A11103 Qfi7be'=i

,^UNSTOPSTANpABOS.af;;-

V) NBS SPECIAL PUBLICATION 400-48

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

Semiconductor Measurement Technology:

Spreading Resistance Analysis!

for Silicon Layers WitN

Nonuniform Resistivity
NATIONAL BUREAU OF STANDARDS

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and the Institute for Computer Sciences and Technology.

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The National Bureau of Standards was reorganized, effective April 9, 1978.
JUN 4 1979

Semiconductor Measurement Technology:

Spreading Resistance Analysis for Silicon Layers

With Nonuniform Resistivity

David H. Dickey

i

Solecon Laboratories

3001 Red Hill Avenue

Costa Mesa, CA 92626

and

James R. Ehrstein

Electron Devices Division

Center for Electronics & Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Washington D.C. 20234

This activity was sponsored by The Defense Advanced Research Projects Agency

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued May 1979
Library of Congress Cataloging in Publication Data

Dickey, David H. Semiconductor measurement technology.

(National Bureau of Standards special publication ; 400-48) "This activity was supported by the Defense Advanced Research

Projects Agency."

Supt. of Docs, no.: 013.10:400^8.

1. Semiconductors—Testing. 2. Electric resistance, Spreading.

3. Silicon—Electric properties. I. Ehrstein, James R., joint author. II. Title. III. Series: United States. National Bureau of Standards.

Special publication ; 400-48. QC100.U57 no. 400^8 [TK787 1 .85] 602'. 1 s [62 1 .38 1 5'2'028] 79-6070 1

2

National Bureau of Standards Special Publication 400-48

Nat. Bur. Stand. (U.S.), Spec. Publ. 400-48, 72 page,s (May 1979) CODEN: XNBSAV

For sale by For sale by the Superintendent Documents, U.S. Government Printing OfHce ,C. 20402

Washington, D.C. 20402 Stock Number 003-003-02061-3
TABLE OF CONTENTS

Page

Abstract 2

1. Introduction 2_ 2 . Theory 2 2.1 The Boundary Value Problem 2 2.2 An Isolated Layer - Parallel Superposition of

Conducting Sublayers 5

2.3 Layer over Conducting Substrate - Series Superposition of Sublayer Resistances 9

2.4 The Logarithmic Transformation 10

2.5 Local Slope 12

3. Experiment 15

3.1 The Probe Spacing Experiment 15

3.2 Diffusion Profiles 18

3.3 Bevel-Edge Proximity Effect 32

4. Conclusions 32

References 34

Appendix A 35

Appendix B 51

Appendix C 57

Appendix D 59

LIST OF FIGURES

1. Reciplrocal of sampling volume correction factor for an isolated uniform slab plotted vs. normalized slab

thickness 4

2. Multilayer model appropriate for isolated slab, showing

contacts extending through total slab thickness at the

surface - parallel superposition, and at the new surface

after removal of layer thickness At - parallel superposition 6

3. Multilayer model for slab over conducting substrate - series

superposition 6

4. Schematic dependence of conductance and resistance on depth

at v/hich measurement is made below original surface of the

structure 12

5. Schematic representation of slope-dependent terms for the

correction factor F, as derived from limiting case analysis . 13

6. Spreading resistance vs. probe spacing for junctionisolated layers 16

iii
Page

7. Conductivity profile for boron diffusion into nominal

lO-f^'cm n-type substrate. Nominal surface density of

1 q _ o boron, 10^ cm ; junction depth by lap and stain, 0.7 ym . . 19

8. Conductivity profile for boron diffusion into nominal

10-fl*cm n-type substrate. Nominal surface density of

boron, 10^'^ cm ^

; junction depth by lap and stain, 1.6 um . . 20

9. Conductivity profile for boron diffusion into nominal

10-^2* cm n-type substrate. Nominal surface density of

boron, 10-^^ cm ^ ; junction depth by lap and stain, 4 . 3 ym . . 21

10. Conductivity profile for boron diffusion into nominal

150-^2 • cm n-type substrate ' . . . 24

11. Conductivity profile for boron diffusion into nominal

1.0-^2 "cm ri-type substrate 25

12. Conductivity profile for boron diffusion into nominal

. - 0.03-f2*cm n-type substrate 26

13. Conductivity profile for boron diffusion into nominal

100-^2 'cm n-type substrate 27

14. Conductivity profile for phosphorus diffusion into nominal

100-fi*cm p-type substrate 28

15. Conductivity profile for phosphorus diffusion into nominal

1.0-fi*cm p-type substrate 29

16. Conductivity profile for phosphorus diffusion into nominal

0.03-Sl'cm p-type substrate 30

17. Conductivity profile for phosphorus diffusion into nominal

150-fi*cm n-type substrate 31

18. Spreading resistance in the vicinity of the bevel edge ... 33

Al . Schematic conductivity profile for Computer Exercise I . . . 36

A2 . Radial potential distribution on a spreading resistance

contact 36

A3. Schematic conductivity profile for Computer Exercises II

and III 41

Cl . Image array for infinite homogeneous wedge 57

C2 . Sampling volume correction factor for uniform wedge with

local thickness equal to contact radius 58

C3. Sampling volume correction factor for uniform wedge with

local thickness equal to 0.1 contact radii 58

Dl . Spreading resistance data for junction-isolated implanted

specimen illustrating the type of local slope behavior near

the top surface which gives nonphysical values for the cor- rection factor when analyzed by eq (22a) or (22b) 60

iv
LIST OF TABLES

Page

1. Comparison of Diffused Layer Sheet Resistance Values for

Three Boron Diffusions 17

2. Comparison of Diffused Layer Sheet Resistance Values for

One n-Type Diffusion and One p-Type Diffusion 17

Al . Reciprocal Correction, H, for a Single Layer of Thickness

t on Lower-Conductivity, Infinite Substrate 39

A2 . Sampling Volume Correction Factors for Exercise II 42

A3. Sampling Volume Correction Factors for Exercise III 43

A4 . The Ratio of Correction Factor from Parallel Superposition

to that from Schumann and Gardner for Exercise II 45

A5 . The Ratio of Correction Factor from Parallel Superposition

to that from Schumann and Gardner for Exercise III 46

A5. The Ratio of Correction Factor from the Local Slope Method

to that from Schumann and Gardner for Exercise II 47

A7. The Ratio of Correction Factor from the Local Slope Method

to that from Schumann and Gardner for Exercise III 48

A8. Reciprocal Correction Factors for a Single Isolated Layer

of Thickness t 50

D. Correction Factor F Based on Local Slope Using K3 for Various

Values of Rq/Rs and Also Using K1K2 and Unity to Replace K3 . 61

Dl, Case of Local Slope, m = -0.5 61

D2. Case of Local Slope, m = -0.1 61

D3. Case of Local Slope, m = 0 - . 62

D4. Case of Local Slope, m = 0.1 52

D5. Case of Local Slope, m = 0.5 63

V
PREFACE

This study was carried out in part at Solecon Laboratories and in pai

at the National Bureau of Standards as part of the Semiconductor Technology Program in the Electron Devices Division. The Semiconductor

Technology Program serves to focus NBS research to enhance the performance, interchangeability , and reliability of discrete semiconductor

devices and integrated circuits through improvements in measurement

technology for use in specifying materials and devices in national and

international commerce and for use by industry in controlling device

fabrication processes. This entire study was funded by the Defense Advanced Research Projects Agency through ARPA Order 2397. The work at

Solecon Laboratories was funded through NBS Contract No. 5-35881. This

contract was monitored by R. L. Raybold as the Contracting Officer's

Technical Representative (COTR) . Drs. W. M. Bullis and J. R. Ehrstein

provided technical review of this contract for the National Bureau of

Standards

.

Certain commercial equipment, instruments, or materials are identified in

this report in order to adequately specify the experimental procedure.

In no case does such identification imply recommendation or endorsement

by the National Bureau of Standards, nor does it imply that the material

or equipment identified is necessarily the best available for the purpose.

vi
Semiconductor Measurement Technology:

Spreading Resistance Analysis for Silicon Layers

with Nonuniform Resistivity

by

David H. Dickey and James R. Ehrstein

Abstract

A simple mathematical algorithm is developed for the

calculation of resistivity depth profiles from spreading resistance measurements on sectioned silicon device structures. It is applicable to structures con- sisting of one or more layers of the same or differing

conductivity types. The algorithm accounts for modification of the sampling volume of the spreading re- sistance probes arising from nearby variations, in

depth, of specimen resistivity whether resulting from

graded dopant distribution or electrical boundaries,

either insulating or conducting.

The algorithm is based on limiting case one- and twodimensional models of the conduction between the

probes. Compared with the traditional spreading re- sistance analysis algorithm based on the threedimensional boundary value solution of Schumann and

Gardner, the present work offers greatly reduced

execution times even with a microcomputer, making

real time analysis for process control possible.

An experiment to test the accuracy of one of the

limiting models used is described. Profiles of

several diffused layers generated from spreading

resistance measurements analyzed by this algorithm

are compared with profiles obtained on replicate

specimens using another electrical technique. In

addition, computer experiments are used for simple

layer models to compare results based on this algorithm with results from the Schumann-Gardner approach.

Key Words: Dopant profiles; resistivity; resistivity

profiles; semiconductor; silicon; spreading resistance.

1 . INTRODUCTION

The spreading resistance technique is a powerful method for determining dopant distributions in silicon device structures. Much of the

usefulness of the technique can be attributed to the relatively small

sampling volume involved. However, modern silicon devices commonly have

dimensions within their structure which are smaller than even this sam1
I pling volume, so it is often necessary to make calculations which remove

the effect of specimen boundaries or large resistivity gradients from the

measurements. Such calculations can be made only approximately and

generally rely on various simplifying assumptions about the structure

being considered. Of the approaches used for these calculations, that

given by Schumann and Gardner (SG) [1] is most generally applicable

because it is based on an approximate solution to the full threedimensional boundary value problem. Although several schemes [2-5] have

been presented to reduce the complexity and computation time of the calculations associated with the SG approach, computer size and time re- quired are still factors which inhibit its widespread usage.

In cases where the structure can reasonably be modeled in one or two dimensions, other approaches allowing vastly simpler calculations

are possible. The work reported here is an investigation of the viability of some simpler approaches and has culminated in a scheme for

making very rapid sampling volume corrections to spreading resistance

measurements. The scheme is applicable to all commonly encountered

structures and, in many cases, has an accuracy with reference to SG-based

results which exceeds the reproducibility of the basic measurements.

Briefly stated, the scheme uses differences between successive measurements on a profile as a basis for calculating sampling volume correction

factors. During the course of the work, the theoretical relation between

profile slope (difference in successive measurements) and required cor- rection factor was developed. For the general case, there is not a one-to-one correspondence between slope and correction factor; but, for

limiting cases, the relation can become exact. Computer exercises have

been performed to test the "accuracy" of the theoretical results as defined by comparison with the SG approach. Laboratory experiments have

been performed to test the slope-based scheme by comparing corrected

spreading resistance profiles with profiles as measured by the technique of anodic oxidation-incremental sheet resistance. The limitingcase theoretical results have been combined in an empirical relation

which in turn has been incorporated into a BASIC computer program for

use in on-line data reduction.

The scheme presented here has a shortcoming which is shared by all

other known methods: it does not account for bevel-angle and bevel-edge

proximity effects [6] . These effects are sometimes important, yet virtually impossible to deal with theoretically. This report describes one brief theoretical consideration of the effect of the bevel angle and

also includes the results of an experimental investigation of the beveledge proximity effect. Together, these results provide an estimate of

the possible errors one makes by ignoring the effects.

2 . THEORY

2.1 The Boundary Value Problem

The physical model assumed for a two-probe spreading resistance ex- periment consists of two circular equipotential regions on one surface

2
of an infinite slab of resistive medium. The -disks have radius, a; their centers are separated by a distance, s; and they are held at equal

but opposite potentials, +V and -V. The problem is to find the current

between the disks. The most general case considered is that of a resistive slab having a variation in resistivity but only in the direction

normal to its surface. The slab may have infinite thickness, or it may

be bounded below by a layer of either infinite or zero resistivity.

For a slab of uniform resistivity and infinite thickness, the re- sistance R between the disks is given in terms of measured voltage, V,

between the disks, and current, I, by the well-known relation [7]:

R = 2V/I = p/2a , (1)

where p is the electrical resistivity of the slab. If the slab has finite thickness, eq (1) must be corrected by a factor, F, which depends

on thickness and contact separation, so that the resistance is given by

R = pF/2a . (2)

In the following sections, discussion will be in terms of conductance as well as resistance, so it is convenient at this point to write the equivalent of eq (2) for the conductance between the disks:

G = 2aaH = 2aa/F , (3)

where a is the (uniform) conductivity of the slab and H is the reciprocal of the factor F

.

Exact calculation of the factor F (or its reciprocal) for a uniform

slab requires the solution of a potential-theory problem having mixed

boundary conditions. Moreover, it is a two-center problem; but this as- pect can be safely handled by superimposing the potential distributions

around each disk treated alone as long as their separation is a few

times the disk radius. The mixed-boundary-condition aspect formally

requires the solution of a pair of dual-integral equations which do not

have a general analytical solution. With the assumption of a particular current distribution under each disk appropriate to a semi-infinite

specimen, SG used a known solution to the dual-integral equations to

evaluate F. Although their assumption is not generally valid, its use introduces errors which (see Appendix A, sec. A. 4) do not exceed a few

percent. It should be noted that even the most streamlined version [5]

of their approach requires at least one numerical integration over a fine scale grid with an integrand involving several transcendental functions .

It is desirable to make certain simplifying assumptions in the model

used in order to formulate a sampling volume correction scheme which is

operationally simpler than that resulting from the boundary value solution of SG.

3
o

esc Oo

0.1

0.02

1 1 n i Ti r I 1 I 1 1 n| I I n TTM

-

- 1

1

i

1

1

1

1

II

1 1

1 M 11 ll 1

1 1 1 t 1 1 ll 1 1 1 1 1 in

0.1 1 10

NORMALIZED SLAB THICKNESS, t/a

100

Figure 1. Reciprocal of sampling volume correction

factor for an isolated uniform slab plotted vs. nor- malized slab thickness. Probe spacing equal to 40

contact radii.

The method of images (detailed in sec. A. 2 of Appendix A) was used

to find numerical values of F for a two-layer structure to a very high

accuracy. This is done for a more general current distribution under the

contact than that assumed by SG, but one that is shown to come much

closer to meeting their assumed boundary condition of constant potential

under the contact. The image calculation results in the dependence of

H, the reciprocal of the correction factor F, on slab thickness shown in

figure 1.

For a slab consisting of a large number of layers of differing re- sistivities, the method of images approach becomes very complicated.

The solution can be simplified in cases which can be modeled in fewer

than three dimensions.

In practice, there are a great number of structures in which the

total effective thickness is less than the contact radius. In these

cases, models can be constructed in which current is exclusively parallel

to coordinate directions . The most common case is that of a shallow diffusion or implant where a p-n junction isolates the material being probed

from the substrate. In such cases, the current is two-dimensional and

predominantly parallel to the surface of the layer. A useful model is one in which the equipotential contacts are cylinders extending through

the thickness of the layer. This model is used in section 2.2 as a starting point to derive a more general solution according to the principle of parallel superposition. Another limiting two-dimensional cur- rent is approximated by a thin, relatively high resistivity structure on

a low resistivity substrate. Although the substrate may be very thick.

4
the dominant contribution to measured resistance in this case is in the

high resistivity layers where current is predominantly normal to the surface. A simple solution involving differences in successive measurements

is obtained from this model in section 2.3. Because the concepts to follow are developed in terms of removing physical layers, a process which

normally proceeds from the top of the structure, a position coordinate

system is defined whose origin is at the original specimen surface and

which increases into the depth of the specimen's structure; see figures

2 and 3.

2.2 An Isolated Layer — Parallel Superposition of Conducting Sublayers

For a thin slab with uniform conductivity on an insulating substrate, the conductance between two circular contacts on the surface

is given by eq (3) , where H depends on slab thickness t as derived in

Appendix A and shown in figure 1. For thickness less than the contact

radius, H(t) is always less than unity, see figure 1, and has for an asymptotic limit at very small thicknesses [8]+

H(t) = -

/ ^

• 2a ln(s/a)

In this asymptotic region, the current is predominantly in the plane

of the slab and the model is equivalent to one in which the contacts

are cylinders extending throughout the thickness of the slab. If a thin slab having nonuniform resistivity with depth is now subdivided,

as shown in figure 2a, into N layers each with different conductivity,

the conductance between the cylindrical contacts is given by

N

G, = —/ , > a. At , 1 ln(s/a) 1

i=l

where At is the common thickness of each sublayer. In keeping with

the position coordinate convention above, the index, i, increases into

the depth of the structure. Equation (5) is the extension of eq (3) using eq (4) for H(t) . To fit the assumption of independent layers con- ducting in parallel, one may consider the layers separated by virtual

insulators. Therefore, H is evaluated for the thickness to the nearest

insulator, namely. At. If the top sublayer is now removed, the conductance becomes

N

Go = -, // N a. At (6a)

2 ln(s/a) Z—/ i

i=2

= G - . /, , a. At . (6b)

1 ln(s/a) 1

t ITt The result stated in the reference incorrectly gives H(t) = ;— — r- . 2a ln(2s/a)
(a)

Oti tN Depth

(b) (c)

Figure 2. Multilayer model appropriate for isolated slab, showing contacts (a) extending through total slab thickness, (b) at the surface - parallel superposition, and (c) at the new surface after removal of layer

thickness At - parallel superposition.

Pi

^

00

Pn

Ps

DISTANCE FROM SURFACE,!

Figure 3. Multilayer model for slab over con- ducting substrate - series superposition.

6
Rearranging eq (6b) , one finds for this limiting case of small total

thickness that the surface conductivity is

ln(s/a)

^1 = —

The conductivity of the next sublayer can be similarly found by removing that layer and measuring a third conductance. Thus, the entire slab

can be profiled by successively removing each sxablayer and using the

difference in measured conductance before and after removal of the sublayer to calculate each conductivity according to the generalized form

of eq (7) . This equation is closely analogous to the relation used with

a four-probe incremental sheet resistance experiment [9].

The preceding analysis assumed a thin slab so that eq (4) would be

valid. For thicker slabs, where probes cannot be represented as extending through the structure, a similar analysis can be made by subdividing

the slab not into sublayers of equal thickness but into layers of fractional conductivity and incrementally increasing thickness as depicted

in figure 2b. As before, the layers are separated by virtual insulators

to assure conduction in parallel. Although the layers act as resistors

conducting in parallel, parts of the resistors happen to occupy the same physical space. This model is then a mathematical construction utilized

to investigate the ramifications of extending the concept of parallel

superposition beyond the limiting case of very thin layers. Writing the

conduction of the slab as the sum of N individual conductances:

- G,

At

(7)

N

s = E K

- ^i+1

i=l

H(t.) , (8)

1

where H{t.) is the uniform- layer correction factor for thickness t. . In

this model, each layer extends to the surface so the assumption of con- tacts extending through the slab is no longer needed. If a physical layer

of thickness At is now removed from the top of the slab, the top layer

(t. = At) of the mathematical representation disappears and all other

layers become smaller by At; see figure 2c. The new conductance is then

N

^2 = 2^ E pi - ^i+l] «^^i - ' i=2

where t. is retained as the original thickness of the ith layer. Since

t. - At''"is just t. , eqs (8) and (9) can be rewritten in equivalent

form after shifting the indices on the terms involving cr :

7
N

^ = O. H(t.) + V a. fH(t.) - H(t. .) 2a 1 1 1 iL^ 1 [ 1 1-1

- a , H(t ) N+1 N

(10)

i=2

N

2a 2 2 i=2

The first term on the right in eq (11) is included for algebraic accuracy to balance an equal term included in the sum, although both terms are zero because t^ is zero. Similarly, in the model (fig. 2b) is zero,

so these terms are dropped and the difference in conductance is taken as:

N

^

(G - G„ ) = a. H(t.) + a. [H(t.) - 2H(t. .) + H(t. . 2a 1 2 11 1 [ 1 1-1 . 1-.. (12)

i=2

If the thickness of the top layer, t^ (= At) is small enough that H(t^)

is in the asymptotic region, H(t ) can be replaced with its value as given by eq (4) and eq (12) can Be rearranged to find a ^:

G - G

^ ^ln(sA). _1_ 2 _ 2a ln(s/a) y, ,2^^^^ , , (13)

IT At uAt

i=2

where the symbol A^^H denotes the second difference appearing in the

square brackets in eq (12) [10]

.

This second difference is always negative (note from fig. 1 that

H(t) is concave downward for the case of insulating substrate), but it

approaches zero for either very small or very large values of t. Equations (7) and (13) are thus seen to agree for the case of small total

slab thickness. The second term in eq (13) may be regarded as a correc- tion to account for the contribution from subsurface layers in a thick

slab. To use this formula for a thick slab, one must first apply it to

the deepest points on a profile, and then work out toward the surface.

For more detail see Appendix B. When used in this way, the formula

gives the correct result (at least for points far from the insulating

substrate) even for a very thick slab of uniform conductivity. In this

case, the first term is negligible and the second can be shown to sum

to a since the a.'s are constant and can be taken outside the summation.

The exact form o^ H (t) has not been specified at this point because it

is not critical: H(t) needs only to asymptotically approach unity for

large thickness and the value given by eq (4) for small t.

The analysis leading to eq (13) is based on the concept called

Parallel Superposition: that the conductance of a nonuniform slab can be written as a suiu of parallel conductances. This model was expected

8
to work well for thin layers with insulating substrates; but at the

time this work was begun, it was not known whether the concept would

yield exact results for intermediate cases of thicker slabs having a wide variation in conductivity. A computer experiment was performed to

test the concept for a few of these intermediate cases with the conclusion that it is not exact for the general case. The results of the com- puter experiments are described in Appendix A.

2.3 Layer over Conducting Substrate — Series Superposition of SubLayer Resistances

For a thin, high resistivity structure on a low resistivity substrate, most of the resistance measured between two disk contacts on

the surface arises in the high resistivity layers. In the limiting case where the total thickness of this region is much less than the contact

diameter, the current through the region is predominantly normal to the

surface of the structure. Figure 3 is an illustration of such a structure. The resistance measured at the surface in this case can be written, approximately, as the sum of the incremental resistances in the

cylinders below each contact disk and the spreading resistance between

virtual contacts on the substrate:

(t. - t. ) p R = 2 > p

. <r + — 1 Z-< 1 TTa^ 2a

i=l

where p is the resistivity of the siabstrate. This formula would be a closer approximation if the radii used in each term in the sum were progressively larger to account for slight spreading with depth, but the

equation is quite exact for the limiting case of small t. 's and large

values of p./p • If the layer nearest the surface is removed, the new

resistance wilf be smaller by just the first term in the siam, and the

resistivity at the original surface can be found from

Pi =

TTa' R, - R^

At

Tra' AR

At

(15)

where R is the measured resistance after removal of the layer and At

is the (positive) thickness of the removed layer.

The important conclusion reached from this limiting case analysis

is that the surface resistivity can be found from the difference in suc- cessive measurements, just as it was for the case of current parallel to

the surface . In the parallel case , the measurements were in terms of

conductance, and in the present case resistance measurements are made.

In actual practice with common spreading resistance electronics, measurements are made in terms of the logarithms of resistance. In the

following section a convenient transformation is described which puts

both eqs (13) and (15) in terms of logarithms of measured resistance.

9
2 . 4 The Logarithmic Transformation

The basic spreading resistance equation for a two-probe experiment

is given in eq (2) , if F in that equation is the sampling volume correction factor. Formulas for F can be extracted from eqs (13) and (15)

since they both contain factors of the type AY/At. If AY and At are both small so that the differential limit may be assumed, these factors

can be replaced as follows:

AY/At Y Aln(Y)/At .

With this substitution, eq (13) and eq (15) become:

N

a ^ ln(s/a) AlnG 2a

G IT At

i=2

(13a)

and

R

P

At

TTa2 AlnR (15a)

The following is then extracted from eq (3) and eq (13a)

:

N

F = 2a ln(s/a) | AlnG 2a

At G At

V/

a.1

A2H(t.1-1J

i=2

and from eq (2) and eq (15a)

F =

4 At

TTa AlnR

(16)

(17)

Further substitutions in eq (16) are made. In G = - In R and G =

1/R, to make eq (16) comparable in form with eq (17), i.e., based on the

directly measured quantity, log of resistance. This has the effect of

formally changing the sign of the first term in eq (16) and then cancel-

^ , . I- • ^ .1 -, ^In G ^ Aln R mg the effect of this change, since for the same data —— and

have opposite signs: At At

F =

-2a ln(s/a)

IT Aln R 2 aR

At At Sa.A2H(t. J

1 1-1

i=2

(16a)

In practice, it is preferable to establish the relationship between spreading resistance and resistivity with a calibration curve rather than with a calculation based on eqs (2) , (13) , or (15) . An appealing feature of the logarithmic transformation is that it preserves

the linear relation between resistance and resistivity so that the cali10
bration curve can still be used. Equations (16a) or (17) can be used

to find the correction factor, and resistivity can then be found from

the calibration curve using the corrected resistance, R/F . Note that

without the logarithmic transform, eqs (13) and (15) relate resistivity

to terms such as AR/At which are not necessarily linear in R.

Converting the results of the initial derivation to a form compatible with data as measured requires several transformations or changes

of reference point, each entailing possible sign changes. Remembering

that the correction factor, F, must be positive since it is the ratio of

positive quantities R, a, and p, it is useful to consider the effect of

sign convention on the component terms. Because of the definition of

the coordinate, t, in the derivation of eqs (13) and (15) , At = t. - t.

^ is positive although it corresponds to the removal of a physical iayer'l"

The differences AG = G . ^ - G^ and AR = R. ^ ~ ^i' constructed, are also positive because ^or the'^physical situations described: those of

parallel conductances and series resistances, respectively, the respective total conductance and resistance are larger for positions near the

surface than for positions into the depth of the structures. The ratios,

AG AR -— [from eq (16)] and — [from eq (17)], are positive as defined for the

AR

the coordinate t starting at the top surface; also, — [from eq (16a)] is

negative. It is to be noted, however, that AG and AR are defined in the

opposite sense from At for the interval being considered, -for example,

G — G

_ initial final , -, ^ ^- AG ^ AR

AG = — . As a result, the ratios, —- and —- are the

t^. - t. .. , . 'At At final initial

AG AR

negative of the ratios —^ and -r— taken from the slopes of the measured

^ At At

conductance, G vs. t, and the measured resistance, R vs. t, relations

as. they would Be taken directly from acquired data, see figure 4. The

equations for F are rewritten using R to reflect the evaluation of the

AR interval in the same sequence as tRe At interval, the top surface

still being taken as the origin:

2a ln(s/a)

F = Aln R

m

At

2a R

At

m

N

a. A2H(t. J

1=2

(16b)

F =

- 4 At

Tra AlnR

(17a)

For the parallel superposition model, illustrated in figure 4a, the

first term in eq (16b) is positive; the second is the negative of a term,

A^H, itself negative, as discussed subsequent to eq (13). The value for

F in eq (17a) appropriate to the series superposition model, is the negative reciprocal of the slope of the R v-s. t relation in figure 4b, itself negative. Hence, in both cases,

^

is clearly positive as it must

be, since it is the ratio, 2aR/p of positive quantities.

11
0 Distance from top surface, t Q Distance from top surface, t —

a. Case of isolated slab: parallel b. Case of conducting substrate

superposition model. not insulated from slab: series

superposition model.

Figure 4. Schematic dependence of conductance and resistance on depth

at which measurement is made below original surface of the structure.

Reverse coordinates, referenced to layer-substrate interface also shown.

Slopes, tangent to curves, as shown, are referred to coordinates originating at top surface. Schematic values of Rq and Rg are shown for ref- erence in interpreting the substrate factor K.3

.

2.5 Local Slope

The equations developed in the preceding section provide limiting

case descriptions of the behavior of the sampling volume correction factor. In both cases the behavior is governed by the slope of the spreading resistance data profile. The local slope of the measured resistance

profile, AlnR /At, is written as m, and the thin layer limiting forms

for F become:

and

where

and

F = K^m, for m >> 0 (18)

F = -K^/m, for m << 0 (19)

. 2a ln(s/a)

^^O)

1 TT

K = — . (21)

2 fra

12
— <»—— 0 Local slope, m —-l-^

Figure 5. Schematic representation of slope-dependent terms for the cor- rection factor F, as derived from limiting case analysis: solid lines —

regions where the terms are applicable; dashed lines — regions of questionable validity; dotted line — hypothesized form for F in region of

intermediate slope.

It is to be noted that the natural unit for the local slope, m, is decades (of resistance change) per micrometer of depth into the specimen.

In the asymptotic form for large positive slope, the second term in

eq (16) is ignored because it is always negligible relative to the slope

term under these conditions. Since the second difference of H(t) is always negative (or zero) , this second term will always make a positive

(or zero) contribution to F.

For near zero slope, indicative of remote lower boundary, nearly

uniform material, or both, F should be on the order of unity and neither

of the above forms is realistic. Figure 5 illustrates the asymptotic

forms given above and the expected approximate form of F for intermediate values of m. The two asymptotic forms above can be combined into a single hypothesized form which displays the correct asymptotic behavior

for large positive and negative slope

F = K^m/2 + /(K^m/2)2 + K^K^ • (22)

When the slope is zero, this equation reduces to a value near unity.

F = /k^K^ = /8 ln(s/a)/7T2 . (23)

Equation (22) is a reasonable approximation to the sampling volume

correction factor for any profile, regardless of whether the slope is

positive, zero, or negative. The error resulting from the formula is

largest for profiles on bulk samples of uniform, resistivity, for which
«

the slope is zero and the sampling volume correction factor should be

unity. For example, with a probe spacing, s, equal to 40 contact radii,

a, the factor K^K^ on a profile with zero slope is approximately 3.0

which results in a value for F of about 1.7, as opposed to unity. However, for either large positive or large negative slope, eq (22) rapidly

approaches the correct limiting value.

An improvement can be made in the overall applicability of the

local-slope formula in the range of intermediate m values by replacing

the constant K in eq (22) by a variable factor which depends on the

measured resistance approximately one contact radius deeper on the profile. This variable factor is called the substrate factor, K . For the

bulk specimen case mentioned above, the resistance one contact radius

deeper is equal to the resistance at the point being corrected, so for

this condition the variable factor K should have the value of unity.

The formula tentatively adopted for the substrate factor is

where R and R are the measured, i.e., uncorrected, resistances at the

o s point being corrected and at a point approximately one contact radius

deeper on the profile, respectively; the arctan function is expressed in

radians. This form, admittedly arbitrary, was chosen because it is a continuous , monotonic function in the interval O < R /R < °° and because

9 OS it contributes to F in the same sense that the A'^H term does in eq (16) . This formula gives values for the substrate factor of

= K for R /R >> 1 (conducting substrate, m << 0),

3 1 2 o s = 1 for R /R =1 (bulk specimen, m - 0) , and

3 OS

= 2-K, for R /R << 1 (insulating substrate, m >> 0).

3 12 o S

The values for the first two conditions lead to accurate sampling volume correction factors, and since the value of has negligible influence on the magnitude of F in the case of large positive slope (the

third condition above) , acceptably accurate results are expected for all

conditions if the term K^K^ in eq (22) is replaced by . (See also the

discussion in Appendix B and Appendix D for operational application to

real data.)

The final operating equation for F which is expected to have general application in depth profiling by spreading resistance is, therefore,

assuming the data are analyzed from the near surface side of the layer

working into its depth, as was done in the derivation. However, since

the form K requires evaluation of the resistance at point below the one

= 1 + (K K - 1) (2/tt) tan ^ [log (R /R ) ] , 3 12 OS (24)

(22a)

14
for which the factor F is being evaluated, it becomes more convenient to

proceed with the analysis from the bottom of the structure. For analysis

starting at the deepest part of the layer and working towards the shallowest, as is done in Appendix B, the computed slope has the opposite

sense from the foregoing, resulting in:

Km /UnV

^ = - — " J\-r) * S <22'='

3 . EXPERIMENT

3.1 The Probe Spacing Experiment

For the case of a uniform layer, thin enough that H(t) is in the

asymptotic region, on an insulating substrate, the resistance measured

between the two contacts is [from eqs (2) and (4)]:

R = p ln(s/a)/TTt = ^

[ln(s) - ln(a)] . (25) ut

If ln(s) is the independent variable, this is an equation of the type

y = m(x-b) (26)

where the slope, m, is p/rrt and the x-axis intercept, b, is ln(a). The

experiment suggested by this equation is to measure spreading resistance

on a thin layer at various values of probe spacing and to plot the re- sults with R on a linear scale and s on a logarithmic scale. The plotted

points should fall on a straight line with slope 2.3 p/TTt (ohms per decade) and with an intercept on the horizontal axis (R = 0) , at an extrapolated spacing equal to the contact radius. This experiment was performed

with various layers and probe loadings, with consistent results. Typical

results for three thin p-type diffused layers with different junction

depths are shown in figure 6a and results are summarized in table 1. The

resistivity in these layers is not uniform, but if p/t is interpreted in

eq (25) as being the sheet resistance, close agreement is found between

the slopes in figure 6a and sheet resistances measured on the specimens

by four -probe, see table 1. Note that in figure 6a the same value of intercept is found for all three p-type layers, indicating a common value

of the contact radius, a. Figure 6b and table 2 show the results of a similar experiment performed on both n- and p-type diffusions with two

probe loads. Note that the value of slope does not depend on probe load

for either p- or n-type specimens, and consistent values of sheet resistance are obtained. The difference in intercepts for n- and p-type layers

at a given load implies either a difference in contact radius for the two

types of material or the presence of a type dependent series resistance.

Values of a, derived from the x-axis intercept. In (a), eq (25), in such

probe spacing experiments, were used in calculating correction factors,

F, used in profile analyses in section 3.2.

The important result of these experiments is that the measured

spreading resistance on an isolated, thin, nonuniform layer is indeed

15
600

500

-c

400

LUzo

300

200

100

T 1

I I I I I I I

1 1—I I I

I M I1

I I I I M I

I

-O-

/

.A-A- .A-A- A—A-;

i/^^g^^TiTi I I

I I I I I I I I I

1 1—I I M I 1

1

10 100 PROBE SPACING [Mm]

1000

(a)

200

150

lOG

^ 50

a.

n—I I I M I 1 1—I I I I 1 1

I

1 1—I I I M 1

1

1 1—I I M I 11

0.1

I ^-1 I I I 1 1 1

1

1 I I Mini I I I I I I 1

1

1 10 PROBE SPACING Wm\

100 1000

(b)

Figure 6. Spreading resistance vs. probe spacing for junction-isolated

layers. Layer-type and four-probe sheet resistance values are: (a)

O, p-type, 284 fi/D; p-type, 126 S^/D; a, p-type, 56.4 f^/D; (b) O and A,

p-type, 76.4 ,- and O, n-type, 52.0 fi/o . All data for figure 5a and

for triangle and diamond in figure 6b taken at 20-g probe load; data for

circle and square in figure 6b taken at 10-g load.

16
Table 1. Comparison of Diffused Layer Sheet Resistance Values for

Three Boron Diffusions.

Specimen

Sheet

Four Probe

Resistance, Q,/n

Two Probe

ln(s) Dependence^

Boron diffusion. X . J 0. 7 ymb 284 250

Boron diffusion, X . = 1. 6 ym^ 126 126

Boron diffusion.

J X . J = 4. 3 ym*^ 54.6 52

Two-probe data are shown in figure 6a.

Profile data for this diffusion are shown in figure 7.

Profile data for this diffusion are shown in figure 8.

Profile data for this diffusion are shown in figure 9.

Table 2. Comparison of Diffused Layer Sheet Resistance Values for One

n-Type Diffusion and One p-Type Diffusion.

Sheet Resistance, Q/o

Specimen Four Probe Two Probe, ln(s) Dependence^

(average of 10 g and 20 g data)

Boron diffusion'^ 76.4 81.9

Phosphorus diffusion^ 52.0 53.3

Two-probe ln{s) dependence is shown in figure 6b.

Profile data for this diffusion are shown in figure 10.

Profile data for this diffusion are shown in figure 14.

17
found to be closely related to the sheet resistance of the layer. The

result is a verification of eq (8), and by implication, the difference

equations which lead to eq (16)

.

It should be noted that the consistency of the data down to the

smallest spacing shown corresponding to s/a ratios as small as 7 or 8, implies that the basic model for isolated layers is valid down to s/a ratios

of this order. Hence, any volume correction factors derived from the model should be applicable to data taken with such s/a values. However, preliminary data, not shown, indicate a breakdown of the linear dependence

of R on ln(s) when using probe spacing on the order of 6 to 8 ym, corre- sponding to s/a values as small as 4 or 5. Use of volume correction factors as derived here are questionable under such s/a measurement conditions .

3.2 Diffusion Profiles

The sampling volume correction schemes described above were tested

in a series of experiments. Test specimens containing relatively shallow diffusions were prepared, and spreading resistance profiles of the

diffused regions were generated. Two sets of specimens were used in

these experiments, each prepared to test a particular volume correction

equation. The conductivity profiles corrected as discussed below were

then compared with profiles generated using the incremental sheet resistance method, ISR, on adjacent sections of the same specimens. The re- sults generally substantiate the validity of the correction schemes, but

a number of experimental effects (primarily angle effects, see sec. 2.3)

not accounted for in the theory here are evident in the comparisons

.

The first set was a series of boron diffusions having a wide range

of junction depths (0.7, 1.6, and 4.3 ym, as measured by angle lap and

stain) and nominally constant surface concentration (about 1 x 10^^

atoms cm~^) , the substrate being 10 ^'cm n-type silicon. This set was intended to provide a range of conductivity gradients and layer thick- nesses to test the range of validity of the concept of parallel superposition .

The conductivity profiles obtained on this set of specimens are shown in figures 7 to 9 . The points shown were derived from spreading

resistance data using the parallel superposition formula, eq (16) . The

procedure is to first estimate the slope at the deepest point still

above the junction, calculate a correction factor using only the first

term in eq (15) , adjust the measured resistance at that point by the factor F, and look up the conductivity corresponding to the adjusted resistance on the calibration curve generated from a sequence of bulk silicon

specimens with known resistivity (conductivity) , having the proper crystallographic orientation. The conductivity value thus obtained is plotted on the final graph for the datum in question and is also used in the

second term of eq (16) when the correction factor is calculated for each

succeeding shallower point. For each succeeding point (working toward

the surface) , an additional term appears in the summation term of eq

(16) , so when the point at the surface is considered, the summation is

18
10-2 1 \ \ \ J I 0 0.5 1.0 1.5 2.0 2.5 DEPTH iMm]

Figure 7. Conductivity profile for boron diffusion into

nominal 10-fi*cm n-type substrate. Nominal surface density

of boron, 10-^^ cm~^; junction depth by lap and stain, 0.7

ym. Open circles, spreading resistance data corrected using

parellel superposition, eq (16b); solid circles, uncorrected

spreading resistance data; solid line, smoothed incremental

sheet resistance data.

19
DEPTH (pm)

Figure 8. Conductivity profile for boron diffusion into

nominal 10-^2 •cm n-type substrate. Nominal surface density

of boron, 10-^^ cm~^; junction depth by lap and stain, 1.6

ym. Open circles, spreading resistance data corrected using

parallel superposition, eq (15b); solid circles, uncorrected

spreading resistance data; solid line, smoothed incremental

sheet resistance data.

20
DEPTH Ipm)

Figure 9. Conductivity profile for boron diffusion into

nominal lO-fi'cm n-type siibstrate. Nominal surface density

of boron, 10^^ cm~^; junction depth by lap and stain, 4.3

ym. Open circles, spreading resistance data corrected using

parallel superposition, eq (16b); solid circles, uncorrected

spreading resistance data; solid line, smoothed incremental

sheet resistance data.

21
over all points between the junction and the surface. Appendix B contains further details of the calculations. Simple analytic expressions

are used as approximations for H and its second derivative and are described near the end of Appendix A.

In each case for these first three specimens, the spreading resistance profile lies below the incremental sheet resistance (ISR) profile

near the surface but coincides closely at deeper points. The discrepancy

is thought to be related to bevel-angle effects [5]. Such effects, which

are absent in the ISR experiments, are not accounted for in the data re- duction scheme used here. The general agreement between the two methods

is taken as proof of the viability of the superposition scheme for specimens with an isolating junction.

The second set of diffusions was prepared on substrates of widely

different resistivity but with nominally constant profile shape and

depth. This set was used to determine what effects the presence of a low resistivity substrate below the junction would have on the corrected

profiles. Profiles for this set of specimens are shown in figures 10 to

17 and a summary of profile parameters is given in table 3. This set of

specimens consisted of four boron-diffused wafers and four phosphorusdiffused wafers. The boron diffusions were made under the same diffusion conditions into three n-type substrates of nominal 0.03, 1.0, and

150 fi'cm resistivity and into one p-type substrate of nominal 100 Q,'cm

resistivity. The phosphorus diffusions were made under the same diffusion conditions into p-type substrates of nominal 0.03, 1.0, and 100 f2*cm

resistivity and into one n-type substrate of nominal 150 f2*cm resistivity. Each of the spreading resistance-based profiles was generated, using

the local slope formula, eq (22) . This formula yields results almost indistinguishable from those obtained from parallel superposition, not

shown, particularly for points less than one or two contact radii from a junction. The agreement with the ISR profiles was generally the same as was found with the first set of specimens: the spreading resistance profiles tend to be low near the surface, especially on p-type layers. The

depth scales do not coincide exactly in all cases, owing to experimental

error in both measurements and to the fact that slight variations in

junction depth occurred in the different regions of the specimen analyzed.

Comparing figures 10 and 13 for boron diffusions, and figures 14 and

17 for phosphorus diffusions, a very good agreement is seen in each pair

for profile shape and absolute value from the surface to within a few

points of the junction. This is taken as an indication that the local

slope method works quite well even when the substrate does not approach

one of the limiting cases in the starting models.

The computer program used for reduction of the spreading resistance

data is described and listed in Appendix B. It incorporates both eqs

(16) and (22) as separate routines and includes data logging and calibration curve look-up routines . The program does not include raw data

smoothing so the results may suffer from random variations in the data.

Execution times for the local slope routine are on the order of three

22
I

oiCPc

•H

fO

(UMo

0) P

nJM

4-> tn

0) pO

-PU

0) mmw

0)

-P

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CO

0) EHO

-P

(1) COw

0)>1

(0

T!

(U

CO IW

IW

-rH Q

U-l • O W w cn U >i H

-P rd w d H <;

-pu

rd rd u -p

(dM

X!

Eh

-pft

(D QOH

+JUCUC

rd -P

(fl •H

U)

(U Pi

-P

0) mu

0)>i

rd

(U0 —,

-ri rH 0

rd -p +J

+J U) 03 H rH w 0 g CD ft

rd

u -p P G OJ X H Q) W

4::

—

(UuC

H rd P

tdm

0)

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(fl ft

cn Di

dJ

C

a

rdH

rd

q ft +J < rd CO

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Q)0

rH rd T3

rd -P CU +J -P C •H rd

(D U)U e Q) tn

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SH P 0 P c CU(U

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CU c c H rd t3 -P

rdW

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in cn ft CU

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cn cn • ^ -H C5

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LT)

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CO 00

LD in CM

en rH

rH iH rH

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ro

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ft CQ PQ PQ pa m CQ Oj CU

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rd

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23
DEPTH iMm)

Figure 10. Conductivity profile for boron diffusion

into nominal 150-J^*cin n-type substrate. Nominal boron

surface density, 10-^^ cm~^. Open circles, spreading

resistance data corrected by the local slope method,

eq (16b); solid circles, uncorrected spreading resistance data; solid line, smoothed incremental sheet

resistance data.

24
DEPTH (Mm)

Figure 11. Conductivity profile for boron diffusion

into nominal 1.0-fi*cm ri-type substrate. Nominal boron

surface density, 10-^^ cm~^. Open circles, spreading

resistance data corrected by the local slope method;

solid circles, uncorrected spreading resistance data;

solid line, smoothed incremental sheet resistance data.

25
J I L

1 2 3 DEPTH iMm)

Figure 12. Conductivity profile for boron diffusion

into nominal 0.03-^2 •cm n-type substrate. Nominal

boron surface density, 10^^ cm~'^. Open circles,

spreading resistance data corrected by the local

slope method; solid circles, uncorrected spreading

resistance data; solid line, smoothed incremental

sheet resistance data.

10°

26
DEPTH Ipm)

Figure 13. Conductivity profile for boron diffusion

into nominal 100-^2 • cm p-type substrate. Nominal boron

surface density, 10^ cm . Open circles, spreading

resistance data corrected by the local slope method;

solid circles, uncorrected spreading resistance data.

Compare with figure 10 for effect of noninsulating

boundary on profile interpretation.

27
DEPTH (Mm)

Figure 14. Conductivity profile for phosphorus diffusion into nominal 100-f^*cm p-type substrate. Nominal

phosphorus surface density, 10^ cm" . Open circles,

spreading resistance data corrected by the local slope

method; solid circles, uncorrected spreading resistance;

solid line, smoothed incremental sheet resistance data.

28
DEPTH (Mm|

Figure 15. Conductivity profile for phosphorus diffusion into nominal 1.0-J^*cm p-type substrate. Nominal

phosphorus surface density, 10^^ cm~^. Open circles,

spreading resistance data corrected by the local slope

method; solid circles, uncorrected spreading resistance;

solid line, smoothed incremental sheet resistance data.

29
0 1 2 3 4 5 DEPTH (pml

Figure 16. Conductivity profile for phosphorus diffusion into nominal 0.03-J^*cm p-type substrate. Nominal

phosphorus surface density, 10^^ cm~^. Open circles,

spreading resistance data corrected by the local slope

method; solid circles, uncorrected spreading resistance

solid line, smoothed incremental sheet resistance data.

30
10^

102

EoG

B 100

10-

10-2

10-3

2 3 DEPTH (Mm)

Figure 17. Conductivity profile for phosphorus diffusion into nominal 150-fi*cm r2-type substrate. Nominal

phosphorus surface density, 10-^^ cm~^. Open circles,

spreading resistance data corrected by the local slope

method; solid circles, uncorrected spreading resistance,

Compare with figure 14 for effect of noninsulating

boundary on profile interpretation.

31
seconds per point for the Hewlett-Packard Model 9830A programmable calculator. For the parallel superposition routine, times run as high as

10 s for the last few points on a 40-point profile. As can be judged

from the simplicity of the program listed in Appendix B, a relatively

inexpensive machine can be used for data reduction and so can be con- sidered as a dedicated accessory to the spreading resistance probe.

3.3 Bevel-Edge Proximity Effect

When the original top surface of a beveled specimen is probed, an increase in measured spreading resistance is often noted as the bevel

edge is approached. The effect is most noticeable on shallow diffusions

and depends on the spacing between the two probes. The effect is an obvious illustration of asymmetry introduced by beveling away one side of

the specimen. A theoretical treatment of the effect is an extremely

complicated task involving conductivity gradients below the surface,

bevel angle and location, and probe spacing. The interrelation of angle

and spacing was investigated briefly on one of the specimens used for

the profiling experiments. The results are shown in figure 18 where

spreading resistance on the original surface is plotted versus distance from the bevel edge for two bevel angles and two values of probe

spacing

.

A partial understanding of the bevel-edge proximity effect can be

gained by considering the simple case of a uniform-resistivity wedgeshaped specimen. A theoretical treatment of this problem is given in

Appendix C. The results of this investigation show how the uniformlayer correction factor is affected by nonparallel top and bottom surfaces of the classic infinite slab. As in the experimental investigation above, the effect is minimized with small probe spacing and, of

course, vanishes as the bevel angle approaches zero.

4. CONCLUSIONS

The exact solution of the boundary value problem associated with a spreading resistance experiment is an exceedingly difficult task. Experimental problems, such as barrier resistance effects, probe calibration, and probe stability, introduce errors which make the accuracy of

computations difficult to verify. This work has shown that simplifying

the theoretical model can accrue computational savings which far outweigh the loss of accuracy involved. The mathematical treatments of

limiting-case parallel and series models have served to provide proportionality constants which tie the empirical local slope scheme to a physical model.

The probe spacing experiment has two main values: it demonstrates

that spreading resistance on a thin, insulated structure is closely re- lated to sheet resistance , and it provides an experimental value for

contact radius which is a necessary parameter in the calculations.

The diffusion profile experiments, while demonstrating the viability of the correction schemes, serve better to illustrate the need for

32
GO

LU

C9

oc

to

CD

10

2

DEPTH

a. Bevel angle, 0.05 rad; upper

trace probe spacing, 200 urn; lower

trace probe spacing, 12 ym.

10^ Top I Bevel

DEPTH

b. Bevel angle, 0.01 rad; upper

trace probe spacing, 200 ym; lower

trace probe spacing, 12 ym.

Figure 18. Spreading resistance in the vicinity of the bevel edge.

Specimen same as shown in figure 10. Original top surface extends to

left of demarcator. Step size is 10 ym.

33
further understanding of bevel-angle effects. A recent investigation

[5] has shown that if spreading resistance data are taken on an anodically sectioned (rather than beveled) specimen, much of the discrepancy of

the type observed near the top of the layer will disappear.

REFERENCES

1. Schijimann, P. A., and Gardner, E. E., Application of Multilayer Potential Distribution to Spreading Resistance Correction Factors, J. Electrochem. Soc . 116 , 87-91 (1969).

2. Yeh, T. H., and Khokhani , K. H., Multilayer Theory of Correction Factors for Spreading Resistance Measurements, J. Electrochem. Soc. 116 , 1461-1464 (1969)

.

3. Hu, S. M., Calculation of Spreading Resistance Correction Factors,

Solid-state Electronics LS, 809-817 (1972)

.

4. Choo, S. C, Leong, M. S,, and Kuan, K, L. , On the Calculation of

Spreading Resistance Correction Factors, Solid-state Electronics 19 , 561-565 (1976).

5. D'Avonzo, D. C, and Button, R. W. , Spreading Resistance for Impurity

Profiles, Technical Report No. 5013-2, Stanford Electronics Laboratories, Stanford University (February 1977)

.

6. Pinchon, P. M., On the Validity of Correction Factors Applied to

Spreading Resistance Measurements on Beveled Structures, Semiconductor Measurement Technology: Spreading Resistance Symposium, J. R.

Ehrstein, Ed, NBS Special Publication 400-10, pp. 51-61 (December

1974)

.

7. Holm, R,, Electric Contacts, 4th ed., p. 16 (Springer-Verlag, New

York, 1967)

.

8. Dickey, D. H., Two-Point Probe Correction Factors, Semiconductor Measurement Technology: Spreading Resistance Symposium, J. R. Ehrstein,

Ed, NBS Special Publication 400-10, pp. 45-50 (December 1974).

9. Tannenbaum, E., Detailed Analysis of Thin Phosphorus Diffusions in

p-type Silicon, Solid-State Electronics 2_, 123-132 (1961).

10. Spiegel, M. R. , Calculus of Finite Differences and Difference Equations, p. 3, Schaum's Outline Series (McGraw-Hill, New York, 1971).

11. Jackson, J. D., Classical Electrodynamics , p. 64 (Wiley, New York,

1962)

.

12. Ibid., p. 92.

34
APPENDIX A

COMPUTER EXERCISES TO COMPARE CORRECTION FACTOR METHODOLOGIES

A.l Introduction

The exercises described here were designed to test the two schemes

developed: parallel superposition for isolated layers only and local

slope for general layer configurations

.

In Exercise I, section A. 2, the correction factor from parallel su- perposition is compared with factors calculated using the method of images. Here a single thin layer of uniform conductivity on an infinitely

thick substrate of lower conductivity is considered. In Exercises II

and III, section A. 3, both parallel superposition and local slope are compared with the results from the Schumann-Gardner (SG) approach. In

Exercise II, they are compared for two uniform layers on an infinite substrate and, in Exercise III, for three layers on an insulating siibstrate

.

(The second exercise, wherein the bottom-layer thickness is taken as very large, is a special case of the third.)

A. 2 Exercise I — Parallel Superposition versus Image Solution — Based

on Current Admixture

For this exercise, the correction factor evaluated using the parallel superposition formalism is compared with a correction factor derived

directly from the method of images. This latter correction incorporates

finite size contacts and a more general current distribution than used by

SG. This more general current distribution allows calculations that

reasonably guarantee, not merely assume, constant potential value under

the contacts. In this way, parallel superposition is tested with the most

generalized physical parameters yet developed which meet the basic boundary

conditions. In order to keep the image solution tractable, however, the

model is limited to two layers.

The image solution using current admixture is constructed as follows:

the radial potential distribution for one of the contact disks is derived

for two types of current distribution over the disk, one being a ringshaped distribution in which the current is a delta function at the periphery of the disk (correct for a very thin, isolated single layer) , and

the other being the classical distribution appropriate for a semi-infinite

uniform solid. The radial potential for unit current of each distribution

is calculated and admixed to achieve an approximately constant radial potential. The model used for this first exercise is shown in figure Al

.

By taking the substrate conductivity as zero in this model, the result

is the correction factor for an isolated slab of arbitrary thickness,

and subsequent results (for various thicknesses) are then used to calculate factors for arbitrary substrate conductivity from parallel su- perposition .

Given a ring-shaped current of radius, a, imbedded in a medium with

uniform resistivity and located a distance, z, above the origin, the po35
t

02

0

0 t DEPTH

Figure Al . Schematic conductivity profile for Computer Exercise I,

0 0.2 0.4 0.6 0.8 1.0

NORMALIZED DISTANCE FROM CONTACT CENTER, r/a

Figure A2 . Radial potential distribution on a spreading resistance contact. Structure of figure Al with t = 0.3a and 6 = 0.5 (ai = 302).

Curve A: from the ring-shaped delta function current distribution, eq

(A3); curve B: from current distribution appropriate to semi-infinite

substrate, eq (A4) (this is the distribution used by SG) ; curve C: from

admixture of current distributions above. The admixture coefficient is

chosen to force V^(0) to equal (r/a = 0.9) with the result for this

structure = (1 - 0.659) + 0.659 Vg

.

36
tential V(r,z) on the z = 0 plane may be written in terms of Legendre

polynomials [11]

:

V(r,0) =

Tr/a^

(Al)

For a ring-shaped current on the surface of the model of figure Al , the

potential inside the ring can be written as a sum of terms, each of the same form as eq (Al) , where each term of the sum corresponds to an image source. If the real source has strength I, the first image pair

(at a distance z = ±2Nt) will have strength 31/ where

(A2)

Each succeeding image pair will be weaker by the factor 3. If the substrate has zero conductivity, 3 is unity and each image has the same strength as the source. The second contact, displaced a distance s, also

has a series of images of similar strength associated with it. For simplicity, using s = 40a, the second contact and its images can be assumed

to be point-like and a constant potential is subtracted to account for

them. The radial potential then, inside the first current ring, is given

by

V(r,0) =

IP

27Ta 1=0 L J

(A3)

^

/a2 + (2Nt)2 f-'V/a2 + (2Nt)2/ \^ 7^2 + (2Nt)2/

N=l )i=0 ^

/s2 + (2Nt)2

This expression has been summed numerically for various values of r, t,

and 3. The results for t = 0.3a and 3 = 0.5 are shown as curve A in fig- ure A2.

The second current distribution to be considered is that appropriate to an equipotential disk contact on a semi-infinite uniform solid,

namely:

J(r) =

2-na' 1 - (r/a)

]-l/2

(A4)
Given this distribution, the potential at any point in the solid is [11]

V(r,z) =

2Tra sin- i 2a

_/(r-a) 2 + z2 + /(r+a)2 + z2_

(A5)

This current distribution is applied to the two-layer model structure,

using the same image strengths as before; the radial potential on the

first contact, z = 0, is written as

V(r,0) = 12.

4a

SinN=l

_ 2a _ 2

ITS TT y -t N=l

2a

/(r-a)2 + (2Nt)2 + /(r+s)2 + (2Nt)2

/s2 + (2Nt)2

(A6)

where the second contact and its images are again assumed to be point

sources. This expression has also been summed numerically; the results

for t = 0.3a and 3 = 0.5 are shown as curve B in figure A2 . (It is this

radial potential which is averaged over the surface of the disk in the

SG approach .

)

There is not a linear current admixture, using the two types of cur- rent distribution considered above, which will result in a constant radial potential under the contact for any particular values of t and 3

Nevertheless, the requirement of constant potential can be approximated:

mixtures of the two (keeping unit total current) have been taken which

force the potential at r = 0.9a to equal the potential at r = 0 . The

result of such admixture for t = 0.3a and 3 = 0.5 is shown as curve C in

figure A2 . The difference between curve B, the SG distribution, and curve C suggests that noticeable errors may occur in the SG procedure as

a result of considering only the current distribution appropriate to a semi-infinite solid, when evaluating more general structures. As will

be shown in section A. 4 of this appendix, this error is small, at least

for the representative structure chosen.

A more complete representation of the results of the above current

admixture procedure is given as the "exact" entries in table Al for various values of top-layer thickness, t, normalized to contact radius, a, and of the conductivity ratio, 3- To derive the values in the table,

after determining the admixture coefficients in each case for which

V(r = 0) equals V (r = 0.9a), the potential at r = 0 was taken as the effective potential on the disk contact. An equal but opposite potential

was assigned to the second contact. The reciprocal correction factor H was then determined for selected cases using appropriate admixtures of

eqs (A3) and (A6) to calculate the resulting differences, AV, of the

potential between the probes, with H being given by:

38
pi

2aAV (A7)

The results for 3=1 (corresponding to a = 0) are the most accurate

values known for the isolated single layer correction factor and are therefore referred to as exact in this appendix. These results are used

in section A. 4 of this appendix to compare with the results of the SG

assumptions on current distributions and are used below to calculate

values for nonzero substrate conductivity using parallel superposition.

Table Al — Reciprocal Correction Factor, H, for a Single

Layer of Thickness t on Lower-Conductivity, Infinite Substrate

t/a

0.1

exact

approx

1.016173

1.016173

0.1 0.5 0.9

0.874233

0.839153

0,442556

0.367099

0.134553

0.093805

1.0

0.042562

0.042562

1.0

exact 1.016173 0.963419

approx 1.016173 0,898496

0.745523 0.478176 0.363947

0.584689 0,403011 0.368947

10

exact 1,016173

approx 1,016173

1,012463 0.995510 0. 972602 0.964729

1,006820 0.981877 0.967437 0.964729

To derive the results of parallel superposition analysis of the

structure in figure Al for comparison with the foregoing, the reciprocal

correction factor H for measurements on the top of the structure is taken

from eq (3)

:

H = Gi/2aoi (A8)

Note that H is shown with no layer thickness dependence when it is the

correction factor for the entire underlying structure. By parallel su- perposition analysis of the structure

G = 2a [(01 - 02) H(t) + a2H(«>)] , (A9)

where H(t) and H(°°) are the values of H appropriate to the two layers of

the structure, see figure Al. Then, combining eqs (A8) and (A9) , the

factor, H, for the entire structure is given in terms of the individual

H values for the two component layers by:

Oi - O2 ^2

H = H(t) + — H(°°) (AlOa)

39
Each component term, H(t) and H(oo), is most accurately evaluated, in

turn, as a case of the isolated single layer, using the current admixture

formalism above. The factor H from parallel superposition is thus evaluated for the same values of t and 6 and current admixture formalism as was the image-derived solution for H above; the results from parallel

superposition are given as the "approx" entries in table Al

.

Table Al compares values of the correction factor, H, derived from

an image calculation, with values of H calculated by assuming that the

structure could be represented as independent parallel conducting layers

.

Both calculations feature the same treatment of the boundary value potential under the contacts as determined from the generalized current distribution used in this work, and appropriate to the same infinite thickness of the top layer. The discrepancies seen in table Al between the results from parallel superposition and the "exact" results obtained by admixing eqs (A3) and {A5) are not serious, and yet they provide proof that

the parallel superposition concept is not in general valid.

A. 3 Exercises II and III — Parallel Superposition and Local Slope versus

the Schumann and Gardner Boundary Value Solution

The model used for the following calculations consists of two uniform layers of differing conductivity on a third layer of still different

conductivity. In Exercise II the third layer has infinite thickness,

and in Exercise III the third layer has a thickness equal to that of

the first two and is bounded below by an insulating surface. The structure is shown in figure A3. The SG method is used to calculate reference two-probe correction factors for the structures , and then the parallel superposition and the local slope methods are employed to calculate equivalent factors for comparison.

The correction factors calculated in the SG approach are found by

numerical integration of the expression, [1]

:

F = — 26 (k)

Jl(k) Jo(sk)

"k^ 2k sin(k) dk (All)

The integrals were evaluated using a trapezoid method with an interval

dk = 0.01 and are truncated at the upper limit k = 10.* A probe spacing.

* An interval dk = 0.005 and an upper limit k = 20 were used for several

integrations, with insignificant change in the results.

40
tl t2 t3 DEPTH BELOW SURFACE

Figure A3. Schematic conductivity profile for

Computer Exercises II and III. The layer with

conductivity 03 is infinitely thick for Exercise

II

.

(A13)

s, equal to 40 contact radii was assumed for all the calculations. The

integration factor for the first structure (Exercise II) is

1 + 3 E + ^ E + B^a^E/E

and for the second structure, it is

1+26 (K) =

(1 + E3)E^E^ + B^E^(E3 + E^^) + B^E^ (E3 + e/) + B^B^ (e/ + E^^E^)

(1 - E3)E^E2 . 3^E^(E3 - E^^) , ^^^^^^^ _ ^^2) ^ 3^3^ (^^2 _ ,,^2^^)

where

E. = e~^^^^ (A14)

1 and

(a. - a._^,)

B. = , " ,

. (A15)

1 (a. + a.^^)

The correction factors obtained from the SG technique for the various

combinations of thickness and conductivity ratios are listed in table A2

for the first structure and in table A3 for the second structure. These

41
Table A2 — Sampling Volume Correction Factors for Exercise II*.

^1

6,z 0.9 0.6 0.3 0.0 -0.3 0.6 -0.9

0.9 34.532 19.355 12.401 8.008 4.878 2 .495 0.601 = 0. 05a 0.6 21.488 8.875 4.976 2.959 1.708 0 .851 0.225

0.3 15.078 5.298 2.805 1.622 0.927 0 .467 0.141 = 0. 10a 0.0 10.669 3.326 1.704 0.976 0.560 0 .291 0.103

-0.3 7.210 2.042 1.029 0.591 0.346 0 .190 0.081 = 00 -0.6 4.286 1.128 0.570 0.335 0.206 0 .124 0.067

-0.9 1.683 0.438 0.235 0.152 0.106 0 .077 0.057

0.9 19.499 12.277 8.326 5.601 3.533 1.879 0 509 = 0 10a 0.6 13.846 6.652 3.996 2.493 1.502 0.791 0 253

0.3 10.525 4.360 2.474 1.502 0.899 0.486 0 185 = 0 20a 0.0 8.008 2.959 1.622 0.976 0.591 0.335 0. 152

-0.3 5.866 1.970 1.062 0.644 0.402 0.244 0 132 — CO -0.6 3.901 1.217 0.659 0.412 0.272 0.182 0 119

-0.9 1.990 0.614 0.354 0.241 0.178 0.138 0. 109

0.9 7.335 = 0. 30a 0.6 6.157

0.3 5.237 = 0. 60a 0.0 4.424

-0.3 3.650 = oo -0.6 2.872

-0.9 2.045

5.372 3.980 2.831

3.703 2.513 1.738

2.820 1.846 1.265

2.202 1.417 0.976

1.717 1.102 0.773

1.310 0.854 0.619

0.955 0.651 0.496

1.966 1.179 0.487

1.173 0.734 0.379

0.866 0.571 0.340

0.686 0.477 0.319

0.563 0.415 0.305

0.472 0.370 0.295

0.401 0.335 0.288

0.9 2.245

= 1.2a 0.6 2.137

0.3 2.025

= 2.4a 0.0 1.906

-0.3 1.774

-0.6 1.623

-0.9 1.444

0.9 1.138 = 5a 0.6 1.132

0.3 1.126 = 10a 0.0 1.118

-0.3 1.108 = oo -0.6 1.096

-0.9 1.081

1.918 1.633 1.376

1.690 1.394 1.170

1.523 1.247 1.055

1.386 1.138 0.976

1.267 1.051 0.916

1.160 0.978 0.867

1.061 0.915 0.826

1.097 1.058 1.020

1.081 1.039 1.002

1.066 1.023 0.988

1.052 1.009 0.976

1.037 0.996 0.966

1.023 0.984 0.958

1.009 0.973 0.950

1.143 0.927 0.727

0.988 0.835 0.703

0.909 0.791 0.693

0.857 0.763 0.686

0.818 0.743 0.681

0.788 0.727 0.678

0.763 0.714 0.675

0.984 0.949 0.915

0.969 0.940 0.913

0.958 0.933 0.911

0.950 0.929 0.910

0.943 0.925 0.909

0.938 0.922 0.908

0.933 0.919 0.908

* Obtained using the method of Schumann and Gardner for two uniform layers

on an infinite substrate.

42
Table A3— Sampling Volume Correction Factors for Exercise III*.

^2 0.9 0.6 0.3 0.0 -0.3 -0.6 -0.9

0.9 45.483 38.847 31.329 23.926 16.636 9.457 2.385 = 0 05a 0.6 46.031 37.390 29.344 21.834 14.808 3.221 2.032

0.3 45.337 35.449 26.860 19.363 12.762 6.905 1.674 = 0 10a 0.0 44.393 32.732 23.659 16.398 10.455 5.502 1.309

-0.3 42.660 28.660 19.378 12.774 7.835 4.001 0.940 = 0 15a -0.6 38.878 21.880 13.363 8.247 4.834 2.394 0.564

-0.9 24.142 8.347 4.289 2.428 1.361 0.668 0.183

0.9 23.270 19.460 15.709 12.015 8.378 4.795 1.266 = 0 10a 0.6 23 . 046 18 . 736 14.722 10. 975 7.469 4 .180 1 .091

0.3 22.725 17.772 13.488 9.746 6.451 3.526 0.912

= 0 ,20a 0.0 22.233 16.424 11.898 8.273 5.304 2.828 0.731

-0.3 21.376 14.407 9.775 6.475 4.003 2.082 0.547 = 0 .30a -0.6 19-516 11.058 6.796 4.229 2.512 1.283 0.360

-0.9 12.373 4.404 2.312 1.346 0.788 0.426 0.171

0.9 7.854 6.609 5.382 4.174 2.984 1.811 0.656 = 0.30a 0.6 7.783 6.378 5.068 3.842 2.694 1.615 0.600

0.3 7.682 6.073 4.676 3.452 2.370 1.406 0.543 = 0.60a 0.0 7.528 5.650 ^.175 2.986 2.006 1.184 0.485

-0.3 7.264 5.024 3.511 2.419 1.593 0.947 0.426 = 0.90a -0.6 6.707 3.999 2.586 1.715 1.123 0.693 0.366

-0.9 4.631 2.008 1.211 0.817 0.580 0.421 0.306

0.9 2.278 2.014 1.753 1.496 1.242 0.992 0.745 = 1.2a 0.6 2.265 1.970 1.693 1.433 1.187 0.954 0.734

0.3 2.246 1.913 1.620 1.359 1.125 0.914 0.723 = 2.4a 0.0 2.218 1.336 1.527 1.272 1.057 0.872 0.712

-0.3 2.172 1.724 1.406 1.167 0.979 0.827 0.701 = 3.6a -0.6 2.079 1.546 1.241 1.039 0.892 0.779 0.689

-0.9 1.772 1.215 1.002 0.877 0.792 0.728 0.678

0.9 1.140 1.101 1.063 1.026 0.989 0.953 0.916 = 5a 0.6 1.138 1.096 1.056 1.018 0.982 0.948 0.915

0.3 1.136 1.089 1.047 1.009 0.975 0.943 0.914 = 10a 0.0 1.133 1.080 1.037 0.999 0.967 0.938 0.912

-0.3 1.128 1.068 1.023 0.987 0.957 0.932 0.911 = 15a -0.6 1.119 1.050 1.005 0.972 0.947 0.927 0.910

-0.9 1.097 1.020 0.981 0.955 0.936 0.921 0.908

* Obtained using the method of Schumann and Gardner for three uniform layers

on an insulating substrate.

43
factors are taken as a point of reference and assumed to be accurate;

the factors calculated using the parallel superposition and local slope

approaches are compared with them in succeeding tables

.

For the parallel superposition calculations, eq (8) is combined with

eq (A8) to derive an expression for F, the reciprocal of H, for the en- tire structure:

F = — = S Pi/^1 H(t. ) 1

(A16)

where the terms H(t.) are those appropriate to component sublayers of

thickness t.. In contrast to Exercise I where the sublayer H(t) values

were evaluated from the current admixture formalism, eq (AlO) ff., here

they are evaluated from the SG formalism applied to single isolated layers of thickness t. . The results of the calculations based on eq (A16) are listed for the two structures in cables A4 and A5 . They are shown

there, for each combination of thickness and conductivity ratio, as the

ratio of F derived from parallel superposition, eq (A16) , to the F derived directly from the SG approach, using eqs (A12) or (A13) , as appropriate „ Note that for all tables (A2 to A7) in Exercises II and III the

values for t^, t^, as listed are cumulative thickness values, in keeping with the formalism of SG [1], and not single layer thickness values.

Comparisons of the results of local slope calculations for F with

the SG values for F on the same two structures are given in tables A6

and A7 . To generate these results, one needs to simulate real data so

as to calculate local slopes. To do this, one calculates "true" factors

(using SG) for values of t slightly smaller and larger than nominal.

For example, in the case for t = 0.1a, SG factors have also been generated using t^ = 0.09a and t^ ='^0.11a (keeping t = 0.2a and t^ = 0.3a or

°°, as appropriate) . The "local slope" is then round from

d(logF) logF(t = 0.11a) - logF (t = 0.09a) m = = —— . (A17) dt 0.02a

Having defined a profile slope for each case, one uses eq (22) to calculate correction factors for the two structures. The substrate factor,

K K , in eq (22) is replaced by unity for the purpose of these calculations. This modification should improve the comparison based on eq (22)

in the vicinity of = Q = 0 where the structure resembles a large

uniform slab, but it simultaneously causes poorer comparison with SG in

the direction of lower values for F (from local slope) for cases of 3^

and Q^, both negative, where the structure resembles the conducting substrate limit.

* The substrate factor given in eq (24) was developed subsequent to the

generation of tables A6 and A7

.

44
Table A4 — The Ratio of Correction Factor from Parallel Superposition to that

from Schumann and Gardner for Exercise II .

Q^2 U.y U . Dn 0 U . JA A

U . 0 -0A O. J A C -\J • D

C\ Q — u . y

u. y 1 . ID J 1 . JU / 1 . J JZ 1 o o c 1 . 3z5 1 O A ij 1 . jUo 1 0 Q A i . zoO 1 1 = 0.05a 0.6 1.318 1.281 1.219 1.173 1.133 1.080 0.874

A 'i U . J1 nA 1 . J20 1 1 OQ 1- lyy 1 • 12.5 1 ATQ 1 • 01 A 1 C

1 . OJb 0 • yD4 u . D /y = o.lUa U . (J 1 .26d 1 1 AC 1-105 1 f\ AC 1 •04b 1 AAA 1 . 000 A A j4 T 0 . y4 / A OC 1 0 . obl A CIA U . DiU

A D -U . J1 ITT 1.1/2 1.015 A OCT 0* yb / 0 • yob A O O "7 A T 1 A A 0 c: 0 U • jDZ = 00 A C -O.D 1 . 018 A 0"7 A 0.879 0.81 /

f\ TC 1 0. 751 0.661 A C 1 1 0. bil A 1 QQ u.iyy

A Q 0.592 0.486 A jl A 1 0.421 A 1 C A 0. 352 A ATT 0.2/2 A y 1 A 0.1/4 A A >! Q u . U4y

A Q u . y 1.082 1.244 1 . 307 1.330 1 . 330 1 A 1 1 . 301 1 1 A 1 1 . lUl = 0.10a 0.6 1.264 1.344 1.305 1.255 1.195 1.098 0.743

U.J 1 111 1 . 331 1 '^'7*7 1.2// 1 1 0*7 1 . ly /

1 1 T7 1 • Iz /1 A y! O i . U4y A n u • y/u n CI 7 — u.^ua A n 1.325 1 . 173 1 A"? Q

1 . 0 /y 1 AAA 1 . OOU A QAC u . yoo A Tc: 1 U. /Di U . J JZ

A T 1.248 1.03 / A QO T

(J . yj / A O AQ 0. o4y A ~i A n U. /4U U • J /i n 0 9 7 u . ZZ J —U.o 1 ACT 1 . Ob / A OOO U. /ZD A "3 1 u • bi / U • JDU nu • i117i / A Q —u. y

r\ CIA 0.510 A 0£ A 0. 360 A O Q 0 A ATI A 1 0- loy A 1 ni U . iUi n no7 U . Uz /

A Q u.y 1.024 1.136 1.203 1 ^ O "7 1.23 /

1 AAA 1.223 1 IOC 1.135 U . Dbb = 0.30a 0.6 1.149 1.333 1.362 1.321 1.215 0.997 0.438

0 . J 1.246 1.368 1 . 308 1 1 A >! 1 . iy4 1 A 0 O 1 . 02o A TT\ U • / /i U . zo4 = O.oOa 0.0 1.312 1.298 1.158 1 AAA 1.000 A 0 1 O 0.812 A ceo U . bb J

A 1 QO

A O -O.J 1.319 1.124 /^ A^ A 0.934 A Id O 0. IbZ 0 . 584 U . J /b n 1 no u . iuy = oo A ^ -O.D 1.175 0.813 A C\ A 0.624 U . 4 /o A 0 VI J( U. j44 U . ZUd U . U J J -0.9 0.534 0.270 0. 187 A 1 O 0 0.133 A AOQ u . UBy A AX Q u . U4y n ni 7

0.9 1.019 1.070 1 .093 1.0/3 0.9/8 A Tc: A 0 • /bU A 0 U . zb J = 1.2a 0.6 1.061 1.171 1.197 1.141 0.984 0.693 0.215

0.3 1.106 1 A O C 1 . 235 1 • III 1 1 AQ u . oyo U . joo nU .

1 (\i J.D / = 2.4a 0.0 1.152 1.257 1 - 1 /b 1 AAA 1 - OUO A Tc: Q U . 4D^ nu . IZ1 7TJ -0.3 1.195 1 .210 1 A O O

1 . 038 A O 1 C

0 . OlD U . J /D n 0 TO U . J ju n r»R7 U . UoZ = CO -0.6 1.201 1.016 0. /53 A C yl O 0 o b4 J

A 0 c: /I U . J j4 u . i yu U . Ul J -0.9 0.863 0.424 0.254 r» 1 c A 0.159 n A A c 0 . 095 A A <1 Q U • U48 r»U . miUii

0.9 1.021 1.032 1.033 1.013 0.955 0.807 0.364 = 5a 0.6 1.025 1.044 1.045 1.019 0.950 0.786 0.340

0.3 1.030 1.054 1.050 1.016 0.933 0.753 0.309 = 10a 0.0 1.036 1.060 1.048 1.000 0.900 0.702 0.269

-0.3 1.042 1.059 1.031 0.962 0.837 0.621 0.217 = CO -0.6 1.045 1.037 0.974 0.866 0.708 0.481 0.147

-0.9 1.005 0.849 0.672 0.499 0.337 0.185 0.045

* Two uniform layers on an infinite substrate,

45
Table A5 - The Ratio of Correction Factor from Parallel Superposition to that

from Schumann and Gardner for Exercise III*.

t, =

t^ =

t, =

t-, =

t^ =

0,9 0.6 0.3 0.0 -0.3 -0,6 -0 .9

0.9 1.000 1.000 1,000 1.000 0.999 0.997 0. 981 = 0.05a 0,6 1.000 1.000 1.000 1.000 0.999 0.997 0. 978

0.3 1.000 1.000 1.000 1.000 0.999 0.996 0. 973 = 0.10a 0.0 1.000 1.001 1.001 1.000 0,999 0,994 0. 964

-0.3 1.000 1.000 1,000 0.999 0.997 0,991 0. 949 = 0.15a -0.6 1.000 0.999 0.998 0.996 0.992 0.981 0. 911

-0,9 0.993 0.984 0.977 0.953 0.951 0.912 0. 703

0.9 1.000 0.993 0.997 0.994 0,990 0.981 0. 921 = 0.10a 0,6 1.000 0.999 0.998 0,996 0.992 0,982 0. 913

0.3 1.000 1.001 1.000 0.998 0.994 0.982 0. 899 = 0.20a 0.0 1.001 1.002 1.002 1.000 0.995 0.979 0. 875

-0.3 1.001 1.003 1.003 1.000 0,991 0.969 0. 830 = 0.30a -0.6 1.002 1.003 0.999 0,991 0.975 0.936 0. 730

-0.9 0.984 0,957 0.932 0.893 0.845 0.738 0. 391

t^ =

t^ =

0.9 1.001 1,004 1.004 0.998 0.980 0,924 0.639 = 0.30a 0.6 1.002 1,008 1,008 1.002 0.980 0.915 0.604

0.3 1.004 1.011 1.012 1.003 0.976 0.898 0.557 = 0,60a 0.0 1.005 1.015 1.014 1.000 0.962 0,864 0.492

-0.3 1.007 1.016 1,009 0.982 0.926 0.800 0,403 = 0.90a -0.6 1.008 1.002 0.972 0.917 0,827 0,662 0.276

-0.9 0.941 0.796 0.684 0.572 0,446 0.290 0.085

t, =

to =

t, =

t^ =

0.9 1.008 1.025 1.025 0.996 0.911 0.712 0.261 = 1.2a 0.6 1.011 1.035 1.033 1.004 0.903 0.693 0.243

0,3 1.015 1.047 1.050 1,008 0.894 0.662 0.221 = 2,4a 0,0 1.021 1.061 1.059 1,000 0.862 0,611 0.191

-0.3 1.030 1.075 1.054 0.962 0.790 0,527 0.153 = 3,6a -0,6 1.044 1.070 0.988 0.836 0.631 0,384 0.102

-0,9 1.013 0.775 0.555 0,333 0.244 0,129 0.030

0,9 1.006 1.020 1.025 1.016 0.974 0.851 0.419 = 5a 0,6 1.007 1.022 1.027 1.015 0.956 0.832 0.394

0.3 1.008 1.025 1.028 1.010 0.952 0.804 0.363 = 10a 0.0 1.009 1.027 1.027 1.000 0.928 0.761 0,323

-0.3 1.011 1.028 1.019 0,976 0.831 0.691 0.267 = 15a -0.6 1.013 1,020 0.935 0,903 0.776 0.560 0.186

-0.9 0.995 0.893 0.747 0.585 0.414 0.238 0.060

Three uniform layers on an insulating substrate,

46
Table A6 - The Ratio of Correction Factor from the Local Slope Method to that

from Schumann and Gardner for Exercise II*.

t, =

t^ =

t, =

t^ =

0.50a

0.10a

t^ =

0.10a

0.20a

t^ =

t^ =

0.30a

0.60a

t^ =

^2 0.9 0.6 0.3 0.0 -0.3 -0.6 -0.9

0.9 1.031 1.197 1.308 1.384 1.416 1.302 0.509

0.6 1.234 1.577 1.711 1.709 1.447 0.737 0.435

0.3 1.414 1.795 1.830 1.551 0.888 0.513 0.425

0.0 1.595 1.931 1.707 1.000 0.579 0.458 0.420

-0.3 1.804 1.945 1.204 0.629 0.483 0.436 0.417

-0.6 2.105 1.573 0.662 0.493 0.445 0.426 0.415

-0.9 2.921 0.683 0.484 0.443 0.428 0.419 0.413

0.9 0.986 1.082 1.156 1.202 1.200 1.021 0.494

0.6 1.107 1.368 1.478 1.459 1.203 0.684 0.457

0.3 1.237 1.563 1.601 1.362 0.851 0.542 0.448

0.0 1.384 1.709 1.551 1.000 0.629 0.493 0.443

-0.3 1.570 1.787 1.257 0.717 0.534 0.469 0.440

-0.6 1.861 1.679 0.830 0.569 0.490 0.456 0.438

-0.9 2.631 1.081 0.595 0.502 0.467 0.448 0.437

0.9 0.943 0.965 0.979 0.972 0.916 0.766 0.585

0.6 0.981 1.091 1.131 1.086 0.931 0.719 0-577

0.3 1.036 1.205 1.217 1.087 0.863 0.676 0.571

0.0 1.110 1.310 1.249 1.000 0.791 0.645 0.567

-0.3 1.216 1.410 1.227 0.935 0.733 0.624 0.564

-0.6 1.392 1.504 1.151 0.847 0.689 0.608 0.561

-0.9 1.777 1.579 1.036 0.773 0.657 0.596 0.559

t, =

t^ =

t-, =

t, =

t^ =

t^ =

0.9 0.844 0.847 0.856 0.874 0.902 0.943 1.003 = 1.2a 0.6 0.850 0.879 0.910 0.940 0.968 0.995 1.020

0.3 0.861 0.917 0.958 0.987 1.007 1.020 1.027 = 2.4a 0.0 0.877 0.958 1.001 1.000 1.034 1.036 1.031

-0.3 0.903 1.003 1.042 1.055 1.055 1.046 1.033 = 00 -0.6 0.945 1.054 1.081 1.081 1.070 1.054 1.035

-0.9 1.019 1.113 1.119 1.104 1.083 1.059 1.036

0.9 0.926 0.945 0.966 0.989 1.014 1.042 1.071 = 5a 0.6 0.929 0.955 0.979 1.004 1.027 1.051 1.074

0.3 0.932 0.964 0.991 1.015 1.037 1.057 1.075 = 10a 0.0 0.936 0.974 1.002 1.000 1.044 1.061 1.076

-0.3 0.942 0.984 1.012 1.033 1.050 1.065 1.077 = 00 -0.6 0.949 0.994 1.021 1.040 1.055 1.068 1.073

-0.9 0.959 1.005 1.030 1.047 1.060 1.070 1.079

* Two uniform layers on an infinite substrate,

47
Table A7 - The Ratio of Correction Factor from the Local Slope Method to that

from Schumann and Gardner for Exercise III .

t, =

t^ =

t. =

t. =

t^ =

t, =

t, =

t-, =

t. =

t-. =

t^ =

0.9 0.6 0.3 0.0 -0.3 -0.6 -0.9

0. y 0 - 969 0 . 965 0 962 0. 958 0. 954 0.943 0.765 = 0. 5a 0.6 0 . 969 0 . 964 0 . 961 0. 957 0. 953 0. 938 0.715

0. 3 0 968 0 . 963 0 . 960 0 . 955 0. 950 0. 930 0.650 = 0.10a 0.0 0. 968 0 . 962 0 - 958 0. 954 0. 946 0. 915 0.574

-0.3 0. 967 0 . 951 0 . 956 0 . 950 0. 937 0.879 0.501 = 0.15a -0.6 0-965 0-958 0.951 0.939 0.903 0.768 0.447

A Q

0 . 959 A A >l 1 0 . 943 0 . 896 0 781 A CAT 0. 593 0 . 463 0.420

u . y 0 . 958 9 . 956 0 • 954 0 . 949 0 - 940 0 . 905 0. 585 = 0.10a 0.6 0 . 958 A AC el 0 . 956 A AC 1 0. 953 0 . 943 A A "5 C 0 ~ 93b 0 . 889 0.549

0 .

3

0 958 0 . 955 0 952 0 . 945 0.928 0 . 865 r\ c 1 "7 0. 517 = 0.20a A A 0.0 0 . 958 0 . 954 0 . 949 0 - 940 A A1 C 0 • 915 0.822 0 . 488

-0.3 0 . 957 0 . 953 0 . 945 0.929 0 . 886 A 1 A 0 . 742 0 . 465 = 0.30a -0.6 0.956 0.949 0.934 0.396 0.800 0.597 0.449

-0.9 0. 955 0 . 925 0. 818 A C A C 0.545 0.520 A A ^ 0 462 0.439

0.9 0.933 0.931 0.920 0. 900 0 . 859 0 . 760 0 . 582 = 0.30a 0.6 0. 933 0. 930 0 . 917 0 . 893 0. 344 0 736 0 . 577

0.3 0.937 0.928 0 . 912 0 . 883 0.824 0 . 708 0 . 572 = 0.60a A A 0.0 0 . 937 0 925 0 - 904 0 . 867 0. 796 0 . 675 A C ^ O

0 . 568

A T -0 .

3

0 . 936 0 . 920 0.893 0 . 842 A T C C 0 - 755 A e: A 0 - 543 A C A 0 . 554 -n 6 U . 7 J 4 U . -7 J. *1 . O U W 0 701 U . U J. 0 . 561

-0.9 0.952 0.975 0.746 0.652 0-595 0-559

0.9 0.843 0.839 0.839 0.847 0.868 0.910 0.983 = 1.2a 0.6 0.843 0.840 0.843 0.855 0.879 0.924 0 995

0.3 0.843 0.842 0.849 0.866 0.895 0.939 1.002 = 2.4a 0.0 0.843 0.846 0.859 0.883 0.916 0.958 1.009

-0.3 0.844 0.855 0.879 0.911 0.946 0.981 1.017

= 3.6a -0.6 0.847 0-881 0.922 0.960 0.990 1.011 1.024

-0.9 0-883 0.994 1.041 1.057 1.056 1.048 1.033

t, =

t., =

t., =

0.9 0.925 0.943 0 . 963 0.9P5 1-010 1.038 1.070 = 5a 0.6 0.926 0.946 0.967 0.991 1.016 1.042 1.071

0.3 0.927 0.949 0.973 0.997 1.022 1.047 1.073 = 10a 0.0 0.928 0.955 0-980 1.005 1.029 1.052 1.074

-0.3 0.931 0.962 0.990 1.014 1.037 1-057 1.075 = 15a -0.6 0.935 0.974 1.003 1.027 1-046 1.063 1.077

-0.9 0.948 0.996 1.024 1-043 1.057 1.069 1.078

* Three uniform layers on an insulating substrate.

48
A. 4 An Empirical Expression for H(t)

In Exercise I, above, the method of images was used to find correction factors for a single isolated layer by admixing two possible cur- rent distributions on the contacts. In Exercise II, equivalent single

isolated layer factors were found using the SG approach [by setting 3^^ = 1

and ~ 0-in eq (A12) ] . Both of these methods require lengthy calculations. For the purpose of making rapid calculations of correction factors using parallel superposition, an empirical expression for H(t) has

been developed. The expression used is hyperbolic, having asymptotes

for small and large slab thicknesses which coincide with the exact form

of H(t) calculated by the admixture of currents and subsequent application of the method of images as done in Exercise I above. The hyperbolic

expression, having an easily calculated second derivative for use in the

parallel superposition method, is:

Numerical values given by this empirical expression are listed in table

A8 for a single isolated slab with various values of t, using s = 40a.

The same table lists values, for these same cases calculated using admixed

current distributions under the contacts, i.e., the exact results from

A. 2 of this appendix and, additionally, values for H calculated using the

SG method. Ratios of empirical to exact values, and of SG to exact

value are also given. It is here seen that the results from the SG as- sumption regarding current distribution under the contact are not significantly different from those with the more general current distribution

used in section A. 2 to better satisfy the requirement of equipotential

under the contact. The values listed for admixed current distributions

(the first column of table A8) were used to plot figure 1 of this report.

H(t) (A18)

49
Table A8 — Reciprocal Correction Factors for a Single Isolated Layer of Thickness t*.

loq, n ( t/a) exact

-0 ft 0 0673

-0.6 0.1058

-0.4 0-1645

-0.2 0.2503

0.0 0.3689

0-2 0 5166

0.4 0-6726

0.6 0 8084

0.8 0.9058

1.0 0.9647

SG empir

0 0407 0.0425

0 0644 0-0673

0.1016 0.1064

0.1590 0.1672

0.2450 0.2597

0.3656 0.3927

0 5162 0.5622

0.6753 0.7367

0.8134 0.8716

0 9125 0.9505

0.9723 0.9884

Ratios

/ SG \ /empir

exact

/

V

'

1 exact

\

0.9569 0.9996

0.9572 1-0007

0.9598 1.0052

0.9669 1.0168

0.9788 1.0377

0.9909 1.0645

0.9993 1.0883

1.0039 1-0953

1.0062 1.0782

1.0074 1-0493

1.0079 1 0246

* Factors are calculated: exact - from admixed current distributions

using eqs (A3) and (A6) , SG - from eq (All) , and empir - from eq (A18)

.

50
APPENDIX B

Universal Data Reduction Program

A general program utilizing both the parallel superposition and local slope techniques for interpreting spreading resistance data is listed

at the end of this appendix. Also provided is the list of variables for

the program. As listed, the program is written for the Hewlett-Packard

9830A programmable calculator. The program contains three semiindependent subprograms: a data logging routine, a data reduction routine, and plotting and graph scale subroutines for deriving the associated printer/plotter. The function of the program is to log raw spreading resistance values as they appear at the digital output of the

spreading resistance probe; correct them for sampling volume if desired;

convert them to either carrier concentration, resistivity, or conductivity; and plot them on a profile graph. (The plotting and graph scale

subroutines are not included in the listing because they are specific

* to the graph format and plotter used, but are available on request. ) The program requires interaction from the operator; when it is run, the

calculator display shows a prompt, or message, whenever an input is re- quired from the keyboard.

The normal flow of the program begins with the loading of carrier

concentrations and resistivities for 32 standard calibration specimens

(through line 50) . Lines 50-220 read five standard resistances and up

to 140 data points from the digital output of the spreading resistance

probe. If fewer than 140 points are run, the data logging loop must be

interrupted by pressing the STOP key, and the program must be continued

at line 230. Lines 230-250 determine and execute a branch to a number

of different places in the program.

The data reduction section begins at line 260. Prestored spreading

resistance values appropriate to the probe loading and specimen surface

orientation are loaded through line 300. Lines 310-350 define the contact

radii to be used in the following calculations . The values given are those used for data reductions contained in this report. They were obtained from the probe spacing experiment on diffused specimens. New values as appropriate would be inserted by the user. Lines 380-430 input

experimental parameters, including bevel angle and location information.

A loop begins at line 440 which is cycled for each layer of different

conductivity type in the profile. Conductivity type, correction scheme,

and last (shallowest) point for the layer in question are specified

through line 510, and three constants needed later are precalculated at

lines 520-540. At line 550 another loop begins which is cycled for each

point in the layer. The uncorrected spreading resistance is calculated

from raw data at line 560, and line 570 temporarily sets the sampling

volume correction factor to unity (in the event that no correction has

been specified) . The local slope of the profile is determined at line

* Available from D. H. Dickey, Solecon Laboratories, 3001 Red Hill

Avenue, Bldg. 3, Suite 106, Costa Mesa, California 92626.

51
590, and the desired correction subroutine is called out at line 600.

The spreading resistance is corrected for sampling volume at line 620,

and the calibration curve look-up subroutine is called at line 640. This

subroutine returns interpolated values of resistivity, carrier concentration, and conductivity based on the corrected spreading resistance of

the point in question. The depth for the point is calculated at line

660, and the plot subroutine is called out at line 570. After data for

each point specified in the layer is reduced and plotted, a test is made

at line 720 to determine whether the bevel edge has been reached. If not,

the program branches back to line 440 to input the conductivity type of

the next layer. When the bevel edge is reached, the graph scales are printed (line 750), and the program branches back to line 230. From line

230, the operator can branch to the data logging routine or can restart

the data reduction from the same data after changing some of the parameters. When line 230 is accessed after logging new data, the operator can branch directly to line 360 in the event that the same probe load and orientation are appropriate.

I In the local slope subroutine, lines 800-860, the substrate point

number is first calculated at line 820. It is the shallower of the deepest point in the layer or a point N3 increments deeper than the point in

question. The uncorrected spreading resistance for the substrate point

is calculated from raw data at line 830, and the substrate factor is calculated [according to eq (24)] at line 840. The value of the sampling

volume correction factor is calculated [according to eq (22) , using

for K^K^] at line 850 and is returned to the main program.

In the parallel superposition subroutine, lines 880-940, the second

term of eq (16) is first calculated. The second derivative of H(t),

from eq (A15) , is multiplied by the conductivity of each point and

summed through line 920. The sum Si is multiplied by its coefficient to

complete the second term of eq (16) , and this is added to the slope term

to give the sampling volume correction factor at line 930.

For routine use, the local slope scheme is used almost exclusively

except when no correction is desired. The parallel superposition scheme

is only used as a check on simple diffusions, where surface resistivity

is lower than subsurface or substrate resistivity. It cannot be used in cases where subsurface resistivity is lower than that at the surface,

whereas the local slope scheme handles these cases easily. The accuracy

of the local slope scheme is not particularly sensitive to the choice of

substrate factor except in a few special cases, but an improved formula

for it is probably desirable. The substrate point has been specified

N3 points deeper on the profile because the second derivative of H(t)

peaks at this depth. In effect, the substrate factor approximates the

summation in eq (16) with a single point.

52
10 DIM AS[2,2] ,NS[2,16] -PS [2,16] ,RS[2,16] ,SS[140] .UI[5] , VI [140] ,YS(3]

20 LOAD DATA 21,

P

30 REM P ARRAY CONTAINS LOGS OF CALIBRATION SAMPLE RESISTIVITIES

40 LOAD DATA 22,

N

50 REM N ARRAY CONTAINS LOGS OF CALIBRATION SAMPLE CCff^CENTRATIONS

60 FOR 1=1 TO 5 70 REM READING PROBE OUTPUT FOR 5 STANDARD RESISTANCES

80 DISP "PRINT FOR R=10'~ "2*1-2

90 WAIT 1000

100 ENTER (1,*)A,V

110 U[I]=V*lE+04

120 DISP I,U[I]

130 miT 1000

140 NEXT I 150 FOR 1=1 TO 140

160 REM READING PROBE OUTPUT FOR DAtA POINTS

170 REM INTERRUPT LOOP AND CONTINUE AT LINE 230 IF LESS THAN 140 POINTS RUN

180 ENTER (1,*)A,V

190 V[I]=V*lE+04

200 DISP I,V[I]

210 WAIT 1000

220 NEXT I 230 DISP "WHAT NEXT"; 240 INPUT D 250 GOTO D OF 60,260,360,440,750,780

260 DISP "PROBE LOAD, ORIENTATION";

270 INPUT L1,Q

280 L=INT(L1/10)

290 LOAD DATA {L+2*Q+22),R

300 REM R ARRAY CONTAINS LOGS OF CALIBRATIC»I SAMPLE RESISTANCES

310 A[l,l]=l.l

320 A[l,2]=2.1

330 At2,l]=0.5

340 A[2,2]=l.l

350 REM A(T,L) IS CONTACT RADIUS FOR TYPE(T) AND LOAD(L)

360 Fl=l

370 Rm Fl IS A FLAG USED IN PLOT SUBROUTINE

380 DISP 'PROBE SPACING";

390 INPUT S 400 DISP "BEVEL ANGLE, X-STEP";

410 INPUT A,B

420 DISP 'BEVEL EDGE, START PT^'; 430 INPUT N0,N2

440 DISP "TYPE(P OR N)";

450 INPUT B$

460 T=POS(B$, "P")+2*P0S(B$, "N ') 470 IF T=0 THEN 230

480 DISP "CORRECTION";

490 INPUT C

500 DISP "STOP POINT';

510 INPUT Nl

520 Kl=2*A[T,L]*L0G(S/A{T,Ll )/PI

530 K=K1/A/B

540 K4=(1-2*A[T,L]/PI/S)~2

550 FOR N=^2 TO Nl STEP -1

560 R0=4+4*(V[N]-U[3] )/(U[4]-U[2]

)

53
570 F=l

580 IF C=0 THEN 620

590 M=4.6/A/B*(1+(N^1) + (N^2) )*(V[N-1+(N=4^1)I-V[N+1-(N^2)] )/(U(41-U[2] ) 600 G0SU3 C OF 800,880

610 REM BRANCHING TO EITHER LOCAL SLOPE OR PARALLEL SUPERPOSITION SUBROUTINE

620 R=RO-LGTF

630 REM RO HAS NOW BEEN CORRECTED FOR SAMPLING VOLUME

640 GOSUB 960

650 REM BRANCHING TO CALIBRATION CURVE LOOK-UP SUBROUTINE

660 X=(N-N0)*A*B

670 GOSUB 1120

680 REM BRANCHING TO PLOT SUBROUTINE

690 NEXT N

700 N2=N1-1

710 REM SET NEW START POINT TO (OLD STOP POINT - 1)

720 IF N2 <= NO THEN 750

730 GOTO 440

740 REM RETURN TO REDUCE NEXT LAYER

750 GOSUB 1620

760 REM BRANCHING TO SCALE GRAPH

770 GOTO 230

780 END

790 REM

800 RET'l LOCAL SLOPE

810 N3=INT(K)

820 N4=(N+N3)*( (N+N3) <= N2)+N2*( (N+N3)>N2)

830 R1=4+4*(V[N4]-U[3] )/(U[4]-U[2]

)

840 K3=l+(8*I;OG(S/A[T,L] )/PI/PI-l ) *2/PI*ATN (RO-Rl

)

850 F=-Kl*M/2+SQR(Kl*M*Kl*M/4+K3)

860 RETURN

870 REM

880 REM PARALLEL SUPERPOSITION

890 S1=0

900 FOR 1=1 TO N2-N

910 S1=S1+3*I*K*K*K4*S[N+I]/(K4*I*H-K*K)~2.5

920 NEXT I 930 F=-K1*M+0.0002*A[T,L]*10"RO*K*S1

940 RETURN

950 REM

960 REM CALIBRATION CURVE LOOK-UP

970 J=16

980 IF R>R[T,J] THEN 1020

990 IF J=l THEN 1020

1000 J=J-1

1010 GOTO 980

1020 IF J=16 THEN 1060

1030 Y[1]=(R-R[T,J] )*(N(T,J+l]-NtT,J] )/(R [T, J+1] -R [T, J] )+N[T,J]

1040 Yt2]=(R-R[T,J] )*(P[T,J+1]-P(T,J] )/(R[T,J+l]-R[T,J] )+PtT,J]

1050 GOTO 1080

1060 Y[l]=RlT,16]-R+N[T,16]

1070 Y[2]=R-R[T,16]+P[T,16]

1080 Y[3]=-Y[2]

1090 S[N]=10'-Y[3]

1100 RETURN

1110 REM

1120 REM PLOT SUBROUTINE BEGINS

54
Variable List

Units

I. Dimensioned Variables — Integer Precision

U(5) - Digital voltmeter reading at output of

probe, for five standard resistances 10~^ V

V(140) - Digital voltmeter reading at output of

probe, for data points 10"^ V

II. Dimensioned Variables — Split Precision

A (2, 2) - Contact radius, for two conductivity

types and two probe loadings ym

N(2,16) - Logarithm of carrier concentration for

calibration sample, for 15 specimens of

each conductivity type

P(2,16) - Logarithm of resistivity for calibration

sample, for 16 specimens of each conductivity type

R(2,16) - Logarithm of spreading resistance for

calibration sample, for 16 specimens of

each conductivity type

S(140) - Conductivity for each data point (^^•cm)"-'-

Y(3) - Logarithm of (1) carrier concentration,

(2) resistivity, and (3) conductivity,

for data point

III. Simple Variables

A - Bevel angle rads

B - X-step ym

F - Sampling volume correction factor

K - Defined constant (see program line 530)

Kl - Defined constant (see eq (20) ) ym

K3 - Substrate factor (see eq (24)

)

K4 - Defined constant

Ll - Probe loading g

M - Profile slope (decades) /ym

NO - Bevel edge point number

Nl - Stop point number

N2 - Start point number

55
-1

III. simple Variables (continued)

Units

N3 - The number of points to substrate point

N4 - Point number for substrate point

R - Logarithm of corrected spreading resistance

RO - Logarithm of uncorrected spreading resistance

Rl - Logarithm spreading resistance for substrate point

S - Probe spacing ym

Si - Sum of conductivity x A^H (see eq

(16)) (n-cm)

IV. Flags and Pointers

C - Correction scheme pointer

0 - no correction

1 - local slope

2 - parallel superposition

D - Branching pointer, range 1-5

Fl - Flag used to set up plot scales

T - Conductivity type flag

1 - p-type

2 - n-type

L - Probe load flag

1 - 10-g load

2 - 20-g load

Q - Orientation flag

0 - (100) orientation

1 - (111) orientation

56
APPENDIX C

CORRECTION FACTOR FOR UNIFORM-RESISTIVITY WEDGE

The effect of bevel angle on spreading resistance measurements has

been explored by solving the potential distribution problem represented

by two disk contacts on one surface of an infinite homogeneous wedge. If

the wedge angle is taken as an odd-integral submultiple of tt , the problem

can be solved using the method of images with a finite array of images.

The solution is approximated by taking the images as point sources rather

than disk-like, although it is straightforward (though laborious) to

write the exact solution in terms of Legendre polynomials. The model as- sumed is shown in figure CI, where the two contacts are aligned parallel

to the axis of the wedge at a place where the local thickness is t. The

correction factor in the point-image approximation is

F = 2a

(2N+1)

N-1

M=0 COS

(CI)

where X is given by

A

ITS 2(2N+l)t '

and N is the number of images, related to the wedge angle according to

(C2)

2N+1 • (C3)

Correction factors were computed from eq (Cl) for a variety of wedge angles, for two values of local thickness, and two values of probe spacing,

The results are shown in figures C2 and C3.

* *

*

Figure Cl. Image array for infinite homogeneous wedge.

57
0.02 0.04 0.06 0.08

WEDGE ANGLE (radj

0.10

Figure C2 . Sampling volume correction factor for uniform wedge with

local thickness equal to contact radius. Curve A, probe spacing equal

to 400 contact radii; curve B, probe spacing equal to 40 contact radii.

O

0.02 0.04 0.08 0.08

WEDGE ANGLE (radj

0.10

Figure C3. Sampling volume correction factor for uniform wedge with

local thickness equal to 0.1 contact radii. Curve A, probe spacing

equal to 400 contact radii; curve B, probe spacing equal to 40 contact

radii

.

58
APPENDIX D

SPECIAL OPERATIONAL PROBLEMS OF THE LOCAL SLOPE METHOD

AND SUGGESTED ALTERNATIVES

A problem was encountered when using the local slope method for

analyzing seemingly anomalous data which occurred near the bevel edge of

isolated layers. The problem was the calculation of nonphysical (negative or imaginary) values for F. It is shown below that the anomalous

data are not the prime cause of the unacceptable values for F, but they

do complicate analysis of the problem. The existence of such anomalous

data, which are more obvious on thin isolated layers, is believed due

to the asymmetry in the conduction path between the probes when in proximity to the bevel edge. This same asymmetry in potential distribution

near the bevel edge should cause similar, if less recognizable, data

anomalies in other layer structures and will affect the accuracy of analysis for any correction factor algorithm chosen [5]

.

Whatever the cause of such anomalous data, they are typically seen

on measurements of a thin isolated layer as nonmonotonic or turnaround behavior for data near the bevel edge. Abnormally high spreading resistance

values are seen at the bevel edge, followed by decrease to some minimum

value, then rapidly increasing values of spreading resistance following

the normal behavior for thin isolated layers as shown in figure Dl . This

data turnaround is generally found to increase with increasing s/a values

in a manner similar to that shown in figure 18 for the uniform top surface of a layer. The result of the nonmonotonic data is that local slope

values, calculated for coordinates beginning at the bevel edge, are negative for a few points, then increase through zero to normal positive

values

.

It is noted that for thin isolated layers such nonmonotonic data are contradictory to the assumptions of the parallel superposition model upon

which the isolated layer limit of the local slope model is founded. According to the parallel superposition model, as supported by the results

of the probe spacing experiment in section 3.1, the spreading resistance

measured at any point on a sectioned isolated layer is proportional to

the sheet resistance of the layer beneath the point of measurement.

Since the concept of parallel superposition is supported by the ex- perimental results of section 3.1, it was necessary to question whether

the anomalous data or the local slope extension of parallel superposition was responsible for the nonphysical values for F. Tables Dl to D5 were constructed for various values of local slope. Equation (22a) using the function K was used to calculate F for various values of s/a

and of R /R . For the same values of local slope and of s/a, F was also

o s calculated using K^K^ or unity in place of K^.

Two features found in the tables indicate that anomalous data are not the primary cause of nonphysical F values. The first is that use of

K^K^ or unity always results in positive F values regardless of local

59
0 12 3 DISTANCE FROM SURFACE |Mm|

Figure Dl . Spreading resistance data for junction-isolated implanted

specimen illustrating the type of local slope behavior near the top sur- face which gives nonphysical values for the correction factor when analyzed by eq (22a) or (22b)

.

slope value. The second is that use of results in some nonphysical F

values even for small positive local slope values.

The only exception to the general pattern of nonphysical F values,

when using the form K. with small R /R in eq (22a) , occurs for very

small values of s/a.

3 This result, OSonly seen near the limit of physically attainable s/a values, is regarded as an accident. It is noted that

the values for F under such circumstances, although positive, are no- ticeably different from values calculated using the forms K^K^ or unity.

The fact that nonphysical values can be calculated for F for certain

combinations of local slope and measurement condition does not invalidate

the derivations or algorithm construction embodied in this work. For many

layers of this type, the strict formalism of parallel superposition, eq

(16) , could always be used as outlined in Appendix B; no such spurious results are encountered for this formalism. However, it is assumed that the

local slope formalism is more appropriate for multilayer structures, even

if slight modifications must be made for application to isolated layers.

It must be remembered that the term K was originally hypothesized as a replacement for K^K^ in eq (22) to make the equation for F more generally

60
Tables D - Correction Factor F Based on Local Slope Using K3 for Various Values

of Rq/Rj and Also Using K^K2 and Unity to Replace K^: Case of a = 2 ym.

Table Dl

:

Case of Local Slope, m = -0.5

V. Rp/Rs

1 0.5 0.1 0,01 0.001 0.0001 K1K2 1

10 .51 .44 .32 .23 .19 .14 . 3 \

20 .43 .32 .14 0 -.03 -.17 .87 .43

40 .37 .23 0 -.18 - 29 -.41 . 0 /

60 .34 .19 -.05 -.26 -.39 -.52 .94 .34

80 .32 .17 -.10 -.32 -.44 -.59 .95 .32

Table D2: Case of Local Slope, m = -0.1

sRq/Rs

s/a\^ 1 0.5 0.1 0.01 0.001 0.0001 1

10 .86 .77 .62 .49 .42 .34 1 .23 .86

20 .83 .68 .38 0 i 1 .38 .S3

40 .79 .57 .01 i i 1 .51 .79

60 .77 .52 i i i 1 .58 .77

80 .76 .47 i i i 1.62 .76

61
Table D3: Case of Local Slope, m = 0

1 0.5 0.1 0.01 0.001 0.0001 KiK2 1 1

10 1 .91 .75 .52 .55 .47 1 .35

20 1 .34 .54 0 i 1.55

40 1 .78 0 i i 1 .73

60 1 .73 i i i 1.82

80 1 .70 1 i i 1.86

Table D4: Case of Local Slope, m = 0.1

1 0.5 0.1 0.01 0.001 0.0001

10 1.15 1 .07 .92 .79 .72 .64

20 1.21 1.06 .76 .33 1

40 1.27 1.05 .49 i i

60 1.29 1.04 i i i

80 1.32 1.03 i i i

K1K2 1

1.53 1 .16

1 .76 1 .21

1 .99 1 .27

2.10 1 .29

2.18 1 .32

62
Table D5: Case of Local Slope, ni = 0.5

1 0.5 0.1 0.01 0.001 0.0001 K1K2 1

in I . 30 1 . /U 1 . 00 1 .61 2.23 1 .97

? '-id 1 Ql 1 . y 1 1 .00 1 . /4 2.73 2

40 2.72 2,57 2.35 2.17 2.05 1.94 3.25 2 .71

60 2.95 2.79 2.55 2.35 2.22 2.09 3.54 2 .94

80 3.11 2.97 2.69 2.47 2.35 2.20 3.74 3 .12

applicable. In particular, it was intended to make the local slope formalism applicable to a wider variety of layer and substrate resistivities

and thicknesses than were embodied in the two limiting forms used as starting points. The term in R /R had the effect of smoothing the transition between the limiting (large negative slope and large positive slope)

forms for F. In addition, for the case of a general isolated layer, the

term in R /R was intended to incorporate the contribution due to sublayers lying deeper than one contact radius as was done by the second term

in the pure parallel superposition formalism [eq (16) ]

.

Accounting for such deep sublayer effects is clearly not necessary

for thin, t < a, isolated layers, and it makes only a small difference in

the analysis of moderately thicker layers. Hence, the added flexibility

built into is clearly expendable for analysis of many well-defined

isolated layers and the use of a form other than in eq (22) is advisable .

For moderate to large positive values of local slope, little difference results from use of , K^K^ , or unity as the second term under the

radical sign in the equation for F. A meaningful difference among these

choices only occurs for small values of local slope, i.e., near the

bevel edge of an isolated layer. Since no analytical model is available

for bevel-edge effects, the correct manner of analyzing data taken near

a specimen bevel edge is not established. Any procedure used to calculate the sampling volume factor, F, will be an approximation. This limitation also occurs for all other methods of calculating the sampling volume factor which are based on models with lateral symmetry, including

the method of SG.

Some manner of calculating F, other than that using the term must

still be chosen. The choices mentioned, K^K^ and unity, are not the only

63
possible replacements, but they are seen to be reasonable choices. As

mentioned, for moderate to large local slope values, little difference results from any of the choices. For small values of local slope, use of

either K^K^ or unity gives values for F near unity. Such a value is appropriate since, when the true correction is uncertain, the least correction possible, i.e., unity, is the safest. Further, the tendency of the

data to show a near zero value of local slope near the surface indicates

that the effect of the underlying junction is no longer dominant in the

conduction between the probes. Hence, a correction near unity is not unreasonable. It is noted from the tables that attempts to retain in

the equation for F and to modify the data set instead, as by replacing

negative values of local slope by zero or by extrapolating the minimum

measured resistance to the surface, simply result in replacing a negative F, for m < 0, by an imaginary F, for m = 0. When correct nearsurface data analysis is particularly important, replacement of K also

appears preferable for somewhat thicker isolated layers despite the possible need to account for the deeper layer contributions that are built

into . Such a case exists when the sheet resistance of moderate depth

diffusions needs to be evaluated from the integration of the resistivity

profile, where near-surface data make the dominant contribution to the

sheet resistance calibration. Then replacement of results in relative differences ranging from a fraction of a percent to about 15 percent for data on the steeper part of the profile where m > 1. The effect of these changes on the contribution to the layer sheet resistance

will generally be small compared to the contribution from reasonably determined values of near-surface points, particularly if m < 0 is en- countered .

Since the terms K^K^ and unity come from hybrid forms more general

than the simple isolated layer model, neither is strictly correct. ^1^2

gives somewhat larger values for F than does unity; the difference in- creases with increasing s/a values. The volume correction factor for an isolated layer is normally greater than unity to account for an in- crease in measured resistance resulting from a foreshortened conducting

volume. Near the bevel edge, the measured resistance again increases,

apparently due to a missing portion of the specimen, as seen in figure

18, and a correction factor greater than unity can again be argued. It

is perhaps judicious, then, to choose the form K^K^ as a replacement for

K^, since it seems to give a scale of volume correction factors in rea- sonable agreement with what is expected from consideration of the physical problem.

One possible exception to the recommendation for replacing occurs

in the analysis of isolated layers of thyristors and related deep structures. Here the cumulative effect of the loss of deeper layer considerations built into may be more severe than the advantage gained from

better interpretation of a few points near the surface. A possible re- course here is simply to retain in the calculation for F until and unless negative or zero values are calculated for F. If zero pr negative

values are calculated for F for a few points near the surface, the re- sistivity of those affected points is replaced with the value extrapo64
lated from points just below the affected region. Again, the strict

parallel superposition formalism could be used instead.

The discussion in this appendix assumes ideal, or noise-free, data

so that the local slope evaluated from successive pairs of data is a smoothly varying function. In the event of noisy data, it is possible

to encounter difficulties of the type described if given pairs of data

result in local slope values which run counter to the trend in that part

of the profile. Data smoothing is advisable for most effective use of

these algorithms if data tend to be noisy.

65
S.T14A (WEv. ».7e)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBS Spec. Publ . 400-^*8

4. TITLE AND SUBTITLE Semioonductor Measurement Technology:

Spreading Resistance Analysis for Silicon Layers with

Nonuniform Resistivity

S. Publication Date May 1979

7. AUTHOR(S)

David H. Dickey and James R. Ehrstein

i. FsMffOmritig 6i9(miz>lSoA Cede

8. Performing Organ. Report No.

9, PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

Solecon Laboratories

and 3001 Red Hill Avenue

Costa Mesa, CA 92626

11. Contract/Grant No. 5-35881

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS rSfreef, city, state. ZIP) ARPA

l^tOO Wilson Boulevard

Arl ington, VA 22209

IS.jJy^e <|f Report & Period Covered 10/1/75 - 9/30/76

' TCodi

^

15. SUPPLEMENTARY NOTES

This work was funded by the Defense Advanced Research Projects Agency under

f5''l^cSirent^cresai^s'^a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or leaa tactual auwmery of moat algntUcant Inlormatlon. If document Ineludea a algnltlcant bibliography or

literature aurvey. merxtion it here.) i i ^. r - - : ^ < ; : ^,^n4-k

A simple mathematical algorithm is developed for the calculation of resistivity depth , profiles from spreading resistance measurements on sectioned silicon device structures

It is applicable to structures consisting of one or more layers of the same or differ--

ing conductivity types. The algorithm accounts for modification of the sampling volum,;

of the spreading resistance probes due to nearby variations in depth of specimen 1 resistivity whether resulting from graded dopant distribution or electrical boundaries

either insulating or conducting.

[ The algorithm is based on limiting case one- and two-dimensional models of the conduc-j^^

tion between the probes. Compared with the traditional spreading resistance analysis »

algorithm based on the three-dimensional boundary value solution of Schumann and Gardn

the present work offers greatly enhanced execution times even with a microcomputer,

making real time analysis for process control possible.

An experiment to test the accuracy of one of the limiting models used is described.

Profiles of a number of diffused layers generated from spreading resistance measurements analyzed by this algorithm are compared with profiles obtained on replicate

specimens using another electrical technique. In addition, computer experiments are

used for simple layer models to compare results based on this al gor i thm w 1 th results

from the Schumann-Gardner approach.

17. KEY WORDS (aix to twelve entriea: alphabetical order; capitalize only the firat letter of the firat key word unleaa a proper name;

aeparated by aemicolona) • i . J

• Dopant profiles; resistivity; resistivity profiles; semiconductor; silicon; spreading

resistance.

18. AVAILABILITY [3 Unlimited

I I For Official Distribution. Do Not Release to NTIS

fE Ofder From Sup. of Doc„ UA_GeYernment Printing Office, Washington, DC

20402, SD Stock No. SN ooj^oj^jgogjcL--- pq Order From National Technical Information Service (NTIS), Springfield, VA. 22161

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