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Applications of Extended Operators to Diffusive Systems
APPLICATIONS O F EXTENDED OPERATORS T O DIFFUSIVE SYSTEMS E. M. Stafford and G. Dowrick University of Southampton, Southampton, Hunts SO9 5NH
ABSTRACT An o p t i m a l c o n t r o l problem i n semi-conductor d i f f u s i o n forms t h e major a p p l i c a t i o n t o t h i s p a p e r . Arguments a r e p r e s e n t e d as t o why t h e e x t e n d e d o p e r a t o r d e s c r i p t i o n i s u s e f u l . A t a b l e of combinations of boundary c o n d i t i o n s i s p r e s e n t e d which i n c o n j u n c t i o n w i t h t h e h e a t d i f f u s i o n e q u a t i o n i n one dimension i s d e s c r i b e d by v a r i o u s e x t e n d e d o p e r a t o r e q u a t i o n s . A t a b l e o f t h e c o r r e s p o n d i n g modal c o e f f i c i e n t s i s g i v e n . R e f e r e n c e i s made t o o t h e r u s e s of t h e e x t e n d e d o p e r a t o r d e s c r i p t i o n of d i f f u s i v e s y s t e m s , and a f i r s t o r d e r c a n o n i c a l form outlined.
1
.
INTRODUCTION
T h i s p a p e r forms p a r t of a s t u d y i n t o t h e c o n t r o l o f semi-conductor d i f f u s i o n , and hence t h e manufacture of t r a n s i s t o r s , d i o d e s and o t h e r d e v i c e s . Because t h e i n t e n d e d p r a c t i c a l a p p l i c a t i o n i s i n t e r e s t i n g i n i t s e l f and a l s o t o p l a c e t h e m a t h e m a t i c a l t r e a t m e n t t h a t f o l l o w s i n a r e l e v a n t c o n t e x t , some p h y s i c a l d e s c r i p t i o n of a p a r t i c u l a r c o n t r o l problem i n t h e p r o d u c t i o n of h i g h f r e q u e n c y d e v i c e s w i l l f i r s t b e g i v e n . The d i s t r i b u t i o n o f i m p u r i t y c o n c e n t r a t i o n w i t h i n a semi-conductor m a t e r i a l d e t e r m i n e s t h e c h a r a c t e r i s t i c s of a d e v i c e . F o r example, t h e c o n c e n t r a t i o n p r o f i l e s shown i n t h e s i m p l i f i e d t r a n s i s t o r d i a g r a m F i g . 1 may be r e l a t e d i n a l a r g e l y r i g o r o u s manner t o p a r a m e t e r s such a s C o l l e c t o r b a s e breakdown v o l t a g e Emitter efficiency Current gain P a r a s i t i c capacitance e t c . I m p u r i t i e s a r e i n t r o d u c e d i n t o a semi-conductor m a t e r i a l by means of a two-stage p r o c e s s : a ) by f i r s t l y d e p o s i t i n g a h i g h l y c o n c e n t r a t e d , t h i n , i m p u r i t y r e g i o n a t t h e s u r f a c e o f t h e semi-conductor b ) by t h e s u b s e q u e n t o x i d a t i o n o f t h e semi-conductor a t a r a i s e d t e m p e r a t u r e (> 1000°C). T h i s second s t a g e i s termed t h e " d r i v e - i n " p r o c e s s . The i m p u r i t y becomes d i s t r i b u t e d between t h e semi-conductor b u l k m a t e r i a l and a growing o x i d e l a y e r , F i g . 2. L a t e r t h e o x i d e l a y e r i s removed by chemical e t c h i n g , l e a v i n g t h e o r i g i n a l semi-conductor w i t h a r e q u i r e d c o n c e n t r a t i o n p r o f i l e of t h e i m p u r i t y atoms. P r a c t i c a l l y , t h e r a t e of o x i d a t i o n i s v a r i e d by s w i t c h i n g between w e t , o r s t e a m , o x i d a t i o n and d r y a i r o x i d a t i o n , t h e t e m p e r a t u r e i n each c a s e r e m a i n i n g i d e n t i c a l . E m p i r i c a l t i m e s ( t y p i c a l l y 5 m i n u t e s ) a r e chosen i n t h e manufacture p r o c e s s t o a t t a i n good y i e l d s , and by i n f e r e n c e i m p u r i t y c o n c e n t r a t i o n p r o f i l e s . We n o t e h e r e t h a t t h e w e t / d r y c y c l i n g i s r e m i n i s c e n t of minimum time boundary c o n t r o l o f d i f f u s i o n p r o c e s s e s ( 1 ) and d e s p i t e t h e n o n - l i n e a r i n f l u e n c e of t h e c o n t r o l v a r i a b l e s , i t becomes u s e f u l t o d e r i v e t h e s w i t c h i n g t i m e s by a n a l y t i c a s w e l l a s e m p i r i c a l means. The problem t h e n i n summary t u r n s o u t t o b e a s f o l l o w s ( a ) t h e r e e x i s t s a d i f f u s i v e problem, d e f i n e d f o r a composite g e o r e t r y of two m a t e r i a l s : o x i d e and b u l k s i l i c o n . ( b ) t h e geometry of b o t h m a t e r i a l s i s changing a t a r a t e d e t e r m i n e d by two r a t e s of o x i d e growth (assuming o t h e r v a r i a b l e s e .g. t e m p e r a t u r e remain c o n s t a n t ) ( c ) t h e d i f f u s i o n c o n s t a n t s , which a r e d i f f e r e n t f o r t h e c a s e of o x i d e from t h a t of t h e s i l i c o n , a r e n o n - l i n e a r l y dependent on c o n c e n t r a t i o n . A t t h e o x i d e / s i l i c o n i n t e r f a c e t h e i m p u r i t y c o n c e n t r a t i o n s a r e unequal ( u n l i k e t h e h e a t d i f f u s i o n c o u n t e r p a r t ) and d e f i n e d by a s e g r e g a t i o n c o e f f i c i e n t ( 2 , 3 ) ( d l t h e c u r r e n t s t a t e of 2D and 3D m o d e l l i n g of t h e d i f f u s i v e p r o c e s s does n o t a l l o w
.
E. M. S t a f f o r d and G. Dowrick
an a n a l y t i c d e r i v a t i o n of the d i f f u s i o n constants: these have t o be i d e n t i f i e d empirically ( 4 ) . (e) t h e r e i s no means a t present of on-line monitoring of impurity p r o f i l e s . ( f ) v a r i a t i o n s i n the sequence timing of the two oxidation procedures have most e f f e c t on shallow junction devices, with f i n e geometries, as f o r example i n high frequency t r a n s i s t o r s (of the order of 10 GHz). Such s e n s i t i v i t i e s a r e p a r t i c u l a r l y pronounced f o r high power devices. For t h i s highly non-linear problem, the major aim i s t o propose an i d e a l s e t of switching times f o r the a l t e r n a t i v e forms of oxidation. I n the following treatment, the extended d e s c r i p t i o n of d i f f e r e n t i a l operators i s applied t o d i f f u s i v e systems f o r combinations of boundary conditions. The a p p l i c a t i o n above r e l a t e s t o the d e s c r i p t i o n of d i f f u s i o n i n a composite slab: hawever the extended operator method of a r r i v i n g a t modal c o e f f i c i e n t s i s considered t o be useful i n the a n a l y s i s of many d i f f u s i v e systems, of f i x e d o r v a r i a b l e geometry. 2.
DERIVATION OF THE EXTENDED OPERATOR: AN EXAMPLE
The extended operator i s obtained by i n t e g r a t i o n by p a r t s , i n much the same way as the d e r i v a t i v e of a l i n e a r functional i s defined. Following t h i s treatment which i s d e t a i l e d i n the appendix and a l s o i n reference (5,6), f o r the h e a t conduction equation, we can e s t a b l i s h an equivalence between two s e t s of system d e s c r i p t i o n , f o r example:
+
inhomogeneous boundary conditions
+ homogeneous boundary conditions
The domain of t h e extended operator version of t h e d i f f u s i v e system i s seen t o c o n s i s t of functions with homogeneous boundary conditions. I n c o n t r o l problems, where a s p e c i f i e d s t a t e of a d i s t r i b u t e d system i s required by manipulat i n g t h e boundary conditions, a much simpler formulation is p o s s i b l e when the extended operat o r formulation i s used (7,8). Additionally i f feedback c o n t r o l i s applied t o t h e boundary conditions, the open and closed loop eigenvalues and eigenfunctions d i f f e r considerably: t o t r u n c a t e an i n f i n i t e s e t of open loop modal c o e f f i c i e n t s , and t o compute t h e subsequent closed loop behaviour based on these c o e f f i c i e n t s i s t o approximate the problem badly ( 7 ) .
3.
EXTENDED OPERATOR TABLE FOR I D DIFFUSION EQUATION
It may be noted t h a t everything derives from a knowledge of the Green's function f o r a nonhomogeneous equation with homogeneous boundary conditions; t h i s e s t a b l i s h e s t h e equivalence immediately s t a t e d above. There now follows a t a b u l a t e d l i s t of t h e extended operator form of t h e d i f f e r e n t i a l operator.
( t h e d i f f u s i o n equation), f o r various s e t s of boundary conditions.
4.
(Table I)
EXPANSION COEFFICIENTS FOR DIFFUSION EQUATION
I f an eigenfunction expansion t o an inhomogeneous d i f f u s i o n equation, of form
Applications of Extended Operators t o D i f f u s i v e Systems
i s attempted, then i t i s e a s i l y shown t h a t t h e expansion c o e f f i c i e n t s a l a2 i n f i n i t e expansion
... e t c .
i n the
y i e l d a r e l a t i o n of t h e form
The same form of expansion h o l d s f o r t h e t h r e e dimensional d i f f u s i o n equation. This expansion c o e f f i c i e n t equation i s d i r e c t l y o b t a i n a b l e from t h e extended o p e r a t o r equations (Table I ) , g i v i n g Table 11, s o t h a t providing t h e a p p r o p r i a t e eigenvalues and f u n c t i o n s a r e used ( t h o s e s a t i s f y i n g t h e homogeneous boundary conditions) t h e extended operat o r provides a u n i f i e d method of providing a truncated s e t of expansion c o e f f i c i e n t s s u i t a b l e f o r s i m u l a t i o n f o r c o n t r o l purposes, ( 9 ) . 5.
HEAT DISTRIBUTION I N A COMPOSITE SLAB
Consider t h e case of a s l a b of h e a t conducting m a t e r i a l , of f i n i t e l e n g t h , f i n i t e h o r i z o n t a l t h i c k n e s s , and e f f e c t i v e l y i n f i n i t e width (Fig. 3a). The boundary c o n d i t i o n s a r e of form u ( o , t ) = f o ( t ) : u(9.,t) = f ! (x) Kncwing t h e c o n d u c t i v i t y ( a 2 ) i t i s p o s s i b l e t o w r i t e an extended o p e r a t o r d e s c r i p t i o n of t h e system, f o r a u n i t width.
.
Now consider a second s l a b of o v e r a l l l e n g t h 112, conductivity B 2 , placed end-on t o t h e f i r s t s l a b , (Fig. 3b). The extended o p e r a t o r d e s c r i p t i o n of t h i s second s l a b may be p u t i n terms of t h e conditions a t each end, F o ( t ) and F l ( t ) , and transforming t h e p o s i t i o n of t h e i n t e r f a c e t o x = 0 (Fig. 3 c ) , t h e two extended o p e r a t o r equations apply f o r t h e temperature u, v i n t h e respective slabs.
&! at
=
fi + a26'(x+9. 1) f 0( t ) - a 2 6 ' ( x ) f l ( t ) ax2
A non d i m n s i o n a l equation can be now derived over t h e range -1
where
S
"
"a
=--
The i n t e r f a c e conditions
now become
and 5 i s t h e normalised range.
f
u = v
2
5
Is
s e e Fig. Id, giving
E . P I . S t a f f o r d and G. Dowrick
where
M
=
I2 -
I n making t h i s t r a n s f o r m a t i o n we f o l l o w t h e p r o c e d u r e of Concus and O l l a n d e r ( l o ) , who p r o v i d e e i g e n f u n c t i o n s o l u t i o n s t o t h i s problem. The aim h e r e i s t o s e e whether t h e extended o p e r a t o r m a n i p u l a t i o n s s u c c e s s f u l l y l e a d t o t h e same r e s u l t . Examining
where L i s t h e o p e r a t o r
a - - , and a~ ac2 -
L* may be d e r i v e d as
[
y i s an a s y e t undefined f u n c t i o n , t h e a d j o i n t o p e r a t o r +
$1
[see
(hppendix)]
where t h e c o n j u n c t terms a r e chosen t o make a z e r o c o n t r i b u t i o n .
1 s employed, where w(r,-1)
via
Then t h e r e l a t i o n
= f o ( ~ ) ;W ( T , S ) = El ( T ) ; W(T,O-) = w ( T , o + )
The extended o p e r a t o r e q u a t i o n i s now f i n a l l y s e e n t o b e
F i n d i n g t h e b i o r t h o g o n a l s e r i e s @;:Sj, $ i ( ~ ) of f u n c t i o n s s a t i s f y i n g t h e a p p r o p r i a t e e i g e n f u n c t i o n e q u a t i o n , t h e Green's F u n c t i o n i s found t o b e
and t h e r e f o r e t h e s o l u t i o n found t o be
w i t h t h e a p p r o p r i a t e meaning a s e a r l i e r d e f i n e d g i v e n t o t h e symbolic f u n c t i o n s . a g r e e s w i t h t h e a l t e r n a t i v e d e r i v a t i o n ( 10)
.
This r e s u l t
E q u a t i o n s i n t h e modal c o e f f i c i e n t s may be a l s o o b t a i n e d v i z
dam + dr
A2 am(r)
m
- A
sin A
(T)
m -s sin X s m
where
s o t h a t t h e Kitamori method ( 9 ) i s seen t o f o l l o w n a t u r a l l y i n t h i s c a s e a l s o . The d e r i v a t i o n s of such modal c o e f f i c i e n t e q u a t i o n s a r e thought t o be u s e f u l i n t h e c a s e of moving boundary d i f f u s i o n s y s t e m s , and a r e c u r r e n t l y b e i n g a p p l i e d i n t h e semi-conductor d i f f u s i o n example r e f e r r e d t o e a r l i e r . There i s a l s o a s t r o n g r e l a t i o n between t h e d i s t r i b u t e d forms of Krons' t e a r i n g method , e . g . r e f . ( I 1) and t h e development i n t h i s s e c t i o n . The major d i f f i c u l t y i n t h e moving boundary c a s e i s t o c o r r e c t l y s p e c i f y t h e i n t e r f a c e c o n d i t i o n s
6.
FOURIER APPROXIMATIONS TO PERIOD
I t i s p o s s i b l e t o i n s e r t an a r t i f i c i a l p e r i o d i c i t y i n t o t h e extended o p e r a t o r e q u a t i o n , t h u s c a u s i n g t h e a n a l y t i c s o l u t i o n t o b e made p e r i o d i c , w i t h o u t u p s e t t i n g t h e c o n d i t i o n s o v e r the f i n i t e r a n g e of i n t e r e s t . T h i s is p a r t i c u l a r l y u s e f u l i n t h e c a s e of f i r s t o r d e r s p e c i a l o p e r a t o r s , where no e i g e n f u n c t i o n s e x i s t , and where no g u i d e l i n e s may b e used t o choose c o n t i n u o u s f u n c t i o n s t h a t might a s s i s t G a l e r k i n approximation methods. Once such
A p p l i c a t i o n s o f Extended O p e r a t o r s t o D i f f u s i v e Systems s y s t e m s a r e known t o b e p e r i o d i c , t h e n s i n u s o i d a l approximating f u n c t i o n s p r o v i d e a measure of modal independence, and hence s i m p l i c i t y i n t h e e x p a n s i o n c o e f f i c i e n t e q u a t i o n s . Thus an e q u a t i o n
becomes
where k , i n p r a c t i c e , h a s t o be a d j u s t e d t o a l l o w f o r t h e i n i t i a l c o n d i t i o n u ( o , t ) = a ( t ) : t h i s means t h a t t h e nodes a r e n o t i n d e p e n d e n t , as i n t h e p r e v i o u s e i g e n f u n c t i o n a n a l y s i s of the simple d i f f u s i o n equation (7,8). However, s e t t i n g a s i d e c l a s s i c a l e i g e n f u n c t i o n a n a l y s i s , i t i s p o s s i b l e t o pose t h e h e a t e q u a t i o n i n s i m u l t a n e o u s f i r s t o r d e r form
where u
u2 r e q u i r e boundary c o n d i t i o n s , g i v e n a t e i t h e r end of t h e s p a t i a l domain.
T h i s h a s been computed (12) and i s found t o be s u c c e s s f u l f o r an o r d i n a r y boundary c o n d i t i o n and a d e r i v a t i v e boundary c o n d i t i o n , b u t becomes more c o m p l i c a t e d f o r p u r e l y d e r i v a t i v e , o r p u r e l y f u n c t i o n v a l u e s , boundary c o n d i t i o n s . The same approach c o u l d a l s o b e made t o t h e flow d i f f u s i o n e q u a t i o n
T h i s e q u a t i o n s u g g e s t s t h a t one boundary c o n d i t i o n i s t h e v a l u e of ul a t t h e s p a t i a l boundary, w h i l e a n o t h e r i s u2 i.e. a u l + a u l / a x a t t h e boundary. T h i s i s i n f a c t a n a t u r a l boundary c o n d i t i o n f o r t h e system, when flow p l u s d i f f u s i o n i s t a k i n g p l a c e : n o t , a s o f t e n i n t h e l i t e r a t u r e , a boundary c o n d i t i o n au/ax = b ( t ) . The b a s i c e q u a t i o n o f f o u r t h s p a t i a l o r d e r and same o r d e r i n time can a l s o b e p u t i n t o t h i s c a n o n i c a l form, and i t i s h e r e n o t e d t h a t o n l y c e r t a i n combinations of boundary c o n d i t i o n s e x i s t i n p r a c t i c e , r e f . ( 1 3 ) . T h i s i s e x p e c t e d t o r e l a t e t o t h e work of Evans and o t h e r s on system s t r u c t u r e s ( 1 6 ) . The u n i f y i n g f u t u r e of t h e e x t e n d e d o p e r a t o r w i t h a r t i f i c i a l p e r i o d i c i t y i s p a r t i c u l a r l y c o n s t r u c t i v e i n t h e c a s e of boundary c o n d i t i o n s s p e c i f i e d a t e i t h e r end of t h e s p a t i a l domain. Recent work u s i n g e x t e n d e d o p e r a t o r f o r m u l a t i o n h a s been used t o d e t e r m i n e t h e o p t i m a l p o s i t i o n of h e a t s o u r c e s i n a d i f f u s i v e h e a t s y s t e m , employing attainability/contro?ability c o n c e p t s ( 1 7 ) : s i t i n g of o b s e r v e r s e n s o r s may be s i m i l a r l y d e r i v e d .
CONCLUSION P a r a l l e l r e s u l t s t o t h o s e o b t a i n e d i n d i f f u s i v e systems u s i n g c l a s s i c a l a n a l y s i s can b e o b t a i n e d u s i n g t h e e x t e n d e d o p e r a t o r . I n t h i s r e s p e c t t h e p a p e r i s p r e s e n t e d w i t h much t h e same p h i l o s o p h y a s t h a t of r e f . ( 1 8) A t a b l e of e x t e n d e d o p e r a t o r s f o r d i f f e r e n t boundary c o n d i t i o n s is immediately a p p l i c a b l e t o a s e t of chosen b a s i s a p p r o x i m a t i n g f u n c t i o n s including eigenfunctions. A r t i f i c i a l p e r i o d i c i t y introduced i n t o extended operators favours a G a l e r k i n - F o u r i e r e x p a n s i o n , which can r i v a l e i g e n f u n c t i o n a n a l y s i s whenever t h e e i g e n functions are d i f f i c u l t t o derive.
.
I n d i s t r i b u t e d feedback c o n t r o l , t h e open and c l o s e d l o o p e i g e n f u n c t i o n problems can o n l y be p r o p e r l y d e s c r i b e d i n a n o p e r a t o r , and most c o n v e n i e n t l y an e x t e n d e d o p e r a t o r s e n s e . Match-
E . M. S t a f f o r d and G. Dowrick
i n g of boundary c o n d i t i o n s a t t h e j u n c t i o n s of d i s s i m i l a r b u t l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n systems a p p e a r s t o b e a i d e d by extended o p e r a t o r methods.
APPENDIX
-
DERIVATION OF THE EXTENDED OPERATOR
-d2/dx2 b e t h e d i f f e r e n t i a l o p e r a t o r and S t h e s e t of f u n c t i o n s i t a c t s on. 1 1 S = {u(x) : u 2 ( x ) d x < +(D} , S O t h a t t h e i n n e r p r o d u c t < u,v > = uv dx, V u , v E S i s w e l l
Let L
5
I
I
0
0
= 5 s h a l l have a s o l u t i o n , i t i s n e c e s s a r y t o defined. In order t h a t the equation L i n t r o d u c e t h e n o t i o n of t h e a d j o i n t o p e r a t o r L*.
The a d j o i n t o p e r a t o r L* i s d e f i n e d by
Vu, v E S. L e t M , M* b e t h e domains of L, L* r e s p e c t i v e l y . I f v E M*, u E M t h e n b o t h s i d e s of [I] a r e d e f i n e d . However, what happens i f u i s r e p l a c e d i n [I] by a f u n c t i o n w E S which i s n o t i n M? Then < L* ,w > i s s t i l l d e f i n e d , b u t < v,L w > i s n o t . The i d e a of an extended o p e r a t o r means t h a t we wish t o f i n d a meaning f o r a p p l y i n g t h e o r i g i n a l o p e r a t o r t o a s e t of f u n c t i o n s n o t i n i t s domain.
A meaning (and d e f i n i t i o n ) i s r e a d i l y a v a i l a b l e from t h e above, simply by p u t t i n g < L*v, w > = < v , Le w > where L i s now t o be understood i n i t s e x t e n d e d s e n s e ( w r i t t e n L e ) , Also, s i n c e by r e l a t i n g i t t o something which i s a l r e a d y w e l l - d e f i n e d i . e . < L*v, w > < L*v, w > w i l l b e a c o n t i n u o u s l i n e a r f u n c t i o n a l , t h e n we may u s e t h e s e t of f u n c t i o n s v a s t e s t i n g f u n c t i o n s f o r Le.
.
I t now remains t o f i n d Lew, and by way of an example we s h a l l r e t u r n t o t h e o r i g i n a l problem. Equation 2 i s rewritten
v l L e w dx =
-
i
v
w dx - ; ( I )
w(1) + ;(o)
w(0)
~ [ 2 ]
S i n c e t h e right-hand s i d e i s a continuous l i n e a r f u n c t i o n a l i t d e f i n e s a symbolic f u n c t i o n Lew. E q u a t i o n 2 i s r e w r i t t e n
REFERENCES Butkovsky, D i s t r i b u t e d C o n t r o l Systems ( t r a n s l . ) E l s e v i e r (1969).
1.
A.G.
2.
J . Huang, L.C. W e l l i v e r 'On t h e r e - d i s t r i b u t i o n of boron i n t h e d i f f u s e d l a y e r d u r i n g thermal o x i d a t i o n ' J . Electrochem Soc 117 (1970).
3.
S. M a r g o l i t , A. Neugroshel, A . B a r l e v , 'The r e d i s t r i b u t i o n of Boron and Phosphorous a f t e r t h e second o x i d a t i o n s t a g e and i t s thermal e f f e c t on MOST t r a n s i s t o r p r o p e r t i e s ' I n s t i t u t e of P h y s i c s Conference, L a n c a s t e r U.K. S e p t . (1971)
4.
C.R.
5.
B . Friedman, P r i n c i p l e s of Applied Mathematics, Wiley ( 1 9 5 6 ) .
.
6.
J e s s h o p e , E . J . Z a l u s k a , H.A. Kemhadjian, Comparison of numerical s o l u t i o n methods f o r two dimensional b i p o l a r t r a n s i s t o r a n a l y s i s a l g o r i t h m . E l e c t r o n i c s L e t t e r s 11,14(1975).
W.L. Brogan,
Theory and A p p l i c a t i o n of Optimal C o n t r o l f o r D i s t r i b u t e d Parameter Systems.
Automatics 4 (1967).
ILD~I
7.
D.M. Wiberg,
Feedback c o n t r o l of l i n e a r d i s t r i b u t e d s y s t e m s .
8.
E.D. G i l l e s ,
Systeme m i t V e r t e i l t e n Parametern, Oldenbourg V e r l a g
9.
T. K i t a m o r i , ' T r a n s f o r m a t i o n of d i s t r i b u t e d parameter systems i n t o lumped parameter systems f o r t h e s t u d y of o p t i m a l c o n t r o l ' Paper 24(a) IFAC London (1966).
T r a n s A.S.M.E.
' T r a n s i e n t D i f f u s i o n i n a Composite S l a b ' 10. P. Concus, D.R. Olander, T r a n s f e r I1 pp.610-613 (1970).
1973
.
89 (1967).
I n t . J . Heat Mass
'The p i e c e by p i e c e s o l u t i o n of e l l i p t i c boundary 1 1 . G . J . Rogers, G.K. Cambrell, problems' J . Phys. D: Appl. Phys. 8 (1975).
value
A p p l i c a t i o n s o f Extended O p e r a t o r s t o D i f f u s i v e Systems 'A G a l e r k i n - F o a r i e r method o f s o l u t i o n of d i f f u s i v e s y s t e m s ' C h a p t e r i n 12. E .M. S t a f f o r d , P r o c e e d i n g s of I n t . Conf. on V a r i a t i o n a l Methods i n E n g i n e e r i n g ( e d . B r e b b i a ) S.U.P. 1973. 13. F. Church, Mechanical V i b r a t i o n s (Appendix B: C h a r a c t e r i s t i c s f u n c t i o n s of beams,) Wiley ( 1 9 6 0 ) . 14. F.J. Evans, J.J. Van Dixhoorn ( e d s ) 1974.
P h y s i c a l S t r u c t u r e s i n System Theory, Academic P r e s s
15. M. Amouroux, On a method of d e t e r m i n i n g t h e o p t i c a l p o s i t i o n o f ' p o i n t s of a c t i o n ' f o r a c l a s s of l i n e a r d i s t r i b u t e d p a r a m e t e r s y s t e m s , C .R. Acad. Sc. P a r i s S g r i e A ( i n F r e n c h ) J u l y (1973). ' D i s t r i b u t e d P a r a m e t e r C o n t r o l Theory a p p l i e d t o t h e W i l d l a n d F i r e S u p p r e s s 16. W.L. Brogan, i o n Problem. IFAC Symposium on t h e C o n t r o l of D i s t r i b u t e d P a r a m e t e r S y s t e m s , B a n f f , Canada ( 1 9 7 1 ) .
E. M. Stafford and G. Dowrick
TABLE I
EXTESDED OPERATOR (A,)
BOUNDARY CONDITIOKS
u(o,t) = fo(t)
au -
1
at
u(L,~) = fl(t) u'(0.t)
fo(t)
=
au -
2
fo(t)
=
ul(o,t) = fo(t)
au -
4 =
fl(t)
u(0,t)
+
llul(o,t) = f (t)
5
u(0,t)
=
fl(t)
fo(t)
=
6
u(L,t) u'(0,t)
cu'(L,t)
+
=
=
fo(t)
7
u(e,t) + cul(L,t) u(%,t)
=
=
fl(t)
=
f (t)
fl(t)
8 u(o,~) + Sul(o,t) ul(L,t) = fl(t) u(o,~) + Sul(o,t) = f (t) u(o,~)
=
a2
fo(t)
a2a'(x)f0(t)
a26(x)f0(t)
+a26(x-k) f 1 (t)
a26(x)fo(t)
+ a26' (x-!.)
ax2
fi- a26'(x-e)fl(t)
aU -
a2u .2-+al(x-e)a2r1(t)
-
ax2
f 1 (t)
a26'(x)fo(t)
-a2st(x)f (t)
ax2
au at
a '
"at
a2
au at -
10
a2, -
2 .
au at
9
-
(t)
2 e a26' (x) fo(t)
au at
at
fl(t)
fi + a26(x-~)f
ax2
at
u(e,t)
-
a261(x-e)rl(t)
a x2
at
u'(L,~) = f +t)
u(e,t) + cul(e,t)
a2
& - 2.
3
a2 u + ax2
at
ul(k,t) = fl(t) u(o,t)
a2
fi +a26'(x-L)
f,(t) + a26(x)f0(t)
ax2
fi + a26'(x-!L)f
(t)
-
a26' (x)fo(t)
ax2
a2fi+ ax2
a2
f2i -
az6(x-()fl(t)
a26(x)fl(t)
-a26'(x)fo(t)
-
ci26'(x)fo(t)
ax2
ul(o,t) = fl(t)
u(L,t)
=
fo(t)
au -
11 uq(e,t) = f,(t)
at
I
a2fi+a26'(x-i)f0(t) ax2
+a26(x-l)f,(t)
I
Applications of Extended Operators to Diffusive Systems
TABLE I1
DERIVED EQCAT IONS
BOUNDARY CONDITIONS u(o,t> = fo(t>
I u(L,t) ut(o,t)
am
+
am
-
a2$r(l)fl m (t)
+ ~ : a ~+ a2$m(L)f,(t)
-
a2$rn(o)fo(t)
m
+ i2a + a2$i(o)fo(t) m m
-
a2$m(~)f, (t)
am
+ hiam + a2bm(o)fo(t)
-
a2$'(e)f m
am
+
am
+ h2a m m + a2bi(o)fo(t)
+
a2$;(o)fo(t)
fl(t)
=
fo(t)
=
2
ut(e,t) = fl(t) u(0,t)
f (t)
=
3 ~'(a,t)
fl(t)
=
ut(o,t) = fo(t) 4
u(L,t)
u(0,t) + r,uf(o,t) = fo(t) 5
u(k,t)
u(0,t) = fo(t)
u1(o,t)
a2a m m + a2@;(t)f
(t) + a26i(o)fo(t)
+ Sul(L,t) = f,(t)
6
u(e,t)
I (t)
fl(t)
=
-
a2@;(e)f
-
~~@~(o)f~(t)
l(t)
+ cul(e,t) = f ,(t) =
fo(t) m + h2a m m
7 u(e,t)
+ cuf(t,t) = fl(t)
u(e,t)
=
u(0,t)
+ ~u'(0,t)
=
fo(t)
u(0,t)
+ Su'(0,t)
=
fo(t)
fl(t)
8
9
ut(2,t) = f ,(t)
-
a2@A(i)f,(t)
-
a2@,(e)f
am
+ i2a m m
am
+ h2a + a2@m(~)fl (t) + a2@A(o)fo(t) m m
,(t) + a2@;(o)fo(t)
Where Am, @m are the eigenvalues and eigenvectors for the diffusion equation with homogeneous boundary conditions
E. M. Stafford and G. Dowrick
t Impurity concentration
I n type silicon
emitter n-type dominates
I 1 base
7
TzE?
collector
P-type dominates
Fig. I
Impurity profiles for npn transistor
Stage a)
T
time
interface
~
diffusion by two i 2~ Semiconductor . stage oxidation
Applications o f Extended Operators t o D i f f u s i v e Systems
ABSTRACT An o p t i m a l c o n t r o l problem i n semi-conductor d i f f u s i o n forms t h e major a p p l i c a t i o n t o t h i s p a p e r . Arguments a r e p r e s e n t e d as t o why t h e e x t e n d e d o p e r a t o r d e s c r i p t i o n i s u s e f u l . A t a b l e of combinations of boundary c o n d i t i o n s i s p r e s e n t e d which i n c o n j u n c t i o n w i t h t h e h e a t d i f f u s i o n e q u a t i o n i n one dimension i s d e s c r i b e d by v a r i o u s e x t e n d e d o p e r a t o r e q u a t i o n s . A t a b l e o f t h e c o r r e s p o n d i n g modal c o e f f i c i e n t s i s g i v e n . R e f e r e n c e i s made t o o t h e r u s e s of t h e e x t e n d e d o p e r a t o r d e s c r i p t i o n of d i f f u s i v e s y s t e m s , and a f i r s t o r d e r c a n o n i c a l form outlined.
1
.
INTRODUCTION
T h i s p a p e r forms p a r t of a s t u d y i n t o t h e c o n t r o l o f semi-conductor d i f f u s i o n , and hence t h e manufacture of t r a n s i s t o r s , d i o d e s and o t h e r d e v i c e s . Because t h e i n t e n d e d p r a c t i c a l a p p l i c a t i o n i s i n t e r e s t i n g i n i t s e l f and a l s o t o p l a c e t h e m a t h e m a t i c a l t r e a t m e n t t h a t f o l l o w s i n a r e l e v a n t c o n t e x t , some p h y s i c a l d e s c r i p t i o n of a p a r t i c u l a r c o n t r o l problem i n t h e p r o d u c t i o n of h i g h f r e q u e n c y d e v i c e s w i l l f i r s t b e g i v e n . The d i s t r i b u t i o n o f i m p u r i t y c o n c e n t r a t i o n w i t h i n a semi-conductor m a t e r i a l d e t e r m i n e s t h e c h a r a c t e r i s t i c s of a d e v i c e . F o r example, t h e c o n c e n t r a t i o n p r o f i l e s shown i n t h e s i m p l i f i e d t r a n s i s t o r d i a g r a m F i g . 1 may be r e l a t e d i n a l a r g e l y r i g o r o u s manner t o p a r a m e t e r s such a s C o l l e c t o r b a s e breakdown v o l t a g e Emitter efficiency Current gain P a r a s i t i c capacitance e t c . I m p u r i t i e s a r e i n t r o d u c e d i n t o a semi-conductor m a t e r i a l by means of a two-stage p r o c e s s : a ) by f i r s t l y d e p o s i t i n g a h i g h l y c o n c e n t r a t e d , t h i n , i m p u r i t y r e g i o n a t t h e s u r f a c e o f t h e semi-conductor b ) by t h e s u b s e q u e n t o x i d a t i o n o f t h e semi-conductor a t a r a i s e d t e m p e r a t u r e (> 1000°C). T h i s second s t a g e i s termed t h e " d r i v e - i n " p r o c e s s . The i m p u r i t y becomes d i s t r i b u t e d between t h e semi-conductor b u l k m a t e r i a l and a growing o x i d e l a y e r , F i g . 2. L a t e r t h e o x i d e l a y e r i s removed by chemical e t c h i n g , l e a v i n g t h e o r i g i n a l semi-conductor w i t h a r e q u i r e d c o n c e n t r a t i o n p r o f i l e of t h e i m p u r i t y atoms. P r a c t i c a l l y , t h e r a t e of o x i d a t i o n i s v a r i e d by s w i t c h i n g between w e t , o r s t e a m , o x i d a t i o n and d r y a i r o x i d a t i o n , t h e t e m p e r a t u r e i n each c a s e r e m a i n i n g i d e n t i c a l . E m p i r i c a l t i m e s ( t y p i c a l l y 5 m i n u t e s ) a r e chosen i n t h e manufacture p r o c e s s t o a t t a i n good y i e l d s , and by i n f e r e n c e i m p u r i t y c o n c e n t r a t i o n p r o f i l e s . We n o t e h e r e t h a t t h e w e t / d r y c y c l i n g i s r e m i n i s c e n t of minimum time boundary c o n t r o l o f d i f f u s i o n p r o c e s s e s ( 1 ) and d e s p i t e t h e n o n - l i n e a r i n f l u e n c e of t h e c o n t r o l v a r i a b l e s , i t becomes u s e f u l t o d e r i v e t h e s w i t c h i n g t i m e s by a n a l y t i c a s w e l l a s e m p i r i c a l means. The problem t h e n i n summary t u r n s o u t t o b e a s f o l l o w s ( a ) t h e r e e x i s t s a d i f f u s i v e problem, d e f i n e d f o r a composite g e o r e t r y of two m a t e r i a l s : o x i d e and b u l k s i l i c o n . ( b ) t h e geometry of b o t h m a t e r i a l s i s changing a t a r a t e d e t e r m i n e d by two r a t e s of o x i d e growth (assuming o t h e r v a r i a b l e s e .g. t e m p e r a t u r e remain c o n s t a n t ) ( c ) t h e d i f f u s i o n c o n s t a n t s , which a r e d i f f e r e n t f o r t h e c a s e of o x i d e from t h a t of t h e s i l i c o n , a r e n o n - l i n e a r l y dependent on c o n c e n t r a t i o n . A t t h e o x i d e / s i l i c o n i n t e r f a c e t h e i m p u r i t y c o n c e n t r a t i o n s a r e unequal ( u n l i k e t h e h e a t d i f f u s i o n c o u n t e r p a r t ) and d e f i n e d by a s e g r e g a t i o n c o e f f i c i e n t ( 2 , 3 ) ( d l t h e c u r r e n t s t a t e of 2D and 3D m o d e l l i n g of t h e d i f f u s i v e p r o c e s s does n o t a l l o w
.
E. M. S t a f f o r d and G. Dowrick
an a n a l y t i c d e r i v a t i o n of the d i f f u s i o n constants: these have t o be i d e n t i f i e d empirically ( 4 ) . (e) t h e r e i s no means a t present of on-line monitoring of impurity p r o f i l e s . ( f ) v a r i a t i o n s i n the sequence timing of the two oxidation procedures have most e f f e c t on shallow junction devices, with f i n e geometries, as f o r example i n high frequency t r a n s i s t o r s (of the order of 10 GHz). Such s e n s i t i v i t i e s a r e p a r t i c u l a r l y pronounced f o r high power devices. For t h i s highly non-linear problem, the major aim i s t o propose an i d e a l s e t of switching times f o r the a l t e r n a t i v e forms of oxidation. I n the following treatment, the extended d e s c r i p t i o n of d i f f e r e n t i a l operators i s applied t o d i f f u s i v e systems f o r combinations of boundary conditions. The a p p l i c a t i o n above r e l a t e s t o the d e s c r i p t i o n of d i f f u s i o n i n a composite slab: hawever the extended operator method of a r r i v i n g a t modal c o e f f i c i e n t s i s considered t o be useful i n the a n a l y s i s of many d i f f u s i v e systems, of f i x e d o r v a r i a b l e geometry. 2.
DERIVATION OF THE EXTENDED OPERATOR: AN EXAMPLE
The extended operator i s obtained by i n t e g r a t i o n by p a r t s , i n much the same way as the d e r i v a t i v e of a l i n e a r functional i s defined. Following t h i s treatment which i s d e t a i l e d i n the appendix and a l s o i n reference (5,6), f o r the h e a t conduction equation, we can e s t a b l i s h an equivalence between two s e t s of system d e s c r i p t i o n , f o r example:
+
inhomogeneous boundary conditions
+ homogeneous boundary conditions
The domain of t h e extended operator version of t h e d i f f u s i v e system i s seen t o c o n s i s t of functions with homogeneous boundary conditions. I n c o n t r o l problems, where a s p e c i f i e d s t a t e of a d i s t r i b u t e d system i s required by manipulat i n g t h e boundary conditions, a much simpler formulation is p o s s i b l e when the extended operat o r formulation i s used (7,8). Additionally i f feedback c o n t r o l i s applied t o t h e boundary conditions, the open and closed loop eigenvalues and eigenfunctions d i f f e r considerably: t o t r u n c a t e an i n f i n i t e s e t of open loop modal c o e f f i c i e n t s , and t o compute t h e subsequent closed loop behaviour based on these c o e f f i c i e n t s i s t o approximate the problem badly ( 7 ) .
3.
EXTENDED OPERATOR TABLE FOR I D DIFFUSION EQUATION
It may be noted t h a t everything derives from a knowledge of the Green's function f o r a nonhomogeneous equation with homogeneous boundary conditions; t h i s e s t a b l i s h e s t h e equivalence immediately s t a t e d above. There now follows a t a b u l a t e d l i s t of t h e extended operator form of t h e d i f f e r e n t i a l operator.
( t h e d i f f u s i o n equation), f o r various s e t s of boundary conditions.
4.
(Table I)
EXPANSION COEFFICIENTS FOR DIFFUSION EQUATION
I f an eigenfunction expansion t o an inhomogeneous d i f f u s i o n equation, of form
Applications of Extended Operators t o D i f f u s i v e Systems
i s attempted, then i t i s e a s i l y shown t h a t t h e expansion c o e f f i c i e n t s a l a2 i n f i n i t e expansion
... e t c .
i n the
y i e l d a r e l a t i o n of t h e form
The same form of expansion h o l d s f o r t h e t h r e e dimensional d i f f u s i o n equation. This expansion c o e f f i c i e n t equation i s d i r e c t l y o b t a i n a b l e from t h e extended o p e r a t o r equations (Table I ) , g i v i n g Table 11, s o t h a t providing t h e a p p r o p r i a t e eigenvalues and f u n c t i o n s a r e used ( t h o s e s a t i s f y i n g t h e homogeneous boundary conditions) t h e extended operat o r provides a u n i f i e d method of providing a truncated s e t of expansion c o e f f i c i e n t s s u i t a b l e f o r s i m u l a t i o n f o r c o n t r o l purposes, ( 9 ) . 5.
HEAT DISTRIBUTION I N A COMPOSITE SLAB
Consider t h e case of a s l a b of h e a t conducting m a t e r i a l , of f i n i t e l e n g t h , f i n i t e h o r i z o n t a l t h i c k n e s s , and e f f e c t i v e l y i n f i n i t e width (Fig. 3a). The boundary c o n d i t i o n s a r e of form u ( o , t ) = f o ( t ) : u(9.,t) = f ! (x) Kncwing t h e c o n d u c t i v i t y ( a 2 ) i t i s p o s s i b l e t o w r i t e an extended o p e r a t o r d e s c r i p t i o n of t h e system, f o r a u n i t width.
.
Now consider a second s l a b of o v e r a l l l e n g t h 112, conductivity B 2 , placed end-on t o t h e f i r s t s l a b , (Fig. 3b). The extended o p e r a t o r d e s c r i p t i o n of t h i s second s l a b may be p u t i n terms of t h e conditions a t each end, F o ( t ) and F l ( t ) , and transforming t h e p o s i t i o n of t h e i n t e r f a c e t o x = 0 (Fig. 3 c ) , t h e two extended o p e r a t o r equations apply f o r t h e temperature u, v i n t h e respective slabs.
&! at
=
fi + a26'(x+9. 1) f 0( t ) - a 2 6 ' ( x ) f l ( t ) ax2
A non d i m n s i o n a l equation can be now derived over t h e range -1
where
S
"
"a
=--
The i n t e r f a c e conditions
now become
and 5 i s t h e normalised range.
f
u = v
2
5
Is
s e e Fig. Id, giving
E . P I . S t a f f o r d and G. Dowrick
where
M
=
I2 -
I n making t h i s t r a n s f o r m a t i o n we f o l l o w t h e p r o c e d u r e of Concus and O l l a n d e r ( l o ) , who p r o v i d e e i g e n f u n c t i o n s o l u t i o n s t o t h i s problem. The aim h e r e i s t o s e e whether t h e extended o p e r a t o r m a n i p u l a t i o n s s u c c e s s f u l l y l e a d t o t h e same r e s u l t . Examining
where L i s t h e o p e r a t o r
a - - , and a~ ac2 -
L* may be d e r i v e d as
[
y i s an a s y e t undefined f u n c t i o n , t h e a d j o i n t o p e r a t o r +
$1
[see
(hppendix)]
where t h e c o n j u n c t terms a r e chosen t o make a z e r o c o n t r i b u t i o n .
1 s employed, where w(r,-1)
via
Then t h e r e l a t i o n
= f o ( ~ ) ;W ( T , S ) = El ( T ) ; W(T,O-) = w ( T , o + )
The extended o p e r a t o r e q u a t i o n i s now f i n a l l y s e e n t o b e
F i n d i n g t h e b i o r t h o g o n a l s e r i e s @;:Sj, $ i ( ~ ) of f u n c t i o n s s a t i s f y i n g t h e a p p r o p r i a t e e i g e n f u n c t i o n e q u a t i o n , t h e Green's F u n c t i o n i s found t o b e
and t h e r e f o r e t h e s o l u t i o n found t o be
w i t h t h e a p p r o p r i a t e meaning a s e a r l i e r d e f i n e d g i v e n t o t h e symbolic f u n c t i o n s . a g r e e s w i t h t h e a l t e r n a t i v e d e r i v a t i o n ( 10)
.
This r e s u l t
E q u a t i o n s i n t h e modal c o e f f i c i e n t s may be a l s o o b t a i n e d v i z
dam + dr
A2 am(r)
m
- A
sin A
(T)
m -s sin X s m
where
s o t h a t t h e Kitamori method ( 9 ) i s seen t o f o l l o w n a t u r a l l y i n t h i s c a s e a l s o . The d e r i v a t i o n s of such modal c o e f f i c i e n t e q u a t i o n s a r e thought t o be u s e f u l i n t h e c a s e of moving boundary d i f f u s i o n s y s t e m s , and a r e c u r r e n t l y b e i n g a p p l i e d i n t h e semi-conductor d i f f u s i o n example r e f e r r e d t o e a r l i e r . There i s a l s o a s t r o n g r e l a t i o n between t h e d i s t r i b u t e d forms of Krons' t e a r i n g method , e . g . r e f . ( I 1) and t h e development i n t h i s s e c t i o n . The major d i f f i c u l t y i n t h e moving boundary c a s e i s t o c o r r e c t l y s p e c i f y t h e i n t e r f a c e c o n d i t i o n s
6.
FOURIER APPROXIMATIONS TO PERIOD
I t i s p o s s i b l e t o i n s e r t an a r t i f i c i a l p e r i o d i c i t y i n t o t h e extended o p e r a t o r e q u a t i o n , t h u s c a u s i n g t h e a n a l y t i c s o l u t i o n t o b e made p e r i o d i c , w i t h o u t u p s e t t i n g t h e c o n d i t i o n s o v e r the f i n i t e r a n g e of i n t e r e s t . T h i s is p a r t i c u l a r l y u s e f u l i n t h e c a s e of f i r s t o r d e r s p e c i a l o p e r a t o r s , where no e i g e n f u n c t i o n s e x i s t , and where no g u i d e l i n e s may b e used t o choose c o n t i n u o u s f u n c t i o n s t h a t might a s s i s t G a l e r k i n approximation methods. Once such
A p p l i c a t i o n s o f Extended O p e r a t o r s t o D i f f u s i v e Systems s y s t e m s a r e known t o b e p e r i o d i c , t h e n s i n u s o i d a l approximating f u n c t i o n s p r o v i d e a measure of modal independence, and hence s i m p l i c i t y i n t h e e x p a n s i o n c o e f f i c i e n t e q u a t i o n s . Thus an e q u a t i o n
becomes
where k , i n p r a c t i c e , h a s t o be a d j u s t e d t o a l l o w f o r t h e i n i t i a l c o n d i t i o n u ( o , t ) = a ( t ) : t h i s means t h a t t h e nodes a r e n o t i n d e p e n d e n t , as i n t h e p r e v i o u s e i g e n f u n c t i o n a n a l y s i s of the simple d i f f u s i o n equation (7,8). However, s e t t i n g a s i d e c l a s s i c a l e i g e n f u n c t i o n a n a l y s i s , i t i s p o s s i b l e t o pose t h e h e a t e q u a t i o n i n s i m u l t a n e o u s f i r s t o r d e r form
where u
u2 r e q u i r e boundary c o n d i t i o n s , g i v e n a t e i t h e r end of t h e s p a t i a l domain.
T h i s h a s been computed (12) and i s found t o be s u c c e s s f u l f o r an o r d i n a r y boundary c o n d i t i o n and a d e r i v a t i v e boundary c o n d i t i o n , b u t becomes more c o m p l i c a t e d f o r p u r e l y d e r i v a t i v e , o r p u r e l y f u n c t i o n v a l u e s , boundary c o n d i t i o n s . The same approach c o u l d a l s o b e made t o t h e flow d i f f u s i o n e q u a t i o n
T h i s e q u a t i o n s u g g e s t s t h a t one boundary c o n d i t i o n i s t h e v a l u e of ul a t t h e s p a t i a l boundary, w h i l e a n o t h e r i s u2 i.e. a u l + a u l / a x a t t h e boundary. T h i s i s i n f a c t a n a t u r a l boundary c o n d i t i o n f o r t h e system, when flow p l u s d i f f u s i o n i s t a k i n g p l a c e : n o t , a s o f t e n i n t h e l i t e r a t u r e , a boundary c o n d i t i o n au/ax = b ( t ) . The b a s i c e q u a t i o n o f f o u r t h s p a t i a l o r d e r and same o r d e r i n time can a l s o b e p u t i n t o t h i s c a n o n i c a l form, and i t i s h e r e n o t e d t h a t o n l y c e r t a i n combinations of boundary c o n d i t i o n s e x i s t i n p r a c t i c e , r e f . ( 1 3 ) . T h i s i s e x p e c t e d t o r e l a t e t o t h e work of Evans and o t h e r s on system s t r u c t u r e s ( 1 6 ) . The u n i f y i n g f u t u r e of t h e e x t e n d e d o p e r a t o r w i t h a r t i f i c i a l p e r i o d i c i t y i s p a r t i c u l a r l y c o n s t r u c t i v e i n t h e c a s e of boundary c o n d i t i o n s s p e c i f i e d a t e i t h e r end of t h e s p a t i a l domain. Recent work u s i n g e x t e n d e d o p e r a t o r f o r m u l a t i o n h a s been used t o d e t e r m i n e t h e o p t i m a l p o s i t i o n of h e a t s o u r c e s i n a d i f f u s i v e h e a t s y s t e m , employing attainability/contro?ability c o n c e p t s ( 1 7 ) : s i t i n g of o b s e r v e r s e n s o r s may be s i m i l a r l y d e r i v e d .
CONCLUSION P a r a l l e l r e s u l t s t o t h o s e o b t a i n e d i n d i f f u s i v e systems u s i n g c l a s s i c a l a n a l y s i s can b e o b t a i n e d u s i n g t h e e x t e n d e d o p e r a t o r . I n t h i s r e s p e c t t h e p a p e r i s p r e s e n t e d w i t h much t h e same p h i l o s o p h y a s t h a t of r e f . ( 1 8) A t a b l e of e x t e n d e d o p e r a t o r s f o r d i f f e r e n t boundary c o n d i t i o n s is immediately a p p l i c a b l e t o a s e t of chosen b a s i s a p p r o x i m a t i n g f u n c t i o n s including eigenfunctions. A r t i f i c i a l p e r i o d i c i t y introduced i n t o extended operators favours a G a l e r k i n - F o u r i e r e x p a n s i o n , which can r i v a l e i g e n f u n c t i o n a n a l y s i s whenever t h e e i g e n functions are d i f f i c u l t t o derive.
.
I n d i s t r i b u t e d feedback c o n t r o l , t h e open and c l o s e d l o o p e i g e n f u n c t i o n problems can o n l y be p r o p e r l y d e s c r i b e d i n a n o p e r a t o r , and most c o n v e n i e n t l y an e x t e n d e d o p e r a t o r s e n s e . Match-
E . M. S t a f f o r d and G. Dowrick
i n g of boundary c o n d i t i o n s a t t h e j u n c t i o n s of d i s s i m i l a r b u t l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n systems a p p e a r s t o b e a i d e d by extended o p e r a t o r methods.
APPENDIX
-
DERIVATION OF THE EXTENDED OPERATOR
-d2/dx2 b e t h e d i f f e r e n t i a l o p e r a t o r and S t h e s e t of f u n c t i o n s i t a c t s on. 1 1 S = {u(x) : u 2 ( x ) d x < +(D} , S O t h a t t h e i n n e r p r o d u c t < u,v > = uv dx, V u , v E S i s w e l l
Let L
5
I
I
0
0
= 5 s h a l l have a s o l u t i o n , i t i s n e c e s s a r y t o defined. In order t h a t the equation L i n t r o d u c e t h e n o t i o n of t h e a d j o i n t o p e r a t o r L*.
The a d j o i n t o p e r a t o r L* i s d e f i n e d by
Vu, v E S. L e t M , M* b e t h e domains of L, L* r e s p e c t i v e l y . I f v E M*, u E M t h e n b o t h s i d e s of [I] a r e d e f i n e d . However, what happens i f u i s r e p l a c e d i n [I] by a f u n c t i o n w E S which i s n o t i n M? Then < L* ,w > i s s t i l l d e f i n e d , b u t < v,L w > i s n o t . The i d e a of an extended o p e r a t o r means t h a t we wish t o f i n d a meaning f o r a p p l y i n g t h e o r i g i n a l o p e r a t o r t o a s e t of f u n c t i o n s n o t i n i t s domain.
A meaning (and d e f i n i t i o n ) i s r e a d i l y a v a i l a b l e from t h e above, simply by p u t t i n g < L*v, w > = < v , Le w > where L i s now t o be understood i n i t s e x t e n d e d s e n s e ( w r i t t e n L e ) , Also, s i n c e by r e l a t i n g i t t o something which i s a l r e a d y w e l l - d e f i n e d i . e . < L*v, w > < L*v, w > w i l l b e a c o n t i n u o u s l i n e a r f u n c t i o n a l , t h e n we may u s e t h e s e t of f u n c t i o n s v a s t e s t i n g f u n c t i o n s f o r Le.
.
I t now remains t o f i n d Lew, and by way of an example we s h a l l r e t u r n t o t h e o r i g i n a l problem. Equation 2 i s rewritten
v l L e w dx =
-
i
v
w dx - ; ( I )
w(1) + ;(o)
w(0)
~ [ 2 ]
S i n c e t h e right-hand s i d e i s a continuous l i n e a r f u n c t i o n a l i t d e f i n e s a symbolic f u n c t i o n Lew. E q u a t i o n 2 i s r e w r i t t e n
REFERENCES Butkovsky, D i s t r i b u t e d C o n t r o l Systems ( t r a n s l . ) E l s e v i e r (1969).
1.
A.G.
2.
J . Huang, L.C. W e l l i v e r 'On t h e r e - d i s t r i b u t i o n of boron i n t h e d i f f u s e d l a y e r d u r i n g thermal o x i d a t i o n ' J . Electrochem Soc 117 (1970).
3.
S. M a r g o l i t , A. Neugroshel, A . B a r l e v , 'The r e d i s t r i b u t i o n of Boron and Phosphorous a f t e r t h e second o x i d a t i o n s t a g e and i t s thermal e f f e c t on MOST t r a n s i s t o r p r o p e r t i e s ' I n s t i t u t e of P h y s i c s Conference, L a n c a s t e r U.K. S e p t . (1971)
4.
C.R.
5.
B . Friedman, P r i n c i p l e s of Applied Mathematics, Wiley ( 1 9 5 6 ) .
.
6.
J e s s h o p e , E . J . Z a l u s k a , H.A. Kemhadjian, Comparison of numerical s o l u t i o n methods f o r two dimensional b i p o l a r t r a n s i s t o r a n a l y s i s a l g o r i t h m . E l e c t r o n i c s L e t t e r s 11,14(1975).
W.L. Brogan,
Theory and A p p l i c a t i o n of Optimal C o n t r o l f o r D i s t r i b u t e d Parameter Systems.
Automatics 4 (1967).
ILD~I
7.
D.M. Wiberg,
Feedback c o n t r o l of l i n e a r d i s t r i b u t e d s y s t e m s .
8.
E.D. G i l l e s ,
Systeme m i t V e r t e i l t e n Parametern, Oldenbourg V e r l a g
9.
T. K i t a m o r i , ' T r a n s f o r m a t i o n of d i s t r i b u t e d parameter systems i n t o lumped parameter systems f o r t h e s t u d y of o p t i m a l c o n t r o l ' Paper 24(a) IFAC London (1966).
T r a n s A.S.M.E.
' T r a n s i e n t D i f f u s i o n i n a Composite S l a b ' 10. P. Concus, D.R. Olander, T r a n s f e r I1 pp.610-613 (1970).
1973
.
89 (1967).
I n t . J . Heat Mass
'The p i e c e by p i e c e s o l u t i o n of e l l i p t i c boundary 1 1 . G . J . Rogers, G.K. Cambrell, problems' J . Phys. D: Appl. Phys. 8 (1975).
value
A p p l i c a t i o n s o f Extended O p e r a t o r s t o D i f f u s i v e Systems 'A G a l e r k i n - F o a r i e r method o f s o l u t i o n of d i f f u s i v e s y s t e m s ' C h a p t e r i n 12. E .M. S t a f f o r d , P r o c e e d i n g s of I n t . Conf. on V a r i a t i o n a l Methods i n E n g i n e e r i n g ( e d . B r e b b i a ) S.U.P. 1973. 13. F. Church, Mechanical V i b r a t i o n s (Appendix B: C h a r a c t e r i s t i c s f u n c t i o n s of beams,) Wiley ( 1 9 6 0 ) . 14. F.J. Evans, J.J. Van Dixhoorn ( e d s ) 1974.
P h y s i c a l S t r u c t u r e s i n System Theory, Academic P r e s s
15. M. Amouroux, On a method of d e t e r m i n i n g t h e o p t i c a l p o s i t i o n o f ' p o i n t s of a c t i o n ' f o r a c l a s s of l i n e a r d i s t r i b u t e d p a r a m e t e r s y s t e m s , C .R. Acad. Sc. P a r i s S g r i e A ( i n F r e n c h ) J u l y (1973). ' D i s t r i b u t e d P a r a m e t e r C o n t r o l Theory a p p l i e d t o t h e W i l d l a n d F i r e S u p p r e s s 16. W.L. Brogan, i o n Problem. IFAC Symposium on t h e C o n t r o l of D i s t r i b u t e d P a r a m e t e r S y s t e m s , B a n f f , Canada ( 1 9 7 1 ) .
E. M. Stafford and G. Dowrick
TABLE I
EXTESDED OPERATOR (A,)
BOUNDARY CONDITIOKS
u(o,t) = fo(t)
au -
1
at
u(L,~) = fl(t) u'(0.t)
fo(t)
=
au -
2
fo(t)
=
ul(o,t) = fo(t)
au -
4 =
fl(t)
u(0,t)
+
llul(o,t) = f (t)
5
u(0,t)
=
fl(t)
fo(t)
=
6
u(L,t) u'(0,t)
cu'(L,t)
+
=
=
fo(t)
7
u(e,t) + cul(L,t) u(%,t)
=
=
fl(t)
=
f (t)
fl(t)
8 u(o,~) + Sul(o,t) ul(L,t) = fl(t) u(o,~) + Sul(o,t) = f (t) u(o,~)
=
a2
fo(t)
a2a'(x)f0(t)
a26(x)f0(t)
+a26(x-k) f 1 (t)
a26(x)fo(t)
+ a26' (x-!.)
ax2
fi- a26'(x-e)fl(t)
aU -
a2u .2-+al(x-e)a2r1(t)
-
ax2
f 1 (t)
a26'(x)fo(t)
-a2st(x)f (t)
ax2
au at
a '
"at
a2
au at -
10
a2, -
2 .
au at
9
-
(t)
2 e a26' (x) fo(t)
au at
at
fl(t)
fi + a26(x-~)f
ax2
at
u(e,t)
-
a261(x-e)rl(t)
a x2
at
u'(L,~) = f +t)
u(e,t) + cul(e,t)
a2
& - 2.
3
a2 u + ax2
at
ul(k,t) = fl(t) u(o,t)
a2
fi +a26'(x-L)
f,(t) + a26(x)f0(t)
ax2
fi + a26'(x-!L)f
(t)
-
a26' (x)fo(t)
ax2
a2fi+ ax2
a2
f2i -
az6(x-()fl(t)
a26(x)fl(t)
-a26'(x)fo(t)
-
ci26'(x)fo(t)
ax2
ul(o,t) = fl(t)
u(L,t)
=
fo(t)
au -
11 uq(e,t) = f,(t)
at
I
a2fi+a26'(x-i)f0(t) ax2
+a26(x-l)f,(t)
I
Applications of Extended Operators to Diffusive Systems
TABLE I1
DERIVED EQCAT IONS
BOUNDARY CONDITIONS u(o,t> = fo(t>
I u(L,t) ut(o,t)
am
+
am
-
a2$r(l)fl m (t)
+ ~ : a ~+ a2$m(L)f,(t)
-
a2$rn(o)fo(t)
m
+ i2a + a2$i(o)fo(t) m m
-
a2$m(~)f, (t)
am
+ hiam + a2bm(o)fo(t)
-
a2$'(e)f m
am
+
am
+ h2a m m + a2bi(o)fo(t)
+
a2$;(o)fo(t)
fl(t)
=
fo(t)
=
2
ut(e,t) = fl(t) u(0,t)
f (t)
=
3 ~'(a,t)
fl(t)
=
ut(o,t) = fo(t) 4
u(L,t)
u(0,t) + r,uf(o,t) = fo(t) 5
u(k,t)
u(0,t) = fo(t)
u1(o,t)
a2a m m + a2@;(t)f
(t) + a26i(o)fo(t)
+ Sul(L,t) = f,(t)
6
u(e,t)
I (t)
fl(t)
=
-
a2@;(e)f
-
~~@~(o)f~(t)
l(t)
+ cul(e,t) = f ,(t) =
fo(t) m + h2a m m
7 u(e,t)
+ cuf(t,t) = fl(t)
u(e,t)
=
u(0,t)
+ ~u'(0,t)
=
fo(t)
u(0,t)
+ Su'(0,t)
=
fo(t)
fl(t)
8
9
ut(2,t) = f ,(t)
-
a2@A(i)f,(t)
-
a2@,(e)f
am
+ i2a m m
am
+ h2a + a2@m(~)fl (t) + a2@A(o)fo(t) m m
,(t) + a2@;(o)fo(t)
Where Am, @m are the eigenvalues and eigenvectors for the diffusion equation with homogeneous boundary conditions
E. M. Stafford and G. Dowrick
t Impurity concentration
I n type silicon
emitter n-type dominates
I 1 base
7
TzE?
collector
P-type dominates
Fig. I
Impurity profiles for npn transistor
Stage a)
T
time
interface
~
diffusion by two i 2~ Semiconductor . stage oxidation
Applications o f Extended Operators t o D i f f u s i v e Systems