ic o n d u c to r s a m p le .
F in a lly , tw o ty p e s o f d e t e c t o r s a r e u s e d , o n e fo r lig h t p u ls e s
and o n e f o r th e tr a n s m itte d m ic r o w a v e .
N ote th at o n ly o n e o f
th e d e t e c t o r s ca n b e in o p e r a tio n a t a n y g iv e n t im e .
T h e output
o f e a c h d e te c to r i s c o u p le d in to a d ig i t i z e r and th en fed in to
g r a p h ic s d is p la y and c o p ie r .
P r e s e n t e d n e x t , i s a d e ta ile d
d e s c r ip t io n o f th e a p p a ra tu s and s o m e o f th e m e th o d s u s e d f o r
a lig n m e n t and tu n in g .
38
a
LASER PULSE
MICROWAVE
DETECTOR
OPTICS
LASER
OPTICAL DETECTOR
MICROWAVE
WAVEGUIDE
TEXTRONICS
DIGITIZER &
MICROWAVE
SYSTEM
F ig u r e 2 .1
:
B lo c k d ia g r a m illu s t r a t in g th e e x p e r im e n ta l a p p a r a tu s .
GRAPHIC DISPLAY
T h e o p tic a l s y s t e m
c o n s is t e d o f
1)
Q?s w itc h e d n eo d in iu m YAG l a s e r
2)
o n e - q u a r t e r 04) to tw o (2 ) in ch b ea m
expand er
3)
o p t ic a lly fla t m ir r o r
4)
c o n v e x le n s
5)
o p tic a l b e n c h e s , m o u n ts , f i l t e r s , and
o th e r h a r d w a r e .
T h e o p tic a l p o r tio n o f th e a p p a ra tu s i s sh o w n in F ig u r e 2 . 2 .
T h e l a s e r w a s p u lse d o n c e e v e r y s e c o n d , p ro d u cin g a o n e q u a r te r 04) in c h b e a m o f lig h t fo r a d u ra tio n o f 2 0 n a n o s e c o n d s ,
h a v in g a p p r o x im a te ly 3 0 m illij o u le s o f e n e r g y a t a w a v e le n g th o f
1 .0 6 m ic r o m e t e r s .
F i l t e r s w e r e p la c e d in th e b e a m 's path to
r e d u c e it s to ta l e n e r g y c o n te n t.
T o d is tr ib u te th e b e a m 's e n e r g y
s o th at it w o u ld not h a rm c e r ta in p ie c e s o f th e o p t i c s , th e b e a m
w a s exp an d ed to tw o in c h e s in d ia m e t e r w ith a b ea m e x p a n d e r . A
m ir r o r w a s u s e d f o r c o n v e n ie n c e in th e ta b le s e t - u p to r e f le c t
th e l a s e r b e a m th ro u g h a o n e t h ir t y - s e c o n d in ch h o le a t a 90
d e g r e e bend in th e w a v e g u id e , th u s illu m in a tin g th e s e m ic o n d u c to r
s a m p le .
T h e fo c a l le n g th o f th e le n s w a s c h o s e n to illu m in a te
th e e n t ir e s e m ic o n d u c to r s a m p le .
41
SEMICONDUCTOR
SAMPLE
CONVEX
LENS
MIRROR
u
1/32 INCH HOLE
IN WAVEGUIDE
BEAM EXPAND
WAVEGUIDE
FILTERS
CP
i
LASER
?
F ig u r e 2 . 2
:
Y
O p tic a l p o r tio n o f th e e x p e r im e n ta l a p p a r a tu s .
42
T h e m ic r o w a v e a p p a r a tu s , u s e d to m e a s u r e th e r e f le c t io n
and t r a n s m is s io n c o e f f ic ie n t s , i s sh ow n in F ig u r e 2 . 3 .
A n x -b a n d
(8 -1 1 G ig a h e r tz ) s ig n a l w a s g e n e r a te d by a k ly s tr o n o s c i l l a t o r and
c o u p le d in to th e w a v e g u id e and th en gu id ed th ro u g h a c a v it y - t y p e
fr e q u e n c y m e t e r , an a tte n u a to r and a u n ife ed lin e to e n te r a
c a v ity fo r m e d by EH tu n e r N o . 1 and th e s a m p le . T h e uni fe e d
lin e w a s u s e d to p r e v e n t an y e n e r g y fr o m c o u p lin g o u t th e c a v ity
in to the w a v e g u id e a p p ro a c h in g th e k ly s tr o n .
S in c e u n ife e d lin e s
a r e not an id e a l im p e d a n c e m a tch to w a v e g u id e s ,
EH tu n e r N o . 1
w a s tuned to e lim in a t e an y sta n d in g w a v e s in th e c a v it y .
The
d ir e c t io n a l c o u p le r d ir e c t s r e f le c t e d e n e r g y fr o m th e s a m p le to
th e R F d e te c to r a t p oin t B w h e r e EH tu n e r N o . 2 i s u s e d to
m a x im iz e th e c o u p lin g .
T w o ty p e s o f d e t e c t o r s w e r e p la c e d a t p oin t
A .
An
o p tic a l d e t e c t o r w a s p o s itio n e d to c a p tu r e a ll o f th e lig h t th a t
p a s s e d th rou gh th e w a v e g u id e s y s t e m s o the to ta l in c id e n t and
tr a n s m it te d lig h t e n e r g ie s c o u ld b e m e a s u r e d .
O n ce th e lig h t
e n e r g ie s w e r e d e t e r m in e d , th e o p tic a l d e t e c t o r w a s r e p la c e d by
a b r o a d -b a n d m ic r o w a v e d e te c to r to m e a s u r e th e t r a n s ie n t
r e s p o n s e o f th e tr a n s m it te d m ic r o w a v e e n e r g y w h en th e lig h t
p u ls e fr o m th e l a s e r illu m in a te d th e s e m ic o n d u c to r s a m p le . T h e
ou tp u ts o f th e d e t e c t o r s w e r e fed in to a d ig it iz e r w h ic h a llo w e d
/
LENS
EH TUNER #1
EH TUNER #2
KLYSTRON
CAVITY TYPE
FREQUENCY METER
ATTENUATOR
LASER
BEAM
SEMICONDUCTOR
/
SAMPLE
UNIFEED
LINE
MICROWAVE
DETECTOR
I
1
SHORT CIRCUIT
50fl LOAD
MICROWAVE DETECTOR
OPTICAL DETECTOR
F ig u r e 2 . 3 :
I llu s tr a tio n d e p ic tin g th e m ic r o w a v e a p p a r a tu s .
co
44
th e r e s u lt s to b e p r e s e n te d in a g r a p h ic s d is p la y and r e p r o d u c e d
on a c o p ie r s y s t e m .
B.
A p p a ra tu s C a lib r a tio n and D ata C o lle c tio n
B e fo r e any e x p e r im e n ts co u ld b e p e r fo r m e d , th e o p tic a l
sy ste m
had to be a lig n e d w ith th e m ic r o w a v e w a v e g u id e s o th at
th e s e m ic o n d u c to r s a m p le w o u ld b e p r o p e r ly illu m in a t e d .
The
f i r s t s te p tak en w a s to a lig n th e l a s e r c a v it y , th e c e n t e r o f th e
s a m p le and th e m ic r o w a v e w a v e g u id e s o th a t th e y w e r e on th e
s a m e p la n e .
T h e n , w h ile th e l a s e r w a s b e in g p u ls e d , a p h o sн
p h o r e s c e n t c a r d w a s u s e d to d e te r m in e th e d ir e c tio n o f th e b e a m .
T h e p h o s p h o r e s c e n t c a r d g lo w e d y e llo w w h en e x p o s e d to th e l a s e r
beam .
U sin g th e m i r r o r , th e b ea m w a s d ir e c te d to th e c o n v e x
le n s and f o c u s s e d th ro u g h a h o le in th e w a v e g u id e .
w a s a c c o m p lis h e d b y s lig h t l y m o v in g th e l e n s .
F in e tuning
The card w a s
a ls o u s e d to m a k e s u r e th at th e c r o s s - s e c t i o n o f th e w a v e g u id e
at the s a m p le -m o u n t w a s illu m in a te d .
O n ce a lig n m e n t w a s
a c h ie v e d v is u a lly , a pin d io d e w a s m o v ed a c r o s s th e c r o s s s e c t io n a l a r e a o f th e s a m p le m ou n t to c h e c k fo r u n ifo rm
illu m in a t io n .
T o p r e p a r e th e m ic r o w a v e a p p a ra tu s fo r th e m e a s u r e m e n t
o f th e r e f le c tio n and t r a n s m is s i o n c o e f f ic i e n t s , th e fo llo w in g p r o н
c e d u r e w a s fo llo w e d :
1)
S e t th e k ly s t r o n o s c i l l a t o r a t 10 G ig a H z.
2)
P la c e a s lid in g s h o r t c ir c u i t a t p o in t A a s
sh o w n in F ig u r e 2 . 3 .
3)
A d ju st EH tu n e r N o . 2 u n til a p eak r e a d in g
is a c h ie v e d a t p o in t B .
4)
M ove th e s lid in g s h o r t c ir c u i t a t p o in t A
and r e c o r d th e m a x im u m and m in im u m
v a lu e s a t p o in t B .
A d ju st EH tu n e r N o . 1
to r e d u c e th e d if fe r e n c e b e tw e e n th e m a x im u m
and m in im u m .
5)
R e p e a t s t e p s 3 and 4 u n til th e m a x im u m
and m in im u m r e a d in g s a r e w ith in
.3 db o f
e a c h o th e r .
O n ce th e a p p a ra tu s w a s c a lib r a t e d , th e in c id e n t and t r a n s н
m itte d o p tic a l e n e r g ie s a lo n g w ith th e c o r r e s p o n d in g m ic r o w a v e
t r a n s m is s io n and r e f le c tio n c o e f f ic ie n t s c o u ld b e m e a s u r e d . T h e
in c id e n t o p tic a l e n e r g y w a s m e a s u r e d b y p la c in g an o p tic a l
d e te c to r a t p oin t A ,s h o w n in F ig u r e 2 . 3 , w h ile th e l a s e r w a s
p u lsin g a t a r a te o f o n e p u ls e p e r s e c o n d .
T h e output o f th e
d e te c to r w a s c o u p le d in to a d ig it iz e r w h e r e th e s ig n a l w a s c o n н
v e r te d to a d ig ita l fo r m w h ic h c o u ld th en b e p r o c e s s e d by a
g r a p h ic s s y s t e m .
T h e g r a p h ic s s y s t e m
in te g r a te d th e p u ls e to
46
g iv e th e c o r r e s p o n d in g e n e r g y o f th e p u ls e .
T h e tr a n s m itte d
e n e r g y m e a s u r e m e n t w a s p e r fo r m e d m u ch th e s a m e w a y , e x c e p t
th at th e s a m p le w a s p la c e d b e tw e e n th e o p tic a l d e te c to r and th e
w a v e g u id e a s illu s t r a t e d in F ig u r e 2 . 3 .
A lth o u g h th e a p p a r a tu s w a s s e t to m e a s u r e th e m ic r o w a v e
t r a n s m is s io n and r e f le c t io n c o e f f ic i e n t s , c o n s id e r a b le tr o u b le w a s
e n c o u n te r e d m e a s u r in g th e l a t t e r .
S in c e th e d ir e c t io n a l c o u p le r
a tten u a te d th e s ig n a l 10 d b , and th e b r o a d -b a n d d e te c to r w a s not
v e r y s e n s i t i v e , th e m e a s u r e d s ig n a l w a s w e ll in to th e r a n g e o f
th e b a ck g ro u n d n o i s e .
D ue to t h e s e ty p e s o f p r o b le m s , m o s t o f
th e e f f o r t s w e r e c o n c e n tr a te d on m e a s u r in g th e t r a n s m is s i o n
c o e f f ic ie n t .
A l s o , it w a s found to b e m o r e f e a s ib le to m e a s u r e
th e c h a n g e in tr a n s m it te d f ie ld s tr e n g th in s te a d o f th e t r a n s н
m is s io n c o e f f ic ie n t .
E ith e r m e a s u r e d q u a n tity ca n b e c o r r e la t e d
to th e t h e o r e t ic a l c a lc u la t io n .
C.
P r e s e n t a tio n o f C o lle c t e d D a ta and C o r r e la tio n w ith
T h e o r e t ic a l R e s u lts
A s in g le s i l i c o n w a f e r w a s u se d a s a s a m p le to e v a lu a te
th e v a lid ity o f th e t h e o r e t ic a l d e r iv a tio n d is c u s s e d in C h a p ter I.
T h e s a m p le had a t h ic k n e s s o f 2 8 5 m ic r o m e t e r s and w a s p o lis h e d
on o n e s u r f a c e w h ile b e in g la p p ed on th e o t h e r .
w a s in t r in s ic a t r o o m t e m p e r a t u r e .
A l s o , th e s a m p le
E x p e r im e n ta lly , fo u r c u r v e s
47
w e r e g e n e r a te d illu s t r a t in g th e m ic r o w a v e t r a n s m is s io n c h a r a c t e r н
i s t i c s a s a fu n c tio n o f t im e a t d if fe r e n t in c id e n t lig h t e n e r g i e s .
E ig h t c o m p u ta tio n a l s t e p s d e s c r ib e th e n u m e r ic a l c a lc u la н
tio n s a s sh o w n in F ig u r e 2 . 4 .
S t e p 1 c a lc u la t e s the in c id e n t
lig h t e n e r g y p e r m e t e r s q u a r e d p e r p u ls e w h ic h i s g iv e n b y
R e [ S Q]
w here
г
i
=
(2
A
i s th e in c id e n t e n e r g y and A
in a te d s u r f a c e .
T a b le 2 .1
s
i s th e a r e a o f th e i llu m -
l i s t s th e r e s u lt s o f E q . 2 - 1 .
T a b le
2 .1
R e s u lts o f S t e p 1
M easu rem en t
N u m b er
R e[ S ]
o
As
( j o u le s )
(m e te r )
2
jo u le s (m e te r )2/ p u l s e
1
7 . 1 5 x 1 CT3
2 . 8 8 x 10"4
3 .9 9
2
3 . 0 4 x 10 - 3
2 .8 8 x 10 ? 4
1 .0 6 x 1 0 1
3
1 . 7 2 x 10- 2
2 . 8 8 x 10 ? 4
3 .8 9 x 101
4
4 . 9 9 x 10~ 2
2 .8 8 x 10 ? 4
1 .7 3 x 102
S t e p 2 d iv id e s th e tr a n s m itte d e n e r g y
energy
г
b y th e in c id e n t
г ? to g e t th e o p tic a l t r a n s m is s io n c o e f f ic ie n t T Q ;
48
C o m p u ta tio n o f th e in c id e n t
o p tic a l e n e r g y p e r ( m e t e r ) 2
p e r p u ls e
F ig u r e 2 .4 :
^
C o m p u ta tio n o f th e o p tic a l t r a n s m is s io n c o e f f ic ie n t T q
S tep
2
C om p u ta tio n o f th e a b s o r p tio n
c o e f f ic ie n t
S tep
3
C o m p u ta tio n o f th e o p tic a l
r e f le c t io n c o e f f ic ie n t R q
S tep
4
C om p u ta tio n o f th e m a g n itu d e s o f
th e e l e c t r i c f ie ld s sq u a r e d
S tep
5
C om p u ta tio n o f th e f i r s t q u e s t f o r
th e s p a c e - t i m e m a t r ix
S tep
6
G e n e r a liz e d c o m p u ta tio n o f th e
s p a c e - t im e m a tr ix
S tep
7
C om p u ta tio n o f th e m ic r o w a v e
t r a n s m is s io n c o e f f ic ie n t
S tep
8
B lo c k d ia g r a m d e s c r ib in g th e c o m p u ta tio n a l
s t e p s u s e d in th e n u m e r ic a l a n a ly s i s .
49
T0
=
лt /
|,
(2 -2 )
T h e r e s u lt s a r e ta b u la te d in T a b le 2 . 2 .
T a b le
2 .2
R e s u lts o f S t e p 2
M easu rem en t
г^
j o u le s
N u m b er
jo u le s
u n it le s s
1
1 . 1 5 x 10 ' 3
1 .2 4 x 10~4
.1 0 8
2
3 . 0 4 x 10"3
3 . 0 7 x 10"4
.101
3
1 . 1 2 x 10 - 2
1 .0 9 x 10"3
.0 9 7
4
4 .9 9 x 10-2
1 .4 0 x 10~2
.281
S t e p 3 im p le m e n ts E q s .
v a lu e s f o r
agree,
To
?"
1 -7 th ro u g h 1 - 1 3 .
B y g u e s s in g
u n til th e co m p u te d and m e a s u r e d v a lu e s
fo r T
o
ca n b e d e te r m in e d , w h ic h g iv e s th e a b so r p tio n c h a r -
a c t e r i s t i c s fo r th e s i l i c o n - w a f e r .
T a b le 2 . 3 g iv e s th e v a lu e s o1
th e c o n s ta n ts u s e d and th e r e s u l t s .
S t e p s 4 and 5 c o m p u te th e o p tic a l r e f le c t io n c o e f f ic ie n t
R q and th e m a g n itu d e s o f th e e l e c t r i c f ie ld s q u a r e d
| E ? | 2 , r e s p e c t iv e ly .
i "t 12
|
|
and
T a b le 2 . 3 l i s t s th e c o n s ta n ts u s e d in
t h e s e e q u a tio n s w h ile T a b le 2 . 4 g iv e s th e r e s u l t s .
50
T a b le 2 . 3
R e s u lts o f S t e p 3 and a L is t o f C o n sta n ts
U se d T h ro u g h o u t th e C a lc u la t io n s .
V a lu e
C o n sta n ts
(se c ) 1
1 .7 7 7 x 1 0 15
CO
S
U n its
8 .8 5
X
1 0 " 12
f a r a d /m e t e r
1 .2 6
X
1 0 ?6
h e n r y /m e t e r
1 .0 4
X
1 0 ' 10
f a r a d /m e t e r
?"
2
3 .7 0
X
10" 14
f a r a d /m e t e r
d
2 .8 5
X
1 0 -5
m e te r
2 .0 7
X
107
(m e te r ) 1
2 .9 7
X
102
( m e t e r ) -1
2 .8 1
X
10"1
u n it le s s
* e3
Mi , M2 , M3
e2
*2
tt2
T
o
T a b le 2 . 4
R e s u lts o f S t e p s 4 and 5
|r q
N u m b er
u n it le s s
( v o lt s /m e t e r )
1
.3 6 4
5 . 6 9 x 102
2
.3 5 8
1 . 5 2 x 103
3
.3 5 6
5 . 6 0 x 103
Is-
4
.5 4 3
2 . 0 8 x 1 04
CO
I
M easu rem en t
\ 4 \ s
| e2 |2
o
( v o lt s /m e t e r )
9 .7 3
2 . 2 4 x 101
2
o
X
Si?
o
CM
X
CD
CD
51
S t e p 6 is p e r fo r m e d in tw o p a r t s .
T h e f i r s t p a r t c o m p u te s
th e c a r r i e r c o n c e n tr a tio n a t t = 0 , o r im m e d ia te ly a ft e r th e lig h t
p u ls e h a s p a s s e d .
E q u ation 1 -2 3 d e s c r ib e s th e e le c t r o n - h o le p a ir s
c o n c e n tr a tio n w h ic h i s p lo tte d fo r an in c id e n t e n e r g y o f 2 . 8 4 x 10
_p
j o u le s (m e a s u r e m e n t n u m b e r 1) a s illu s t r a t e d in F ig u r e 2 . 5 .
P a r t 2 c o m p u te s an e s t im a t e o f th e e n tir e s p a c e - t im e m a t r ix .
T h e e q u a tio n u s e d i s a c lo s e d fo r m s o lu tio n o f E q . 1 - 4 8 w h ig h
a s s u m e s D a is a c o n s ta n t
which^ in g e n e r a l, i s not th e c a s e .
H o w e v e r , if o n e c o m p u te s th e c o n c e n tr a tio n a t s o m e s m a ll i n c r e н
m e n t o f tim e a ft e r t = 0 , A t ,
and th en r e c o m p u te s a new in itia l
b ou n d ary c o n d itio n , and u s e s an u p dated v a lu e Da , a v e r y good
e s t im a t e o f th e s p a c e - t i m e m a tr ix c a n be a c h ie v e d .
T h e fo r m o f
th e s o lu tio n u s e d i s
-o /r +
C ( x ,t )
m =0
(2 -3 )
w here
X
m
and
(2m +1 )17
2d
(2 -4 )
CL
CO
0.8
EI
QJ O ? ??-I?I/)
0 +-> s- n e.
ZC. as <U 0 . 0
S- 4->
1 -M <U
E
e C
ai?P u,
0.4
4-> O CM
U
OO
(U
rH
X
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1.0
Normalized Spacial Displacement Through Semiconductor
F ig u r e 2 . 5
:
E le c tr o n - h o le p a ir c o n c e n tr a tio n th ro u g h th e s i l i c o n w a fe r
a ft e r th e lig h t p u ls e p a s s e d .
ui
io
53
A
2
?
d
110
V/
_yd
\h * e
( " y c o s [X
1
~ 2 -- r p
y2 + A
|_ \
- rd
A s i n [ A m d ] ) + r) d
-
(-7 ;+ y +
j)
y)
m
d] +
\
( y c o s [ A m d] + A s in [A ^ d ])j
J
(2 -5 )
w h ile
2tt2 ,
(2-6)
+
╗?
=
2 tt2
. + .2
? 5--------- l E2 l
03 jU0 h
77"
=
i2
? ---------- | E2 |
03 M2 h
.
(2 -7 )
and
S t e p 7 im p le m e n ts E q s .
1-51
(2 -8 )
th ro u g h 1 -6 7 w h ic h , w h en ite r a te d
a s u f f ic ie n t n u m b e r o f t i m e s , c o n v e r g e s to a s e l f c o n s is t e n t s o lu н
tio n g iv in g a g e n e r a l s o lu tio n f o r th e s p a c e - t i m e m a t r ix . A p lo t
d e s c r ib in g
th e d e c a y p r o c e s s i s g iv e n in F ig u r e 2 . 6 fo r m e a s u r e н
m ent N o. 1 .
S tep 8 u tiliz e s E qs 1 - 7 5 ,
1-77,
1-78,
1 -8 4 ,
1 - 8 9 , and
1 - 9 0 w h ic h c o m p u te th e m ic r o w a v e t r a n s m is s io n c o e f f ic ie n t and
Carrier Concentration
23
21
19
17
15
0 .0
0 .1
0 .2
0 .3
0.4
0 .5
0 .6
0 .7
0 .8
0 .9
1.0
Normalized Displacement
F ig u r e 2 . 6 :
P lo t o f th e s p a c e - t im e m a tr ix w h ic h i s th e p lo t o f th e r e s u lt s
o f s te p 7 .
01
th e n o r m a liz e d tr a n s m it te d e n e r g y .
F ig u r e 2 . 7 c o n c lu d e s th e
r e s u lt s fo r m e a s u r e m e n t s 1 th ro u g h 4 and c o m p a r e s th em to the
e x p e r im e n ta l r e s u l t s . T o o b ta in s u c h a c l o s e m a tc h b e tw e e n
th e o r y and e x p e r im e n t s , th e l i f e - t i m e o f th e m a t e r ia l had to be
d e te r m in e d .
F i r s t , a s e r i e s o f c u r v e s w e r e g e n e r a te d w h ic h
illu s t r a t e d th e t r a n s m is s i o n c h a r a c t e r is t ic s a s a fu n c tio n o f tim e
f o r a ra n g e o f d if fe r e n t
T a s d is p la y e d in F ig u r e 2 . 8 .
N e x t, th e s lo p e w a s c o m p u te d fo r e a c h c u r v e a t th e .5
t r a n s m is s i o n
in F ig u r e 2 . 9 .
le v e l and p lo tte d a s a fu n c tio n o f T a s illu s t r a t e d
B y d e te r m in in g th e c o r r e s p o n d in g s lo p e fr o m th e
e x p e r im e n ta l r e s u l t s , a lif e t i m e o f 7 /is e c
w a s c o m p u te d fr o m
F ig u r e 2 . 9 and u s e d th ro u g h o u t th e c a lc u la t io n s .
T w o o th e r b ou n d a ry c o n d itio n s w e r e c o n s id e r e d d u rin g the
th e o r e t ic a l c a lc u la t io n .
T h e f i r s t b o u n d a ry c o n d itio n a s s u m e d
in fin ite s u r f a c e r e c o m b in a tio n v e lo c it y at both s u r f a c e s w h ic h is
a p p r o x im a te ly th e s itu a tio n in a s e m ic o n d u c to r w ith both s u r f a c e s
la p p e d .
F ig u r e s 2 . 1 0 and 2 . 1 1
illu s t r a t e th e r e s u l t s .
The
o th e r lim itin g b o u n d a ry c o n d itio n w a s z e r o s u r f a c e r e c o m b in a tio n
v e lo c i t y a t th e s u r f a c e s .
T h is c o n d itio n i s d iffic u lt to a c h ie v e in
p r a c t ic e but can b e a p p r o a c h e d w ith c a r e fu l p o lis h in g and c h e m ic a l
tr e a tm e n t o f th e s u r f a c e .
T h e r e s u l t s o f t h e s e c o m p u ta tio n s a r e
illu s t r a t e d in F ig u r e s 2 . 1 2 and 2 . 1 3 .
Normalized Transmitted
Microwave Energy
.0
Experimental
Theoretical
0.8
0.6
0 .4
.2
0.0
10
20
30
40
50
60
70
80
90
100
Time (microseconds)
F ig u r e 2 . 7
:
C o m p a r iso n o f th e e x p e r im e n ta l and th e o r e tic a l r e s u lt s
.
oi
O)
Normalized Transmitted
Microwave Energy
0.8
100
0. 6
0 .4
0.0
10
20
30
40
50
60
70
80
90
100
Time (microseconds)
F ig u r e 2 . 8 :
P lo t o f n o r m a liz e d tr a n s m itte d m ic r o w a v e e n e r g y v e r s u s tim e
w ith in fin ite s u r f a c e r e c o m b in a tio n v e lo c it y on o n e s id e o f
a
s ilic o n w a fe r and z e r o s u r f a c e r e c o m b in a tio n v e lo c it y on th e o th e r .
01
Slope of Normalized
Transmitted Energy at
Half Power
?n
c
IQ
?5
(D
o
IV )
INI
CD
<c
3
(0
3
IQ
u
<<
3
0)
?5
X
CO
to
0
c
(D
O
C
-3
<
(0
to
3
O
if
o
13
O
-h
if
co
c
c
(0
O
-3
ro
n
o
3
If
CO
If
(D
a
r-*
3
?5
O
m
tn
CL
in
n
(Q
C
3
(D
IO
co
00
0>
3"
fl)
01
8S
o
Normalized Transmitted
Microwave Energy
0.8
0 .4
0.0
10
20
30
40
50
60
70
80
90
100
Time (microseconds)
F ig u r e 2 . 1 0 :
P lo t o f n o r m a liz e d tr a n s m itte d m ic r o w a v e e n e r g y v e r s u s
tim e w ith in fin ite s u r fa c e r e c o m b in a tio n v e lo c it y on both
s u r f a c e s o f a s ilic o n w a f e r .
01
(0
Slope of Normalized
Transmitted Energy at
Half Power
?
n
H*
IQ
C
3
0)
ro
o
ro
(t>
cn
(0
o
?Q
3
3
IQ
<
3
p>
3
X
P)
(fl
p)
?
3
nr t
o
3
ft
p>
(D
a
ro
5
(D
O
c
3
fl>
u>
to
c
u
)
r+
3
ft
<t>
a
c
O
-s
o
cn
0)
o
o
3
Q.
10
CTl
to
to
c
3
to
IO
o
p>
T
(D
0)
09
o
Normalized Transmitted
Microwave Energy
.0
0.8
0.6
0 .4
0.2
0.0
10
20
30
40
50
60
70
80
90
100
Time (microseconds)
F ig u r e 2 . 1 2 :
P lo t o f n o r m a liz e d tr a n s m itte d m ic r o w a v e e n e r g y v e r s u s
tim e w ith z e r o s u r f a c e r e c o m b in a tio n v e lo c i t y on both
s u r f a c e s o f a s i lic o n w a f e r .
0)
Slope of Normalized
Transmitted Energy at
Half Power
0
0.8
0. 6
0 .4
0.2
0.0
10
20
30
40
50
60
100
x (microseconds)
F ig u r e 2 . 1 3 :
S lo p e o f th e c u r v e s illu s t r a t e d in F ig u r e 2 . 1 2 a t t he .
e n e r g y m a r k a s a fu n c tio n o f ta u .
CH APTER
D E V IC E
III
A P P L IC A T IO N S
T w o p o s s ib le a p p lic a tio n s b e c o m e a p p a r e n t d u rin g th e
e x p e r im e n ta l p h a se o f t h is d is s e r t a t io n .
O ne d e v ic e c o u ld b e
u s e d a s a h ig h -p o w e r o p tic a l d e t e c t o r , m e a s u r in g th e to ta l e n e r g y
and th e s h a p e o f an in c id e n t lig h t p u ls e .
T h e o th e r a p p lic a tio n
d e a ls w ith an e x is t in g p e r io d ic a r r a y w h ic h i s tr a n s p a r e n t to a
n a r r o w ra n g e o f m ic r o w a v e f r e q u e n c ie s .
B y in c o r p o r a tin g a
s e m ic o n d u c to r d e v ic e w ith th is ty p e o f a r r a y , e v e n th e fr e q u e n c ie s
w h ic h n o r m a lly p a s s th ro u g h th e d e v ic e c a n b e sh u t o ff fo r a
s h o r t p e r io d o f t i m e .
S e c t io n A o f th is c h a p te r w i l l d e a l w ith
th e o p tic a l d e te c to r le a v in g th e p e r io d ic a r r a y s t r u c t u r e to
s e c t io n B .
A.
O p tica l D e te c to r
M any o p tic a l d e t e c t o r s a r e p la g u ed w ith th e in a b ility to
h an d le la r g e a m o u n ts o f p o w e r w ith o u t in c u r r in g p e r m a n e n t
d am age.
B y p la c in g a s e m ic o n d u c to r s la b in a w a v e g u id e and
u s in g m ic r o w a v e s to d e te r m in e th e n u m b er o f f r e e c a r r i e r s
63
64
w h ic h , in tu r n , i s r e la te d to th e in c id e n t o p tic a l e n e r g y , o p tic a l
d e te c tio n c a n b e a c h ie v e d .
R e s p o n s e tim e and s e n s it iv it y a r e
th e tw o p a r a m e te r s th a t m u s t be c o n s id e r e d .
If th e w a fe r th ic k н
n e s s and th e d e v ic e b u lk lif e t im e a r e la r g e , th e s e n s it iv it y w i l l
be h i g h .
T h e r e w i l l a ls o b e a f a s t r i s e t i m e .
H o w e v e r , s in c e
th e d o m in a n t d e c a y p r o c e s s fo r the f r e e c a r r i e r s is s u r f a c e
r e c o m b in a tio n , th e c a r r i e r s co u ld ta k e te n s o f m illis e c o n d s to
r e tu r n to t h e ir in itia l s t a t e s .
B y r ed u c in g the w a fe r t h ic k n e s s ,
th e n u m b er o f f r e e c a r r i e r s p e r s q u a r e c e n t im e t e r d e c r e a s e s ,
th u s d e c r e a s in g th e s e n s i t i v i t y .
W ith th e r ed u c tio n in w a fe r
t h ic k n e s s , th e s p e e d o f th e d e v ic e i n c r e a s e s .
U sin g t h e s e id e a s , a la r g e a r e a d e te c to r ( 1 x 2
cm ) w as
fa b r ic a te d u sin g a s il i c o n on s a p p h ir e w a f e r .
T h is ty p e o f w a fe r
w a s u s e d to a c h ie v e a v e r y th in a c tiv e la y e r .
T h e s il ic o n th ic k н
n e s s , in th is c a s e , w a s a p p r o x im a te ly o n e m ic r o m e t e r , and th e
s a p p h ir e w a s u s e d o n ly a s a c a r r i e r .
B y illu m in a tin g th e w a fe r
w ith a 2 0 n a n o se c o n d p u ls e o f lig h t, a s sh o w n in F ig u r e 3 . 1 ,
a
r e s p o n s e c o m p a r a b le to th a t o f a known d e te c to r w a s o b ta in ed a s
sh ow n in F ig u r e 3 . 2 .
It w a s a ls o found th at c r e a tin g a p la s m a
b y fo c u s s in g e x c e s s i v e o p tic a l p o w er on th e w a fe r did not ch a n g e
th e r e s u l t s .
Normalized Light
Intensity
0.8
0.6
0.4
0.2
0.0
10
20
30
40
50
60
70
80
Time (microseconds)
F ig u r e 3 .1
:
T h e r e s p o n s e o f an e x is t e n t o p tic a l d e t e c t o r .
90
100
Normalized Microwave
Intensity
0.8
0.6
0.4
0. 2
0.0
10
20
30
40
50
60
70
80
90
Time (microseconds)
F ig u r e 3 . 2
:
T h e R F r e s p o n s e o f th e s ili c o n on s a p p h ir e w a f e r .
100
67
If th e e n e r g y o f an
in c o m in g o p tic a l p u ls e i s th e o n ly
p a r a m e te r to b e m e a s u r e d , an in t r i n s ic w a fe r on th e o r d e r o f
2 5 0 m ic r o m e t e r s th ic k w o u ld g iv e g o o d s e n s it iv it y w ith a b u ilt - in
s t o r a g e m e c h a n is m .
A lth ough th e o p tic a l p u ls e m ig h t b e in th e
n a n o se c o n d r a n g e in d u r a tio n , th e e le c t r o n - h o le p a ir s th at a r e
g e n e r a te d fr o m th e in c id e n t p h o to n s o c c u r in s ta n ta n e o u s ly .
O n ce
th e e le c t r o n - h o le p a ir s a r e g e n e r a t e d , th e p u ls e w i l l r e q u ir e te n s
o f m ic r o s e c o n d s to d e c a y .
T h e r e f o r e , s lo w and in e x p e n s iv e
e le c t r o n ic s c a n b e u se d to m a k e th e u ltim a te m e a s u r e m e n t .
F ig u r e 3 . 2 il l u s t r a t e s th e r e la tio n s h ip o f the in c id e n t o p tic a l p o w er
and an e a s i l y m e a s u r e d p a r a m e te r o f th e tr a n s m itte d m ic r o w a v e
energy.
T o im p le m e n t th e o p tic a l m e a s u r e m e n t s d is c u s s e d s o fa r
in a p r a c tic a l f a s h io n , a d iffe r e n t m ic r o w a v e c o n fig u r a tio n w o u ld
h a v e to b e d e v e lo p e d .
T h e c o n fig u r a tio n w o u ld h a v e to e x p o s e
m u ch o f th e s i l i c o n d ir e c t ly to th e in c id e n t o p tic a l e n e r g y
in s te a d o f d ir e c t in g lig h t th ro u g h a pin h o le .
T h e s e d e v e lo p m e n ts
a r e le f t to fu tu r e s t u d i e s .
B.
C o n tr o l o f M ic r o w a v e T r a n s m is s io n
T h rou gh a P e r io d ic S u r f a c e
P e r io d ic s t r u c t u r e s h a v e b e e n u s e d to t r a n s m it a n a r r o w
band o f m ic r o w a v e fr e q u e n c ie s and to r e f le c t a ll o t h e r s .
If
in t r in s ic s i l i c o n i s p la c e d o v e r th e p e r io d ic s tr u c tu r e a s sh o w n in
68
F ig u r e 3 - 3 , lig h t p u ls e s c a n b e u s e d to g e n e r a t e e le c t r o n - h o le
p a ir s , th u s p r e v e n tin g m ic r o w a v e s fr o m p a s s in g th ro u g h th e
s tr u c tu r e fo r a s h o r t p e r io d o f t i m e , e v e n a t the b a n d p a s s .
E > p e r im e n ts w e r e p e r fo r m e d to d e te r m in e w h e th e r th e s ili c o n
w ou ld a f f e c t th e p r o p e r tie s o f th e p e r io d ic s t r u c t u r e .
F ir s t, a
p e r io d ic s tr u c tu r e w a s m o u n ted in w a v e g u id e and th en th e
tr a n s m itte d m ic r o w a v e e n e r g y w a s r e c o r d e d w h ile v a r y in g th e
in c id e n t m ic r o w a v e f r e q u e n c y .
N e x t, a low doped s i lic o n w a fe r
w a s p la c e d n e x t to th e p e r io d ic s t r u c t u r e w h ile th e e x p e r im e n ts
w e r e rep ea ted .
A s illu s t r a t e d in F ig u r e 3 . 4 ,
fa c to r w a s not a lt e r e d .
th e a tten u a tio n
H o w e v e r , th e r e w a s a s h if t in th e bandн
p a s s f r e q u e n c ie s , w h ic h w a s e x p e c te d b e c a u s e o f the p e r m itt iv ity
o f th e s i l i c o n .
A lth ou gh o p tic a l e x c it a tio n i s n o t p r a c t ic a l, tw o im p o r ta n t
p o in ts h ave b e e n b r o u g h t fo r t h .
T h e f i r s t i s th a t in t r in s ic s il ic o n
w i l l not a ffe c t th e b a n d p a ss p r o p e r t ie s o f th e p e r io d ic s t r u c t u r e ,
e x c e p t fo r a s lig h t s h if t in s p e c t r u m w h ic h c a n b e c o m p e n s a te d
b y ch a n g in g th e p h y s ic a l d im e n s io n s o f th e s t r u c t u r e .
S e c o n d ly ,
en ou gh e le c t r o n - h o le p a ir s ca n b e g e n e r a te d to c u t o f f the m ic r o н
w a v e t r a n s m is s i o n .
B y d e v e lo p in g an e le c t r o n ic m e a n s o f
g e n e r a tin g the e le c t r o n - h o le p a i r s , a p r a c tic a l s w itc h co u ld b e
d e v e lo p e d .
S u c h a d e v ic e m a y m e r it fu r th e r s tu d y .
69
Waveguide
Wafer
S
7
F ig u r e 3 . 3
:
I llu s tr a tio n sh o w in g th e p e r io d ic a r r a y and
w a f e r m ou n ted in w a v e g u id e .
Amplitude
(decibels)
20
15
10
???? No Sample
- - Array Only
? Wafer and Array
5
0
3
8
9
10
11
12
Frequency (gigahertz)
F ig u r e 3 . 4 :
P lo ts illu s t r a t in g th e fr e q u e n c y r e s p o n s e o f no s a m p le ,
p e r io d ic a r r a y o n ly , and p e r io d ic a r r a y w ith s ilic o n
w a fe r .
o
CH APTER IV
SU M M A R Y
In th is d is s e r t a t i o n , a t h e o r e t ic a l m o d e l i s d e v e lo p e d
d e s c r ib in g
th e m ic r o w a v e t r a n s m is s io n and r e f le c t io n p r o p e r tie s
o f s e m ic o n d u c to r w a f e r s a ft e r th e y h a v e b e e n illu m in a te d w ith a
s h o r t p u lse o f lig h t fr o m a l a s e r .
T h e p u r p o se o f th is w o r k i s
to d e v e lo p a w o r k a b le m o d e l w ith a m in im u m o f unknow n v a r ia b le s
w h ile o b ta in in g r e a s o n a b le c o r r e la t io n w ith an a s s o c ia t e d e x p e r iн
m e n t.
T h e m o d e l h a s th r e e fu n d a m en ta l p a r t s .
T he f ir s t part
d e s c r ib e s th e a b s o r p tio n m e c h a n is m o f th e in c id e n t lig h t p u ls e
th ro u g h o u t the s e m ic o n d u c t o r , g iv in g th e e le c t r o n - h o le p a ir c o n н
c e n tr a tio n a s a fu n c tio n o f p o s itio n .
In th e s e c o n d p a r t th e e l e c t r o n -
h o le p a ir c o n c e n tr a tio n d u rin g th e d e c a y p r o c e s s i s d e s c r ib e d .
P a r t th r e e d e s c r ib e s th e m ic r o w a v e p r o p e r tie s o f th e
s e m ic o n н
d u c to r a s th e e le c t r o n - h o le p a ir s r e tu r n to t h e ir e q u ilib r iu m
sta te .
*
An e x p e r im e n t w a s d e v e lo p e d to t e s t th e t h e o r e t ic a l m o d e l.
A l a s e r g e n e r a tin g a 2 0 n a n o -s e c o n d lig h t p u ls e illu m in a te d a
71
s i l i c o n w a f e r c la m p e d b e tw e e n tw o f la n g e s in a w a v e g u id e .
The
lig h t p u ls e w a s d ir e c t e d and f o c u s e d th ro u g h a p in h o le a t a bend
in th e w a v e g u id e s o a s to m in im iz e m ic r o w a v e le a k a g e .
The
m ic r o w a v e e n e r g y tr a n s m itte d th rou gh th e s ili c o n w a fe r w a s
m e a s u r e d w ith r e s p e c t to tim e and c o r r e la t e d w ith th e r e s u lt s
fr o m th e t h e o r e t ic a l m o d e l.
A.
T heory
S e v e r a l p lo ts w e r e p ro d u ced illu s t r a t in g th e r e s u lt s o f th e
t h e o r e t ic a l in v e s t ig a t io n .
F ig u r e s 2 . 8 ,
2 .1 0 ,
and
2 .1 2
d e s c r ib e th e m ic r o w a v e r e s p o n s e o f a s il ic o n w a f e r f o r th r e e
d if fe r e n t b ou n d a ry c o n d itio n s a ft e r it w a s illu m in a te d w ith a
s h o r t p u ls e o f lig h t .
E a ch c u r v e on th e r e s p e c t iv e p lo ts c o r r e s н
pon d s to a d e v ic e w ith a d if fe r e n t l i f e t i m e .
A s e x p e c t e d , it w a s
found th a t a s th e s u r f a c e r e c o m b in a tio n v e lo c it y w a s i n c r e a s e d ,
th e tim e n e e d e d f o r th e s ilic o n to r e tu r n to s te a d y s t a t e d e c r e a s e d .
A ls o , a s th e b u lk l i f e - t i m e in c r e a s e d , th e s u r f a c e c o n d itio n s
b e c a m e d o m in a n t in th e d e c a y p r o c e s s .
2 . 1 3 c o r r e s p o n d in g to F ig u r e s 2 . 8 ,
F ig u r e s 2 . 9 ,
2 .1 0 ,
and
2 .1 2 ,
2 . 1 1 , and
r e s p e c t iv e н
l y , illu s t r a t e th e s lo p e o f th e tr a n s m itte d m ic r o w a v e in te n s ity a t
th e 5 0 p e r c e n t le v e l a s a fu n c tio n o f ta u , th e bu lk l if e t i m e .
w a s found th a t e a c h o f th e c u r v e s f it s the e q u a tio n
It
73
(4 -1 )
K
S
r e m a r k a b ly w e l l , w h e r e :
S
i s th e s lo p e
K
is s o m e c o n s ta n t
T,
B
i s th e b u lk lif e t im e
T
S
is th e e f f e c t iv e s u r f a c e lif e t im e
T h is r e s u lt a llo w s m e a s u r e m e n t o f th e bu lk and e f f e c t iv e
s u r f a c e lif e t i m e s w ith o u t a lte r in g th e w a fe r o r m a k in g any
p h y s ic a l c o n ta c t w ith i t .
B y k eep in g th e s u r f a c e c o n d itio n s th e
s a m e f o r s e v e r a l w a f e r s w ith d iffe r e n t b u lk l i f e t i m e s , th e s lo p e
o f th e tr a n s m it te d in te n s ity co u ld b e fitte d to E q . 4 - 1 , g iv in g
1 /T
and k .
e a c h d e v ic e .
F rom th e se r e s u lts ,
Tq m a y b e d e te r m in e d fo r
S i m i l a r p r o c e d u r e s co u ld b e u se d on a s in g le
w a f e r b y v a r y in g th e s u r f a c e c o n d it io n s , d e te r m in in g f i r s t 1 / T g
and K and th en
B.
Ts .
E x p e r im e n ta l
F ig u r e 2 . 7 i llu s t r a t e s th e c o m p a r is o n b e tw e e n th e t h e o r e t iн
c a l and e x p e r im e n ta l r e s u l t s .
In th is f ig u r e , th e n o r m a liz e d
tr a n s m it te d m ic r o w a v e e n e r g y is p lo tte d w ith r e s p e c t to t im e .
The
d is c r e p a n c y
or
s lig h t
s h if t
b e tw e e n
th e
th e o r e tic a l
and e x p e r im e n ta l
r e s u lt s
i s a ttr ib u te d to an u n c e r ta in ty in th e
am o u n t o f o p tic a l
en ergy
a b s o r b e d and c o u p le d to th e e le c t r o n
h o le p a ir g e n e r a t io n .
T h e r e a r e s e v e r a l p o s s ib le e x p la n a tio n s
f o r s u c h an u n c e r ta in ty .
T h e m o s t o b v io u s w o u ld b e an e r r o r
in th e m e a s u r e m e n t o f th e in c id e n t o p tic a l e n e r g y .
A l s o , it is
u n d ou b ted ly tr u e th at th e quantum e f f ic ie n c y o f th e p r o c e s s o f
c r e a tio n o f e le c t r o n - h o le p a ir s i s l e s s than u n ity ..
A quantum
e f f ic ie n c y l e s s than u n ity co u ld r e s u l t , f o r e x a m p le , fr o m o th e r
photon a b s o r p tio n p r o c e s s e s s u c h a s in tr a -b a n d t r a n s i t io n s . S u c h
t r a n s it io n s a r e r e la t iv e ly u n lik e ly in m a t e r ia ls th a t h a v e low
c o n c e n tr a tio n s o f h o le s and e l e c t r o n s .
H o w e v e r , a s th e c o n н
c e n tr a tio n o f c o n d u ctio n e le c t r o n s i n c r e a s e s th e r e i s an i n c r e a s н
in g p r o b a b ility o f t r a n s it io n s fr o m th e b o tto m o f th e c o n d u ctio n
band to h ig h e r l e v e l s in th e s a m e b a n d .
T he d ecay of su ch
e x c it a tio n i s p r e d o m in a n tly n o n r a d ia tiv e and th u s w ill n o t r e s u lt
in d ir e c t ly in e le c t r o n - h o le p a ir g e n e r a t io n .
th e v a le n c e band
i s a ls o
known to o c c u r .
A s i m i l a r p r o c e s s in
W h a te v e r th e c a u s e , a
c o r r e c t io n o f th e e f f e c t iv e in c id e n t o p tic a l e n e r g y w o u ld
r e s u lt in
e x c e lle n t a g r e e m e n t b e tw e e n th e th e o r e tic a l and e x p e r im e n ta l
c u r v e s sh o w n in F ig u r e 2 . 7 .
75
C.
F in a l R e m a r k s
S e v e r a l d ir e c t io n s f o r fu r th e r r e s e a r c h w e r e o p en ed d u r н
in g th e c o u r s e o f t h is in v e s tig a tio n .
O ne p a r tic u la r ly in t e r e s t in g
a r e a w o u ld b e r e f in e m e n t o f th e m e a s u r e m e n t te c h n iq u e f o r bu lk
lif e t im e and e f f e c t iv e s u r f a c e l i f e t i m e .
W hen m e a s u r in g the
t im e r a te o f c h a n g e (s lo p e ) o f th e tr a n s m itte d m ic r o w a v e e n e r g y ,
th e m e a s u r e m e n t i s ta k en a t th e 5 0 p e r c e n t l e v e l ,
w h ic h c o r r e s н
p on d s to a p a r tic u la r le v e l o f c a r r i e r c o n c e n tr a tio n .
If the
s l o p e s a r e s tu d ie d e x p e r im e n ta lly f o r d iffe r e n t s a m p le s w ith tr a n s н
m itte d m ic r o w a v e e n e r g ie s o th e r than th e 5 0 p e r c e n t l e v e l ,
a c c u r a t e m e a s u r e m e n t s o f lif e t im e a t d iffe r e n t c o n c e n tr a tio n s
co u ld b e o b ta in e d .
S u c h m e a s u r e m e n t o f th e b u lk and e f f e c t iv e
s u r f a c e l if e t i m e s w o u ld be p a r tic u la r ly a d v a n ta g e o u s s in c e th ey
c o u ld b e m a d e w ith o u t h avin g to s u b j e c t th e s a m p le to undue p r o н
c e s s i n g , s u c h a s c r e a tin g a p n -ju n c tio n , o r g r o w in g an o x id e ,
w h ic h in tu r n , c o u ld a lt e r th e c o e f f ic ie n t s w h ic h w e r e in it ia lly to
be m ea su red .
M o s t a ll
te c h n iq u e s a t p r e s e n t fo r m e a s u r in g
l i f e t i m e s o f m a t e r i a l s , p r o c e s s th e m a t e r ia ls to s o m e e x te n t. T h u s ,
th e d y n a m ic r e s p o n s e o f lif e t im e w ith r e s p e c t to c a r r i e r c o n c e n н
tr a tio n c o u ld b e s tu d ie d in d e t a il.
A d d itio n a l w o r k in th e a r e a o f
o p tic a l d e te c tio n and m ic r o w a v e s w itc h in g a s d is c u s s e d in
C h a p ter III a p p e a r s to b e p r o m is in g .
A P P E N D IX
A
B e lo w i s a s a m p le lis t in g o f o n e o f th e e le v e n c o m p u te r
p r o g r a m s u s e d in th e t h e o r e t ic a l c o m p u ta tio n s .
C a r e sh o u ld be
u s e d w h en tr y in g to im p le m e n t t h is p r o g r a m on a n y m a c h in e
o th e r than an IBM 3 7 0 o r e q u iv a le n t s i n c e s p e c ia l fu n c tio n s a r e
n e e d e d in o r d e r to r e s o lv e o v e r flo w and u n d e rflo w p r o b le m s .
76
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00410
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00470
0 04 30
0 0 4 90
00500
00510
00520
00530
00540
00550
00550
00570
00530
00590
00600
cii= l . d : 6
m=: i x - i
i i = ii
HN = 2 . 5 9 2 D - 4
17= 2 . * P 1 * 1 0 . 0 D 9
E=C0*11.70 0
H= 1 M / 0 F L O A T ( M)
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FORMAT ( I X , / , / , 1 X , ' T H , D , TAII, !I, MX , NT , HM , 5 CALK ' , / , IX , 7 ( 0 1 2 . 4 , 2X ) )
WRITE( 6 , 3 9 )
;
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DO 7 0 0 I T I = 1 , TIT
T=T! I * ( I T I - 1 )
DO 1 0 1 = 1 , n X
TCMP = COil ( I , I T D + C3
XCii = C3
SCO MX ( I ) = T M H P * ( SrlODK (T CMP , TKM P , XC 3 , 3 0 0 . ) + 3 M 9 R P ( TEMP, TEMP , XCR, 3 0 0 . )
X ) *1 . 6 0 2 D - 1 9 / 1 . D 4
CONTi ri UC
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1=1,11
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00970
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00990
01000
500
01010
01020
01030
01 040
01050
01060
01 070
01080
110
01090
01 100
01110
01120
01130
01140
01 150
01 160
01 170
01180
01190
01 2 00
M= 1
S9T 2 = (0 SQflT ( (T*D * 1 . 0 0 - 1 2 ) / 3 600 . 0 0 ) ) *2 .
CONTINUE
3 i r , ; i A=s C0NX( ; i +l )
Fl iPl = (SC'11 + 2 ) * F' I- H*H * W*II * S IGMA*FiTC-FMM 1
F11P 1C= S CR1 * F. IC+ 1i * H* tJ* U * S 1 0 MA* FTi+ 2 * F:IC- F'1! 110
I F ( M . L T . I I ) GO TO 500
WRITE( 6 , * ) CN, SIGMA, T IP 1 , FMP1C , r[
CONTINUE
I P ( i-I+1 . E O . i l ) GO TO 110
FI!Ml=Fil
F?iM1C=F!1C
F?I= FMP 1
FMC=F:(P 1C
II = 11+1
GO TO 100
CONTINUE
ED=DCMPLX(FMP1 . Fi tPl C)
EDMl=DCilPLX(FN,F'IC)
DS9=(EI)-EDM1 ) /DCUPLXCl , 0 . 0 0)
S C R 3 = ( ( ? I / R ) **2
*2 *U*EO)
SCR4=DCilPEX(SCR3 , O.D0)
T ?1AN=(2 . / (гU+onn/CDS<╗RT(SCR4) ) )
R EFL = ( CDS QtlT ( 3 CR4 ) - 9 E 0 / ET)) / (CDSORT (SCR4)+OEO/HO)
MT,1AN=CDARS (TRAN)
HUE FL=C 1) AR S ( AEFL )
0 UTT ( IT I ) = ? ITMA N* r! TUAM
OiJTR (ITI)=:tREFL*'lREFL
01210
01220
01230
01240
01 250
01260
01 270
01 230
01 290
01 300
01310
01320
01330
01 340
01350
01 360
01370
01 330
01 390
01 400
01410
01420
01430
01 440
01450
01460
01470
01430
01490
01500
39
40
C
C
700
C
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750
UE:)=CDAB 3 (CD)
.rIDCD = CDA3S (TED)
D3TTTA=DL0C1 0 ( MTUAN) * 2 0 . D0
DSMREF=OLOGl O(MRCFL)*2 0 . DO
F ORilAT ( I X , / , / , / , IX, 'TIlfK.TRAN , REFL' , / )
WRITS( 6 , 4 0 ) T , DBUTUA , DB.IREF
FORMAT( I X , 3 (G 1 1 . 4 , 3 X ) )
CONTINUE
DO 750 I T I = 1 , N T
T=T I!* (ITT ?1 )
0 U T T ( I T I ) = 0 ! J T T ( I T I ) /OUTT ( IT)
0 UT R ( I T I ) = 0 UT R ( IT I ) /Q UT R ( 1 )
U R I T E ( o , * ) T , 0 U T T ( I T I ) , OUTR( ITI )
CONTINUE
CALL EXIT
CUD
C
C
C
C
c
C I
FUNCTION D E L I I C ( C X , C ?> )
c::+cu<o THEN n - t y p k , e l s e p - t y p e .
D O U B L E P R E C I S I O N B ( 5 ) , A ( 6 ) , A i T S W E R , P A T (5 ) , D C X, D C
DOUBLE P R E C I S I O N PAR 1( 4 ) , T 1 ( 4 ) , A 1 ( 4 ) , D T , DABS
D A T A P A R / 2 3 S D 2 0 , G. O D 1 9 , 9 . 50 1 3 , 1 . O D 7 , 3 . 5P 1 5 /
3, D ELNC
00
01510
01520
01530
01540
01550
01 560
0 15 70
01530
01590
01600
01610
01620
01630
01640
01650
01660
01670
01630
01690
01700
01710
01720
01730
01740
01750
01760
01 770
01780
01 790
01 300
DATA 11/ 1 . 0 4 D - 6 , 1 . 43D-1 2, 2 . QD-1 6, 6 . 9.3D-9, 6 . 9 7 0 - 1 4, 2 . 0D-1 6 /
DATA A/ . 4 5 6 , . 7 4 4 , . 9 4 0 , . 5 4 3 , . 8 3 7 , 1 . 0 /
DATA PARI / I . 500 1 0 , 2 . 4 0 0 13, 1. 50T116/
DATA 3 1 / 4 . 0 0 0 - 1 7 , 1 . 4 7 D - 1 4 , 3 . 3 0 0 - 1 1 , 7 . 2 0 0 - 1 7 /
DATA A1 / . 9 6 6 , . 8 3 2 , . 6 5 0 , 1 . 0 0 0 /
DC X=C X
DC!1=C3
DT = 0 AP) S ( DCX) + DA3 S (DC 3)
1=1
IF ( 9CX+DC 3 . GT . 0 . OD0 )C. 0 TO 20
5 COOTIIIUE
I F ( D T . G E . P A R ( I ) )GO TO 10
1=1 + 1
I F ( I . E O . 6) CO TO 10
GO TO 5
1 0 DELNC=DA3S(DCX+DC3)* 3 ( I ) > ( (DT) * * ( A( I ) - 1 ) )
P.3TORN
20
I F ( DT. GE. PA3 1 ( I ) ) CO TO 30
1=1 + 1
I F ( I . EQ? 4 ) GO TO 3 0
CO TO 2 0
30 DCL:;C = DA3S (DCX+DC3) *P. 1 ( I ) '? ( (DT) * * ( A1 (I ) - l ) )
RETlJRfl
E'JD
C
C
C
C
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FJMCTKHJ IFAC(O)
00
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01810
01820
01 330
0134 0
01 850
01860
01870
01880
01 390
01900
01910
01920
01930
01 94 0
01950
01960
01 970
01980
01 990
02000
02010
02020
02030
02040
02 050
02060
02070
02 030
02090
02100
10
20
00U3LP. P11EC IS 10 U IFAC.X
iw.\n = i . n o
IP('T. EQ.O)GO TO 20
DO 10
1=1,N
X= I
IFAC=IFAC*X
CONTINUE
RETURtl
I FAC=1. DO
RSTURM
EMD
C
C
C
C
C
FUNCTION DCFAC(I)
DOUBLE PRECISION IFAC
COMPLEX*16 DCFAC
DCF AC** I FAC ( I )
RETURN
END
C
C
C
C
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FUNCTION DC?URM( I , K)
DOUBLE PE r.CTSIOU IFAC
COMPLEX*! 6 nCPUH?f
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03170
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0 31 90
03200
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03240
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XIN P= ( 2 . 4 5 E 2 1 * C) / (XC*ALOG ( I . +1 . 4 1 X 2 0*D / (XN+XP ) ) )
p ) / ( x ti f.+;i i mp )
Z=( 1 3 5 0 . * A ) / ( 1 3 5 0 . * A+XA)
F1=.0954+.473*Z-.383*(Z*Z)
F2=.095 6+.8 66*Z-.776*(Z*Z)
F=F 1 / F2
S H0 8 ?'!=F * (1 3 3 0 . * A* XA ) / ( 1 350 .*A+XA)
XN= XN * 1 . F6
XP=MP*1. 0 6
C=C*1 . E6
RETURN
END
xa= { h r *; a
C
C
C
C
C
FUNCTION SMOG P ( XN , XP , XC , T )
C
C
C
C
C
C
C
c
C
C
THIS FUNCTION CALCULATES THE HOLE MOBILITIES FOR SILICON.
N.P.AHD CARE III ATOM/( METER* '3 )
T IS IN DECREES KELVIN
N IS THE ELECTRON CONCENTRATION
P EC THE MOLE CONC ENTRATION
C IS THE IMPURITY CONCENTRATION
N:I=X:J*1 . E?
XP=X? * 1. r - h
CD
?vl
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0 0 U;J L ?, PRECISION CNO , CPO, UP, UN, SCALP.
DIl-iCnSION C( 201 , 201 )
DA ( CX) = (CX+CX+C P0+CP0) *TJP*UN* . 02 59/ ( ( CX+C PQ ) *?J P+ ( CX+C NO ) *T
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04060
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WRIT г (6 , * ) TH, I LOO
DO 4 0 1 = 1 , !IT
I p r i =( ; i x - i ) / 1 0
URI TE( 6 , 100) ( C ( I I , I ) , 1 1 = 1 , NX, I PHI)
FORMAT ( 1 X, ' 1 ' , 1X, 1 IE 1 1. 4 )
COTIT I HUE
RETURN
END
C
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S U!J ROUT INF A V? ( C , \?X, MT , ?1IJM)
c
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DOUBLE PRECIS ION C(.IX, NT)
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iJT-il=;iT-l
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DO 2 0 KX=2, NXM1
DO 20 KT = 2 , rlT l 1
C ( KX, XT ) = ( C( XX, NT ) +C ( XX, ST +1 ) +C ( X.X, flT
XKT)) / 5 . 0 0
CONTINUE
UR IT E( 6 , * ) I
CONTINUE
RETURN
END
04210
04220
04230
04240
042 50
04260
04270
04280
04290
04300
04310
04320
04330
04340
04350
04360
04370
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04390
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04410
04420
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04450
04460
04470
04480
04490
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S U3 ROUT IKE S INT (NX , NT , Til, C , TAIJ, CIi, 0 , 'I, S A.TH, SCALE )
DOUBLE PREC IS 1 0 M T H, C, TAU , CW, D, H, SATH , XHX, 0 1 S PIrl , XI T, XITM 1
DIMENSION C ( .1X, NT )
DOUBLE PRECISION C P , S P , T i I , A l ! ( 5 0 ) , X N P ( 5 0 ) , GAN, ETM , ETP , ET, SUM
DOUBLE PR EC IS 1 0 II B 2 , A2 , W, U2 , E2P , E2 M Z , P I 2
DOUBLE PRECISIOH CNO, CPO, UP, UN, DAX, CX,SCALE
DA(CX)= (CX+CX+CNQ+CPO) *?JP*UM * . 02 59 / ( (CX+CPO) *T
J P+ ( CX+C MO) *'JN )
CN0=CB
CP0=CB
XNX=NX
3 2 = 2 .0275E7
A2 = 2 9 7 . 19
U = 1 . 777E15
U2 = 1 . 2 5 6 6 E - 6
E 2 ? = l 3 2 1 . 7 1 * 1 . 76*3CALE
E2!l = 294 . 4 1 *1 . 76*SCALE
P 1 2 = 2 . DO*DARCOS( 0 . D0) / 2 . DO
GA;1 = 2 .D0*A2
.ET=B 2 * A2 /U/iJ/U 2 / 1 . 0 5 4D - 3 4
ET?=ET*E2P
ETH = ET*E2?i
WRITE (6 , *) ETP, ETH, ET, GA:1
NTil 1 =MT?1
NX:i i =u:: -i
00 10 1 = 1 , 5 0
T 1=1-1
92
H
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CP=DCOS(XMP(I) *S ATH)
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RETURK
CUD
(0
CO
A P P E N D IX
B
I llu s tr a te d b e lo w a r e th r e e g r a p h s d e p ic tin g th e raw d a ta
p ro d u ced b y a T e c h tr o n ix d i g i t i z e r , c o m p u te r , and c o p ie r s y s t e m .
In th e t e x t , th e raw d a ta i s r e p lo tte d in a m o r e c o n v e n ie n t f o r m .
F ig u r e B .1 :
P lo t o f th e in c id e n t in t e n s it y o f th e l a s e r p u ls e
illu m in a tin g th e s ili c o n w a f e r .
F ig u r e B .2 :
P lo t o f th e in t e n s it y o f th e l a s e r p u ls e e x itin g
s ilic o n w a fe r .
F ig u r e B .3 :
P lo t o f th e tr a n s m itte d m ic r o w a v e e n e r g y in
r e s p o n s e to th e in c id e n t lig h t p u ls e .
94
th e
20
18 v o 1 t s
U nax
T r 1 3 . 2 8 risec
P u 2 4 . 2 1 Ttsec
P o w e r 5 . 7 7 2 K u a t l\J
?*"PtPu 1 3 9 .7 4 n ic r-o jo u le s
Energy 172 .98 n ic r o jo u le s
F ig u r e B .1
r u iT io s e c /d iv .
HOW MANY a - 6 4 > ?
MSI + 1 0 0 . E - 0 3 !
HS1 + 2 0 . E - 0 9 !
1
0.1
i
!"? ?' i ????" r
Uh q x 0 . 1 7 8 v o l t s
Tr 1 4 . 0 6 nsec
Pw 2 5 n s e c
P o w e r 0 . 5 9 8 Kwattfs"
P*Pu 1 4 .9 4 Microjoules
Energy 1 7 .4 5 Microjoules
r
20 n ?i n o s e c
volts/div.
?I
F ig u r e B .2
.4
╗??
di v.
╗ , ?? t
10000
Unax 0 . 0 5 8 v o l t s
Tr 0 nsec
Pu 2 0 5 0 7 . 8 1 n s e c
Power 0 . 1 9 4 K w a tr
PT-Pw 3 9 7 8 . 5 1 n i c r o j o u l e s
Energy 4297.61 M icro jo u le s
F ig u r e b . 3
r ia r i G s e c - ' d i v
B IB L IO G R A P H Y
1.
D e b , S . and N a g , B . R . ,
" M e a s u r e m e n t o f L ife tim e o f
C a r r ie r s in S e m ic o n d u c t o r s th ro u g h M ic r o w a v e R e f le c t io n ,"
J o u r n a l o f A p p lie d P h y s i c s , V o l. 3 3 , N o . 4 ,
2.
B ro u ssea u , M .
and
S c h u t t le r , R . ,
pp. 1 6 0 4 , 196 2 .
" U se o f M ic r o w a v e
T e c h n iq u e s f o r M e a s u r in g C a r r ie r L ife tim e and M o b ility in
S e m ic o n d u c t o r ,"
S o l i d - S t a t e E le c t r o n i c s , V o l.
12, pp. 4 1 7 -
423, 1969.
3.
W h ite , R . and B e r n s t e in , J . , "A S im p le C o n ta c tle s s M ic r o н
w a v e M eth od f o r M e a s u r in g C a r r ie r L ife tim e D u rin g S o l a r C e ll F a b r ic a tio n , "
V o l. E D -2 6 , N o .
4.
B row n , H . K . ,
11,
IE E E T r a n s a c tio n s on E le c tr o n D e v i c e s ,
1979.
" P r o p a g a tio n o f E le c tr o m a g n e t ic W a v e s T h ro u g h
S il ic o n w ith an A r b itr a r y C o n c e n tr a tio n D is tr ib u t io n ,"
T h e O hio S t a t e U n iv e r s it y ,
5.
S tr a tto n , J . A . ,
T h e s is ,
1977.
E le c tr o m a g n e t ic T h e o r y , M c G r a w -H ill B o o k
C o m p a n y , 1941 .
98
99
6.
B erz,
F ., C ooper, R . W . ,
and
th e End R e g io n s o f PIN D io d e s ,"
V o l. 2 2 , p p . 2 9 3 - 3 0 1 ,
7.
W estm a n , H . P . ,
Fagg, S .,
" R e co m b in a tio n in
S o lid S t a t e E le c t r o n i c s ,
1979.
" R e fe r e n c e D ata fo r R a d io E n g in e e r s ,
" F ifth E d itio n , H ow ard W . S a n s and C o . ,
1973.
e s e e q u a tio n s a r e u s e d to it e r a t e th ro u g h th e s p a c e - t im e m a tr ix
u n til a s o lu tio n i s o b ta in ed fo r th e e l e c t r o n - h o le p a ir c o n c e n tr a н
tio n w ith a t o le r a b le e r r o r .
N e x t, th e c o n d u c tiv ity o f th e s e m ic o n d u c to r is co m p u te d a s
a fu n c tio n o f s p a c e and t im e .
U tiliz in g E q s . 1 -5 7 th rou gh 1 -6 7
e a c h e le m e n t o f th e s p a c e - t im e m a t r ix i s c o n v e r te d fr o m c a r r i e r
c o n c e n tr a tio n to c o n d u c tiv ity .
C o n d u c tiv ity i s th e d om in an t
v a r ia b le in th e c o m p u ta tio n o f th e m ic r o w a v e t r a n s m is s io n and
r e f le c t io n c o e f f i c i e n t s .
C o m p u ta tio n o f th e m ic r o w a v e r e f le c t io n and t r a n s m is s io n
c o e f f ic ie n t s i s a c h ie v e d w ith th e im p le m e n ta tio n o f E q s . 1 -7 5 ,
1 -7 7 ,
1 -7 8 ,
1 -8 4 ,
and
1 -8 9 th ro u g h 1?9 1 .
E q u ation 1 -7 5 i s
th e e q u a tio n th at d e s c r ib e s th e n o r m a liz e d e l e c t r i c fie ld in th e
s e m ic o n d u c t o r .
T o s t a r t th e ite r a tio n p r o c e s s E q s .
1 -7 7 ,
1 -7 8 ,
and 1 - 8 4 a r e u s e d in a c o m b in e d fa s h io n w h e r e th e p a r tia l d e r i v it iv e in E q . 1 - 8 4 i s co m p u te d n u m e r ic a lly fr o m th e s p a c e - t im e
37
m a t r ix .
A fte r E q . 1 -7 5 i s ite r a te d th ro u g h th e s a m p le , E q s .
1 - 8 9 th rou gh 1-91 a r e u s e d to m a tc h th e b o u n d a ry c o n d itio n s and
c o m p u te th e m ic r o w a v e t r a n s m is s io n and r e f le c t io n c o e f f i c i e n t s .
S a m p le c a lc u la tio n o f th e a b o v e e q u a tio n s a r e illu s t r a t e d
in C h a p ter 2 , S e c t io n C .
A l s o , c o m p a r is o n o f e x p e r im e n ta l and
th e o r e t ic a l r e s u lt s i s p r e s e n t e d .
CHAPTER
E X P E R IM E N T A L
A.
II
R ESU LTS
D e s c r ip tio n o f A p p a ra tu s
T h e a p p a r a tu s ca n b e d iv id e d in to th r e e s e c t io n s ; o p t ic a l,
m ic r o w a v e , and d e te c tio n .
T h e in itia l s t im u lu s i s a lig h t p u ls e
w ith a w a v e le n g th o f 1 .0 6 m i c r o m e t e r s , g e n e r a te d b y a Q s w itc h e d n e o d im iu m VAG l a s e r .
A n o p tic a l s y s t e m
is u sed
to
c o u p le th e lig h t in to a m ic r o w a v e w a v e g u id e , th u s illu m in a tin g
th e s e m ic o n d u c to r s a m p le a s illu s t r a t e d in F i g . 2 . 1 .
A run o f
x -b a n d w a v e g u id e i s u s e d to su p p o r t p r o p a g a tio n o f a 10 g ig a н
h e r tz c o n tin u o u s w a v e (CW ) m ic r o w a v e w h ic h i s g e n e r a te d by a
k ly s tr o n o s c i l l a t o r , and gu id ed to th e s e m ic o n d u c to r s a m p le .
F in a lly , tw o ty p e s o f d e t e c t o r s a r e u s e d , o n e fo r lig h t p u ls e s
and o n e f o r th e tr a n s m itte d m ic r o w a v e .
N ote th at o n ly o n e o f
th e d e t e c t o r s ca n b e in o p e r a tio n a t a n y g iv e n t im e .
T h e output
o f e a c h d e te c to r i s c o u p le d in to a d ig i t i z e r and th en fed in to
g r a p h ic s d is p la y and c o p ie r .
P r e s e n t e d n e x t , i s a d e ta ile d
d e s c r ip t io n o f th e a p p a ra tu s and s o m e o f th e m e th o d s u s e d f o r
a lig n m e n t and tu n in g .
38
a
LASER PULSE
MICROWAVE
DETECTOR
OPTICS
LASER
OPTICAL DETECTOR
MICROWAVE
WAVEGUIDE
TEXTRONICS
DIGITIZER &
MICROWAVE
SYSTEM
F ig u r e 2 .1
:
B lo c k d ia g r a m illu s t r a t in g th e e x p e r im e n ta l a p p a r a tu s .
GRAPHIC DISPLAY
T h e o p tic a l s y s t e m
c o n s is t e d o f
1)
Q?s w itc h e d n eo d in iu m YAG l a s e r
2)
o n e - q u a r t e r 04) to tw o (2 ) in ch b ea m
expand er
3)
o p t ic a lly fla t m ir r o r
4)
c o n v e x le n s
5)
o p tic a l b e n c h e s , m o u n ts , f i l t e r s , and
o th e r h a r d w a r e .
T h e o p tic a l p o r tio n o f th e a p p a ra tu s i s sh o w n in F ig u r e 2 . 2 .
T h e l a s e r w a s p u lse d o n c e e v e r y s e c o n d , p ro d u cin g a o n e q u a r te r 04) in c h b e a m o f lig h t fo r a d u ra tio n o f 2 0 n a n o s e c o n d s ,
h a v in g a p p r o x im a te ly 3 0 m illij o u le s o f e n e r g y a t a w a v e le n g th o f
1 .0 6 m ic r o m e t e r s .
F i l t e r s w e r e p la c e d in th e b e a m 's path to
r e d u c e it s to ta l e n e r g y c o n te n t.
T o d is tr ib u te th e b e a m 's e n e r g y
s o th at it w o u ld not h a rm c e r ta in p ie c e s o f th e o p t i c s , th e b e a m
w a s exp an d ed to tw o in c h e s in d ia m e t e r w ith a b ea m e x p a n d e r . A
m ir r o r w a s u s e d f o r c o n v e n ie n c e in th e ta b le s e t - u p to r e f le c t
th e l a s e r b e a m th ro u g h a o n e t h ir t y - s e c o n d in ch h o le a t a 90
d e g r e e bend in th e w a v e g u id e , th u s illu m in a tin g th e s e m ic o n d u c to r
s a m p le .
T h e fo c a l le n g th o f th e le n s w a s c h o s e n to illu m in a te
th e e n t ir e s e m ic o n d u c to r s a m p le .
41
SEMICONDUCTOR
SAMPLE
CONVEX
LENS
MIRROR
u
1/32 INCH HOLE
IN WAVEGUIDE
BEAM EXPAND
WAVEGUIDE
FILTERS
CP
i
LASER
?
F ig u r e 2 . 2
:
Y
O p tic a l p o r tio n o f th e e x p e r im e n ta l a p p a r a tu s .
42
T h e m ic r o w a v e a p p a r a tu s , u s e d to m e a s u r e th e r e f le c t io n
and t r a n s m is s io n c o e f f ic ie n t s , i s sh ow n in F ig u r e 2 . 3 .
A n x -b a n d
(8 -1 1 G ig a h e r tz ) s ig n a l w a s g e n e r a te d by a k ly s tr o n o s c i l l a t o r and
c o u p le d in to th e w a v e g u id e and th en gu id ed th ro u g h a c a v it y - t y p e
fr e q u e n c y m e t e r , an a tte n u a to r and a u n ife ed lin e to e n te r a
c a v ity fo r m e d by EH tu n e r N o . 1 and th e s a m p le . T h e uni fe e d
lin e w a s u s e d to p r e v e n t an y e n e r g y fr o m c o u p lin g o u t th e c a v ity
in to the w a v e g u id e a p p ro a c h in g th e k ly s tr o n .
S in c e u n ife e d lin e s
a r e not an id e a l im p e d a n c e m a tch to w a v e g u id e s ,
EH tu n e r N o . 1
w a s tuned to e lim in a t e an y sta n d in g w a v e s in th e c a v it y .
The
d ir e c t io n a l c o u p le r d ir e c t s r e f le c t e d e n e r g y fr o m th e s a m p le to
th e R F d e te c to r a t p oin t B w h e r e EH tu n e r N o . 2 i s u s e d to
m a x im iz e th e c o u p lin g .
T w o ty p e s o f d e t e c t o r s w e r e p la c e d a t p oin t
A .
An
o p tic a l d e t e c t o r w a s p o s itio n e d to c a p tu r e a ll o f th e lig h t th a t
p a s s e d th rou gh th e w a v e g u id e s y s t e m s o the to ta l in c id e n t and
tr a n s m it te d lig h t e n e r g ie s c o u ld b e m e a s u r e d .
O n ce th e lig h t
e n e r g ie s w e r e d e t e r m in e d , th e o p tic a l d e t e c t o r w a s r e p la c e d by
a b r o a d -b a n d m ic r o w a v e d e te c to r to m e a s u r e th e t r a n s ie n t
r e s p o n s e o f th e tr a n s m it te d m ic r o w a v e e n e r g y w h en th e lig h t
p u ls e fr o m th e l a s e r illu m in a te d th e s e m ic o n d u c to r s a m p le . T h e
ou tp u ts o f th e d e t e c t o r s w e r e fed in to a d ig it iz e r w h ic h a llo w e d
/
LENS
EH TUNER #1
EH TUNER #2
KLYSTRON
CAVITY TYPE
FREQUENCY METER
ATTENUATOR
LASER
BEAM
SEMICONDUCTOR
/
SAMPLE
UNIFEED
LINE
MICROWAVE
DETECTOR
I
1
SHORT CIRCUIT
50fl LOAD
MICROWAVE DETECTOR
OPTICAL DETECTOR
F ig u r e 2 . 3 :
I llu s tr a tio n d e p ic tin g th e m ic r o w a v e a p p a r a tu s .
co
44
th e r e s u lt s to b e p r e s e n te d in a g r a p h ic s d is p la y and r e p r o d u c e d
on a c o p ie r s y s t e m .
B.
A p p a ra tu s C a lib r a tio n and D ata C o lle c tio n
B e fo r e any e x p e r im e n ts co u ld b e p e r fo r m e d , th e o p tic a l
sy ste m
had to be a lig n e d w ith th e m ic r o w a v e w a v e g u id e s o th at
th e s e m ic o n d u c to r s a m p le w o u ld b e p r o p e r ly illu m in a t e d .
The
f i r s t s te p tak en w a s to a lig n th e l a s e r c a v it y , th e c e n t e r o f th e
s a m p le and th e m ic r o w a v e w a v e g u id e s o th a t th e y w e r e on th e
s a m e p la n e .
T h e n , w h ile th e l a s e r w a s b e in g p u ls e d , a p h o sн
p h o r e s c e n t c a r d w a s u s e d to d e te r m in e th e d ir e c tio n o f th e b e a m .
T h e p h o s p h o r e s c e n t c a r d g lo w e d y e llo w w h en e x p o s e d to th e l a s e r
beam .
U sin g th e m i r r o r , th e b ea m w a s d ir e c te d to th e c o n v e x
le n s and f o c u s s e d th ro u g h a h o le in th e w a v e g u id e .
w a s a c c o m p lis h e d b y s lig h t l y m o v in g th e l e n s .
F in e tuning
The card w a s
a ls o u s e d to m a k e s u r e th at th e c r o s s - s e c t i o n o f th e w a v e g u id e
at the s a m p le -m o u n t w a s illu m in a te d .
O n ce a lig n m e n t w a s
a c h ie v e d v is u a lly , a pin d io d e w a s m o v ed a c r o s s th e c r o s s s e c t io n a l a r e a o f th e s a m p le m ou n t to c h e c k fo r u n ifo rm
illu m in a t io n .
T o p r e p a r e th e m ic r o w a v e a p p a ra tu s fo r th e m e a s u r e m e n t
o f th e r e f le c tio n and t r a n s m is s i o n c o e f f ic i e n t s , th e fo llo w in g p r o н
c e d u r e w a s fo llo w e d :
1)
S e t th e k ly s t r o n o s c i l l a t o r a t 10 G ig a H z.
2)
P la c e a s lid in g s h o r t c ir c u i t a t p o in t A a s
sh o w n in F ig u r e 2 . 3 .
3)
A d ju st EH tu n e r N o . 2 u n til a p eak r e a d in g
is a c h ie v e d a t p o in t B .
4)
M ove th e s lid in g s h o r t c ir c u i t a t p o in t A
and r e c o r d th e m a x im u m and m in im u m
v a lu e s a t p o in t B .
A d ju st EH tu n e r N o . 1
to r e d u c e th e d if fe r e n c e b e tw e e n th e m a x im u m
and m in im u m .
5)
R e p e a t s t e p s 3 and 4 u n til th e m a x im u m
and m in im u m r e a d in g s a r e w ith in
.3 db o f
e a c h o th e r .
O n ce th e a p p a ra tu s w a s c a lib r a t e d , th e in c id e n t and t r a n s н
m itte d o p tic a l e n e r g ie s a lo n g w ith th e c o r r e s p o n d in g m ic r o w a v e
t r a n s m is s io n and r e f le c tio n c o e f f ic ie n t s c o u ld b e m e a s u r e d . T h e
in c id e n t o p tic a l e n e r g y w a s m e a s u r e d b y p la c in g an o p tic a l
d e te c to r a t p oin t A ,s h o w n in F ig u r e 2 . 3 , w h ile th e l a s e r w a s
p u lsin g a t a r a te o f o n e p u ls e p e r s e c o n d .
T h e output o f th e
d e te c to r w a s c o u p le d in to a d ig it iz e r w h e r e th e s ig n a l w a s c o n н
v e r te d to a d ig ita l fo r m w h ic h c o u ld th en b e p r o c e s s e d by a
g r a p h ic s s y s t e m .
T h e g r a p h ic s s y s t e m
in te g r a te d th e p u ls e to
46
g iv e th e c o r r e s p o n d in g e n e r g y o f th e p u ls e .
T h e tr a n s m itte d
e n e r g y m e a s u r e m e n t w a s p e r fo r m e d m u ch th e s a m e w a y , e x c e p t
th at th e s a m p le w a s p la c e d b e tw e e n th e o p tic a l d e te c to r and th e
w a v e g u id e a s illu s t r a t e d in F ig u r e 2 . 3 .
A lth o u g h th e a p p a r a tu s w a s s e t to m e a s u r e th e m ic r o w a v e
t r a n s m is s io n and r e f le c t io n c o e f f ic i e n t s , c o n s id e r a b le tr o u b le w a s
e n c o u n te r e d m e a s u r in g th e l a t t e r .
S in c e th e d ir e c t io n a l c o u p le r
a tten u a te d th e s ig n a l 10 d b , and th e b r o a d -b a n d d e te c to r w a s not
v e r y s e n s i t i v e , th e m e a s u r e d s ig n a l w a s w e ll in to th e r a n g e o f
th e b a ck g ro u n d n o i s e .
D ue to t h e s e ty p e s o f p r o b le m s , m o s t o f
th e e f f o r t s w e r e c o n c e n tr a te d on m e a s u r in g th e t r a n s m is s i o n
c o e f f ic ie n t .
A l s o , it w a s found to b e m o r e f e a s ib le to m e a s u r e
th e c h a n g e in tr a n s m it te d f ie ld s tr e n g th in s te a d o f th e t r a n s н
m is s io n c o e f f ic ie n t .
E ith e r m e a s u r e d q u a n tity ca n b e c o r r e la t e d
to th e t h e o r e t ic a l c a lc u la t io n .
C.
P r e s e n t a tio n o f C o lle c t e d D a ta and C o r r e la tio n w ith
T h e o r e t ic a l R e s u lts
A s in g le s i l i c o n w a f e r w a s u se d a s a s a m p le to e v a lu a te
th e v a lid ity o f th e t h e o r e t ic a l d e r iv a tio n d is c u s s e d in C h a p ter I.
T h e s a m p le had a t h ic k n e s s o f 2 8 5 m ic r o m e t e r s and w a s p o lis h e d
on o n e s u r f a c e w h ile b e in g la p p ed on th e o t h e r .
w a s in t r in s ic a t r o o m t e m p e r a t u r e .
A l s o , th e s a m p le
E x p e r im e n ta lly , fo u r c u r v e s
47
w e r e g e n e r a te d illu s t r a t in g th e m ic r o w a v e t r a n s m is s io n c h a r a c t e r н
i s t i c s a s a fu n c tio n o f t im e a t d if fe r e n t in c id e n t lig h t e n e r g i e s .
E ig h t c o m p u ta tio n a l s t e p s d e s c r ib e th e n u m e r ic a l c a lc u la н
tio n s a s sh o w n in F ig u r e 2 . 4 .
S t e p 1 c a lc u la t e s the in c id e n t
lig h t e n e r g y p e r m e t e r s q u a r e d p e r p u ls e w h ic h i s g iv e n b y
R e [ S Q]
w here
г
i
=
(2
A
i s th e in c id e n t e n e r g y and A
in a te d s u r f a c e .
T a b le 2 .1
s
i s th e a r e a o f th e i llu m -
l i s t s th e r e s u lt s o f E q . 2 - 1 .
T a b le
2 .1
R e s u lts o f S t e p 1
M easu rem en t
N u m b er
R e[ S ]
o
As
( j o u le s )
(m e te r )
2
jo u le s (m e te r )2/ p u l s e
1
7 . 1 5 x 1 CT3
2 . 8 8 x 10"4
3 .9 9
2
3 . 0 4 x 10 - 3
2 .8 8 x 10 ? 4
1 .0 6 x 1 0 1
3
1 . 7 2 x 10- 2
2 . 8 8 x 10 ? 4
3 .8 9 x 101
4
4 . 9 9 x 10~ 2
2 .8 8 x 10 ? 4
1 .7 3 x 102
S t e p 2 d iv id e s th e tr a n s m itte d e n e r g y
energy
г
b y th e in c id e n t
г ? to g e t th e o p tic a l t r a n
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