cafa: a computer-aided curve-fit and … a computer-aided curve-fit and analysis program ap3 94 by...
Post on 29-Apr-2018
239 Views
Preview:
TRANSCRIPT
CAFA: A COMPUTER-AIDED CURVE-FIT AND
ANALYSIS PROGRAM Ap3 94
by
B. W. Noel
Bureau of Engineering ResearchThe University of New Mexico
Albuquerque, New Mexico
Technical Report EE-187 (71)ONR-005
October 1971
SD CLAIMEI NO)TICE
THIS DOCUMENT IS BEST
QUALITY AVAILABLE. THE COPY
FURNISHED TO DTIC CONTAINED
A SIGNIFICANT NUMBER OF
PAGES WHICH DO NOT
REPRODUCE LEGIBLY.
ACKNCILEDGE14ENTS
The author wishes to thank H. D. Southward and C. E. Barnes
for helpful comments and criticism and S. D. Noel for technical
assistance.
The work was supported by the United States Atomic Energy
Commission, the Department of Health, Education and Welfare
(N.D.E.A.), and the Office of Naval Research through Project
Themis.
li
ABSTRACT
The theoretical basis for the CAFA program is discussed.
An approximate technique evolving from the theory is applied
to the analysis of the current-voltage characteristic of a
hypothetical diode, with good results. A printout of the re-
sultant program and data is included.
iii
CONTENTS
Page
INTRODUCTION 1
THEORETICAL BACKGROUND 3
AN APPROXIMATE APPROACH 12
REFERENCES 36
iv
LIST OF FIGURES
Figure Page
1 Current-Voltage Characteristic of aHypothetical Diode 17
2 The CAFA Program 19
v
INTRODUCTION
The CAFA (Computer-Aided Fit and Analysis; pronounced
"cafg") program was developed for fitting smooth curves to
experimental current-voltage and similar data and for using
the curves obtained to analyze the data. The original goal
was to allow the slope of such curves to be obtained at any
point in order to assist in determining the current mechanisms
existing in solid-state devices. The program was found to be
useful for several other purposes. One, in particular, allowed
interpolation between data points. The preliminary results
obtained have been encouraging. The program was used exten-
sively to analyze the data discussed in Reference 1.
The subroutine which makes the program possible is the
SMOOTH subroutine originally developed by Reinsch (1967),
adapted for Fortran by R. E. Jones of Sandia Laboratories, and
modified by the author to include interpolation. This sub-
routine fits a series of spline (cubic) functions to the data
points and smooths the transitions between functions by re-
quiring continuity of the first and second derivatives within
a certain error chosen by the user. The first and second de-
rivatives are available as printout in addition to the fitted
data points and the coefficients of each cubic equation. The
latter permitted interpolation between data points. Examina-
tion of the program, Fig. 2, reveals the use of these features.
Several general approaches to determining current mecha-
nisms from the CAFA program are discussed below, along with
the specialized version used in this study.
2
THEORETICAL BACKGROUND
Suppose the current in a diode, 1, is given by
I = I exp(OV/m) , (1)0
where I is constant, 8 = q/kT, V is the applied voltage, and0
m has a constant value.* In this case, the current in the
diode is described by one mechanism and the slope of the
ln(I) vs. V curve is
aln(I) = $1 (2)Sm
Since everything in Eq. (2) is known except m, m can be deter-
mined and the current mechanism can often (but not always) be
identified. Unfortunately, total diode currents given by Eq.
(1) usually do not exist in practice. The current in real
diodes is more often given by an expression of the form
I = Ioi exp(OV/mi) + Ioj exp(V/0) , (3)i. J
where mi describes the ith temperature-dependent mechanism
(e.g., diffusion, space-charge region recombination, surface)
and 0. describes the jth nontemperature-dependent mechanism
(e.g., tunneling). Even Eq. (3) does not describe the most
general current form because it neglects nonlinear effects,
such as interactions between current components, plus it
*To be perfectly general, "current" should be replaced by"flux" and "voltage" should be replaced by "force."
3
assumes all the I 's are constant and that V = Vi, the junc-0
tion ,"oltage. Nevertheless, Eq. (2) is often an excellent
approximation. For real diodes, one often does not know what
the current mechanisms are but could deduce the mechanisms if
the mi's in Eq. (3) were known. If, however, the diode cur-
rent consists of only two components and is of the form
I ) I exp(epV) ,(4)
01 m 1 02 m2
then
Dln(I) - 1 1 (5)3V I +m1 m2
and the two values of m cannot be determined easily unless
each current type has a clearly defined region of dominance
and the experimenter has data covering those regions. Even
the simple case of Eq. (5) is not usually seen in practical
diodes. To further complicate matters, the value of m describ-
ing one mechanism may vary; as an example 2 < m < -, depending
on injection level, for donor-acceptor pair (DAP) recombination2
in the space-charge region. A method for finding the value
of m at any point, given a current-voltage curve, is therefore
desirable. A graphical approach suffers on several counts:
It is not accurate enough unless the ln(I) vs. V curve is quite
linear, as will be seen below; and it is extremely tedious and
time consuming to obtain m at many points. The numerical ap-
proach discussed here is quick and gives good results for the
special cases discussed.
4
Given an experimental current-voltage curve in which the
current is presumed to arise from the mechanisms described in
Eq. (3), we assume that the current at any point on the curve
can be written as
I = IO exP(m .) (6)
Thus, our basic assumption is that the experimentally-determined
current given by Eq. (6) is equivalent to the theoretical cur-
rent given by Eq. (3). The parameter that allows Eq. (6) to
describe the current at any point is, of course, m(I). If m(I)
is constant, Eq. (6) reduces to Eq. (1). The slope obtained
from Eq. (6) is
DI3(V\ (7)
But m(I) - m(V) implicitly, since I a V. That is,
a V = a 1 + 1 avW-y V -WmT-T iTT W,
or
[ 1 am(I) DIl+ 1()
WL= m (1) 2 1vJ
The dependence of m on I will be understood unless otherwise
stated.
A differential equation can be obtained that would solve
for m in closed form if a closed form solution exists for the
5
differential equation. By combining Eqs. (8) and (7) we obtain
3I 0I . (9)im + BIV am
By rearranging Eq. (9) we obtain, since am/DI = dm/dI (m is a
function only of the current at constant temperature),
dm+ 1 2 1 0 (10)
U-I ý_Iv Vai/av
This is a very nonlinear d.e. of the form
2
x + -axz =0 ' (11)
where x = x(z) is known and b (= I/Va) is known. The d.e.
might be solvable using numerical techniques.
We shall briefly discuss the errors which result when the
nonlinearity in Eq. (7) is assumed negligible. Considering the
derivative on the right-hand side,
a v V am 17-V (m) -1 -TV•• m '(2m
which is
a v ) (13)
if am/aV is negligible. This requires
6
1 ->> V amm -- TV 'm
so that
amm >> V . (14)
For m i 2, V 1 volt,
1V << 2 units/volt. (15)
The rate of change of m with V must be very small for this
condition not to be violated. Hence, the approximation that
im/3V is negligible is often not valid. This approximation
is numerically identical to that obtained using an incremental
approach which approximates the curve between a series of
closely adjacent data points by a set of straight lines, and
is approximately equivalent to the results which would be ob-
tained using a graphical technique. Therefore, unless a ln(I)
vs. V curve is very linear, a graphical or incremental approach,
or an approach which neglects am/Wv will not suffice for deter-
mining the value of m from an experimental curve.
As a special case of the relationship between Eqs. (3)
and (6) we write the relation as
1exp() = 1 exp( ) + 1 exp(A (16)I 0 ex- 201 i I02 2
The two terms on the right-hand side of Eq. (16) may be con-
sidered as two distinct components or as one distinct component
7
plus a sum of other components, so that the relation is not
necessarily restricted to only two distinct components. This
will be called the "two-process" model. We shall assume Eq.
(16) holds and determine the effective m vs. V for various
ratios R,
R H 101/102 . (17)
R will be constant or nearly so for many situations. Using
Eq. (17) we can rewrite Eq. (16) as
I 02 [R exp1Ei + expjjj- " (18)
The derivative is
[- 1 8 expy.-) + A- exp(E) . (19)aV I02 m 1 m1 m 2 2
Define
f (20)
combining Eqs. (18) and (19) gives
R exp(-) + exp(-)f m 1 m2 (1
R exp (E) + - exp( (±)
Equation (21) can be solved for R:
J8
R 2 exp V( (22)
Since R is constant with V, we must have
aRv 0 (23)
Performing the indicated operations on Eq. (22), realizing that
exp V(m• - L)1 0, and rearranging gives
W m2 1af .(If i) (lf l ) ( 24)
This is the criterion for R to be constant. One way to use
Eq. (24) would be to find f(V) (from the results of curve-
fitting to the I-V characteristic using the SMOOTH subroutine,
since this gives aI/aV also), curve fit to the points f, find
af/3V from the curve fit and plot the right-hand side and
af/DV vs. Of on the same curve, using mI and m2 as parameters;
one could inspect the results to find integer values of m1 and
m2 such that the curves intersect. Another way would be to
find a second expression describing af/WV. By differentiating
Eq. (20) directly, we obtain
3f I I1- I a 2 (25)
7-V (alay)2 = I(•z/av)
Since the second derivatives can also be obtained from the
curve fit of I vs. V, we have enough information to find af/aV;
9
that is, if the second derivatives are reasonably well be-
haved. A third method is to rearrange Eq. (24), if af/WV can
be found, to obtain
"m" +m 2 1 2 •12 -Sf 2 V~m + 2)- 4mlm2(lI - ••) (26)
Both sides of Eq. (26) can be plotted vs. V with mI and m2 as
parameters. The solutions would be obtained at the curve inter-
sections. For the special case when af/aV = 0, Eq. (26) re-
duces to
1f = ml or m2 (27)
Thus, one of the m's can be found if af/aV = 0 somewhere. If
f/•V -= 1,
Bf a m1 + m2 or 0 . (28)
If 3f/aV = 1 and 8f p 0, then
m 2 = Of - m1
if m1 was found at a point where 3f/aV = 0. Still another ap-
proach uses f in the differential equation (10). Using the
definition of f, Eq. (10) becomes
V dm• = m - -2 (29)
If Of is a determinable function, this can be solved to get a
family of curves for various V's. Even if f is not a
10
determinable function, the expression (26) could be substituted
into the d.e. and then one would have a d.e. in terms of m, mi,
and mi2 , so that m could be predicted given mI and m2 (a trial-
and-error approach). For the special case when 9f/aV - 1, the
d.e. becomes
dm 1 (m m(30)I m1--•-2) (0
where we have used Eq. (28) with af • 0. This can be solved,
yielding
2 2 m3 3
V n(Vo} (31)
where mo is a known value of m at V = Vo. This transcendental
equation may also be solved graphically.
A somewhat simpler and less tedious approximate approach
to determining mi1 and mi2 has been developed and will be dis-
cussed next. The approach is useful in a transition region or
any region where superposition of two or more current compon-
ents results in some curvature of the experimental ln(I) vs. V
curve.
11
AN APPROXIMATE APPROACH
If the two-process model, Eq. (16), is valid, and if the
ln(I) vs. V curve is nonlinear, one may obtain a pair of equa-
tions (24) for each two adjacent data points. If the curve is
nearly linear, the pair of equations thus obtained may not be
linearly independent or may not have a solution, so that the
procedure described here will not work in such a region. If a
linearly independent pair of equations is obtained, each pair
has a unique solution from which the m's can be determined ap-
proximately. Since each such pair has a unique solutionwhether or not the two- or one-process model is valid, values
of m can be determined, but these may not be the integer values
expected. We define
af
G 1-• , (32)av
RPM m 1 , (33)m1 2
(the reciprocal product of m's), and
SRM = 2 (34)m1m2
(the sum of reciprocals of m's). These symbols are useful com-
puter words. Let I and I + 1 be adjacent data points and look
for the Jth solution of the pair of equations obtained from
Eq. (24). To be compatible with computer language, let 6f - BF
12
2
and (Bf) * BF2. Also rewrite Eq. (24) as
G - = (ml m 2 1f - 1 02
Then
BF(I)SRM(J) - BF2(I)RPM(J) - G(I) (36)
and
BF(I+I)SRM(J) - BF2(I+I)RPM(J) = G(I+I) (37)
to a good approximation. The solution of this pair of equa-
tions is
= BF2(I+1)G(I) - BF2. (I)G(I+l) (38)SRM(J) = BF(I)BF2(I+I) - BF(I+I)BF2(I)
and
RPM(J) = BF(I+I)G(I) - BF(I)G(I+l)BF(I)BF2(I+l) - BF(I+I)BF2(I)
Letting XM2 H m2 and XM1 - mi.
SRM (J) - 1 S- 4PM(J) (40)
XM2 (J) 2RPM(J) 2RPM(J) -ESRM(J(40
and
XMl(J) SRM - XM2(J) (41)
Once the m's have been obtained, there are nine possible combi
nations (which will not be enumerated) of results, wherein m1
13
and m2 are constants of correct value (i.e., integers), con-
stants of incorrect value, or variables. If one of the m's = 1,
the current component may be injection current. If so, its
correct 10 can be found as described below. If the other m is,
say, 2, where the 2 is found to describe space-charge region
recombination current, the 10 found for it may vary, because
1 for such current is, in general, injection-level dependent.
This will complicate the results somewhat, but judicious in-
spection may help in deciphering the results. For any case
where one of the m's is not constant, the value of 10 obtained
may be treated as a "subtotal" current of the form
, 8V= Ioi exp(d-) , (42)
where m = m(I). It is possible to treat this subtotal current
in the same fashion as the total current: Curve fit to it, as-
sume two current mechanisms and break it up into mi' and mi2 ',
as before, continuing until the currents have been resolved
satisfactorily. The foregoing also applies to any case where
one of the m's is constant but incorrect.
In any case, when values of m1 and m2 have been obtained,
we write
11 exp(-) + I exp(-) ( (43)01 e 1 02 in 2
For two adjacent interpolated data points J and J+l, we have
14
1(J) = I 0 1 (J) exp( m--- + I 0 2 (J) exp( m- ) (44)
and
expBy (Ji-l) expV( +l)'
I(J+l) = I 0 1 (J) exP\ m ) + I 0 2 (J) exp .) (45)01m1 02 ( M2
to a good approximation, because the Ioi'S should be nearly
constant between two closely adjacent data points. This pair
of simultaneous equations can be solved for 1 0 1(J) and 1 0 2 (J).
If 10 1 (J O 1 0 1 (J+l), the answers are exact (this will be the
case for mi = 1, for example) if the m's are exact. If
I 0 1 (J) # I 0 1 (J+l), it does not matter, since a better value
will be found on the second iteration. The solutions of Eqs.
(44) and (45) are
I(J) exp/ -$A I (J + 1)
I 0 1 (J) = expaVW(J) ) [exp(8AV(I..2 _ Ai))] (46)
and
I(J + 1) - I(J) expm•-)
1 0 2 (J) = exp V'(J)) [exp2(1AV 1 (47)
where
AV : V(J + 1) - V(J) (48)
15
and we have assumed m1 and m2 do not change much between J and
J+ 1.
To demonstrate two of the approaches which have been used
successfully and to test their validity, data from a hypotheti-
cal diode were analyzed by the program. This diode had mI =
m = 2, 1001 units and 102 = 1 unit. Its I-V character-
istic is plotted in Fig. 1. The resultant computer printout
is given in Fig. 2.
In the lowermost and uppermost portions of the curve,
where m2 and mi, respectively, dominate, Eq. (2) is an excel-
lent approximation. Hence, XM or XMPRIM describes the current
mechanisms very well and one could deduce the components easily
for this diode. PPRIM is equivalent to I and it is seen to
give the correct values at either end of the range.
For the transition region near the center of the curve,
the value of m as calculated from Eq. (2) does not give the
correct value, as expected. In this region (roughly from sub-
scripts I = 50 to I = 70), the approach using Eqs. (36) through
(41) finds its application and the values of mI1 and mi2 are
determined with good accuracy in this region. Note also that
Eq. (27) can be used to find m1 and mi2 at the two extremes,
since af (called BFP) z 1 at either end of the curve. The ex-
perimental results from real diodes usually do not show large
regions of linearity in which Eq. (2) is useful, so that the
utility of the latter approach in regions of curvature becomes
very apparent. This approach does not work well in the linear
16
40 1 ,
HYPOTHETICAL DIODE
35
30)I-:
I-=
4
25 m=1C
zw: 20--
wa0-015-
IL
0
Z m-2t 10-
5-
I I I I I I I'o 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
DIODE VOLTAGE, V, volts
Figure 1. Current-voltage Characteristic of a HypotheticalDiode
17
regions, as discussed earlier. The values obtained in the non-
linear region for 101 and 102 (called CY01 and CY02 or CY01P
and CY02P in the program) are not as satisfactory as those ob-
tained for m and mi2 . The least error occurs where m1 (or mi2 )
passes through its correct value. In this case the error may
be negligible (less than 1%), but it may be off by a factor of
two or three; however, it is well within an order of magnitude
in t-he worst case, and this would often be adequate. If there
is a linear region where m or mi2 is determined correctly, 101
and 102 will always be calculated within ±20%.
At present, the CAFA program is in a primitive stage of
development. Further work should provide a more useful, ac-
curate and flexible program which will be of considerable
value in experimental research.
18
C u
U)
1 0
04Ga V- s
• -. .4- .. - - -- ,- - 4 1
:- -. - * - W ,
-. * _ "
a ""
.4 2 ...-,• . 2 - " .-: -* :... : . ; .. -'2 - ,. . .,. - 4
.5 t , m : 0-:4 4 a 1 _ =..
-- * a - a . - - a
0 "4
C" \ 5" 21
Z,-
*~~~0 E . . 24.-4
90-
* 4 -- .44~ .4 - * 4.4
* - a x ...s19
NOT REPRODUCIBLE
a . a ~ T
L 4
cr r. c- fi
a1 z ir >
-1 2C
G:
L; L -, L;C
-tr.. -~ 20
- --. 32 I
2 C
- -4 -
- aJ
1
2'�
- ---- ' 1?La .�
a.ft., -�- 4
* L. -
CW1 a - -- �a. - 4 -. =�a. , XL. .3
4 �r - aC *. - - .I �.4 - L. -
� V -2 � -' --
C. - C -�* L.
* Cc. I- a. z - -4- 2
-. '� &. V -. '� L.La.. ac- .- �. V -r.u:�. * 4 4 .24 --
a- � .4* * '�.-2 C X--?C.
.�2Cd� * 2 � S� .12
- - *�4rC('.gP�.�.? - 1� 01=C-*.--. � 4 *4.44% � *% *�*��= **-�4��
-. � - .. C�.*- � U U I *)Z-UUU - *UU - *�...�SUUC..VV
C -.---- U 2 -aN *R. C C. N a *S&atS *h1 2G.fl O.2 44MM a � *e; . r� I�
- a - C q o . -J J ,r � .1 V - C- - - - - - L L L�.U -
21
L tL 4L m
L -c r .;L
I 1* L -2 2
I..LA2 -
M, t I 2
IL t. -.-
- .ac N- tC a: ,-I -
- 0 "% - a c S04I
CC 'o T4 0. i
p a r IL I K
ft I . P. ft 4
CC 1: 1
C I - I 4ft 4
N - - -23
I II NS
* . -"
- - - - -1N •
* NOT RERODUCIBL
--N 24
-~ C
-. - . .x
- - x r- l .a 1M .1..C
c Q. - --
.1 -I - - 4- a 2.o 4G
r~~~ F. lo, *1 -~s~
aN - - v
-C Kh or, cc C2, t
-~~i 46 C 1 x N*C-
- ~ ~ O REPRODUCIBLE 5 aaC -
a C a- a.- ~ a~~h.-c: CC~- a25-
c a :c
CSa.z C... CC Z C C C z C
21~ ~~~~ ZZ..- czcz z c z c
C C .- C
z c-. c z z c
T..S~~~ caa--. ccc c c c
%C 9 t V - r 2~ C cC%, c m C~r
c. -. K r% C. -4 (1,~ c aa 4C.
a. 37 . ,-) cJ cNNc c Cc n cfk-CMM, lN a - 7-- . P.I
--- ~ C &Z c CCoac c ccccc c.
MM S S M SMO S M MS M M W
NOT(d REPRDUCIBL
P~~N a~faW.,NWN~~N4 C ~26a
VN 2.
In -0 2
ct 4 4 I
0 7 1 -~ --.
V, c- cr e.
3. C 2 2 4% & C
V. C -N f'
L 4
Z r C U 2* M 11 0:'..aa.>
7 -T I L 2
I- C . 0 Mj . F.. 4- -S -M 4 2 1. C N.~
I L; am X, 20. >- a. -. Z7
*~~~~4 C C22 4 C *.
L UI L) -ý QC ~ U- L.. U L. L- C -L
2 ~~~~O 4 - Z .. aL SU: -, .. L
2. 1 - 2 *4NW~.. .7 4 2 0 LitN27
* 4
C. --
2 I
N2
-4 I
I-
-2
24 -. - -
1 -. 4, -
- -- I-- 4
� 2 41 - --
- C 4 .2 2
* 4 -- 4 8
- I -- -- I - -.-
1 .N *S -
- %I. �*4 4 4
N - 4 --- --- , 3-.4 -
- �.I -- N- - *-4- �..i -- 4--- 4
-- N - � *--. Nt.. ¶ N-2�X
- I zN>-c 1-�- - �.-.-*-- -
.. ). a - * -4 4
cc-c �IN - -�A' L N444 � N* Z-.- C C�-- -
* I-. - 4, 44 4 4�-� 3' 3.e *�- 4 KR-
Ccc. - -N- 4 ..- 4 4 ---- s - -*r- - �34 *- -4,-,,
I e\- - - - 2.42 I - 4.4�.. - -.
- - - �---. .- --- - I
.. ** 4 - 4* .44 �O,4
N.tU 55 SKEW-------- - ��SUK - .. : m*-a ..
- N N -
L L�L�
�O'
28
Zt , .
C 9
-0 10 -0 -
299
-. c1c:
- ~1 c-N
'o - CC C CC
(xr -rr-ff ýcc c
a c C caC C t .
-~O REPRODUCIBLE-
Sn ~ n M* MS 30'
r - .1 c c z I *a- C. 4
! . .+ - ? I ,• ,-•f, - - I I N
1- 2 7 -
It~ ;r t, cr Cr x
o ° °o o °•4
L L-
C, . z -t --
•% T • : U
=0. c c1' c cz. %
-- N. . .1 .
. .- . . r r. .4 .
31 NOT REPRODUCIBLE
Ut 0-~, Z ~3I
c-CC .C cC cC~.C C C C C C c z cQ cc c c C CC C cC C C C C C CC
a c cc x w N le i ' N C 3' x % 2 4 X' C 7 03 .73 '2 3 3 3IIC CICCNNCC7 -
M cý c% N C m Cl m m c C\ C C\ m m C\ m m C c\ C% C% m m 01 C% M C\ N , C\ C\ m m m m r. C CC 5
z C acccccccc7c
CCkC;ZCOCC3w7-2003t0t3ZC3CZZ3CNNN 7 ZCJ 4- ; -- c :Z -*. '- 7-- 2CCOOC 3CCCCOO ~ O ~ -e-oo-o-cc -,- C- cC-C-I-N - ---
3 -c 7 aa a o a'z acCCaa a a"0C oaC0CC a Cc ac acCC0. ~- -- -- --- -- -- --- -- -- - - q- f - C -C~ C - - S
. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 4~ .C . ........... -..........
r-CCCCC r KC C ICxCCC\ C CC C C 4\C wC cCC CCC C\ QC o
N k1 ý \4,c
e.Z
-05 W er% a C-
I x -M
NNýVV CN 'c 7. -V
:C V q, C A C, 7 C 2C CC C, C ;7 0 C CC CCCC CC CCCC CCC c-C C 7C
c ~ LA& K&. X M- --I \.c. . . . . . ..: r. .
II q3. C N N L c CC N C CC4CW ZV V r
4 Ic c : I K: ,
L;
NOr REPRODUCIBLE 32
C~C CC CC C cc C
NNNq
C -0 -4E
.. ZCC C~c CCCCC:CCCCCCCI CC~ CCCCCCCC C QC C -C C
.L.h . . ... . ... ,. .. .. .. . .h . ..
C% C%~ ft6 \CC ý kf , mf \t ,Nf tN NC ,NMNr %( %C4. C QO C J z ~ C N C CC=C C =CCCCC
-- N \ qt N N O t r 6. 6 6 ~ C ': 1:CL
7 2;w0Z x40~;;ý :
mOW 'C'3 'C v %d x eat\ & !IPCCvZ
B'\-'CNBW - C t-ý'Ntt ftt tt fm
...... ..... .... . . .. . . C..
V ~ B n NrC ft a v %N NC'N &C1 %
C. CT CC Cr CC C C. C'
St C C'
C%~ v 14 j
~I 0I- 7a,,I 'local* C N N C: Q C C-' --
- - - - - -- - - - - - - - - - - - - - -
a C CCzft~iwILN 7 ftp r4 ft
N~Wq LN4C zq ~ N ~ N t N 000000:03C'
33 NO RPODCIL
CCCCCCC. C. C CCCCC cC C C~ .QC CCCCC c C CcC0 Q-~ c. Q CC C
geee. K4Vt tt 7 ;
li s. C sMEeý
a a a CC 00 CCC 2 0 -3 C-CC CC CC CCCCCCCCCCCCCcCCCCC ---------
~NC 9. cC c Q~5 £ C C OcNN c... c c c c C fI c C
e S a, SIn IF c I c
x c cý Cco C0 CC .tS U N E £ £ £" .L NO C c-55 ct = .I(a a-C- 0O- OC- L-0 cC cKOO9.000c Oc 4 4
C:5~ " K , - 11
.C Cý4 .4 .% .0 . % % 9 .~ CNCU CC0CCCCCC
Ca C V,% c er K cv r vc c 9. C,~ .ccW'c a c c e z 4 x-
CZJ~~~~~CC0~w ;Ca C ~ C C 1 £@ U 0 £ 5C NC C 0 C0 oCa0 C CCNC0000C0000W 'C0000C
.. . . .. . -.-. .-.- - -N- - - .- - -. . . . . . . . .
CC~~~~~~~~~~~ -CC C C- C- -OCC -C C -CC C -C - C C C -CC -C C
Z*7zz -z zz9 7 7zz7; 7r-79 ;Z:. . :.Z7. . . . . . . ..7. . . ..ý7277 EZZZ ýZ
C 0C000C00C00£ Caa£aa a a 1 0 ~i 0 .1 a c t.
000..00..00 ..00 ..00 ..00..000.00..00..00..00000 .. .. .OO 0O C C .C .C .C . . U. ..
.44o-----C~-w m zvCZ 4CCC -- -- -- NNetc~£- NC S CN -r-
CC CC CC CC C mcCC C C C @5 £ S C C z m c C ,N . C N W C-~~~~~~~~~~~~~~~ Kew *a C C C C C C C C C C C C OC C C C CC C C CCC C-
-oo# # $ $$ $ ".$$ f lo g s.le s s ,
C ~ ~ ~ ~ ~ f "eCC CCC C C C C C C C c C C C C CC C C CCCCftN~setfeee"eeefteiiion v v v v At% 0 W- in inm r.
----------------------------------- RER DU IL
~ qC -34
• • • • • L °• .. , ° •. ••. CN.5 • oN °~ tN . . (CNN. •. 2
a R$ \- - -
t~~N If I ~~ c I x x t c~ 1,t Cd I"C m -Lx C r x V(c CC- C
11 :z z
cx I:cXCa I: c c c c o
c -c c. C *. .. . . . I .... .. . . . . ... . . . .• .
C\ C, Ný A A .,C - - -- - - - - - - - -1 -- - - -, - -- - -
-.- OCSlllx Cllllllllllllllllllllll
(N 1: ". Nt 11: "
-a - - - - - - - - C -~5 c - - 7
- -PaU B E - ---J . C Z .aN . . ... ..... .... 35 . . ...
ar XN 2 't .2,NN C~- c
StN X N C C N N C IJ X NI t2 -~Z IC 14 aE NCD Cc ' sL
NNNC ~N I - ~~O Ctq Ct 3-~~N~35
C 2C CCC 002 0 II NNNN NNN NNNN SO C (NIC NCC4
REFERENCES
1. ., W. Noel, C. E. Barnes, and H. D. Southward, "NeutronIrradiation Effects on Diffused GaAs Laser Diodes,"' ,.u of Engineering Research, The University of New
MixLcu, Technical Report No. EE-186(71)ONR-005, September
S2. L. W. Aukerman and M. F. Millea, Phys. Rev. 146, 759 (1966).
36
top related