Diffusion of Arsenic in Degenerate Silicon: a Quasi-Static ...

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1977

Diffusion of Arsenic in Degenerate Silicon: aQuasi-Static Approach.Rituparna ShrivastavaLouisiana State University and Agricultural & Mechanical College

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78-7559SHRIVASTAVA, Rituparna, 1951-

DIFFUSION OF ARSENIC IN DEGENERATE SILICON:A QUASI-STATIC APPROACH,The Louisiana State University and Agricultural and Mechanical College, Ph.D., 1977 Engineering, electronics and electrical

University Microfilms International, Ann Arbor, Michigan 48106

© 1978

R I T U P A R N A S H R I V A S T A V A

ALL RIGHTS RESERVED

DIFFUSION OF ARSENIC IN DEGENERATE SILICON A QUASI-STATIC APPROACH

A Dissertation

Submitted to the Graduate Faculty of the Louisiana State University and

Agricultural and Mechanical College in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

inThe Department of Electrical Engineering

byRituparna Shrivastava

M.E., Indian Institute of Science, 1973 December 1977

ACKNOWLEDGEMENT

The author wishes to express his gratitude to Dr. Alan H. Marshak for his continued guidance, assistance, encouragement and friendship during the author's graduate program. He would also like to thank Messrs. Pike R. Green and Farrokh Shokooh for some useful discussions, and Mrs. Martha Prather for typing the manuscript.

The author would like to acknowledge his good fortune in having Shri Kamala Kant and Smt. Priamvada Shrivastava as his parents.

The research reported herein was supported in part by the National Science Foundation under grant DMR75- 18864 and by the Department of Defense under contract No. DAAB07-75-C-1344-2, pursuant to ARPA order No. 2985.

ii

TABLE OF CONTENTSPage

1 INTRODUCTION 12 FIELD-AIDED DIFFUSION 11

2.1 Transport Equations for Mondegenerate Case 112.2 Previous Work 142.3 Transport Equations for Degenerate Case 212.4 Degenerate Case Under Charge Neutrality 312.5 Discussion 35

3 DIFFUSION VIA VACANCIES 373.1 Previous Work 373.2 The Diffusion Model 413.3 Computation of the Fermi Level 473.4 Summary of the Problem 493.5 Form of Impurity Flux With and Without

Vacancies 503.6 Transformation of the Equations 513.7 Discussion 56

4 NUMERICAL ANALYSIS 574.1 Discretization of Independent Variables 574.2 General Quasi-linearization Technique 594.3 Quasi-static Problem 664.4 Boundary Conditions 694.5 Quasi-linearization Technique for a

Scalar Equation 7 8

iii

TABLE OF CONTENTS (cont'd)Page

4.6 Application to the Charge NeutralityApproximation 84

4.7 Application to Vacancy-aided Diffusion 854.8 Computation of the Boundary Condition 874.9 Discussion 88

5 RESULTS 906 CONCLUSIONS 102

APPENDIX A - Nondegenerate Quasi-staticFormulation 107

APPENDIX B - Computation of the FermiIntegrals 111

APPENDIX C - Program Information 119REFERENCES 164VITA 171

iv

LIST OP FIGURESFigures

4.14.25.1

5.2

5.3

5.4

5.5

5.6

Flow Diagram of the Numerical Method Flow Diagram of the Interation Procedure Impurity Profiles of Arsenic in Silicon Using the Quasi-static Model Electric Field for the Constant Source DiffusionComparison of Results Using Charge Neutrality and Vacancy-aided Models for High Surface Concentration Effect of Statistics and Ionization on Impurity Concentration Using the Vacancy-aided Model Impurity Profiles for Low Surface Concentration Using Charge Neutrality, Vacancy-aided and Zero Field Models Comparison Between Experimental Data and Calculated Profiles Using Vacancy-aided Model

Page

7075

93

95

96

98

99

101

v

ABSTRACT

Diffusion under controlled conditions is one of themost important processes employed in the manufacture ofsemiconductor devices. The diffusion of group III andgroup V impurities in semiconductor material has been asubject of considerable work. It has been known for manyyears that diffusion in silicon at high concentrations,

21 -3say 10 cm , produces impurity profiles that differ significantly from those predicted by a simple theory.This difficulty hinders the work of those engaged in the design of modern semiconductor devices, such as transistors, solar cells and integrated circuits, for which accurate process prediction is desirable.

The object of this research is to develop and studythe models describing a constant source diffusion processwhich will accurately and efficiently predict the resultsof such a process. The effect of the internal electricfield produced during the diffusion is analyzed using aquasi-static approximation for the holes and electrons.The use of both Fermi-Dirac and Maxwell-Boltzmann statisticsis discussed. The assumption of charge neutrality isinvestigated under typical diffusion conditions. Atrelatively higher surface concentrations, in addition tothe internal electric field, several other effects mustbe considered. In the present work, a model for arsenicdiffusion in silicon is proposed which takes into account

vi

the degeneracy of the carriers, partial ionization of the impurities, single acceptor level vacancies and the internal electric field.

The transport process for holes, electrons and im­purities is described by the flux equations, the continuity equations and Gauss' law. Although simplifying assumptions are made, the resulting partial differential equations are highly nonlinear, and a numerical scheme must be used to solve the problem. An efficient computer program based on a quasi-linearization technique is written to obtain the impurity profiles from the processing data. Several other computer programs are used to investigate different models.

It is found that the internal electric field enhances diffusion at high concentrations. The electric field profiles are reminiscent of those obtained in a high-low junction. It is noted that the field varies almost linearly near the surface and then reaches a maximum value. Charge neutrality under typical diffusion conditions is found to be an excellent approximation. The results ob­tained using Fermi-Dirac statistics show that the use of classical statistics yields an underestimate of the impurity concentration values. At higher concentrations, incorporation of partial ionization, vacancies and degeneracy significantly affects the results obtained.There is a good agreement between the model and experimental

vii

results based on neutron activation analysis. However, the impurity profiles strongly depend on the value of the in­trinsic impurity diffusion coefficient.

viii

CHAPTER IINTRODUCTION

The term "diffusion", when applied to semiconductor device fabrication, is used loosely to describe impurity atom motion in a semiconductor at elevated temperatures. Diffusion under controlled conditions is one of the most important processes employed in the manufacture of semi­conductor devices. In order to fabricate a device with a certain set of parameters, it is very important to be able to control the impurity diffusion profiles in the semi­conductor wafer. The need for suitable models describing the diffusion phenomenon is evident.

In most practical situations, a constant source diffusion, a drive-in diffusion, or both are used. In the first case, surface concentration is held constant during the diffusion, whereas in the second case, the source is removed and redistribution takes place under the condition that the impurity atoms can neither enter nor leave the semiconductor wafer. In theory, it is possible to synthe­size any given arbitrary profile compatible with the two- step process, by generating a proper control function [1]. However, in practice the direct profile resulting from the above two processes may be acceptable. We will mostly concern ourselves with the constant source diffusion process. In this process, the impurity atoms are intro­duced into a flowing inert gas, which deposits these atoms

1

2

on the surface of the semiconductor wafer. A desired surface concentration of the impurity atoms can be main­tained. In practice, this value very often is equal to the solid solubility value. As the impurity atoms cross the surface and move into the semiconductor, some or most of them ionize depending on the concentration. If the motion of these ions were essentially the same as the motion of neutral particles, the flux f or the number of ions crossing a unit area in a unit time, will be given by Fick’s law. For one-dimensional motion, and assuming parallel plane geometry,

f = -D (1.1)c C 9X

where Dc is the diffusion coefficient or diffusivity and c represents the concentration of ions. The ions also satisfy a continuity equation given by

3fc 3C33r + H = Gc (1-2»

where G is the net generation rate for ions. Assuming cthat all the impurity atoms ionize when they enter the surface, we can equate G to zero. Combining (1.1) andL>(1.2) then yields:

3

= j3 -ft c n 3)3t C 2 ' U,J)oX

This is called the simple diffusion equation. The solution to (1.3) for a semi-infinite solid under the constant source boundary conditions

c (0,t) = CQ (1.4a)

c(«,t) = 0 (1.4b)

c(x,0) =0, x > 0 (1.4c)

is given by the complementary error function

c(x,t) = Cq e r f c [ x / ( 4 D ^ t ) (1.5)

where D, is the value of D for the above diffusion step.1 cThe boundary conditions for the drive-in diffusion

are given by

c(x,0) = c^(x) (1.6a)

Igi-O'-IL = o (1.6b)3x

c(«>/t) = 0 . (1.6c)

4

The solution to the diffusion equation in this case is [2]

- (*-£)2 _ (x+g) 2r00 4D~t 4D9t

c(x,t) = -------------- c (c) [e + e ]dg2 (TrD9t) ' J 1 2 O

(1.7)

where D_ is the value of D for the drive-in cycle.In the two-step diffusion process c^(x) is given by

(1.5). If the diffusion time for the first step t^ is such that D^t^ << t*le delta function approximationfor c^(x) yields the Gaussian solution [3]

2x, .. 2C0 .°ltl.l/2 4D2t nc(x,t) = --- (yy - --) e (1.8)7T D2t

for the two-step diffusion profile.It has been known for many years that the results of

simple diffusion theory do not agree with experiment except at low surface concentrations [4]. This departure, to some extent, can be explained by the presence of an internal electric field which arises because of a mismatch in the diffusion coefficients of the impurity ions and the mobile carriers. For example, when arsenic atoms enter the solid, most of them ionize because of the high temperature, resulting in positive ions and electrons. The electrons tend to diffuse away from the ions due to a much higher diffusion coefficient. Since both species carry electrical

5

charge, an electric field develops which tends to retard the motion of electrons and enhance the motion of ions.The incorporation of this electric field in the model yields impurity density profiles which are closer to experimental results. However, the general differential equations governing field-aided diffusion are quite nonlinear and complex and a need for simple models soon becomes evident. Most of the models found in the literature assume local charge neutrality and are valid for nondegenerate conditions. In Chapter 2, the assumption of local charge neutrality has been examined with reference to a quasi-static approxi­mation for both nondegenerate and degenerate statistics.

The field-aided diffusion theory described above, in itself, is inadequate to explain the experimental observa­tions at relatively higher concentrations. At such high concentrations the effect of defects in the lattice becomes very important. Before considering the defects, it will be instructive to briefly discuss various mechanisms of diffusion in semiconductors [5]. Ring mechanism and direct interchange of neighboring atoms have been con­sidered improbable. A "direct interstitial" mechanism has been suggested in which a lattice atom leaves its regular substitutional site and becomes an interstitial.One of its nearest neighboring substitutional atoms moves into the vacancy left behind by the first atom. Then the first atom, now at the interstitial position, moves into

6

the vacancy left behind by the second atom thus completing the cycle of indirect exchange. However, it can be argued that even such an interchange would be less likely than a vacancy mechanism. Defect-aided mechanisms are more probable in silicon and germanium. Most important of them are vacancy and interstitialcy (or indirect interstitial) mechanisms. In the vacancy mechanism the host atom is missing from its regular site and this enhances the im­purity diffusion. In the interstitialcy mechanism, the interstitial atom chooses to move by pushing one of its nearest neighbors into another interstitial site and it itself takes up the substitutional site. Group III and Group V elements form strong covalent bonds with silicon and germanium atoms. This results in their existence being almost entirely in the substitutional form. A con­sequence of this is that they diffuse predominantly by either a vacancy or an interstitialcy mechanism. A definite statement about the mechanism, however, can not be made. A number of experimental techniques have demonstrated that vacancies and presumably interstitials may exist in different charge states. The effect of strong doping on self and impurity diffusion is closely tied to the acceptor and donor actions of the vacancies

and interstitials.An excellent review on diffusion mechanisms and point

defects in Si and Ge can be found in [6]. It has become

7

customary to classify impurities as "slow" and "fast" diffusors. Diffusion coefficients of slow diffusors are 10-100 times higher than self diffusion coefficients.Fast diffusors usually diffuse several orders of magnitude faster than slow diffusors. Group III and Group V elements are typical representatives of slow diffusors. It is generally assumed that the diffusion mechanism involved in slow diffusors is a simple vacancy mechanism. A quantita­tive model for diffusion of these impurities was proposed by Swalin. His model seems to support vacancy mechanism in Ge and donor impurity diffusion via vacancies in Si. It however does not explain acceptor impurity diffusion in Si, for which Seeger and Chik have proposed the interstitialcy mechanism discussed before. An explanation in favor of donor impurity diffusion in Si, via vacancies, is that the Coulomb interaction between positively charged donors and negatively charged vacancies leads to an increased probability of finding a vacancy near a donor impurity, and therefore enhances the impurity diffusion.

A simple way to see how a vacancy may act as an acceptor is as follows [7]. There are four covalent bonds missing at the sight of a vacancy. This gives rise to a strong change in the valence electron distribution in the vicinity of the vacancy and leads to a lattice distortion. From a scattering theory point of view, it means that an atomic scatterer is missing in the lattice. This results

8

in the introduction of some bound states in the band gap.The vacancy may trap electrons from the valence band in these localized states. When a hole is introduced because of the electron making a transition to one of the localized states from the valence band, the vacancy becomes negatively charged because of the trapped electron and thus acts as an ionized p-type impurity. As discussed by Seeger and Chik, concentration of acceptor type defects is increased by n-type doping and decreased by p-type doping. Thus, diffusion via vacancies should be faster in n-doped material, and slower in p-doped material compared to the intrinsic material.

Evidence indicating that vacancies act as acceptors is found in the data obtained from irradiated Ge by Cleland, Crawford and Holmes [8]. In this study, the effect of y- radiation on electrical properties of Ge was studied.Results indicate that exposure of n-type Ge to y-rays decreases the extrinsic electron concentration. Valenta and Ramasastry [9] have explained the effect of heavy doping on self-diffusion of Ge by assuming that vacancies act as acceptors. Agreement for n-type data was fair, whereas p-type data did not agree that well. The dis­crepancy was not attributed to the above assumption, namely, that vacancies act as acceptors, and it was concluded that Ge self diffusion probably occurs via vacancies. The doping effect on impurity diffusion in Ge

9

is also in general agreement with vacancy model of diffusion. Because of the low concentration of point defects, it has not yet been possible to detect directly the presence of point defects in Si and Ge in thermal equilibrium at high temperatures.

Two classical models of the energy levels of vacancies and interstitials are that of James and Lark-Horowitz [10], and of Blount [11]. In the first model, inter­stitials act as donors and vacancies act as acceptors. In Blount's model on the other hand, interstitials and vacancies may act as both acceptors and donors. This is favored for the interstitialcy mechanism proposed by Seeger and Chik to explain the impurity diffusion of Group III and Group V elements in Si, whereas the fact that double negatively charged vacancies can exist is a point in favor of the James and Lark-Horowitz model.

With the above background in mind, the diffusion phenomenon will be discussed in the subsequent chapters.The object of the present research is to develop the models describing the diffusion phenomenon which include the effects of internal electric field, vacancies, partial ionization, and degeneracy of carriers. In Chapter 2, field-aided diffusion is discussed under degenerate and complete impurity ionization conditions. The assumption of local charge neutrality is investigated. In Chapter 3, Hu's theory of impurity diffusion [12] is applied to

10

arsenic diffusion in silicon under general conditions. Numerical computation of impurity profiles and results are discussed in Chapters 4 and 5, and the conclusions summarized in Chapter 6.

CHAPTER 2FIELD-AIDED DIFFUSION

In an intrinsic semiconductor, at thermal equilibrium, holes and electrons are produced in equal numbers by thermal processes. When impurity atoms are introduced, they ionize and alter the concentration of majority carriers (e.g. electrons for an n-type impurity). As a result, the majority carrier concentration increases and due to re­combination the minority carrier concentration decreases. The product pn remains constant at thermal equilibrium for a nondegenerate semiconductor. When the semiconductor is out of thermal equilibrium, it is necessary to consider the motions of electrons, holes, and impurity ions simultaneously, because the charge density at any point is a function of the concentrations of these species.

The analysis in the present work is restricted to the case of a constant band gap semiconductor at a constant temperature. It is also assumed that the impurities are singly ionized.

2.1 Transport Equations for Nondegenerate CaseFor a semi-infinite solid, a one-dimensional diffusion

process for x>0, t>0 is defined by the flux equations, continuity equations, Poisson's equation, and appropriate boundary conditions.

11

12

The flux equations for an arbitrary carrier are given

where a represents the concentration of an arbitrary carrier (n, p or c for electrons, holes or impurity ions, respectively), Da is the diffusion coefficient, ya is the mobility, E is the electric field, and takes on the value +1 or -1 for a positive or negative carrier charge, respectively.

The continuity equation for the carrier a is

where Ga represents the net generation rate of the carrier. Using the nondegenerate Einstein relation

and assuming 100% impurity ionization, which implies that

by

fa (2.1)

G G G (2.4)P n

the general equations can be written as

Poisson's equation yields

(2.6)

where p represents the charge density. Note that Z = 1cfor donors and Z = -1 for acceptors. The permittivity e will be assumed constant.

The net generation rate G may be represented by the Shockley-Read-Hall model

holes, and t and t are the lifetimes of holes and p nelectrons, respectively.

The boundary conditions are governed by the kind of diffusion process. In most of the work, for simplicity, a constant source diffusion will be assumed in which case the boundary conditions are given by

G (2.7)t (p+n.)+t (n+n.)XI X ^ x

where n. is the intrinsic concentration of electrons andl

14

c(x,0) = 0 x > 0 (2.8a)

p(x,0) = n(x,Q) = , x > 0 (2.8b)

E (00, t) = 0 (2.8c)

c(0,t) = C, (2.8d)

9a .(0, t) maj x '9X Dmaj

9c(0/t) 9x (2.8e)

3om . (0,t)— ---- = 0 (2. 8f)

where a . and a . are majority and minority carrier con- maj m m J J Jcentrations, respectively.

2.2 Previous WorkThe effect of the electric field on the transport

process was first considered by Zaromb [13] and Smits [14]. Their work was based upon two major assumptions.

1. The material is charge-neutral at every point sothat

— = p-n+Z„ c = 0e c c (2.9)

15

22. It is assumed that pn = ru. Strictly, this rela­tion is only valid in thermal equilibrium for a nondegener­ate semiconductor.

The electric field can then be expressed as

E - - zc VT n r 1 , If • (2-10>Vc2 + 4n?1An expression similar to (2.10) was derived by Kurtz

and Yee [15] who neglected the effect of the minoritycurrent. They pointed out that an effective diffusionconstant D can be defined when (2.10) is used in theef f.flux equation for the impurity atoms (2.1) to give

f = -D „ — ■ (2.11)c eff 3 x

where

D = D (1 + - = = = ) . (2.12)eff cV c 2 + 24n7

Substitution of (2.11) into the continuity equation yields

8C - 3 [D (1 + c-- ~ ) |5l ■ (2-13)31 3x c 2 3xl.lJ 2 ± , 2M e + 4n,

16

Lehovec and Slobodskoy [16] have obtained an approxi­mate solution to the above equation for a constant source diffusion into an otherwise intrinsic semiconductor. They also provided the "corrections" to the surface concentra­tion by extrapolating from the tail of the impurity distribution using a complementary error function.Bordina et al. [17] have discussed the influence of the internal electric field by assuming that it may be taken as uniform. They then conclude that an effective doubling of the diffusion coefficient takes place in a region wherec >> n .. i

Vas'kin et al. [18] have treated impurity diffusion into a semiconductor uniformly doped with an impurity of the opposite type under the assumption that the local electric field can be represented by an average field defined in terms of a weighting function. Shaw and Wells [19] have analyzed the same problem without making the above assumption and have obtained numerical solutions for the impurity distributions. Klein and Beal [20] have discussed the case of simultaneous diffusion of oppositely charged impurities.

Nuyts and Van Overstraeten [21] have calculated the impurity diffusion profiles in silicon taking into account the diffusion of the base impurities. They have also discussed the use of degenerate statistics and partial impurity ionization although no computations have been

17

made incorporating these aspects. Hu and Schmidt [22] have also calculated constant source diffusion profiles,

VO**#**'* -and have analyzed the effect of the internal electric field on a sequential diffusion process.

Quasi-static Approach. The general problem of the previous section was investigated by Perritt [23] and later by Widiger [24], without making the two major assumptions discussed earlier in this section. A quasi-static approxi­mation was formulated under the following assumption.

In a semiconductor, even at diffusion temperatures,holes and electrons have a much larger mobility than theimpurity ions. Thus the time required for an impurityion distribution to change to a particular profile is manyorders of magnitude larger than that required for the holes

13and electrons. and Dn are approximately 10 timeslarger than Dc . The electrons and holes therefore re­adjust almost instantaneously, staying in a steady state determined by the impurity ion distribution. In thermo­dynamics this is referred to as "quasi-static equilibrium." The equations governing diffusion under quasi-static approximation were developed by Widiger who assumed the SRH model. A more general way to derive these equations is given in Appendix A. It is first assumed that

l£ = M = o . (2.14)3t 3t

18

It can then be shown that regardless of the form of G in(2.5a) and (2.5b), and using only these two equations,we obtain

n = n. exp(^—) (2.15)T

p = n. exp(- ^—) (2.16)T

which, of course, implies that

2pn = ni

The general problem under quasi-static approximation thus reduces to

If ■ k [Dc If - Z-s^ E) (2-17)2

| [n-p-Zc c] (2.18)3x

where

E = - |± . (2.19)

Note that electrostatic potential <J> has been assumed to be zero at x->~ for convenience, where the material has

19

been chosen to be intrinsic. It may also be noted that using (2.15) and (2.16), (2.18) can be written as

The boundary conditions for the above problem for a constant source diffusion are

model of (2.7) is assumed. This fact, however, is not required in the above model.

The quasi-static problem has been investigated and impurity profiles calculated using numerical techniques for a constant source diffusion [24], drive-in diffusion [25] and two-step diffusion processes [26].

(2.20)

c (0, t) = CQ (2.21a)

3<t> (0/1) 3x 0 (2.21b)

c («, t) = 0 (2.21c)

(2.21d)

c(x,0) =0, x > 0 (2.21e)

It is seen that as a consequence of the law of mass 2action pn = n^, the generation term G becomes zero if SRH

20

It is interesting to note that if local charged2d>neutrality is assumed in Poisson's equation, i.e. — % = 0,

then3X2

-1 cZc4> = VT sinh A (-^p) • (2.22)i

Thus,

E = - If = Z-°-- ff • (2.23)lc2 + «„?X

Substituting (2.23) into (2.17) yields

It = 1“ ff Is ) (2.24)3t 3x eff 3x

where

Deff “ Dc + T f - : 1 ' <2-25>V c 2 + 4n?X

This result has been obtained previously in (2.11) and (2.12).

Some important points may be noted at this stage. Asis evident by (2.15) and (2.16), the law of mass action

2pn = ni still holds during the diffusion, although strictly speaking the system is not in thermal equilibrium. This is a consequence of the assumptions made in quasi-static approximation. In the charge neutrality approximation, the

21

3 d)term — %■ has been neglected only in Poisson's equation.3x

It is easy to see (Section 2.4) that if this term is also neglected in the transport equation, the two equations decouple and the simple diffusion equation is obtained.As pointed out earlier, all the above results are valid only under nondegenerate conditions. Finally it may be noted that (2.24) is in such a form that for a constant source diffusion the variables can be separated, as was shown by Shaw and Wells (Section 4.5). It appears that (2.24) is not separable for drive-in diffusion boundary conditions. An alternative formulation of the charge neutrality approximation of (2.24) in terms of <j> (instead of c) is discussed in Section 2.4.

2.3 Transport Equations for Degenerate CaseIn this section the transport equations for the

degenerate case will be discussed. Holes and electrons in this case are described by Fermi-Dirac statistics and instead of the classical Einstein relation, its generalized form must be used. For concreteness, a donor type diffus­ion will be assumed.

The flux equations are still given by [27]

f = " D + y pE. (2.26a)p p 3x p

22

c „ Bn „f = - D 7T“ - U nE n n 9x n

f = - d ■— + y cE c c 9x c

(2.26b)

(2.26c)

The continuity equation for impurity ions with 100% ionization is

ft + 3x = 0 • (2‘27>

At this stage we make the following two assumptions.1. The flux for minority carriers (holes) is zero,

f = 0 . (2.28)P

Note that the continuity equation for holes then implies

Since holes readjust almost instantaneously, this implieslE. = g =0. This, however, need not be assumed for the 91 pderivation that follows.

2. The flux for the impurity ions equals the flux for the majority carriers (electrons),

(2.29)

23

Again, using the continuity equations for impurity ions_ \

and electrons, this implies that — -„-7— ■ = G ,d u n If quasi- 3cstatic conditions were assumed it would imply -r-r- = -G .^ 1 3t n

It will be shown later that the above two assumptions imply that the quasi-Fermi levels for holes and electrons are equal.

The first assumption, using (2.26a), yields

1 3p _ vp p 3X D E = _ IE i AD 3x P

Integrating, we have

- f ^ dx =3 XX

00 D 1- £ i | £ d Xu p 3x

X(2.30)

where <J>(°°) = 0 has been chosen for convenience. For a parabolic density of states, the generalized Einstein relation gives

= kr Fi / 2 (ye f ^ t /o ( n ~ ) (2.31)

- 1/2

with

n p <*> = V x)_Efp(x)

where Ev (x) is the top edge of the valence band and Ef is

the quasi-Fermi level for holes.

24

Also,

p = Nv pi/2 (y • <2-32>

Therefore,

= Nv p-i / 2 (y • <2 -33>

Dividing (2.32) by (2.33) and substituting in (2.31) we obtain

Using this, (2.30) yields

(j) (x) = |£ dx = — [ti (»)-ti ] e dp 3x e 'p 'p

where

E (co)-E^ , \ _ v £

np (“* kTE

_ _a _kT

with

E-j-E (°°)f c kT (2.34)

25

Note that thermal equilibrium conditions have been assumedat x-v°°. E and E (x) represent the band gap and the g cbottom edge of the conduction band, respectively. Thus,

nP = V " ’ ' It = ‘ kf ’ "i ‘ Et (2‘35)

which gives

EP = Nv Fl/2 kT " kT “ ni) • (2.36)

Turning to the second approximation, we have

-D + y CE = -D - y nE . (2.37)c 3x *c n 3x n

Now, at typical diffusion temperatures, y >> y and n = cXl v»

so that y nE >> y cE. Also, Poisson's equation*.givesn c <■•

a2*c = n - p - — — . (2.38)3x

Thus, (2.37) becomes

-D lH + D IE + d - i-t = - D — + y n . (2.39)c 3x c dx c e * 3 n 3x n 3xd X

The first term in (2.39) drops out because D >> D . Using11 v(2.32) and (2.35) we can write

since is a large negative quantity. Substituting in (2.39) we obtain,

3n . Mn „ 3<f>3x D 3x n

Dc kTwhere — = — has been used. Observing that y n >> y p y e n crcthe above equation simplifies to

yn 3(j> 3n _ Pc £ /3 xD n 3x ” 3x D e *n n ox

Now, considering the quantity on the r.h.s., it is noted3 3 .

that D << D , and if it is assumed that — ^ is not very C n 3x

large, the r.h.s. can be neglected in comparison to otherterms. Physically, this means that the gradient of thecharge density should not be extremely large. Thus, weobtain

Using the generalized Einstein relation

Dn _ kT Fl/2y e F i / o / \ n -1/2(n)

where

E _ (x) -E (x) / x _ f n cn(x) = -----------

and

n = Nc F1/2(ri) / (2.42)

we can integrate the r.h.s. of (2.41) to obtain

n = H + ni (2.43)

which gives

n = Nc Fl/2 (It + "i1 • (2-44)

It may be noted that (2.35) and (2.43) imply that

This result is not surprising because (2.41) could have been written directly if the thermal equilibrium conditions were assumed at the outset. Present analysis, however, gives a better insight of the approximations involved.

Substituting (2.36) and (2.44) back into the Poisson's equation (2.38), and combining the flux and the continuity equations we obtain

3c _ 3 rrv 3c , „ 3<j>, „c.at - sJ IDC i5 * “c c te1 (2-45)

- ? tNc Fl/2 <$; - "i>- Nv fi/2 <- ^ - e? - V- c] . (2.46)

It is noted that under typical diffusion conditions the argument of the second Fermi function in (2.46), which represents the minority carriers, is a large negative quantity and hence the Fermi function can be approximated by an exponential function; thus

where

ni = Nv Fl/2( " kT ' ni> ' Nv exp<- kT ' ni> (2’47)

has been used. Instead of (2.46) , we may then use

= t [No P l/2 (v^ + "i> ' ni exp<- - cl •(2.48)

Note that if the first term is also expressed by Maxwell- Boltzmann statistics (nondegenerate case), we obtain

Nc pi/2 + ni> “ Nc exp(% > exp<ni>

= n. exp(i-)where

has been used. For nondegenerate conditions, therefore,(2.48) reduces to

30

which is the same as (2.20) of the previous section.The boundary conditions and numerical solution to

(2.45) and (2.48) will be discussed in Chapter 4.We now consider the problem of finding Nc, Nv and n *

The effective density of states Nc and Nv are given by

*2TrkT m ~ /0N = 2 (---- T -^) / (2.49a)c

*2irkT m 0/9

N = 2 (----(2.49b)v

* *where m and m are density of states effective masses ofn pelectrons and holes, respectively. In general, knowledge

* *of mn and at typical diffusion temperatures is poor,although it is possible to extrapolate from the resultsobtained at lower temperatures [28]. A way to circumvent

* *the problem is to avoid the direct use of m and m in c n p(2.49) .

For an intrinsic material,

ni = Nc Fl/2 (ni) • (2-50)

Data for the intrinsic carrier concentration n^(T) is known experimentally [29]. It is easy to show that

1 1 NvEf = J [Ec (») + Ev (»)] + J kT In (jp)c

(2.51)

31

since the material is intrinsic as x-*°°. Using (2.34) and (2.51), the value of in (2.50) may be computed as

*E m"i - - + 4 ln • <2'52>

n* i *The ratio of effective masses, m / m , is a relatively weakP n.

function of temperature [30] and therefore N computedcusing (2.50) should give a better value than that using(2.49a). Data for E (T) used in (2.52) is also known9experimentally [31].

2.4 Degenerate Case Under Charge NeutralityThe degenerate quasi-static formulation for a donor

type impurity diffusion yields (2.45) and (2.48), repeated here for convenience.

|£ = |- [D ~ + u c |i] (2.53)at ax c ax c ax

" I" [Nc Fl/2 + - ni exP<- - Cl(2.54)

g 2 .If it is assumed that — = 0, i.e. p(x) = 0, every-

axwhere in the semiconductor, then

a (hE = - — = constant3x

32

and, because of the boundary conditions

E(“,t) = 0

<M°°,t) = 0 ,

we obtain

E(x,t) = 0

<j>(x,t) = 0 .

Thus (2.5 3) and (2.54) decouple and the classical diffusion equation

both (2.53) and (2.54), we neglect it only in Poisson's

8cat

is obtained.92<f>In this section, instead of assuming that — %■ = 0 in3x

9 d)equation, i.e. we assume that ——*• << n, p or c. The equa-3x

tions under charge neutrality thus become

3c 31

(2.55)

33

c = Nc Fl/2 + ni> - ni exP<- • <2-56)

Unlike the nondegenerate case of Section 2.2, E here can not be expressed entirely in terms of c. Nevertheless, it is possible to obtain a single equation in <j>. Differentiat­ing c from (2.56) and substituting in (2.55), we obtain

< H > 2 + D o f2 <*> ' A * <2'57>T u X

where

F-3/2(V + ni} + F-1/2(V + ni}f ($) = ------- i----- ±--------- (2.58a)

F-1/2(V^ + ni) + Fl/2(ni) exP ("

and

F-1/2(V + ni} + F1/2(V + r'i)2 =- £--------------- ------------ . (2.58b)

F-1/2 (V^ + ni} + Fl/2(ni) exP ("

Now, if a normalization

I = t - (2.59)T

is used, (2.57) becomes

34

^ A 2 *It = Dc [fl<*> < H >2 + f2 < « ' M ’1 (2-60>3X

where and f2 (<f>) are the functions defined by (2.58a)A

and (2.58b) with 4>/VT replaced by <f>.Boundary Conditions. Considering a constant source

diffusion, the boundary conditions are given by

c(0,t) = Cq (2.61a)

c(°°,t) = 0 (2.61b)

c ( X f O ) = 0 , x > 0 . ( 2 . 6 1 c )

The boundary conditions in terms-pf d> are easily obtainedas

<{>(0,t) = (2.62a)

4>(°°#t) = 0 (2.62b)

<j>(x,0) = 0, x > 0 (2.62c)

where <j>Q is computed by solving the implicit algebraic

equation

C0 = Nc Fl/2 (*0+r'i) - ni exP (-*0> (2.63)

35

Transformation. It will be shown in Section 4.5 that the partial differential equation of the form (2.60) is separable for the constant source boundary conditions given above. Thus, just like the nondegenerate case of Section 2.2, the degenerate case also yields an equation which is separable although the dependent variable now is <J> instead of c. The electric field here can not be expressed entirely in terms of c but may be easily computed as

E = - || . (2.64)

Reduction of the Equation for Nondegenerate Condi­tions. For nondegenerate case, all the Fermi functions reduce to exponential functions and (2.57) reduces to

Ul. = r _____ -I r ^ (iL£.) ^ + d ( - _^-) 13t ‘ . , , 2*, ‘ VT <3x) c V 1 '1 + exp (- ^-) (2.65)

This equation in terms of <J> is an alternative to the charge neutrality formulation in terms of c discussed earlier in Section 2.2.

2.5 DiscussionIn this Chapter impurity diffusion into an intrinsic

semiconductor was discussed under nondegenerate and degenerate conditions. Formulations resulting from the

36

assumption of local charge neutrality were presented. In each of the cases the diffusion model is described by either a single or a set of differential equations. The numerical procedures to solve these equations are dis­cussed in Chapter 4. In Chapter 5 numerical results are presented and the formulations compared to each other.It is found that under typical diffusion conditions, local charge neutrality turns out to be a good approximation. This provides the basis for the model to be discussed in the next Chapter which includes the effect of vacancies.

CHAPTER 3DIFFUSION VIA VACANCIES

The models for field-aided diffusion discussed in Chapter 2, although applicable to both acceptor and donor type impurities, yield results which do not agree with experiment at higher concentrations. When defects such as vacancies and interstitials are considered, it becomes necessary to specify the kind of impurities. In silicon, vacancies are believed to be responsible for donor type impurity diffusion, whereas interstitialcy mechanism is favored for acceptor type impurities [6]. In this Chapter, diffusion of arsenic in silicon is discussed and a vacancy mechanism is assumed.

3.1 Previous WorkSeveral models for arsenic diffusion in silicon have

been proposed. Hu [12] has considered an impurity-vacancy- semiconductor system. The flux equations have been systematically derived from thermodynamical considerations. Local charge neutrality has been assumed in the theory. Analysis without this assumption becomes very complicated, and does not seem to have been tried in the general case. However, based on the discussion from Chapter 2, it may be expected that local charge neutrality should be an excellent assumption even in the present case. Prior to

37

38

the publication of the above theory, using some other arguments, Hu and Schmidt [22] had analyzed As diffusion in Si. The equations used there were later justified by Hu. Nevertheless, there was some arbitrariness in the computations of Hu and Schmidt because of a factor 8, which was assumed to be 100. As discussed by Nuyts and Van Overstraeten [32], the above value of B is unrealistic. Also, the analysis assumed nondegenerate conditions and complete impurity ionization, although the general theory of impurity diffusion proposed by Hu is not restricted to these conditions. Hu and Schmidt have pointed out that the validity of their model breaks down at high surface con­centrations because there is no limit to the enhancement effect due to vacancies. It should be interesting to find out if the same result is obtained when partial ionization is taken into account and Fermi-Dirac statistics are used.

In the model proposed by Chiu and Ghosh [33], two energy levels have been attributed to the vacancies in an attempt to explain the decrease in the diffusion coefficient of As in Si at very high concentrations. They have reported excellent agreement between the theory and experiment except for short diffusion times. In their analysis, how­ever, as many as four constants were matched numerically, having assumed that the impurity diffusion coefficient ratio in extrinsic to nearly intrinsic silicon is given by

39

§ - = fgh 1

where f, g, and h are contributions due to vacancies, cluster mechanism and internal electric field, respectively. The analysis again assumes nondegenerate conditions and complete impurity ionization. The cluster mechanism mentioned above needs some explanation. In order to explain the retardation of diffusion observed experimentally at higher concentrations, it has been postulated that As atoms start forming clusters as the concentration goes up. Two models have been proposed. The As-complex considered by Fair and Weber [34] consists of two As atoms, whereas in Hu's cluster model [35], it consists of four As atoms. For chemical reasons, only one such complex may dominate in a certain temperature range. However, there is still an uncertainty as to which model actually applies. Hu's model gives a good fit to the experimental vapor pressure data. Fair and Weber have claimed that their model gives better results at shorter diffusion times compared to Chiu and Ghosh who have used Hu's cluster model. It should be pointed out that Fair and Weber have included the influence of partial impurity ionization through an empirical equa­tion. Also, in addition to using nondegenerate equations, they have approximated the electric field by

E kT 1 3 c e c 8x

As can be seen from (2.23), the above equation is valid only for c >> 2n^, and at typical diffusion temperatures, this inequality is easily violated.

Nuyts and Van Overstraeten [32] have applied Hu's diffusion model to the diffusion of phosphorus in silicon. They also restricted their analysis to nondegenerate conditions and assumed complete ionization. Contrary to the comment made earlier in the Chapter that interstitialcy mechanism is favored for acceptor type impurities, they assumed vacancy mechanism to be valid for simultaneous diffusion of boron in silicon.

As the impurity concentration becomes higher and higher, the discrete impurity energy levels separate out and start forming energy bands. This is, of course, a consequence of Pauli's exclusion principle. Under such conditions, strictly speaking, it is not sufficient just to replace Maxwell-Boltzmann statistics by Fermi-Dirac statistics and neglect the impurity band formation. Two of the theories dealing with these impurity bands have been proposed by Kane [36] and Morgan [37]. Jain and Van Over­straeten [38] have used these models and have analyzed the diffusion problem by writing the overall diffusion co­efficient as

41

where the factors on the right hand side correspond to the intrinsic As diffusion coefficient, electric field, vacancies and cluster formation (using Hu's cluster model), respectively. They have claimed a good agreement between the theory and experiment.

Of all the above models, Hu's diffusion model has a very strong point in favor of it, in that it evolves in a very systematic and general way from the fundamentals of thermodynamics. The generality of the results does not seem to have been utilized completely. In this Chapter, As diffusion in Si is analyzed using Hu's theory. The partial ionization of the impurity atoms is taken into account, and Fermi-Dirac statistics are used to describe the carriers.

3.2 The Diffusion ModelThe following major assumptions are made at various

stages in the development of the model.1. The temperature during the diffusion is held

constant.2. Vacancies act as single level acceptors.3. Local charge neutrality is assumed.4. Vacancies have very little effect on the Fermi

level. Conversely, the Fermi level determines the con­centration of vacancies.

5. Fermi-Dirac statistics are used to describe the carrier densities. However, the formation of impurity

42

bands is ignored. Constant band gap and electron affinity are assumed.

6. In order to better understand the influence of carrier degeneracy and partial ionization, cluster forma­tion of As atoms is not considered.

7. Vacancy production due to plastic deformation is neglected.

8. Quasi-thermal equilibrium is assumed so that

Under these assumptions, according to Hu's theory [12], we have

where Dc represents the impurity diffusion coefficient and

vacancy at its thermal equilibrium concentration. Dv ,

and concentrations for vacancies. The flux equations are given by

(3.1)

DV DV (3.2)

Dc is its value at infinite dilution of impurities with

D , v and v* are the corresponding diffusion coefficients

fc (3.3)

f D* v 3 ln Yv 9c p* 3vv c 9 1nc 9x v 3 x (3.4)v

43

where yc and yv are the activity coefficients given by

In writing (3.1)— (3.4), it has been assumed that the con­centration of impurity-vacancy pairs is much smaller than the vacancy concentration. The impurity concentration has also been assumed to be small compared to the concentration of the host lattice atoms. In the above equations, ,Ed, and Ev represent the Fermi level, donor energy level, and vacancy energy level, respectively. E^ represents the value of the Fermi level which would yield equal con­centrations of holes and electrons. Also, gc and gv are degeneracy factors for the donor and vacancy levels. Thus,

*-Yc 1+c (3.5)

Y = ^ v 1+5 (3.6)

where 5, 5/ and 5* are defined by

-1 , D ^fC = % eXP<- k T - (3.7)

5 = gv exP (“icT (3.8)

(3.9)

(3.10)

44

ic

Y = -------- r : T - • (3.11), . - 1 , D f ,1 + gc exP < - w — >

Differentiation w.r.t. x gives

Ef"""ED —1 3n- T j r - S - 11 + 9 „ e x p ( ^ ) J | a ( 3 . 1 2 )

where

Ef-Ec1 5 -Ij-2 • (3.13)

If

gc exP(“TcT— J << 1 '

then (3.12) can be approximated by

3 ln TC . 9JJ.3X 3x (3.14)

It will be shown later that the term ~ is proportional tod Xthe electric field. From (3.12) and (3.3) it is evident that as the Fermi level goes above ED , the term involving the electric field becomes less significant. At 1200 °C,

the above inequality is well satisfied for ED“Ef L 0*38 eV< For donors £ >>1# and (3.6) can be written as

where

and

Ec"EVkT

=Ef-Ec (“}

kT

Note that at typical diffusion temperatures is about0-8 eV for Si and Ev = Ec - 0.4 eV. Therefore, nothing

★can be said about the magnitude of 5 •In computing the flux of total impurities f , thec

theory has taken into account the fact that a certain fraction of donor atoms may remain neutral, some may be charged positively and some of these may form pairs with the charged and neutral vacancies. Within the semi­conductor, there is no generation of the total impurities. The continuity equation thus yields

~ _ 3 f|c + = o . (3.16)31 3x

Substitution of (3.3) in (3.16) gives

46

*3 C C 9 r*- / 1 n \It = — IS [V(IS + C IS11 (3-17)V

*where (3.14) has been used. Note that v is independent of x.

The continuity equation for vacancies is

Is + is1 = '3-18>

In general, Gv may not be zero. This term may be caused, for example, by plastic deformation. A mismatch in the size of diffusing impurities and the host lattice atoms is a major cause of dislocations. Substitution of (3.4) into (3.18) yields

H ■ +k < v 4 + °l Is1 • (3-19)As discussed by Hu and also by Nuyts and Van Overstraeten [32], quasi-equilibrium condition for vacancies under typical diffusion conditions is a good approximation.

8 VThus, assuming - 0, (3.19) yields

!_ [D* v 3 ^ > + D* |H] = -G <x) .ax 1 v ax v ax v

Integrating both sides from x to ®, and noting that as 9 v •x->°°, v->v i.e. -— ->• 0, and y -»-l, we obtaino X *

47

D.*9 I n ( v Y y )

Gv (x)dx .vX

Integrating once again,

>00 y f00v - | (3.20)

' D 'x v x*Note that yv < 1, and Dv is large [32]. The mismatch

in the radii of As and Si atoms is very small resulting in a small generation term G . Thus, if the integral on the r.h.s. can be neglected, we simply have

3.3 Computation of the Fermi LevelThe unknown n appearing in (3.22) can be evaluated

using the condition of local charge neutrality. Then, (3.22) with suitable boundary conditions describes the transport problem. Local charge neutrality implies

*v (3.21)v

Substitution of (3.21) into (3.17) yields

(3.22)

- f “ *n-p-c +v = 0 (3.23)

48

where c+ and v are the ionized donor atom and charged vacancy concentrations, respectively, with v given by [12]

v --------- (3.24)1 + g~ exp(-n-ev)

Since £ >>1, v~ * v. However, this concentration itself is so small compared to other terms in (3.23) that it can be safely neglected [12].

The electron and hole concentrations are given by

n = Nc F ^ y (3.25)

and

P = Nv Fl/2(“n“eg) (3.26)

where e is the normalized band gap. For donor diffusion,9

(-n-e ) << -1

and

p = Nv exp(-n-eg) (3.27)

49

For partial ionization, the ionized donor concentration is given by [12]

c+ = -------^------- — (3.28)d! + gc exp(n + j r)

where E, = E -E^ is the ionization energy. Note that for d c Dsufficiently negative values of n, c+ - c and we approach the 100% ionization case.

Substituting (3.25), (3.27) and (3.28) into (3.23) with g = 2 we obtain

Nc Fl/2<n) ' Nv exp(-n-cg) - i + 2exp(n+e- ) = 0

(3.29)

Thus, (3.29) can be used to evaluate n-

3.4 Summary of the Problem *We must solve the partial differential equation

(3.22) where n is obtained using (3.29). The values ofN and N are found, as in Section 2.3, by using the c vequations

N = F - T rTT (3*30)l/2 (ni)

N = n. exp(Ti.+e ) . (3.31)v l i g

Also, yv and n are given by (3.15) and (3.13).For a constant source diffusion the boundary condi­

tions are

These will be transformed in terms of suitable variables at a later stage.

3.5 Form of Impurity Flux With and Without Vacancies Using (3.3) and (3.14), the impurity flux with

vacancies can be expressed as

c(0,t) = Cq

c(°°,t) = 0

c(x,0) = 0, x > 0 (3.32)

fc (3.33)

When the vacancies are in quasi-equilibrium, using (3.21),

where E is the electric field. Thus, we obtain

*D aC r o c e .

C Y v f3x kT CEl

= — [-D* ~ + y* cE] (3.34)Y C 3x C

* eDSwhere }ic = has been used. Comparing (2.26c) and(3.34) we see that the forms of the flux equation aresimilar except for the factor 1/yv * This can also beviewed as a change in the impurity diffusion coefficient

D*which now becomes — . In other words, the impurity diffusion coefficient is now proportional to the vacancy concentration v. At low donor concentrations yv ■+ 1, and(3.34) reduces to (2.26c).

3.6 Transformation of the EquationsThe equations summarized in Section 3.4 can be

transformed into a simpler form. It is possible to substitute for the derivatives of c in (3.22) using(3.29). An equation entirely in terms of the dependent variable n is then obtained. Noting that

52

5 = 9V exp(ni+ev) ,

(3.22) can be written as

If ■ [2 ff If + c(| ,22 .2

+ 1_| + c -2—9-] (3.35)ax ax

where (3.15) has been used. Also, C3.29) gives

c = [1 + 2 exp(n+ed)] [Nc pjy2 ” Nv exp “T1“eg ^ •

(3.36)

Differentiation of (3.36) yields the following expressions:

££ = ia * (n) (3.37)ax ax yi

where

^1(n) s Nc F_1/2(n) + Nv exp(-n-eg) + 2 Nc exp(n+ed)

x ^Fl / 2 ^ + F-l/2^n^ ' (3.38)

ff - H <3.39,

53

and

i!| = + Cn) + <f£>2 *2(n) (3.40)ax ax

where

<f>2(n) E Nc F-3/2(n) " Nv + 2NC exp(n+ed)

* {Fl/2(n) + 2F-l/2(n) + F -3/2(n)} ‘ (3.41)

Now, substituting (3.37), (3.39), and (3.40) in (3.35),after some lengthy manipulation, we obtain

a ^ 3 <K(n) <t>4(n)9t = D “ Pf"-"!) t <«)* + ] (3-42>

where

♦ 3(n) s Nc F1/2(n) + 2NC F_1/2(n) + Nq F_3/2(n)

+ 2exp(n+ed) {4 Nc F1y2 (n) + 4 Nc f-i/2^

+ Nc F_3^2 (p) - Nv exp(-n-eg)} (3.43)

54

<t,4(n) = Nc Fl/2(Tl) + Nc F-l/2(n) + 2exp(n+e^)

X {2Nc Fl/2(n) + Nc F-l/2(n) " Nv exP (-n-cg)}

(3.44)

and

* *Dc CD = - - V- . (3.45)1+5

At this point, it is convenient to make a transformation of the dependent variable,

n = ifi + , (3.46)

so that the new variable \l> is defined by

* 5 • (3.47)

With this transformation (3.42) becomes

I t = D [ ( ~ ^ - ) 2 f 1 ( i p ) + ( ^ ) f M ) ] ( 3 . 4 8 )d t d X J. 3 x

where

Thus, the problem has been reduced to solving (3.4 8).The boundary conditions for the constant source diffusion, given in Section 3.4, can be easily written in terms of as

<P(0,t) = t|/Q (3.51a)

(°°, t) = 0 (3. 51b)

ip(x,0) = 0 , x > 0 (3.51c)

where is computed by solving the equation

CQ - [1 + 2exp(^0+ni+ed)] [Nc F1/2(^Q+ni)

- Nv exp (-i|/0-ni-eg) ] = 0 , (3.52)

which follows directly from (3.36).

56

3.7 DiscussionIt can be noted that the partial differential equa­

tion (3.48), and the boundary conditions (3.51), are of the same form as those obtained in Section 2.4 except that

A

instead of the variable <p, we now have ip. Therefore, the numerical solution to (3.48) can be found exactly in the same manner. As shown in Section 4.5, a transformation of the variables can be used to separate the variables. The resulting ordinary differential equation can be solved efficiently by using a numerical technique. The results are discussed in Chapter 5.

CHAPTER 4NUMERICAL ANALYSIS

The models discussed in the previous chapters result in a set of partial differential equations. These equa­tions are highly nonlinear, and it may be extremely difficult, if not impossible, to find a closed form solu­tion. A numerical solution is the only viable alternative. There exist a number of methods to solve a boundary value problem [39] . In the present case, a quasi-linearization technique is used. The partial differential equations are first transformed into ordinary differential equations by discretizing the time step. The process then consists of reducing the set of ODE to successive approximate sets of linear equations which can be solved more easily using an iteration scheme. A desired accuracy can be reached by repeating the process of linearization.

In this Chapter, a general method for solving coupled nonlinear ODE is first presented. The treatment closely follows that of Widiger [24]. The procedure is then applied to individual cases discussed in the previous chapters.

4.1 Discretization of Independent variablesTo facilitate the numerical techniques, the independent

variables have to be discretized, thereby yielding sets of57

58

difference equations. The variables in the present caseare t. and x., defined by 1 J ■*

t^ = (i-l)At + tQ, i =

xj = (j —1)Ax j = 1,...,m

where At and Ax are chosen to be fixed for simplicity.The discretization of t transforms the PDE into an ODE at a certain time step. The abruptness of the initial condi­tion at the surface can be dealt with by assuming a non­zero starting time tg.

The time derivative is approximated by a two-point implicit scheme,

3f(t.) f(t.)-f(t. . )31 At

The procedure then is as follows. Once time t is discretized, the PDE is transformed into an ODE at a time t^. If the solution to the ODE is known at time t^_^ • the ODE can be solved to yield the solution at time t^. Beginning with i=l, the above step is performed for each time increment, until the desired final time is reached.The solution of the ODE is discussed in the next Section.

4.2 General Quasi-linearization TechniqueThe quasi-linearization technique is based on Newton's

approximating procedure for finding the roots of an arbitrary function. Given a function f(x), it is desired to find the roots xr of the equation

f(x) = 0 (4.1)

An initial guess x^ to the correct value of the root ismade. Thus

f(x^+Ax) = 0 (4.2)

where

xr

Expanding (4.2) by a Taylor series about x°, we obtain

f(x°) + f'(x°)Ax + i f"(x°)Ax2 + ... = 0

The above equation is to be solved for Ax to determine the true root. The problem can be simplified by truncating the series after the linear term; then

60

The approximate solution to Ax, denoted by Ax\ can thenbe found by solving

f(x°) + f'(x°)Ax1 = 0 ,

thus yielding a better approximation to xr as

f*(xu)(4.3)

One can now start with x^ as the initial guess and find x1which is closer to the true root. The process can then be

iteration until the desired accuracy is achieved.This process, of course, is not guaranteed to work.

The function must satisfy certain properties and the initial guess must be sufficiently close to the desired root. How­ever, if the function has only one root and the iteration procedure converges, the true root will be approached.

Consider now the problem

repeated with x1+ substituted in place of x1 in the next

/ • • • • • • fy^,x) = 0

f • • • • • • / y",x) = o (4.4)

or, expressed in matrix notation,

61

f (y,y',y",x) = o (4.5)

The primes indicate derivatives with respect to x, and represents a known, algebraic function of its arguments. Thus (4.4) is, in general, a set of ordinary, coupled, nonlinear differential equations with dependent variables y^,...,yn and independent variable x. The solution for y is desired. An initial guess, y^, is first made. A set of linear equations with a dependent variable Ay1 can bederived by expanding (4.4) as a power series in terms ofthe dependent variables and their derivatives about the initial guess y^. The resulting equations truncated after the linear term yield for the i'th equation,

n n 3 f ? . n 3f ? , n 3f? .f7 + £ 7— r Ay." + z -r-4- Ay.1 + I Ay. = 0x . , 3y i . , 3y'. . , 3y. J i3=1 3 J 3=1 3 J 3=1 y J J

(4.6)

where f? represents the value of f^ evaluated at the initial guess y°. In matrix notation, the equations can be written as

A ° A y 1 " + B ° A y 1 , + C ° A y 1 = D ° ( 4 . 7 )

where

62

A =

!£ayj3 fn

«- 9yJ

3f3y1n

9 f 0 n3^ir •* n (4.8a)

B =

3f“

3 fn9n

3flwn

3fn3y1 Jn

(4.8b)

Cu =

3f:W -

3f____In

3f3yn

3fn3yn

(4.8c)

and

D° =

< \

= “f (4.8d)

The set (4.7) is a set of coupled, linear, ordinary differential equations, which when solved, yields Ay"*". Once Ay1 is known, the improved initial guess is given by

y1 = y° + A y 1 (4.9)

63

It has been assumed for the time being that the boundary conditions have been taken care of in a similar fashion. They will be dealt with explicitly later on.

It should be pointed out that f is a known function, and hence all of its partial derivatives in (4.8) are known. Thus A^, B®, and are all known functions ofy®, y0', and y®". It is therefore simple, in principle, to solve (4.7). Once y1 is found as in (4.9), the above procedure can be repeated with all the zero superscripts replaced by one superscripts. The process can be repeated until the desired accuracy is attained.

For i'th iteration, (4.7) can be written as

• .2 l+l j l+l . • . iA1 £— £1---- + B1 £-42--- + C1 Ay = D1 (4.10)

dx2 dx

The derivatives involved in A1 , B1, C1 and D1 are evaluated at each point using a five-point polynomial approximation scheme. As mentioned before, the variable x has been discretized. To find the derivatives of Ay1+^, the following three-point approximations are used.

dqk qk+i • qk-idx 2Ax (4.11)

d2qk _ qk+i - 2qk + q*-k-1 (4.12)dx A x

64

Here, k denotes the discretization index of the x coordin­ate. Using (4.11), (4.12) in (4.10) we obtain

& y i+1 - 2 A Y i+1 + AVi+1 AV i+1 - AV'*'+ ^ni k+1 k k-1 , Di k+1 yk-lA ----------- ---------- + B -------------K Ax K 2Ax

+ cj A y j +1 = Dj (4.13)

Simplification of the above equation yields

i . „i+1 i „i Awi+1 . i Al<i+1 ni Ayk+1 + ^ Ay,, + Ay, , = D,k-1 (4.14)

where

a1 + — g1Mk 2Ax k (4.15a)

Ax(4.15b)

Yk = _i_ AiA 2 kAx

1 Ri2 Ax k (4.15c)

If the solution to (4.14) is assumed to have a form

AyJ+1 - g£ hk Ayk+J, (4.16)

then the substitution for Ayj^ in (4.14) gives

65

a"*" AVi+1 + - y 1 h* ) Ay^+ ^ + Y ^ g ‘*’ = D"*-k k+1 lPk k k-1 k Yk yk-l uk

Repeated substitution yields

‘“k - (ek - K hk-i,hk] ^i+lk+1

+ [<ek - K hk - i )gk + K K - i - °k5 = 0

Since is finite, this equation is easily satisfiedby letting each term in the brackets to be zero. Then,

hk = (6k - *k hk-i>'1 4 (4-17)

4 - (6k - K hk-i>"1 (Dk - K 9k-i> • (4-18>

Thus, if and g£ are known for a particular x^, they canbe found for the next by using (4.17) and (4.18).Note that k is the position index whereas i is the itera­tion index. For a particular time step, and for a given i,computations are done for all k. The iteration index i is then changed till a desired convergence is reached. The time index is next changed and the entire process repeated until the final time is reached.

66

4.3 Quasi-static ProblemThe problem stated in Section 2.3 can be simplified

A A A A

by normalization of the variables. Let x, t, c and <t> represent the normalized variables. Then a normalization scheme is as follows.

c = n^ c

♦ = vT *

EVmt = i E * <4-19)C 1

With these substitutions, and defining

c06 = — , (4.20)ni

the quasi-static equations for the degenerate case become

as. = »_ (3£ + a %at ax ax ax

(4.21a)

a2* _ Fi/2 (»+ni)3x2 ' F l / 2 (ni>

exp(-<j>) - c (4.21b)

For notational simplicity the normalized variables will be denoted by c, <j>, x and t in Sections 4.3 and 4.4.

67

With the discretization of the time coordinate (4.21)

where the notation

c = c(x,t^)

<t> = <t>(x, t±)

N = c(x,ti_1)

has been used.Now the function f can be written as

becomes

(4.22a)

(4.22b)

f

, 2 -.2 dxdx At At -dx dx(4.23)

68

y =

1HI <P

1 to 1 c(4.24)

For the nondegenerate case [24] , the second term in the expression for becomes exp(<|>) instead of

fi/2 (‘f,+ni>/Fi/2 (ni> * Corresponding changes for this case can be easily made in the expressions below. For the vector f above, the matrices A1, B1, C1 and D1 are given by

A 1 =3 f i 3fl ] 1 03$"" 3c"

3f2 3*2 -|_ ST*" 3 C " J c 1 (4.25a)

' 9fl 3fl 1 0 0B1 = 3 <J> ' 9c' =

9f2 9 f 2 d«frL 3<f> * 3c' - _dx dx. (4.25b)

C 1 =3f]3c"3 f, 3c"

F-l/2( +T1i)Fl/2(ni)

- exp(-<j>) 1

d24>dx2

1_At-

(4.25c)

69

r d2<f> . Fl/2(*+ni) , 1% n2 f -- UT~)--- exp( ip) cldx^ *1/24V

1CN1

—1

d2c d2<|> dc d(j> c N dx c ^x2 dx dx At At_

(4.25d)

where for simplicity, superscripts i have been suppressed in <j> and c.

The flow diagram of the numerical method is shown in Fig. 4.1.

4.4 Boundary ConditionsNormalized boundary conditions for the constant

source diffusion are

c (0,t) = 3

3<fr(0,t) = 0 3x

c (°°, t) = 0

<|> (“, t) = 0

c (x, 0) = 0 , x > 0 (4.26)

70

Start

NO

Yes

Stop

i+1

i At+t

Has the final time been reached?

Solve (4.22) for new g.(ic) and j> (x) . (see Fig. 4.2)

Select the starting distribution

FIGURE 4.1. Flow Diagram of the Numerical Method

71

Notice that the only parameter in the above conditions, other than the dependent and the independent variables, is the normalized surface concentration 3. A series of solu­tions over a range of 3 will, therefore, give a general solution to the quasi-static problem.

After the time is discretized, the boundary conditions for the ordinary differential equations become

c (0) = 3

d<J> (0) _ „ dx u

c(°°) = 0

<f>(”) = 0 . (4.27)

These boundary conditions can be satisfied in the i'th iteration by requiring that

Ac1+1(0) = 3 - cx (o)

dAj>1+1(0) = _ d4>1 (0) dx dx

, i+1, \ i/ \Ac (°°) = - c (°°)

A<j>1+1(“) = . (4.28)

72

If the initial guess is picked such that

c° (0) * 3

c° (°°) = 0

(°°) = 0

then the desired boundary conditions can be met by merely requiring that

i+1Ac (0) = 0

dA<t>1+1 (0) = _ d<j)1(0) dx dx

Ac1+1(“) = 0

A<J>1+1(») = 0 . (4.29)

Numerically, it is impractical to extend the x-coordinate to infinity. Therefore, a distance L is chosen which is large enough to approximate infinity and the boundary con­ditions are applied at x=L. If the distance L corresponds to the m'th point, the boundary conditions become

73

a nAc^ = 0

Aj.i+1A<J>2 - A ^ d(f)1— -

Ac1+1 = 0 m

A0m+1 = 0 * (4.30)

d<juNote that the quantity ^ — has already been determined while finding the matrices A, B f C and D.

The procedure for solving (4.10) is as follows. If it is selected that

-l(4.31a)

and

-A x

d<j>Jdx" (4.31b)

then Ay^+ given by (4.16) will satisfy the surface boundary conditions of (4.30). Using (4.17) and (4.18) h.1 and g?; can then be generated for k = 2,...,m. Choosing

Ay i+lm00

(4.32)

74

will satisfy the boundary conditions at x=L, given byi+1(4.30), and (4.16) will generate Ay^ for k = m-1,.,.,1.

The technique for numerically solving the problem described here is summarized in the block diagram given in Fig. 4.2. In the actual program many of the functions represented in the block diagram have been combined for better computational efficiency.

The normalized boundary conditions for drive-in diffusion are given by

9c (0, t) _ Q 3x

9<t> (p,t) = 0 •9X

c (°°, t) = 0

<f>(°°/t) = 0

c(x,0) = Nq (x ) , x > 0 (4.33)

where NQ(x) is the normalized starting distribution for the drive-in diffusion. Note that the first boundary condition is obtained because the flux of the impurity

atoms at x=0 is zero. Thus,

75

Enter

Solve the linear difference Eqs. (4.13)

i=i+l

No

YesExit

i=0

i+1 i+1 for all k

Find A and D, for all k

Has sufficient accuracy been reached?

Make an initial guessk=l

Find a and y, for all k using (4.15)

Find -5-— and — x— for all k dx j 2dxusing 5-point method.

Determine Ay

Set Ayi+1

i+1

k=m-l 1 using (4.16)

Determine g, and h

Determine g, and h, using (4.31) or (4.39)

k=2 m using (4.17) and (4.18)

FIGURE 4.2. Flow Diagram of the Iteration Procedure

76

D 80(0, t) c 9x y. cE(0,t) = 0 . (4.34)

The semiconductor material as a whole may be assumed to be charge neutral. The total charge Q per unit area is given by

Q = dx

= £ 9E , dx9x

= e [E (°°, t) - E (0 , t) ]

Equating Q to zero and noting that E(«,t) = 0, we have

E (0 , t) = 0 . (4.35)

Thus (4.34) yields the first boundary condition.Proceeding as we did earlier in the Section, instead

of (4.28), we now obtain

dAci+1(0) = _ dc1 (0) dx dx

dA<f>1+1 (0) = _ d»1(0) dx dx

77

. i+1 , . i , .Ac (°°) = -c (°°)

A 4>1+1 (oo) = -<j,1 (oo) . (4.36)

If the initial guess is picked such that

c^ (°°) = 0

4>° (°°) = 0 ,

then the desired boundary conditions will be met by merely requiring that

dAc1 (0) _ _ dc1 (q) dx dx

dA<ft (0) _ _ d(frX (0) dx dx

A c 1 + 1 (oo) = 0

A<j,i+1(co) = o . (4.37)

Instead of (4.30), in this case, we now have

. i+1 . i+1 , iAC2 - Ac^ dc^Ax dx

Therefore, (4.31) become

-1 0(4.39a)

0 -1

A x — dx

(4.39b)

Note that (4.32) remains the same.

4.5 Quasi-linearization Technique for a Scalar Equation As discussed in Sections 2.4 and 3.6, the partial

differential equations describing the diffusion process can be written in a general form

where f (<J>) and f^ (<P) take different forms. The boundary conditions are of the form

(4.40)

79

<p (<*>, t) = 0

<J)(x,0) = 0 , x > 0 . (4.41)

A transformation of independent variables (x,t) -> (y,T), similar to the one suggested by Shaw and Wells [19], is chosen so that

xy = /4Dt

x = t (4.42)

The dependent variable in terms of these new independent variables then becomes

<{> (x ,t) = v[y (x,t) , t ( x,t) ] . (4.43)

Note that the variable v used here is not to be confused with vacancy concentration, a notation used in Chapter 3. Now,

M. - 2 X iX + Ix Ix3x dy 3x 8 t 3x

= (4.44)/4d 7 3y

80

Similarily/

3 2<f>3X2 ay /4D'

3v , 3y 3y 3x

14 D t ay2

and

M. = IZ iLZ + i*Z ii 31 3y 31 31 31

y 3v . 3v 2 t 3y 3t

Substituting these in (4.40) we obtain

If ■ 2* I? + fi (v) (!?)2 + f2<v> 7 ?ay

and the boundary conditions become

v ( 0 , t ) = (f>Q

V (°° , T ) = 0 .

(4.45)

(4.46)

(4.47)

(4.48)

Notice that the last two conditions in (4.41) reduce to a single condition in (4.48).

81

Because of the form of the equation and the boundary conditions/ it is easy to show that v is independent of t ,

in which case (4.47) reduces to

2* U + fl (v) (i y ) 2 + f 2 (v) 7 7 = 0 ' < 4 ’49)v*jr

with

v (0) = 4>q

v (00) = 0 . (4.50)

Notice that the only parameter in the above problem is <j>Q. Thus, solutions v(y) known for all possible valuesof <f)q constitute a general solution to the problem. Oncev(y) has been computed using a numerical procedure, <f>(x,t) can be obtained for a given x and t by using

4>(x,t) = v (——— ) . (4.51)/ 4Dt

A simple way to find <j>(x,t) from v(y) is to interpolate <{>(x,t) according to the equation x = /4Dt y.

The quasi-linearization technique discussed in Section 4.2 can be used directly to solve (4.49). Since only one equation is involved, the matrices and vectors reduce to scalars. The y coordinate is uniformly

82

discretized and the final point m, corresponding to a distance L sufficiently large to adequately represent infinity, is chosen.

Let i denote the iteration number and k the point in the y direction. Applying the definitions of Section 4.2, we have

For simplicity, superscripts and subscripts have been

in the expression for C, are found by analytic differentia­

tion .

f(v",v',v,y) = 2y + f x (v) (| ) 2 + f2(v) (

(4.52)and

A 3 v" f2(v) (4.53a)

B = f§T = 2y + 2fl(v) (4.53b)

df1dv (4.53c)

(4.53d)

v actually is v^ , etc. Also, ^— - and , which occur

83

Note that for different problems only the quantities A, B, C and D need be changed. The rest of the procedure described here remains unaltered. This, of course, assumes that the problem and the boundary conditions are in the given form. Applying the results of Section 4.3,

(4.54)

where

(4.55a)

and

(4.55b)

where

(4.56a)

(4.56b)

(4.56c)

The boundary conditions chosen for difference variables are

84

hl = gl = 0 (4.57a)

and

Av^+1 = 0 . (4.57b)

This assumes that the initial guess is constrained to satisfy the boundary conditions.

The computation procedure is the same as that shown in Fig. 4.2 except that various expressions are replaced by the ones above. An initial guess v^ for all k is made, the correction Av^+"*" is found, and the new initial guess

vk = vk + Avk ' (4.58)

is used to restart the procedure which is repeated until sufficient accuracy is obtained.

4.6 Application to the Charge Neutrality Approximation The method discussed in the last section can now be

applied to solve the equations of Section 2.4. We have

f , l = F-3/2<V+rii) + F-l/2(v+T1l) __ Al(v) ,,1 f _ i / 2 ( v + ^ i ) + F 1 ^ 2 ( n ^ ) e x p ( - v ) a 2 ( v )

and

85

F_i/o(v+n•) + F 1/9(v+n.) A (v)f2(v) = F_1/2(v+ni) + F1/2(ni)exp("v) = *2 (v) (4*60)

where A^, A^ and A2 represent the numerators and denomina­tors of f^(v) and f2 (v), respectively. Thus,

df-, dA dA „a^r = [V v> av^ - V v) a^r > t V v)I (4-61>

df dA_ dA- -hrt - [V v) wr - V v) srt i ' V v>1 (4-62)

where

dA,3 ^ = F -5/2(v+'1i> + P-3/2(v+,'i) (4-63a)

dA -aV- = P-3/2<v+,’i) ■ F1/2<T'i>eXp<"v) (4.63b)

dA3v- = F-3/2(v+ni> + F-l/2(v+ni) (4.63c)

The quantities A, B, C and D are now known and the procedure of the previous section can be applied.

4.7 Application to Vacancy-aided DiffusionThe diffusion process in this case is defined by the

equation

86

(4.64)

where D, and f2 are given in Section 3.6.The lengthy form of f^ and f2 makes the expressions

for A, B, C and D rather cumbersome, although, inprinciple, it is simple to find them. In order to evaluate

9f. af2these quantities, we need to evaluate -— and --- . After^ s ip a ipsome manipulation, these are given by

4*3 <f>6 (i|»+n )

[ (ljrhr ) ]2

(4.65)

9 f2 - t2W + exp(*) ^ +H) -<f>7 (tp+n i

(4.66)

where

(f)g(X) = Nc tF_i/2 2F—3/2^^ F-5/2

+ 2 expU+e^) { 4f i/2 + 8F- l / 2 ^

+ 5F_ 3/2(X) + F _ 5 / 2 (\)}] (4.67a)

87

4> 6 (X) ~ Nc F-3/2^X “ Nv exP(-x-eg) + 2NC exp(A + ed)

X {F1/2(X) + 2F-1/2(X) + F-3/2(A)} (4.67b)

4>7(X) s Nc [F_i/2 (X) + F_3^ 2 (M + 2exp(X+ed)

x {2F1/2(X) + 3F_1/2(X) + F _ 3/2(X)}] (4.67c )

<t>8(X) 5 Nc F_3/2(X) - Nv exp(-X-eg) + 2Nc exp(X+ed)

x (F1/2(X) + 2F_1/2(X) + F _ 3/2(X)} . (4.6 7d)

Thus A, B, C and D are known, and the numerical procedure of Section 4.5 can be used.

4.8 Computation of the Boundary ConditionWe now consider the computation of for a given Cq

by using (3.52). The Newton-Raphson method for finding the roots of a nonlinear algebraic equation can be used. However, we need a starting guess for i w h i c h is sufficiently close to the true root. One way to find this starting guess is to consider the equation for the non­degenerate case and complete ionization, in which case(3.52) reduces to

c 0 “ <NC exP ('J'q + h^) “ Nv exp (~<J'q “ nj - cg) ) = 0

88

or,

Cif/J = sinh"1 (^--) = sinh"1 (3/2) (4.68)

where the superscript on tj/g denotes the initial guess.The true root can now be obtained by following the itera­tion procedure

i+1 i F <*0> V" <4-69)0 u F 1 (*J>q)

where

Fl/2 ( o+ni)F(i/»q) = [1 + 2exp(i|»0+ni+ed) ] [ — )--- exp(-^0)]-6

(4.70)

and

F-l/?^0+T1i) 2exp(if>n+n-+e JF' (4>Q) = exp(-t|;0) + — P .f- + — d-0 0 1/2(ni' Fl/2(ni)

x Fl/2 (l o+r|i) + F-l/2('l'0+T1in (4,71)

4.9 DiscussionThe results obtained using the numerical techniques

presented in this Chapter are discussed in Chapter 5. To compare the results of vacancy-aided diffusion using Fermi-Dirac statistics to those obtained using Maxwell-

89

Boltzmann statistics, the simplest approach is to replace the Fermi integral subroutine by one where an exponential function is used in place of the Fermi functions. For computations with complete ionization, may be replaced by a negative quantity of large magnitude, effectively negating the effect of the partial ionization term.

CHAPTER 5 RESULTS

The formulations described in the previous chapters, for convenience, are abbreviated here as follows: quasi­static (Section 2.3) as QS, charge neutrality (Section 2.4) as CN, and vacancy-aided (Section 3.2) as VA. In this Chapter, numerical results using the techniques discussed earlier are presented for the case of a constant source diffusion of As in Si.

As indicated earlier, n^ is calculated from the data in [29] using

n± = 7.766xl015 exp(5.528269xl0~3 T) (5.1)

where the units are cm 3 and T is in °K. The above ex­pression is valid in the temperature range of 900-1200 °C. For the intrinsic diffusion coefficient of As in Si several empirical expressions have been suggested [40, 33, 41]. Masters and Fairfield have suggested the expression

D = 6 0 exp(-4.2/kT) . (5.2)

Chiu and Ghosh have proposed

D* = 24 exp(-4.0833/kT) c90

(5.3)

91

whereas Kennedy and Murley have given

D* = 2870 exp(-4.5725/kT) . (5.4)

* 2 The units of Dc and kT in the above expressions are cm /secand eV, respectively. The band gap for Si in eV is givenby [31]

E = 1.205 - 2.8xl0_4 T . (5.5)g

The ratio of effective masses is assumed to be temperature independent [30],

*mp , 0.67818 >2/3 ,c“* = ' 1.19250 1 • l5-6)mn

Using (2.52) then yields

n. = -6.98956*103 T_1 + 1.34193 . (5.7)l

Other physical constants used are:

T = 1050 °Ck = 8.6 2xl0-5 eV/°K

-19e = 1.602x10 coulombe = 11.7 e = 1.0359x10 12 farad/cm o

92

The maximum value of the surface concentration used is 21 -3CQ = 1.6x10 cm , which represents the solid solubility

value at the diffusion temperature. The values of and(E^-Ey) are nominally assumed to be 0.05 and 0.4 eV, respectively.

The profiles resulting from the QS approximation using FD and MB statistics are shown in Fig. 5.1 forvarious diffusion times. An average value from (5.2) and

* -15 2(5.3) of Dc = 6.44x10 cm /sec has been used. Thecomplementary error function profiles, which represent the correct solution for E=0, are also shown. It is observed that the internal electric field enhances diffusion at high concentrations. It is also seen that the use of MB statistics, instead of FD statistics, gives an under­estimate of the impurity density values, typical error being about 4% near the surface and 50% deep in the material. It was found that at low concentrations, use of either statistics yields the same result, which is not unexpected, since all the Fermi functions reduce to exponential functions. The electric field plots for the QS approximation with MB statistics are shown in Fig. 5.2. These profiles are reminiscent of those obtained in a high-low junction. It may be noted that the field varies almost linearly near the surface and then reaches a maximum value. This maximum value decreases as the diffusion time increases. Similar profiles are obtained

9 3

<M't) CONSTANT SOURCE DIFFUSION

— zero field solution•numerical solution

Or-± classical statistics

Fermi-DiracstatisticsO

X(_) o

r-O

wO

3010

2 niinO

0.00 0 . 6 0 1 . 8 01 . 20( MI CRONS)

3 . 0 0-t

FIGURE 5.1. Impurity Profiles of Arsenic in Silicon Using the Quasi-static Model

94

when FD statistics are used.The solution of the CN approximation was compared to

the QS approximation. When the impurity density profiles were plotted on the log scale of Fig. 5.1, no significant difference was observed between 'QS and CN results. The electric field obtained from the CN model with MB statistics was also found to be insignificantly different from that shown in Fig. 5.2. It is concluded from the above analysis that charge neutrality is an excellent approximation for describing a typical diffusion process.The CN model is simpler and computationally very efficient compared to the QS model. It can, therefore, be used con­veniently to find the effect of the internal electric field.

The effect of vacancies can be analyzed by comparing the results of VA model to those of the CN model as shown in Fig. 5.3. It is evident that at high concentrations vacancies substantially enhance impurity diffusion. The profiles show a region of relatively slowly varying density followed by a region where it drops suddenly. The intrinsic diffusion coefficient used has been obtained using (5.2) for both models for a consistent comparison. Note that the VA model includes the effect of partial ionization in addition to the use of FD statistics.

As indicated in Chapter 3, Hu and Schmidt [22] have

computed the results of As diffusion in Si. They have

95oo

■

oo

21 cmooo

mO'o X o ■ooGO

OCD_co

QLUoo

ooC\JC\J

30102 minoo

0 . 5 000 1.00( MI CRONS)

1 . 5 0 2.00 2 . 5 0

FIGURE 5.2. Electric Field for the Constant Source Diffusion

C (X

)

96CM

O

CN VAO

ODo

CN VAO

(£>o

30 mininO

0 . 5 00 . 4 00.20 0 . 3 00.00 0 . 10X ( MI CRONS)

FIGURE 5.3. Comparison of Results Using Charge Neutrality and Vacancy-aided Models for High Surface Concentration

97

used MB statistics and have assumed complete ionization. Also, a factor & was assumed to be 100 which is only a rough approximation [32], In Fig. 5.4, profiles obtained using the VA model are shown for the following cases: FDstatistics with partial ionization, MB statistics with partial ionization, FD statistics with complete ionization and MB statistics with complete ionization. The last case is .similar to that considered by Hu and Schmidt. It is evident that at high concentrations the assumption of complete ionization introduces very large errors. Hu and Schmidt have indicated that the validity of their physical model breaks down at high concentrations because there is no limit to the enhancement effect due to vacancies. It is seen here that with the incorporation of partial ioniza­tion, the VA model continues to remain valid. Note that as in the case of the QS model, use of MB statistics yieldsan underestimate of the impurity density. At relatively

19 -3low surface concentrations, e.g., Cg = 1 0 cm , the use of either statistics and ionization conditions does not make any significant difference.

At low concentrations, the effect of the electric field becomes less significant and the CN model yields results which are very close to the complementary error function profiles as shown in Fig. 5.5. As pointed out in Section 3.5, the form of the impurity flux with vacancies reduces to that of the QS case at near intrinsic conditions.

98t\j•fa

30 min

XooCD O

FD with 100% ionizationCD

MB with 100% ionizationFD with partial ionizationMB with partial ionization

O

LDo

0 . 5 00.20 0 . 3 0 0 . 4 00 . 100.00X ( MI CRONS)

FIGURE 5.4. Effect of Statistics and Ionization on Impurity Concentration Using the Vacancy-aided Model

990)o

Value of D from(1) Masters & Fairfield(2) Kennedy & MurleyOrH

VA (1)

CD erfc

CN

O VA (2)

XinCJ CD

CD

30 min

O

CD

0.00 1 . 20 X ( MI CRONS)

1 . 8 00 . 6 0 2 . 4 0 3 . 0 0

FIGURE 5.5. Impurity Profiles for Low Surface Con­centration Using Charge Neutrality, Vacancy-aided and Zero Field Models

100

However, in the VA model, it was assumed that K >> 1» andtherefore, strictly speaking, the model is not valid atlow concentrations. The range of validity of the model cannot be determined at present because, as is evident fromFig. 5.5, the impurity profiles strongly depend on thevalue of the intrinsic diffusion coefficient used.

Comparison between the experimental results [33] andthe calculated profiles using the vacancy-aided model for

*various values of Dc and the vacancy levels is shown in Fig. 5.6. Note that good agreement exists for the case 3. It was found that the results were relatively insensitive to the variations in donor ionization level.

101

*Dc Ec-Ev (eV)1. Masters & Fairfield 0.42. Kennedy & Murley 0.43. Kennedy & Murley 0.5* Experimental data

*b

OJO

XCOC_) o

o

60 minCOO

0 . 4 5 0 . 6 0 0 . 7 5.00 0 . 3 0X ( MI CRONS)

FIGURE 5.6. Comparison Between Experimental Data and Calculated Profiles Using Vacancy-aided Model

CHAPTER 6 CONCLUSIONS

The object of this study has been to develop and analyze various models describing the diffusion phenomenon. The effects of internal electric field, degeneracy of carriers, partial ionization and vacancies have been con­sidered. Because of the complexity of the models involved, numerical techniques had to be used to obtain the solu­tions of the differential equations describing the trans­port process.

Previous analysis [24] has shown that the assumption of quasi-static equilibrium for holes and electrons in describing a diffusion process is valid. In the present research, formulations considering quasi-static equilibrium and charge neutrality have been examined for degenerate conditions. A vacancy-aided model has been proposed for arsenic diffusion in silicon. A constant source diffusion has been assumed.

The major contributions of this research may be summarized as follows.

1. The quasi-static model used to analyze the effect of internal electric field was extended to degenerate conditions by using the Fermi-Dirac statistics. It was concluded that the use of classical statistics yields an underestimate of the impurity density values. While this

102

103

error is significant at high surface concentrations, at low concentrations the use of classical statistics was con­sidered to be a good approximation. The impurity density profiles were computed by numerically solving the partial differential equations governing diffusion. Due to enormous computation times, a need for simpler models was felt.

2. The assumption of local charge neutrality was examined for the degenerate case. It was concluded that charge neutrality is an excellent assumption to describe impurity diffusion under typical conditions. It was found that due to considerable simplication of the problem fora constant source diffusion, the resulting equations can be solved very efficiently on a digital computer. Whereas it takes hours of CPU time for the quasi-static model, it only takes a few minutes for the charge neutrality model.

3. The results from Hu's diffusion theory [12] were applied to arsenic diffusion in silicon under general con­ditions. The effects of vacancies, electric field and partial ionization of impurities were included and Fermi- Dirac statistics were used to describe the carriers. Quasi­equilibrium for vacancies and local charge neutrality were assumed. It was shown that using a transformation of variables, the problem can be simplified considerably fora constant source diffusion. The numerical solution can thus be obtained very efficiently. The results show very

104

significant enhancement of diffusion at high surface con­centrations when compared to those obtained using the charge neutrality model. At such concentrations, the partial ionization was found to have a very large effect on the density values. The use of classical statistics was found to yield an underestimate of the impurity density values as in the case of quasi-static model.

It was noted that the impurity flux equation in thevacancy-aided model has the same form as that in the

* -1quasi-static model provided that DCYV " is used in place *

of Dc. For low concentrations, Yv^l* However, the equa­tion numerically solved in the model, strictly speaking, is not valid at low concentrations due to an assumption made to simplify the problem at high concentrations. The charge neutrality model, nevertheless, is valid and can be used in such a case.

4. The results obtained using the vacancy-aided model were compared to the experimental data. It was found that using certain suggested values of the intrinsic diffusion coefficient and the energy levels, gives good agreement between the model and experiment. However, the results obtained strongly depend on these values.

Recommendations for further research are as follows.1. It was pointed out that the vacancy-aided model

suggested here may be in error at low concentrations because it was assumed that ? >> 1. This assumption was

105

made to simplify the model, and to avoid unnecessary computations at higher concentrations. The derivations can be modified by relaxing the above assumption so that the vacancy-elided model approaches the charge neutrality model at low concentrations.

2. Further research is needed to determine the correct values of the vacancy and donor levels, and the intrinsic diffusion coefficient of arsenic in silicon.

3. The vacancy-aided model should be re-examined in view of the cluster formation of As-atoms at high con­centrations.

4. The model in the present work was applied to the case of a constant source diffusion. It should also be analyzed for a drive-in diffusion.

5. The model should be re-examined for application to diffusion of phosphorus and boron in silicon [42-46].

6. The problem of diffusion into a doped semi­conductor should be considered.

7. Due to gas-solid interaction at the surface, further examination may show the surface boundary condition to be dependent on the flux and concentration. In such a case, the assumption of a constant surface boundary condition is no longer valid and the problem must be solved in a

different way.In the research presented here an attempt has been

made to enhance the basic understanding of the diffusion

106

process. Such basic research may lead to improvement in device performance through optimization of device parameters.

APPENDIX A

NONDEGENERATE QUASI-STATIC FORMULATION

Under nondegenerate conditions and assuming 100% ionization, the equations governing the holes and electrons are given by

= 2_ [D |iL] + G (A. 1)91 3 x p 9 X V,p 9 x

LEL = <L_ [D 12. g ££-] + G . (A.2)91 3 x n g x VT gx

No assumptions are made regarding the form of the generation term G. According to the quasi-static approxi­mation, we set p = n = 0 in the above equations. Eliminating G then yields

2 _ ID i £ + J s J l s±] = 2 _ [D a n - i£] . (A.3)9 x p 9 x Vt 9Xj 9x n 9 x VT 3X

Integrating both sides from x to °° and noting that ,9 X— , and approach zero as x->-°°, we obtain9 x 3 x

d - d i £n 3x p 9Xf t ^ V T [ " T n T T - p ] * {A*4)0 n p

We note that

107

108

3 (pn) 3n , 3p _.-at— = p + n 35 ' <A-5)

so that

3n = 1 3 (pn) _ n a_£ ( .3X p 3X p 3X

or,

a£ = i Ll£El . E H . (A .7)3 x n 3 x n 3 x

Using (A.6) and (A.7), we can write (A.4) in two equivalent forms:

U l = V [ r- 9 (S.n.) - I ap ] (A 8)3X T p (D n + D p) 3X p 3 X \ • Ia k

= V f ^ _______ M .P.1?.)., ] (A 9)T 1 n 3X n (D n + D p) 3x J lA.y;n

Consider the form (A.9). If it is assumed that pn is a slowly varying function of x so that

, - °P 3 (Pn> | (A 10)3X I I D n + D p 3x I ' (A.±0)n p

we may directly integrate the resulting equation to obtain

109

n = ni exp(|r-) , (A. 11)

where it has been assumed that as x->°°, tp+0, and n-*n . Similarily, it can be seen from (A.8) that if

I >> I------2----- l(Pn> I 12)dx 1 1 D n + D p ax 1 'n p

we can integrate the equation to obtain

p = n. exp (- £-) . (A. 13)T

From (A.11) and (A.13), it follows that

pn = n? . (A.15)

Substituting (A.11) and (A.13) into Poisson's equation we obtain

2= — [2n. sinh(^r-) - c] . (A. 16)

3X e 1 T

Looking back at the inequalities (A.10) and (A.12), we note that at typical diffusion temperatures,

. J L . . i<r19 ,D n + D p n+p n p ^ ^

110

and the inequalities are justified if pn is a slowly varying function of x. It is not necessary to assume that pn is a constant although it is evident from (A. 8) and (A.9) that it forms a self-consistent solution.

It may also be noted that if instead of the in­equalities, we assume n >> p, so that Dn n >> p, we still obtain (A.11) and (A.13). However, at typical diffusion temperatures the inequality n » p is violated over some range of x and this constitutes a higher level of approximation than is really needed.

APPENDIX BCOMPUTATION OF THE FERMI INTEGRALS

The Fermi integrals or Fermi functions arise in the carrier density expressions when a parabolic density of states is assumed. The Fermi integral of order j is defined as

Fj(n) " r c W f exp(xW)Vl • W -10

where r(n) is the Gamma function with the properties

r (n) = (n-1) r (n-1) , n >_ 1 r(l) = lr(1/2) = /7 (b .2)

Gamma function with negative arguments can be avoided by using [47]

r(z) = sin(irz) r (1-z) * (B,3)

In the present work, the functions f i/2 ^ ' F- l / 2 ^ ' F_3/2(n) and F_5/ 2 ^ are required. Some of these are tabulated [48-50] . An excellent discussion on approxi­mations of the Fermi functions can be found in [4 7] . How­ever, these approximations, though useful in analytical

work, are not accurate enough for the present work over

111

112

the entire range of the argument. Brient and Wilson [51] have made accurate computations of these functions by directly integrating the expressions numerically. For the functions F^Cn), j = 1/2 and -1/2, the integrals are obtained in sections using Simpson's rule with 32 points per segment with the lower limit equal to e=d, where d is chosen so that

dx a 1 d^+1 ,0-5 4)exp(j-n)+l l+exp(-n) (j+1) —

0

Each succeeding section is taken as a region equal inlength to all previous regions combined, i.e., d to 2d,2d to 4d etc., until the upper limit of 2nd for n regionsis reached such that 2nd > n+70 for j=i/2, and 2nd > n+50for j = -1/2. The resultant sum of sectional integrations

7is then accurate to five parts in 10 .The derivatives of Fermi integrals are given by

dF. (n)?— = F . . (n) (B. 5)dn j-1

Writing the integrals explicitly,

F („> = 2 - f /S= ,dx- t < B .6 a )1/2 j- J exp(x-n)+1/1T 0

1/2 /ti

“1/2 ,x dx (B.6b)exp(x-n)+1

0

113

f ( \ - "I f°° x dx -3/2 n J exp(x-n)+l

* 0(B.6c)

and

- 5/2(B.6d)

Although the algorithm given by Brient and Wilson is very accurate, it is computationally very expensive to use in an iteration procedure.

Battocletti [52] has proposed a series of polynomial approximations for the Fermi integrals of order 1/2. For arguments less than zero and for large positive arguments analytic expressions can be used [47, 48]. In the range of arguments from -1 to 12, Battocletti"s proposed approxi-

gmations yield an rms relative error of less than 1 in 10 for the function F^y2(x). The polynomials are easily differentiated. Battocletti1s algorithm is as follows.

1. For x < -12.5:

F1/2(x) = exp(x) F _ l / 2 ( x ) = e x p ( x )

F_ 3/ 2 ( x ) = e x p ( x )

F_5/,2(x ) = exp (x) .

114

2. For -12.5 <_ x -2.0:

F . /9(x ) = E (use six terms)1// r=l r

F 1/9(x) = E ---- exPt£.x.) (Use six terms)~1/2 r=l /r

r+1 /-F _,_(x) = E (-1) /r exp(rx) (use eight terms)r=lCO

F c/^(x) = E (-1) r/r exp(rx) (use eight terms)_5/2 r=l

3. For -2.0 < x < 0.0:

6F /9(x) = exp(x) + E a exp[(n+2)x] L/z n=0 n

F n/9(x) = exp(x) + E a (n+2) exp[(n+2)x] ~ L/Z n=0 n

6 2F 0/0(x) = exp (x) + E a_. (n+2) exp [ (n+2) x]n=0

6 3F c/0(x) = exp (x) + E a (n+2) exp [ (n+2) x]~5/2 n=0 n

where

aQ = -0.35353667 a± = 0.19210895 a2 = -0.12236525

115

a3 = 0.78991718x10-1 a4 = -0.43441957X10”1 a5 = 0.16320377 xlO-1 ag = -0.29297496xl0“2

4. For 0.0 < x _< 3.0:

7Pl/2 <x> - bn x

!■ (X, = I b nx'1- 1'' n=l

F 3/2U> - I >n3/2 n=2 n

F e-z-U) = E b n(n-l) (n-2)x(n“3)n=3 n

where

bo 0.76514805

bl = 0.60491025

b2 = 0.18990505

b3 = 0.20131714X10"1

b4 = -0. 39688853x10*”2

b5 = -0.76558999x10”3

b6 = 0.30807837xl0"3

b7 = -0.28395599xl0"4

116

5. For 3.0 < x < 6.0:'

Fw ,(x) = 0.75225278 x3/2 z 1/2 n 2nn=0 x

F_ 7 (x) = 0.75225278 x1/2 E (| - 2n) % - 1/2 n=0 2 x2n

F_3/2 (x ) = 0.75225278 x"1/2 E (| - 2n)(j - 2n)cn=0 ‘ n

1x 2n x

5F c/otx) = 0.75225278 x"3/2 E (| - 2n)(2n - h

' n=0 2 2

“ (2n + 7>°n I n

where

o o II 1.0002404

C1 = 1.1981922

C2 = 2.9838285

C3 = -24.984495

c4 = 96.877279

C5 = -156.23522 .

For 6.0 < x <_ 12:

The equations are the same as in 5 above except that

117

4the summation is 0 through 4 i.e. £ , and

n=0

cQ = 1.0000313 c1 = 1.2248008 c2 = 1.8177898 c3 = -5.0849956 c4 = -3.2409718 .

7. For x > 12.0:

The equations are the same as in 5 above with

co = 1.0

C1 = 1.2337005

C2 = 1.0654119

C3 = 9.7015185

C4 = 242.71502

C5 = 11865.691

In the actual program, the computations have been done more efficiently, e.g.,

118

The second form above is more efficient than the first one, since at the expense of one more multiplication, seven less computations for raising e to a power are made. This subroutine is called millions of times and even a small saving here will reflect in the overall computation time.

Computed values using Battocletti's algorithm were compared with Brient and Wilson's scheme and excellent agreement was found.

APPENDIX CPROGRAM INFORMATION

The computer programs used to obtain the results reported in Chapter 5 are written in FORTRAN-IV and were run on an IBM OS/360 computer system. Typical CPU time required to run the quasi-static program with diffusion time of 30 minutes and starting condition at 2 minutes is about 2 hours. This, of course, depends on the values of Ax and At used in the computation. Typical values are Ax = O.OOly and At = 0.5s. The CPU times required for the charge neutrality and the vacancy-aided models are of the order of few minutes.

A brief description of the "input data" is given below. Wherever necessary, comments have been added in the program to make them self-explanatory. The source deck listing of the various programs follows the input data given below.

Quasi-static ProgramCPUST CPU segmentation time, in minutes

CXEO Surface concentration, -3in cm

TEMP Temperature, in °CDSUBC Diffusion constant, in 2 -1 cm sec

TO Starting time, in secZ = 1 for donors

= -1 for acceptors119

120

TSTOP Diffusion time, in secCKC Convergence factor (accuracy check)DXl Distance increment, in micronDTI Time increment, in secCSTOP Value of concentration below which all the

-3values are equated to zero, m cmCCHK Value of concentration below which the

-3accuracy check is not applied, in cm

Charge Neutrality Program-3CXEO Surface concentration, m cm

TEMP Temperature, in °C2 -1DSUBC Diffusion constant, in cm sec

TO Diffusion time, in secDXl Distance increment at which the solution is

desired, in micron CKC Convergence factorZ = 1 for donors

= -1 for acceptors ALPHA Information to be printed out on the output

Vacancy-aided Program-3CXEO Surface concentration, m cm

TEMP Temperature, in °CTO Diffusion time, in secDXl Distance increment at which the solution is

121

CKCALPHA

desired, in micron Convergence factorInformation to be printed out on the output

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123

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6 ,^. ^ V • y/ I v J V«u * r\ ,*k '■ «* t, r.

IF (Z.Eq I i .BO) PRINT 630 66601050IF]z.EQ.-1.DD) PRINT 631 00C01060PRI NT 602,CXEC 00001070PRINT 619,IRE5ET,KRESEI CCC0108CIF (ITRUNC.EQ.1) PRINT 618 00001090IFjlTRUNC.EQ.O) PRINT 619 0C0611C0PRINT 6C3,NP 00001110ERINT 613 00001120PRINT 6C1,TEMP,NSUEI,ESUEC,BETA,XL,DX1,DT1 0000113CPRINT 620,EK,ECEfiGE,EPSLCN 0 0 0 0 1 1 UGPRINT 621.CKC,CSTOP,CCHK,TO,TSTOF,DTPR N T ,D T P U N ,DTTA P 00001150READ 5,62-) ALEHA CC0C1160PRINT 626, ALPHA 00C01170FRINT 213 0 * rC1180

0 0 0 O 1 190600 FORMAT(//5X,'QUASI-STATIC ANALYSIS PROGRAM FOR FIELD-AIDED DIFFUSI00C01200 10N'////) r _ 0C001210601 FORMAT (//5X,'PHYSICAL C C N D 1 T I O N S - •/10X ,' TEMP = '.F13.1,' DEG C / G C Q C 1 2 2 G 1 10X « NSUBI = ',1PE13.9,» PER C C ' / 1 0 X , » DSOBC = ». 1PD13.9 * CM*C000012302 M / S E C V 1 0 X , ' EETA = ',0PF1 3.2/10X,'LENGTH = • , F13.5.• MICRONS'/ 00001290 3 10X ' DX = ' ,F13.5,' MICRCNS'/10X,» DT = »,F13.3,' SEC*) 00001250603 FORMAT (5X,'TOTAL NUMBER CF FCINIS = *.19) 00011260

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SUBROUTINE MA(A,E,( REAL*8 A,B.C DIMENSION A(U.E(1) IF INS) 10,20,30 10 CONTINUE DC 15 1=1,N15 s m s s * 1*— *1*20 CONTINUE DO 25 1=1,N 25 C{I)=-A(I)RETURN 30 CONTINUE DO 35 1=1,N 35 C (I) =A (I) *B (I) RETURN

END

, N,NS) .C (1)

0000500000005010GG00502C0C00503C0000504000005050C00050600000507000005080C00C509000005100000051100000512000005130000051*100000515000005160

SUBROUTINE MINV2 (A) RE AL*8 A,B,DfDI DIKENSION A{1) ,E(4) B(1)=A(4)E 12/ =** A (2)E < 2) = -A (3)B (U) =A (1)

DO 20 J=1-4 *1)31 = 8 (J)*DI BETDBN END

00005170 00005180 00005190 00005200 00C05210 00005220 00005230 00005240 00C0525C 00005260 00005270 00005280 00005290

SUBROUTINE BM2 (A,B,C,N) REAL*8 A,B.C DIMENSION A (Cc

NSION A ( 1) # E {1) «C(1) = A (1) *B (1 * A l3) *E (2)if (hTeo. i*8]oloAr^)*e I2!

RETURNEND

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n THIS PROGRAM COMPUTES THE RESULT CF THE TRANSPORT PROBLEM UNDER LOCAL CHARGE NEUTRALITY CONDITIONS USING QUA SI-LINEARIZATICN. ********* DFGENtFATE CASE *********

IMPLICIT REAL*8 REALMS N.NSUBI E£AL*4 ALPHA

(A-H,0-Z)

DIMENSION C (2000) . DC (2GC :•) , DDC (2G0v) ,Y (2000) DIMENSION DLTACj2J00) , ALPHA (36) fX (2000) DIMENSION Y 1 (2C 00) ,WCRK(2900) ,A R G (10) ,V A L (1C ) ,NMAX = 2 j'-u

READ INPUT LATAREAD (5,952)CXEC/lEMP,DSUECrTO READ (5,952) EX 1,CKC,CSTCP,2 READ (5,625) ALPHA

COMPUTE VARIOUS QUANTITIES.BK=1.38 ID-23ECHRGE= 1. 602D-19VNCEM=3K*(TEMP+273.15DU)/ECHRGENSUBI = 7 . 766E15*DEXP (5.52 8269D-3 * fTEMP + 27 3 . 15D0)) ETAI=-6.989559165D3/(TEMP*273.15D0)+1.341932688D CALL EEBM1 (£TAI,FETAI,1)BETA=CXEC/NSUEI

C1 (2000)

0

CONC.=•,1PD13.4/

WRITE CUT THE VARIOUS QUANTITIES.WRITE (6fI'v) 1EME.CXEC, DSUEC

1CG FCFHAT(IX,'TFMF='.F13.1 / 1X , 'SURFACE 1 1X ,1 DSUBC= *rlPD13.i|)WR H E (6 .1C 1) IC.DX1

1)1 FORMAT(IX,•T IM H = • F 13.3/1X ,1DX1= *,F13.5) WRITE (6,1j2) EFTA,NSUEI

1)2 FORMAT(1X,'BETA= » , F 13 . 5/1 X , » N SIJBI= *,1PD13.4)THF. PAR A MEIER D Y 1 CORRESPONDS TO NORMALIZED DX 1.

LY1 = D X 1 / (2.Dl*DSQRI (DSUEC*TO))* 1.D-4 DY=0.01 DOCSICPN=CS10P/'NSUEI V3T0P=1.C-31 VSICFN=VSTCP/VNCFM CCHK=VSICPN*1.DOS

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Y (1) =f . DOT H £ INITIAL GUESS AND Y APF DETERMINED

12DU 12 1=2,NMAX Y(I) = Y (1-1) + EY DC 1 1=2 NMAXIF (CXEO. LE*. 1 .D2C ) CE2=!.5 DC *BETA*DERFC (j. 7G7D0*Y (I) )IF (CXSC.GI. 1 .D20)CC2=C.5EG*BETA*DERFC (G. 500D0*Y<I) )C(I) = CD2 + EI,0G ( (Cl 2+ESQRI{CD2**2+1 .DO) ) +DEXF (-CD2) )IF <C (I) . LI . VSIOPN) GC TO 21 CONTINUE

C(I) HAS ALSO BEEN USED TO REPRESENT THE POTENTIAL.2 IS 'I C P = I

FIRSTI=ISIOP-10 HRITE(6.9S5) ISTCE LASTI = ISTCP«-1CIF (LASTI.GT.NMAX) LASTI = NMAX LS1IH1=LASII-1 DC 9 I=ISTOp,NMAX

9 C(I) =C.DOCCMPUIF VEETA AT THE SURFACE ASSUMING NON-DEG. CCNDITICN.

VBEIA=EETA*C.5DCVBETA=VBETA + CLOG ( (V E ET A + DSQR T ( V BET A** 2+1 .DO) ) *D2XP (-VBETA) )

COMPUTE THE TRUE BOUNDARY CONDITION FOR DLG. CONDITIONS.300 CONTINUE

F1=VEETA+ETAI CALL FERMI {F 1,F2, 1)CALL FERMIJF 1 ,F3,2)FU=F2/FETAI-DEXP (-VEE I A)-BETA F5 = F3/FFTAI+DEXP (-VEETA)VB ETA1 = VBETA VBEIA=VBZIA-F4/FFIF (DABS (JVBETA1-VEETA) / VBETA) .LI. 1.D-3) GO TO 301

3'. 1 C (1)=VEETA VFETA1=VEETA*VNOFMHP11E (6,310) V BET A 1

310 F C R M A T ( 1X,»FIC= ",D15.7)

00000 440 c € Q C C 4 6 0 00000 470 C0C0C490 00000500 OC000510 00000520 000 00 530 COLO0540 00C0C542 00000543 0G0GC 550 00000 560 00 C 00 56 1 CCO00 57C 00000560 00000590 00COG 600 OC000610 000CO 620 000OC630 COGCC 640 0C000 641 00000642 00COO643 CO 0 0 0 6 4 4000006450000064600000647 00000649 00C 00 650 00000653 COCOO 654 C C 0 0 C 6 5 5 00000656 C0G00 6 57 CCG00658 00000659 0000066C C00C0 66 1 00000662 00000663 QG0A0664 0 0 G 0 0 6 6 6 00000667 COCOG 66 96700000'

non

non

noon

non

onnono BEGINNING Or ITERATION ICCE

5 K=K+1CCMFUTF DERIVATIVES AND DELTA C

CALL LFSDE8 (C,EC ,EEC,DY,LA3II)CALL CELIA(C ,E C ,tD C ,ELTAC,Y,D Y fLASTI,FTAI,FETAI)

CHECK CONVERGENCE.JF = 0DO 4 I=2,LSTIM1 C (I) =C (I) + ELTAC (I)CABS = DABS (C (I))ERROB=DABS (DLTAC(I)/C (I))IF (C (I) .LT. V3TGPN) GC TC 7 IF (ERROR.GT.CKC.AND.CAES.GT.CCHK)JF=1

4 CONTINUE 7 ISTOP=I

LASTI=ISICP+ 10 IFJLASTI.GT.NMAX)LASTI=NMAX L S U M 1 = LASTI-1 DO 14 I=ISTCE,NMAX

14 C (I) = 0.D0IF (JF. EC. 1) GOTO 5

END OF THE ITERATION LCCF.WRITE (6 , 222)

222 FORMAT (1 X , 'CUT OF THE MAIN LOOP*)FOLLOWING EC LCCF MAKES A MESH WIIH SEPEPATION D Y 1.

DO 20 1=1,NMAX20 Y 1 (I)=DY1*(1-1)

INTERPOLATE THE CCFBE5ECNDING VALUES.N D IM = 8DO 21 1 = 1, NMAXCALL EATSG (x 1 (I) ,Y,C,WCRK, 1000, 1 ,ARG,VAL-NDIM) CALL DACFI(Y1 (I) , AEG , V AL , C 1 (I) , NDIM , CKC, IER)IF (C1 (I) .LI. VS1CFN) GC TO 30

21 CONTINUE

cor0068000 0 0 06 90 00G007CC CO000 7 10 00000720 0000C 730 00000731 00000 740 00000 750 OCCOO 760 00000 770 09CCC760 00000 790 OC000800 00C0C81C 00000820 0000083? 00000840 00000850 00000860 00C0G87C 000 00 8 80 0000089C OCCG0900000 0091C COCOO 920 OOC00930 OCC 00 950 0C00C960 C0000 97C 0000098C 0CCC1001 00OC100 2n 'i * 1 10000109000001091 00 001100 0000 1 110 00001111 0000 111200001 113 00001 130coco 1 m eOCC01150 0000116" CC OC118C 00001220

H*K>

3C IS 1C F =1LASTI = ISiCF + 1r.IF (LASTI.GX.NMAX)LASTI=NMAX DO .31 L=IS1CF / L A S H

31 c i ( i ) = : . c oDO Zl 1= 1 - LASTI

.c. “ g p E i e . j t H Y j i u c i n . j i i i l . c i j i i t .-j i. v B li A i. { iAfi'9#5|fc.Xj^1j»6*^X|-9#5»t./fD15«o22 C (I) = C i (I)(I) =

DC 1 1(I)

1=1, I. AST

o)

F1=C <I) +11 AICALL FERMI (F 1,P2,1iDC (I) = F2/FE1AI-D£X'P I-_ (I) =F2/FElAl-D£XP (-C (I) )

11 CONTINUE KS1 = CCNCRM=NSUEIVNORM=EK* (TEMP + 273. 15DC)/ECHRGE WRITE (6,9/0)

97 2 FO FM AT (5X//1 X, 80 (1H*)/5X,'X (I) * ,12X, *C (X) • , 1oX, 1 V (X) '/DOXzC0 1=1 ,LASTI EEC (I) =DC (x) +CNOFM A2=Z*C (I) *ViNCRM X (I) = DX1*(I- 1)

' WRITE (6 , 9o<:) X (I) , LDC (I) ,A2 96C FORMAT! (1X, F9 . 5, 1F2D15. 6)200 CONTINUE

WEIIEJ6,961)CCCC

961 FCRMAI('I')WRITE RESULTS CM FILE 8OUTPUT FORMAT IS SUITABTE FOR TRUNCATION ST2P IN QS1.

WRITE (6,951) KS1,T0,KST#ISTCP,LASTI

8880/ 60

DO 8 8 8 I=1,LAS1I C (I) = -Z*C (I) *VNCBM DX 1 = 13X1*1. E — 4CALL DFSDER (C,DC ,WORK,EX1,LASTI) EX 1 = EX1*1.DUW P I I E (7,90 C)LA ST I,THIN,CX1 FORMAT <14,F 1C.2,1 PC20. 10)WRITE (7,90 1) (DDC (I) ,1=1 ,IASTI)

00001230 C 0 C 0 1240 00001250 00001260 0CC01270 00001280 000 01290 O C 00 1 JOC 0GC01310 00001390 00^01350 0 0 C 0 1 3 51 00G01352 00001353 OO00137C- 00001380 00001920 00001930 00001990 0 0 0 C 1 950 OCCO 1960 0 3 0 C 1 470 C O C 0 1480 00001490 0CCC15CC 00C0151C 00031520 OCCO 1530 00001540 OOC’O 1 550 00031551 000C156C C00C157C 30001571 00001580 0000159C 00C0160C 000C161C 00001620 O C C O 1630 00C:164C 300016 50 00GC1660 00001691 0 0 0 0 17CC 0CCC171C 00031711

t—■ u>

WBITZ (7,90 1) (DC (I) ,1 = 1,L A S H ) C00G172CSEITE 6. 9 l 1 p C I ,1 = 1 ,L ASTI) CCGC173C901 FO EM AT ( Ffi E1 0 . 3) 0 0 0 M 7 4 CSTCE COCO 1 750

0000176^995 FOFMATf 1X,«NUHEIK CB FCINTS= ’ ,15) 0^091770950 FOBMAT (2 (1X.E25. 18)) 000017809 51 r G EH A i (1 X . 1 1 1 , 1 X ,D25 .1 8 , 1X ,111 , 1 X , 1 12 , 1X , 17) GOT C 179C952 FCBMAT (4C 1v .5) 0CCC18CC953 EG B M AT (911') 0CC01810625 FGRMA1 (18 A V 1 8 A 4 ) 0C00182C

C 0 C 0 1 830

ccr*

ccc

ccc

SOEEOUTINL C i L I A ( C - E C ,EEC-D L 1 A C ,Y ,D Y ,N P , E T A I ,FETAI)THIS SUEECUTINE FOB MS THE DILlAc* 5 USING IHZ QUA5I-

LINEAEIZAIION KFTHCDIMFLICIT EE A L * 8 (A-H,0-Z)PaAL * 8 INVDIKE NSICN C(1) , DC (1) , EEC (1) , DLTAC (1) , ¥(1)D Y BI = 0 . 5 D 1' / D Y DYSI = 1.DCv (D ¥*D Y)DC <1)=j.D0 EDC (1) = v . DC

DETERMINE INITIAL QUANTITIESEC 10 1=2,NF V1 = C (I) + ET AI CALL FERMI (V1,F1,1'CAII FEEMI (V1,F2,2 CALL FSEMI (V 1,F3,3 CALL FEEMI (V1,F4,4 EE =DEXP (-C (I))F5=F2+FETAI*DE F 6 = 1. C C ✓ (F5*F5)F7 = F 3-F ETAI*EE

COMPUTE A, B, C, AND DA=(F2 + F1)/F5B=2.DC*Y (I) + 2. DC ♦DC (I)* (F3+F2)/F5 CC=CDC (I) *F6* <F5* (FS+F2) - (F2 + F1) *F'

CCC

CCC

1 ( F 3 * F 2 ) * t 7 ) ..........................'7) +DC (I) *DC (I) * F6* (?5* (F4 + F3)D=-2.DU*Y (I) *DC (I) - A*EDC (I) - (F3 + F2)*DC (I)*DC (I)/F5AM=A*EYSIBM=B*EYDI

CCKFUTE AL, EE, AND GAA L = A M + E K EE=-AM-AM+CC GA =A M-B M

COMFUTE G AND HS=1. 03/ (BE-GA*DDC (1-1) ) DDC(I)= AI* SEC (I) = ( E-G A*EC (1-1) ) *S

0 0 CO 1840 COC■ 01850 000 0186C 0 0 0 0 137C 00GG188G 00C0189G 0C001900 o r e 01910 C0G0192C 00CG1930 0CC01940 00001950 000 01960 00001970 00001980 000G1981 0C001982 00001983 C C 001984 00001985 0GCG1986000019870000198800001989 00002060 00002070 000G2C8G 00002C 90 00002100 00002110 OOC02111 0DQ0212C 00002150 0000 2160 00002170 000G2180 00002190 GOCC220C 00002210 00002220 0CC0223C 00002240 CC0022 50 00002260 0 C00227C 000 02280 0DC 02290

z'-

c c UTI NUEOT E D ELICL TAC (NPDO 2 J 1 =II = N P-1 +CL TAC (IICO Nil NTJFEE III BNEN C

00C02300 0C0C231C 0000232C OCCO 2 330 C000234Q 00002350 000C-236C 00C02370 0000238C 00002390 00002400 OOOQ241C

SUEriCUTINE DFSDEP (Y. C ¥ , EDY,D X ,NP) 'HIS SUBROUTINE NUMERICALLY COMPUTES TH FIRST AND

SECOND DERIVATIVES BY FITTING A POLYNOMIAL CVEF, 5 POINTS

IMPLICIT REAL * 8 (A-F,C-Z)DIMENSION Y (1) ,DY (1) ,DCY (1)KPM1 = NP- 1 NFM2=NF-2 NPM3=NF-3 NPM4=NF-4C 1 = 1. C O / (12.EC *DX)C2=1.D'V(12.EC*DX*DX)

DETERMINING FOR POINTS 3 TO N-2CO 3r*■“*. I = < NFM2DY (l[=C 1* (1-2) - 8 . DO* (Y (1-1) -Y (1+1) ) - Y (1 + 2) )DDY (I) =C2* (-Y (1-2) +16. DO* (Y (1-1) + Y (1 + 1) ) — 30. DO(I) =C2* (-Y (1-2) +16. DO* (Y (I-1)+Y (1 + 1) )-30. DO*Y (I) -Y (1 + 2) )

ERMINING FOR THE ECINTS 1. 2, N-1, AND N DDY (1) = ]35.Dv*Y (1) -1C4. D)*Y (2) +1 14. DC*Y (3) -56. DO*Y (4)1 +11.DC*Y(5))*02DDY (2) = (11.D'*Y (1) - 20. EO*Y (2) + 6.D0*Y (3) +4. DC *Y (4) -Y (5) ) *C2 DDY(NPM1) = (-Y(NPM4)+4.DO*Y(NPM3)+6.DO*Y(NPM2)-20.DO*Y (NPM1) 1 +11.D.*Y(NP))*C2DDY (NP) = (11. CO *Y (NFM4) -56 . Du*Y (NPM 3) + 114.DC*Y (NPM2)" " .. ... 5.C)*Y (NP))*02DO*Y (2)-36.DQ*Y (3) +16. DO*Y (4)(NFM1)+35.C)*Y (NP))*02= (-25. Ct*Y (1) +48.!------- ----3. CO*Y (5) ) *01

1 ’ - 1 C 4.D G *YDY (1)=(“ 25. EC*DY (2) = (-3. DC*Y'('1) -10. DO*Y (2)+18. DO *Y (3) -6 . DO *Y (4) + Y (5) ) *01 DY{NPMt) = (-Y(NPM4) +6 . DD*Y (NPM3) -18 . DO* Y (NPM2) + 1C . DC *Y (NPM1)

1 +3. DO*Y (NP) ) *01DY (NP) = (3 . DO*Y (NEM4) -16. DO*Y (NPM3) +36. DO*Y (NPM2)

1 -48.D',*Y (NPM1) + 25 . D0* Y (NP) )*C1RETURNENC

00002420 d0C0*4 3C C0CC244G CC00245G 00GG2460 00002470 G0C0248C 00002490 C0C025CG C000251G GC00252C 0000 2 530 CC002540 00002550 C0C0256C C0GO257G 000C258C 000025SO 00CC2600 0000261C 000 C262C 000026 30 00002640 00002650 0 0 0 G 2 6 6 0 uCC02670 0 C C 0 2 6 8 0 0CG0269C 00002700 00002710 CCC0272C 30002730 CCC02740 0CC0 275C C0002760 00002770 00002780

SUBROUTINE FERMI<X,FHALFX,IXX) THIS SUBROUTINE IS BASED CN EATTOCIET IMPLICIT F£AL*8 (A-H,0-Z)DIMENSION C ( 1C)IF(X.L1.-12.5D0.CB.X.GT.-2.D0)GO IF(IXX.EQ.2)GO 10 20IF (IXX. EC. 3) GO TC 18C IF (IXX.EC.MGC TC 1 SC SUK=0.CO A=1.CO DC 3C 1 = 1 r 6 E=I*1.DOCC=A*DEXF{B*X)/(B*DSORT(E))SUM=SUM+CCIF (DABS (CC) .LT. 1.D-3C) GO TO 4C A=A* (-1. DO)3o CONTINUE UO FHALFX=SUM RETURN 20 SUM=C.DC A=1.DC DO 5 0 1=1,8 B = I*1.DOCC=A*E£XP (3*X)/ESQRT (B)S U M = SU M + CCIF (DABS (CC) .IT.1.D-3C)GO TC 60 A = A* (-1.DC)5‘ CONTINUE 6' FHALFX=SUM BE IUFN 18 0 SUM=0.D0a= 1.d:DC 20A 1=1,10 B=I*1.DjCC =A*DEXP(B* X)*DSQST (B)SUN=SUM+CCIF (DABS (CC) .IT. 1.D-3C) GC TO 210 A = A* (- 1 .DO)200 CONTINUE 2 1 C FHALFX=SUM RETURN 190 SUM=0.D0 A=1.DCDO 220 1=1,12I* 1 £CC=A*DtXP(B*X) +B*DSC6T(E) SUK=SUM+CC

QC005680 0 0 u v. 5 6 9 v, 000057CC 000 35710 0C0G572C 00C0573C 00005731 OC0057 32 00 0 3 5740 00005750 OCO0 576C 0 0 C 0 5 7 7 C 00C05780 00005780 00005800 00005810 00005820 00005830 CCGC5840 00005850 00005860 0CCC5870 000C58 80 00005890 0C00590C 0000591C 0CC0592G 00005930 noo05940 OC0059500000595100005952 3CCC5953 0000595a CC005955 03005956 OOC 05957 0 0 C C 5 5 5 6 00C05959 C0005960 CCGC5S61 0CCG59620000596300005964 0GC05965 C C 0 0 5 9 6 6 00005967

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FHALFX=TEME* (0.75D0*C( 1) +SUH)EE1U nN

29c TEM£=0.75225278DC/(X*DSQBT(X) )SUM=O.DC BYX2=1.DU/(X*X)DC 3 1C 1=1,N

310 SUM=3YX2*{SOH+C(N-I + 2)* (1.5DC-2.DO* (N- 1 (N-I + 1)-U.5DC)*(2 .DC* (N-I + 1) +C.5D0) ) FHALFX=TEMF* (-C. *75tG*C (1) +SUM)BETUENEND

1 + 1) )* (2.DO*

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WRITE (6,600)FCFMAT (13 1)WRITE (6 ,8 :- 1) CXEC,IEME,1C WEIIE (6 , 802 NSl’BI , XNC ,XNV - CSUBC WRITE <6,80 3) ES C EG, VS DBG, XISTAR:

4/1X ,' TEMF (DEG C)=' ,

y

801 FORMAT (//1X.'SURFACE CCNC. (/CC .)=',1PD13.11PD13. 4/1X, *DIFF. TIME (SEC.) = • 1 PC1 3. 4)

8 ' 2 FORMAT (1X,?NSUEI (/CC) = • , 1 FD 1 3 . 0/1 X , • NC {/CC .) = • - 1 FD 1 3. H 11 X , * N V i/CC.) = ', 1FD13.4/1X,'DSUBC (CM**2/SEC)=• 1FD13.4)

■3 0 3 FORMAT (IX,'BAND GAF (E V) = • , F 1 3, 5/1 X , • VSUBO= ' , 1 PD 1 3. 4/ 11X,'XISTAR=*, 1FC13.4)

COMPUTE D X1 CCFRESFCNCTNG TC THE REQUIRED DIFFUSION TIME.DY1 = DX1/(2.D3*DSCBT (DSUBC + TO))*1.D-4 DY=C.C1DGIF (EA.LT.-1. CO . ANE.CXEO.GT. 1. C 20) D Y = 0 . 0 5DC'VSTCFN=1.D-15 VCHK=VSICFN*1.DC5

COMPUTE THE STARTING GUESS FCR V (I FCR THE IMPURITY DENSITY AND THEN CONDITION AND COMPLETE ICNIZATICN.

ASSUME ERFC STARTING CONDITION D V<I) ASSUMING NON-DEGENERATE

M= NMAXIF { EA.LE.-1.CO.ANE.CXEO.GT.1.D2C)M=15G0 Y (1) =C.DO DO 10 1=2,NMAX

IF (CXEG.L£.1.D2C)CC2 = C.IF (CXEC.G1.1. D2Q)CD2 = C. _ _IF (EA.LT.-1.CO.ANE.CXEO.GT.1 . D20) CD2=G. 5D(j*BETA V (I) =CD2*CLOG ( (CE2+ESQfiT(CD2*CE2+1.DG) )*DEXP (-CD2) ) IE (V (I) .LI. VSTCFN)GC TC 12 CONTINUE

CXEG.L£. 1.D2C)CC 2 = C .5 EC*BETA*EERFC(O.5DG*Y(I))F (CXEC.G1.1.D20)CD2=C.5E0*BETA*DERFC (0.1DQD0*Y(I))

1 112 ISTCP=I

FIRSTI=ISTCP-1C WRITt (6 , 8C4) ISTCF

8-'4 FORMAT (1X,'NC. OF PCINTS = ',I5)LASII=ISIGF+1C IF(1ASTI.GT.NMAX)LASTI=NMAX LS1IH1=LASTI-1 DO 13 I = IS'IOF,NMAX

13 V (I)=C.D0COMP UIF VEEIA AI THE SURFACE ASSUMING NON-DE’G. CONDITION AND COMPLETE

0CG0044C OC 0 ju450 0C00C460 COOOG47C 0CC0C48C OC000490 C09005GG CC0 C7 5 1 C 00000520 C0C00530 C C 0 0 0 5 4 0 0C00C55C 0C000551 OOCOO 560 C000057C 000Q056C 00000581 0000C590 O O C O C 6 C O 00C0061C CCC00611 C 0 0 0 C 612 0C0C06 13 C000062G 00000621 00000622 00000630 00000640 000 00650 CC000 66C 0C000 670 OOCOC'671 00000672 0 0 0 0 C 6 8 0 00G00690 00 000 7C0 0000071C CC0CC72G 0C000 7 3C 00000740 0007C 750 C0CGC76C 00C0C 770 00C0 078C 00070790 GCC0C791 C000C792

l—■ ui u>

non

non

non

no on

no no ICNIZATICK.

VBETA=C.5CC*EETAVBFTA=VBETA+ELCG{(VEETA+ESQHT(VBETA*VB£TA+1.DO))*D£XP (-VBETA))

CCMPUTE THE TRUE VALUE CF THE BOUNDARY CONDITION VBETA AT THE SURFACE BY ITERATION FCR DEGENERATE CONDITIONS AND INCOMPLETE IONIZATION.1<4

IS8 5

CONTINUE Q1=VEEIA+VSUE0 02=VBETA+VSUB0+EABAR CALL FERMI (Q1,F1,1)CALL FERMI (C 1, F2# 2)F3 = 2.D0*DExP(C2)Fa = DEXP (-VEE1A)F5= (1. d ; + F3) * {F 1/FETA I-P«) - BETA Ffc = F2/FFTAI + FU + F3* (F1 + F2) /FETA I VBETA1=VBETA VBFTA-VBETA-F5/F6IE (DABS ( (VBETA1-VBETA) /VBETA) . LT. 1. D-3) GO TOGO TO 1UV (1) = V3 ET AWRITE (fi, 3'5) VBETAFORMAT(1X,’VBFTA='#1PD15.5)

15

BEGIN ITFRATI ON LOOP.2'' K= K +1

COMPUTE DERIVATIVES O r V AND TH] CORRECTIONS DLT AV.CALL DFSDER( V ,BV,DDV.D Y ,IASTI)CALL DELTA (V ,DV,DDV,DLTAV,Y , D Y ,LAST!r VSTJBO,XNC,XNV,

1EAPAP,EGNM)CHECK CONVERGENCE.

D'l 21 I=2,LSTIM1 V (I) =V (I) +DLTAV (I)A3DL = DABS (V (I))FRPOF = DABS (DLT A V (I) /V (I) )

(V (I) . T."-. VSTOFN) GC TO 22 IF (ERROR.G T .CKC. AND. AD PL. GT. VCHK) -7^=1

C0000793 GOCOG794 000008CC 00000810 C0C00811 CC000812 00000813 C 0 G 0 0 8 14 00000820 00000830o c c o o 8 ac OOOOC05O 00C 00860 00000870 00000880 00000890 n o f n o g r ' - S r r Go g i . Df d c '^ 9 2 rr.f,r DO 9 70 f.A S 3 p (j r>A A -> r- r q r

OPCCC960 DC r r-r qnr0“0'C URO d ~ r r, r ogr r ' r D i D 0 0"> 0 P O 7 0 7 7 r A0 1 a a 2D A 0 A <| A r 30^001' 1C cr>pn 110 r C: o 1 12 r c d 1r- 2^ r f ■ 1 " 3 0nr>c " 1 '■ 40 r-r' 1 u 1 p n r. o 1 * u 2 srpo 1 7U3 0 ^ 1 ' 1 2 5 " « n r r 1 * fi 7 r o r r i " 7 rcA0A1,-‘' r r ' r ' 1' 9r "■''111 c

U1

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21 CONTINUEr o -t> r\ f> _LASTI = ISTrP+1-“IF (I ASTI. G7. NMAX) LASTI=NMAX LSTTM1=LASTI-1 DO 2 3 I=TSTOF,NMAX

2 3 V(T) = ■* . D "I17 {JF. ?Q. 1) SC TO 2’

END OF THE ITERATION T,0CP.W 9 TTF (6 ,8 ' 6 )

S'-6 F-0 F X AT (1 X , 'CUT OF THE MAIN LOOP*)FOFU A MESH COFHFSFCNDINO TO DY1.

DO 26 T=1,NM.AX 26 Y 1 (I) =DY 1* (1 - 1)

IN^FPPOL ATF THE CO P. nES PONDING V (T) VALUES.NDIM=8DO 27 1=1,NMAXCALL DATSG (Y1 (I) ,Y , V , WORK, NH AX . 1 . ABG . V AL, NDIM) CALL CACFI (Y1 (l[,ARG,VAL.C (I) ,NDIM ,CKC ,TER)IF (C(I) . LT. VSTOPN) GO 28

2.7 CONTINUE 2« ISTOP=I

LAS"’I=TSTOP + 1"IF (LASTI.GT. NMAX) L A S TI = N M A X DO 29 I = 1 STOP,LAST!

2° C (I) =" . D“DO 3 ‘ 1 = % LA STII'Tf.pr ic a. a u /tiFFTT" (*, 8, 9)Y(I) » V (I) , Y"1 (I) ,C (I)FORM A- M X , F9. 5,2X,D15.6, 2 X ^ 9 . 5,2X,D15.6) V(I)=C(I)

ELECTRIC FIELD.CALL DFSDEP (V, DV , D D V , DXCM . I, A STI)DO 32 1=1,LA STI

32 DDV (I) =-BKT*DV (I)C O vPUT? C(I) VALUES COFRFSPONDTNG TO V(I). PRINT AND PUNCH IN pappvp FOR'-] AT.

DO 31 1=1,LASTI

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non SUBROUTINE DFSDER (Y , DY , DDY ,DX . NP)

THIS SUBROUTINE NU H F EIC A t L Y COMPUTES THr FIRST AND SECOND P ER T V ATIVFS BY FITTING A POLYNOMIAL OVER c- 101 NTS

IMPLICIT F 5 A L * 8 (A-H.O-Z)DIMENSION Y (1) ,DY (1) ,DDY (1)N?*1=NP-1NPM2=NP-2NPM3=SP-3NPK4=NF-4C1 = 1. ri"/ (12. U'*DX)C2=1. D ‘ / (12. C‘ *DX*UX)

DETERMINING FOR POINTS 3 TO N-2DO 3°" T = 3,NPM 2DY (I) =C1* (Y (1-2) - 8 .PA* (Y (1-1)- Y (1+1)) - Y (1 + 2) )

? " DDY (I) -C2* (-Y (1-2) +16. Dl* (Y (1-1) + Y (1 + 1) ) -30.t)5*Y (I) -Y (1+2) )DETERMINING FCR THE POINTS 1- 2, N-1, AND N

DDY (1) = ( 35. D' *Y (1) -10 4.D**Y (2) +114.Dr*Y (3) -56.D.*Y (4)1 + 11 ,D'*Y (5))*C2DDY (21= 11.D;*Y (1) -2" . DO*Y (2)+6.D'*y (3) + 4. DA*Y (4)-Y(5) ) *C2 DDY (NPM 1) = (-Y (NPMU) +4.D“*Y (NPM 3) + 6.Dr * Y (NPM2)-25.D5*Y (NPK1) 1 +11.DA*Y(NP)) *C2DDY (NP) = (1 1 . D * Y (NPM4) - 5 6 . D>* * Y (N P M 3} + 114.D5*Y (NPM2)1 - H 4. DO *Y (NPM 1) +3 5. DO*Y (NP) ) *C2DY (1) = (-25. D'*Y(1) + 4 8 , D a *Y (2) - 36. DO* Y (3) +16. DO*Y (4)

1 - 3. D ' * Y (5) ) *01-i f M i t i « ! l k A ' o r *y (2) +18.D' *Y (3)-6.D1,v*Y (4) +Y(5) ) *C1

13) - 1 8 . DO *Y (NPM2) +Tr ,DA*Y (NPM1)DY (?) = (-3. D: *Y (1) -1 DY (NPM1) = (-Y (NPPI4) + 6 .D' *Y (NPM 3)

1 + 3 . D *Y (NP) ) *C 1DY (NP) = (3. DA *Y (NPM4) -16. D~*Y (NPM 3) + 36. DO* Y (NPM2)

1 - M . D * Y C ' ---- . . ~ r- " ---- ' * • -

c *Y (NPM*!) +25.D"*Y (NP) ) *C1FSTHFNEND

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163

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VITA

Rituparna Shrivastava was born in Rajnandgaon, India on January 2, 1951. He received his Bachelor of Science degree from Jabalpur University in 1968, and Bachelor's and Master's degrees in Electrical Communication Engineer­ing from Indian Institute of Science, Bangalore, in 1971 and 1973, respectively. Since September 1973, he has been pursuing further graduate studies at Louisiana State University, Baton Rouge, where he has worked as a graduate assistant and as an instructor in the Electrical Engineer­ing Department. He has been a recipient of the Government of India National Schlorship (1965-6 8), Tata Trust Scholarship (1968-71) and Indian Institute of Science Scholarship (1971-73). He is a member of Eta Kappa Nu and Sigma Pi Sigma honor societies and is a student member of Institute of Electrical and Electronics Engineers.

He is presently a candidate for the degree of Doctor of Philosophy in Electrical Engineering.

171

Candidate:

Major Field:

Title of Thesis:

EXAMINATION AND THESIS REPORT

R itu p a rn a S h riv a s ta v a

E le c t r ic a l E n g in ee rin g

D if fu s io n o f A rs en ic in D egenerate S i l ic o n : A Q u a s i-s ta t ic Approach

Approved:

ifessdr and ChairmanM ajor

Dean of the GraduateySehool

E X A M IN IN G C O M M ITTE E :

Date of Examination:

December 1 , 1977