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

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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