SHALLOW IMPURITIES IN SEMICONDUCTOR QUANTUM WELLS

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SHALLOW IMPURITIES IN SEMICONDUCTOR QUANTUM WELLS

Yia-Chung Chang

To cite this version:

Yia-Chung Chang. SHALLOW IMPURITIES IN SEMICONDUCTOR QUANTUM WELLS. Journal

de Physique Colloques, 1987, 48 (C5), pp.C5-373-C5-380. �10.1051/jphyscol:1987580�. �jpa-00226784�

Colloque C5, supplement

au

nO1l, Tome 48, novembre 1987

SHALLOW IMPURITIES

IN

SEMICONDUCTOR QUANTUM WELLS

YIA-CHUNG CHANG

Department of Physics, University of Illinois at Urbana- Champaign, 1110 West Green Street, Urbana, IL 61801, U.S.A.

Theoretical calculations on the binding energies of shallow impurities (donor and acceptors) in semiconductor quantum wells are reviewed.

I. INTRODUCTION

Shallow impurities play important roles in determining the electronic and optical properties of semiconductors quantum wells and superlattices. Bastardtl]

first attacked this problem by considering a hydrogenic impurity in a quantum well with infinite barrier height. The model predicts that the binding energy of an impurity placed at- the center (or edge) of the well decreases from

4

Ro mono- tonically to 1 Ro (or 0.25 Ro) for well width from zero (two-dimensional limit) to infinity (three-dimensional limit), where Ro, the effective rydberg is the binding energy of a three-dimensional hydrogenic impurity. Hailhoit et a1.[2] and Green and Bajaj[31 independently calculated the energies of the ground state and a few excited states of a hydrogenic impurity in the AlxGal-xAs-GaAs quantum well, using varia- tional method. For donors in realistic quantum wells (with finite barrier height), the impurity binding energy as a function of the well width is found to have a maximum at a critical width around 20-50

A,

instead of increasing monotonically from the bulk value to the two-dimensional limit of 4 R . This is becluse the impurity wavefunction begins to leak out the well material ? ~ a ~ s ) into the barrier material

(A1xGal-xAs) as the well width becomes smaller than the critical width, and the impurity binding energy eventually approaches the bulk value of the barrier material (AlxGal-xAs) (which is smaller than 4 Ro) as the well width goes to zero.

Chaudhuril41 examined the case where a hydrogenic impurity is placed at the center of a AlxGal3As-GaAs multiple quantum well. He found that the impurity binding energy as a function of the well width

(=

barrier width) exhibits a double- peak structure, with the two peaks occuring at well widths of approximately

10A

and

100A.

Various experimental measurements of the electronic levels of donors in AlXGal,,As-GaAs quantum wells have recently been reported. These measurements include photoluminescence[5], Raman scattering][6] and far-infrared magneto- spectroscopy[7]. The photoluminescence detects the free heavy-hole to donor transition and by comparing it with the heavy-hole exciton transition, one can deduce the difference between the donor binding energy and the exciton binding energy. Shanabrook and Comas[5] has found that the donor binding energy is

consistently lower than the exciton binding energy by 1-2 meV for quantum wells with widths between 80-450

_A.

Raman scattering[l] measures the energy separation between the ground state (1s) and the first even-parity excited state (2s). The measure- ments are in good agreement with the theoretical predictions of Mailhiot et a1.[21.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987580

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F a r - i n f r a r e d n a g n e t o s p e c t r o s c o p y [ 7 ] n e a s u r e s t h e e n e r g s e p a r a t i o n betireen t h e i s ground s t a t e and t h e f i r s t o d d - p a r i t y e x c i t e d s t a t e (?p*). T h i s neasurernent i s aoc?

s e n s i t i v e and l e s s i a b i g u o u s t h a n t h e p r e v i o u s : s e a s u r e a e n t s . ?he r e s u l t s a r e fouzd i n zood agreement a i e h t h e t h e o r e t i c a l p r e d i c t i o n s by Greene and 3 a j a j l S i .

S h a l l o w a c c e p t o r s i n quantum v e l l s v e r e f i r s t s t u d i e d e x p e r i m e n t a l l y by X i l l e r e t a 1 . [ 9 ] . They o b s e r v e d t h e p h o t o l u m i n e s c e n c e due : o r e c o n b i q a t i o n of n = l f r e e e l e c t r o n s w i t h n e u t r a l a c c e p t o r s . ay measuring t h e z n e r g y separa:i.cn S e t x e e n t h i s peak and t h e heavy-hole e x c i t o n r e c o n b i n a t i o n p e a k , o w c a n d e c e r n i n e t h e a c c e p t o r b i n d i 2 g e n e r g y t o w i t h i n 2 meV ( t h e u n c e r t a i n t y of t h e t h e o r e t i c a l e x c i t o n b i n d i n g e n e r g y ) . D e t e r m i n a t i o n of t h e a c c e p t o r b i n d i n g e 3 e r g y by photoiumines:ence i s a o r e r e l i a b l e t h a n t h a t of t h e donor b i n d i n g e n e r g y , b e c a u s e t h e f r e e e l e c t r o n t o a c c e p t o r t r a n s i t i o n i s w e l l s e p a r a t e d from o t h e r t r a n s i z i o n s s u c h a s t h e Sound e x c i t i o n r e c o m b i n a t i o n s which o f t e n o b s c u r e t h e f r e e h o l e t o donor t r a n s i t i o n . F u r t h e r m o r e t h e r e l a t i v e l y l a r g e b i n d i n g e n e r g i e s of a c c e p t o r s make t h e u n c e r t a i n t y i n d e t e r m i n i n g t h e f r e e e x c i t o n SFnding e n e r g y a l e s s s e r i o u s pro3lem. T h e o r e t i c a l c a l c u l a t i o n s of t h e e n e r g y s p e c t r a of a c c e p t o r s i n AlxGal-,As-GaAs q u a n t u n r e i l s ,vere r e c e n e l y u n d e r t a k e n by X a s s e i i n k e t . a l . [ l O ] . These c a l c u l a t i o n s t a k e i n t s a c c o u n t t h e c o u p l i n g of che t o p Four v a l e n c e bands of b o t h t h e w e l i and b a r r i e r m a t s r i a l s . The t h e o r e t i c a l r e s u l t s a r e found i n good agreement w i t h t h e a v a i l a b l e e x p e r i m e n t a l d a t a , when t i e v a l e n c e band d i s c o n e i n u i t y between GaAs and Al,cCal-xAs i s caken t o Se 357 of t h e e n e r g y pap d i f f e r e n c e . Tor quantxm w e l l s of w i d t h s c o n p a r a b l e t o t h e a c c e p t o r r a d i u s ( a b o u t 30

a ) ,

t h e a c c e p t o r b i n d i n g e n e r g y is q u i z e s e n s i t i v e t o t h e c h o i c e of t h e v a l e n c e band d i s c o n t i n u i t y . Thus, f o r new quantum w e l l s t r u c t u r e s ( s u c h a s SaAs-GaInAs, GaSb-AlSb, e t c . ) , comparing t h e measured a c c e p t o r b i n d i n g e n e r g y w i t h t h e t h e o r e t i c a l p r e d i c t i o n c a n p r o v i d e a c l u e t o t h e i r v a l e n c e band d i s c o n t i n u i t y .

11. DONORS 1 N QUAYCITUM iELL.5

U i t h i n t h e e f f e c t i v e mass a p p r o x i a a t i o n , t h e H a m i l t o n i a n f o r a h y d r o g e n i c donor i n t h e quantum w e l l i s g i v e n by

where m f ( z ) = m* ( e f f e c t i v e mass of t h e w e l l m a t e r i a l ) f o r z i n s i d e t h e w e l l and

*

^W

aB ( e f f e c t i v e mass of t h e b a r r i e r m a t e r i a l ) f o r z o u t s i d e . V(z) i s t h e q u a n t u n Me11 p o t e n t i a l , which is a c o n s t a n t -V,

-

f o r z i n s i d e t h e w e l l and z e r o o u t s i d e .

v(;) i s t h e Coulomb p o t e n t i a l d e s c r i b i n g t h e i n t e r a c t i o n betsreen a n e l e c t r o n and t h e donor i m p u r i t y . v(:) =

- & ^,

where ~ ( z ) is t h e s t a t i c d i e l e c t r i c c o n s t a n t f o r t h e w e l l o r b a r r i e r m a t e r i a l , d e p e n d i n g on where z is. Because t h e d i e l e c t r i c c o n s t a n t s f o r t h e w e l l and b a r r i e r m a t e r i a l s a r e d i f f e r e n t , image c h a r g e s i n d u c e d by t h e d o n o r i m p u r i t y s h o u l d i n p r i n c i p l e be i n c l u d e d . Such image c h a r g e s have been t a k e n i n t o a c c o u n t i n t h e c a l c u l a t i o n s of M a i l h i o t e t a1.121. Here, f o r s i m p l i c i t y , we s h a l l i g n o r e t h e e f f e c t of image c h a r g e s .

The R a y l e i g h - R i t z v a r i a t i o n a l method i s u s e d t o o b t a i n t h e e n e r g y s p e c t r a of bound s t a t e s a s s o c i a t e d w i t h HD. The e i g e n s t a t e s of

%

( i . e . donor s t a t e e n v e l o p e f u n c t i o n s ) a r e expanded i n t e r m s of l i n e a r c o m b i n a t i o n s of a s e t of b a s i s f u n c t i o n s

}

i e . F = cnSn(:)

.

The e x p a n s i o n c o e f f i c i e n t s Cn and t h e e i g e n v a l u e

Sir,ce che e f f e c t i v e masses f o r c'?e w e l l and b a r r i e r n a t e r i a l s a r e d i E f e r e n t , s p e c i a l boundary c o n d i t i o n s f o r che b a s i s f u n c t i o n s and t h e r e f o r e .P(:) must be s a t i s f i e d

* -

such t h a e ~ h e c u r r e n t J

-

^{Y i) i s} c o n t i n u o u s . F a i l c r e oE c h o c s i 3 g b a s i s

3.X!L)

f u n c t i o n s whLch s a t i s f y che bcundarjr c o n d i t i o n s wo11I.d r e s i l l i i n a n l n p h y s i c a l ?on- k e r m i t i a n Xamiltonian m a t r i x , <Sni5iD/ For e v e n - p a r i t y s t a t e s ( s - l i k e ) X a i i h i o t e t a 1 . [ 2 ] have used G a u s s i a n - t y ? e b a s i s f u n c t i o n s of t h e form:

+ 2 2 2 2

3' ( r ) = a ( z ) exp { - : ( z ) ( x + y + z

) j ,

( 3 )

and a~ f o r 121 < W / 1 -

a ( z ) = 2 1

awexp ((5-1) 3 x < W 1 2 ) j f o r Izl > W!2. ( 5 ) The s p e c i a l r e l a t i o n s between G u a s s i a n e x p o n e n t s and p r e f a c t o r s i n s i d e and o u t s i d e t h e w e l l e n s u r e t h e c o n t i n u i t y of t h e wave f u n c t i o n

%(I?)

and 1

a &(I?).

X i s a v a r i a t i o n a l p a r a m e t e r , which i s a d j u s t e d t o minimize t h e e n e r g y .

E

Green and

B a j a j [ l O ] used a b a s i s s e t which i s c a p a b l e of d e s c r i b i n g t h e donor s t a t e s a c c u r a t e l y f o r b o t h narrow and v i d e w e l l s . However, t h e y c o n s i d e r e d t h e prob- lem i n which b o t h t h e w e l l and b a r r i e r m a t e r i a l s have t h e same e f f e c t i v e mass.

T h e r e f o r e i n t h e z e r o - w i d t h l i m i t , t h e i r r e s u l t s a r e i d e n t i c a l t o t h o s e f o r b u l k GaAs i n s t e a d of A1,Gal-xAs. It is p o s s i b l e t o improve t h e c a l c u l a t i o n s of M a i l h i o t e t a 1 . [ 2 ] and t h a t of Green and B a j a j [ 8 ] by c h o o s i n g a b a s i s s e t c o n t a i n i n g two t y p e s of s t a t e s . Type 1 I s i d e n t i c a l :o t h e b a s i s s e t used by Y a i l h L o t et a l . , i . e .

a(')

(:) a s g i v e n i n ( 3 ) . Type 2 h a s t h e form

2

3(2)(t?, = t n ( z ) e - a ( x 2 + ) where f n ( z ) i s t h e n t h e i g e n s t a t e ( n i s t h e p r i n c i p a l quantum number) of t h e quantum w e l l H a m i l t o n i a n ,

H ( ~ ) ( z )

[- --&- ^a$+

^{~ ( z )}

1.

For c a l c u l a t i n g t h e ground s t a t e , u s e n = 2m ( z )

1. For e a s e i n c o m p u t a t i o n , f n ( z ) i s a g a i n w r i t t e n a s a l i n e a r c o m b i n a t i o n of G a u s s i a n - t y p e o r b i t a l s w i t h p r o p e r boundary c o n d i t i o n s , i . e .

f n ( z ) =

1

Ckak(z) e x p

1-

^{$ ( z ) z} 2

}

where i k ( z ) and a k ( z ) t a k e t h e forms d e s c r i b e d i n k ( 4 ) and ( 5 ) . The c o e f f i c i e n t s

Cl,

a r e o b t a i n e d by s o l v i n g t h e S c h r o d i n g e r e q u a t i o n ~ ( O ) j z ) i n ( * ) = ~ ( O ) f ~ ( z ) .

S o l v i n g ( 2 ) w i t h i n t h e b a s i s s e t d e s c r i b e d above g i v e s t h e e i g e n v a l u e s of t h e H a m i l t o n i a n . With s u f f i c i e n t l y l a r g e b a s i s s e t ( a b o u t 9 G a u s s i a n - t y p e o r b i t a l s f o r e a c h t y p e ) and v a r y i n g t h e p a r a m e t e r X t o minimize t h e e n e r g y ,

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a c c u r a t e e n e r g y s p e c t r a f o r t h e low-lying s t a t e s can be o b t a i n e d . Fig. 1 shows t h e ground s t a t e e n e r g y of t h e center-doped donor i m p u r i t y i n

A1xGal-xAs-GaAs quantum w e l l a s a f u n c t i o n of w e l l w i d t h measured w i t h r e s p e c t t o t h e b o t t o m of t h e f i r s t c o n d u c t i o n subband of t h e quantum v e i l (E:')). The c o n d u c t i o n band o f f s e t i s chosen t o b e 65% of t h e e n e r g y gap d i f f e r e n c e between t h e w e l l and b a r r i e r m a t e r i a l s [ l l ] ; whereas i n p r e v i o u s c a l c u l a - t i o n s [ 2 ] a v a l u e of 85% was used. As a consequence, t h e r e s u l t s f o r x = 0.4 o b t a i n e d h e r e c o r r e s p o n d a p p r o x i n a t e l y t o t h e r e s u l t s f o r x = 0.3 o b t a i n e d i n p r e v i o u s c a l c u l a t i o n s . I n t h i s f i g u r e , t h e dashed c u r v e s a r e o b t a i n e d w i t h t y p e 2 b a s i s o n l y and t h e s o l i d c u r v e s a r e o b t a i n e d w i t h combined b a s i s s e t . For w e l l w i d t h between 20A and 200a, t h e d i f f e r e n c e i n b i n d i n g e n e r g y between t h e two s e t s of c u r v e s i s l e s s t h a n 5%.

4

0 50 100 150 200 250 300

Well Width

(A)

Ground State of Donor in A1,Gal-,As-GaAs

Quantum Well

'&pe I

+

Type 2 basis

Fig. 1. Binding e n e r g i e s of d o n o r s a t t h e c e n t e r of A1,Gal-xAs-GaAs quantum w e l l s a s functions of w e l l width.

111. ACCEPTORS I N QUANTUM WELLS

I n t h e p r e s e n c e of quantum w e l l p o t e n t i a l and Couloum i n t e r a c t i o n w i t h a n i d e a l a c c e p t o r i m p u r i t y , we w r i t e t h e e i g e n s t a t e s of t h e s y s t e m a s l i n e a r c o m b i n a t i o n s of b u l k v a l e n c e band Bloch s t a t e s , v i z .

^.

-

^{+ +}

Y(r) =

I+

Fv(k) e i k * r ~ v , b ( 6 )

v k + +

where

I

v,c> a r e c e l l - p e r i o d i c f u n c t i o n s , c o r r e c t t o f i r s t o r d e r i n t h e k * p

...

Fourier transforms of acceptor envelope functions, F,

(:).

Within the effective-mass approximation, it can be shown that F ) : ( V satisfy the equation

where H , , ,

(0)

(-i V) are matrix elements of the Lut tinger-Kohn Hamiltonian[ 131 H(o)(~+) with k+ replaced by the operator -iV. Note that the Luttinger

parameters Y1,

Y

and

Y

take on two different values for z inside and outside

2 3

the well. Three parameters are listed in Table I1 of

Ref.

14 for all materials of interest here. V(z) is the quantum well potential for holes and v ) : ( is the Coulomb potential for the ideal acceptor.

For semiconductors with spin-orbit splitting

( A )

much larger than the acceptor binding energy, the split-off states (J

=

1/2) may be ignored. This is true for A1xGal-xAs-GaAs and GaAs - Gal-,InXAs systems. For Si-Sil-,GeX systems, which are attracting a great deal of interest recently, the split-off states must be included. In the present paper, we shall only consider systems with large spin-orbit interactions. We shall only use the upper-left

4 x 4

block of the matrix H(O) (c).

To solve (7) for the acceptor spectra in quantum wells, we expand the envelope function F ) ( : + in terms of linear combinations of a set of basis

+ V v +

functions ( B : ( : ) 1, ^{i.e., PV(r)}

⁼

2 Cn Bn(r). The expansion coefficients and the eigenvalue E can be found by solving the secular equation

where H , & : - HE!

^(3.)+

^{[ v ( : )}

⁺

^{) : ( v 1}

^bVv,.

The forms of the basis functions are chosen such that the symmetry properties of the Hamiltonian H(*) are fully exploited. In general, all basis

V +

functions Bn(r) can be written as a radial function fs(r) multiplied by a spherical harmonics. Because the quantum well potential V(z) has a prefer- ential direction, it is advantageous to replace the radial function fs(r) by a ellipsoidal function fS(r1), where r' a 4x2+ y2+ X2

z2

with X to be adjusted to produce fastest convergence. If the best value of A is chosen, then we only need a small number of spherical harmonics to produce good results. We shall only include spherical harmonics of

L = 0,

1, and 2 in our calculation.

For center-doped accept0r.s in quantum wells, the effective-mass Hamil- tonian has a DZd symmetry and the double group irreducible representations are

rgf ^and r7* (both doubly degenerate)[lS], where the superscript

+(-)

denotes

parity. Here parity is a good quantum number, so a

=

1 states are decoupled

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From 2 = 0 and 2 s t a t e s . For t h e

n6+

s t a t e s , which a r e p r e d o m i n e n t l y heavy- h o l e l i k e ( u =

*

^3/2), o n l y s i x t y p e s of b a s i s f u n c t i o n s (one s - l i k e and f i v e d - l i k e ) aeed t o be c o n s i d e r e d . % e s e a r e

where f S ( r 1 ) i s a n i l l i p s o i d a i f u n c t i o n , which we choose t o have t h e G a u s s i a n form,

V

+

^v

+

^V

+

Note t h a t 6

( r

) d o e s n o t c o u p l e d i r e c t l y t o a l ( r 6 ) and E 2 ( r 6 ) [dominent terms 6 6

i n t h e g r o u n d and f i r s t e x c i t e d s t a t e s ] . T h i s t y p e of b a s i s f u n c t i o n i s i g n o r e d i n o u r c a l c u l a t i o n . S i m i l a r l y f o r t h e I'f s t a t e s ( p r e d o m i n e n t l y l i g h t - h o l e l i k e , v = i 1 / 2 ) , o n l y f i v e t y p e s of b a s i s f u n c t i o n s ( d e n o t e d )

v + +

S n ( r 7 ; r ) ; n = 1 ,

. . . ^,

5 ) a r e u s e d . They a r e s i m p l y r e l a t e d t o t h e

rgf

b a s i s s t a t e s by e x c h a n g i n g t h e i n d i c e s :

*

^{312 ++ f 1/2.}

The b a s i s f u n c t i o n s d e f i n e d i n ( 9 ) and ( 1 0 ) do n o t s a t i s f y t h e boundary c o n d i t i o n s a p p r o p r i a t e l y . The c a l c u l a t i o n becomes r a t h e r cumbersome, i f we u s e t h e same method a s f o r t h e d o n o r c a s e t o g e n e r a t e b a s i s f u n c t i o n s which s a t i s f y t h e boundary c o n d i t i o n s . The s i m p l e s t method t o c i r c u m v e n t t h i s d i f f i c u l t y i s t o p e r f o r m t h e c a l c u l a t i o n t w i c e , once u s i n g t h e h t t i n g e r p a r a m e t e r s and d i e l e c t r i c c o n s t a n t a p p r o p r i a t e f o r t h e w e l l m a t e r i a l and t h e o t h e r u s i n g p a r a m e t e r s a p p r o p r i a t e f o r t h e b a r r i e r m a t e r i a l . The two s e t s of r e s u l t s a r e t h e n a v e r a g e d a c c o r d i n g t o t h e p r o b a b i l i t i e s of f i n d i n g t h e h o l e i n s i d e and o u t s i d e t h e quantum w e l l [ l O ] .

Fig. 2 shows b i n d i n g e n e r g i e s o f t h e l o w e s t

r6

and

r7

s t a t e s f o r a c c e p t o r s a t t h e c e n t e r of AlxGal-x.4s-GaAs quantum w e l l s c a l c u l a t e d by

X a s s e l i n k e t a l . [ l O ] a s f u n c t i o n s of t h e w e l l w i d t h f o r x = 0.1 and 0.3. The

rb

^{and ,'l} a c c e p t o r b i n d i n g e n e r g i e s a r e measured w i t h r e s p e c t t o t h e t o p of

E f f e c t s o f c h e m i c a l s h i f t s of v a r i o u s d o p a n t s on t h e a c c e p t o r b i n d i n g e n e r g y i n AlxGal-xAs-GaAs quantum w e l l s a r e a l s o s t u d i e d . The c e n t r a l - c e l l p o t e n t i a l i s modeled by t h e s h o r t - r a n g e p o t e n t i a l Vc

-

^{Uo e}

r o = 1 A and Uo i s a d j u s t e d t o r e p r o d u c e t h e a c c e p t o r b i n d i n g e n e r g y i n b u l k GaAs measured e x p e r i m e n t a l l y . It i s assumed t h a t Uo remains unchanged i n t h e quantum w e l l . Fig. 3 shows t h e I'6 ground s t a t e e n e r g i e s of c e n t e r doped b e r y l i u m and c a r b o n a c c e p t o r s a s f u n c t i o n s o f GaAs w e l l w i d t h f o r x = 0.3.

The p a r a m e t e r Uo u s e d i s -5.55 eV f o r b e r y l i u m and 8.00 eV f o r c a r b o n . The e x p e r i m e n t a l d a t a f o r b e r y l i u m ( o p e n s q u a r e s ) [from Ref. 101 and c a r b o n (open c i r c l e s ) [ f r o m Ref. 91 a r e a l s o shown f o r comparison. Very good a g r e e m e n t between t h e o r y and e x p e r i m e n t i s f o u n d .

I , , , , , , I

2 5 ~ 50 100 150 200 250 300 3%

Well Width (A) ,p-w

OA

^ido 140 2&

$;a L

^3JJ

Well Width (A) Y P - 6 ~ )

Fig. 2 B i n d i n g e n e r g i e s of l o w e s t Fig. 3 B i n d i n g e n e r g i e s o f b e r y l i u m

rb

^and

r7

s t a t e s Eor I d e a l accep- and c a r b o n a c c e p t o r s a t t h e c e n t e r o f t o r s a t t h e c e n t e r of A1,Gal-xAs-GaAs AlxGal-xAs-GaAs quantum w e l l s a s quantum w e l l s a s f u n c t i o n s of w e l l f u n c t i o n s of w e l l w i d t h , open

w i d t h . s q u a r e s : d a t a f o r b e r y l i u m from Ref.

10, open c i r c l e s : d a t a f o r c a r b o n from Ref. 9.

JOURNAL

DE

PHYSIQUE

The a u t h o r acknowledges j.alalable d i s c u s s i o n s i r i t h W . T. Y a s s e l i n k , 2 . M e r l i n , C. H a i l h i o r , H. :.(.ork6c, and K. K. a a j a j . T n i s vork was s u p p . > r r e d i n p a r t by t h e O f f i c e of Naval '+search under C o n t r a c t No. 'i0001i-81-K-3*30,

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