On models of phosphorus diffusion in siliconS. M. Hu, P. Fahey, and R. W. Dutton Citation: J. Appl. Phys. 54, 6912 (1983); doi: 10.1063/1.331998 View online: http://dx.doi.org/10.1063/1.331998 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v54/i12 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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On models of phosphorus diffusion in silicon s. M. HU,a) P. Fahey, and R. W. Dutton Stanford Electronics Laboratories, Stanford University, Stanford, California 94305

(Received 14 April 1983; accepted for publication 22 July 1983)

Various phen?~ena ass~ciated with phosphorus diffusion in silicon are reviewed and prominent ~odel~ are cn~lqued. It IS shown that these models are either fundamentally unsound, or are mconslste~t wl!h observed phenomena. A consistent model is proposed in which two mechanisms are operatmg slmult~neous!~, namely, the. vacancy mechanism for the slower diffusing component, and the mterstltlalcy mechamsm for the faster diffusing component. It is assumed that phosphorus exists in silicon in both the substitutional and the interstitialcy species, and that both are shallo~ donors. The c~nversion between the two species is relatively slow, giving rise to the so-called kmked concentratIOn profile. Diffusion via a partial interstitialcy mechanism leads to a supersaturation of self-interstitials.

PACS numbers: 66.30.Jt, 61.70.Bv, 61.70.Ph, 66.30.Lw

I. INTRODUCTION

The efforts of nearly one and a half decades, to date, have resulted in a substantial understanding of diffusion pro­cesses in silicon, not just on important nonlinear phenomena under equilibrium conditions such as the concentration de­pendence of diffusivity, the internal field effect, and the in­teractions among different diffusants,I-5 but also on certain complex phenomena under non equilibrium conditions such as the oxidation enhanced diffusion.4

-12 One can now rather

accurately design and predict diffusion profiles of boron and arsenic to satisfy the exacting demands of the present-day IC device technology-The SUPREM (Stanford University Process Engineering Model) process simulation program is a good example of a predictive tool suitable for process de­sign. 13 Yet within SUPREM there is a notable deficiency in the simulation of phosphorus diffusion. Despite the efforts and progress that have been made in the area just mentioned, the diffusion process of phosphorus in silicon is still poorly understood. It is a very complex process, producing a con­centration profile that often features plateau, kink, and tail, and gives rise to such anomaly as the emitter-push effect. Nearly a dozen models have so far been proposed, yet none has proved satisfactory. Most of the earlier models have al­ready been critiqued by one of us. 14 Models published since then,15-22 however, have yet to be critically reviewed. Among these, the model of Fair and Tsai 5,22 is probably the best known and the most widely used, due in part to its sim­plicity and in part to its adoption in the SUPREM program. Although the Fair-Tsai model has been reviewed, the com­ments are limited and contradictory in many cases. For ex­ample, Kimerling23 considered the model to be accurate, universal, and fully consistent with a first principles ap­proach. Others, e.g., Kroger24 and Panteleev,25 have ex­pressed contrary and incisive opinions about the model; but their comments are not as well known. It seems clear that the phosphorus diffusion in silicon, far from being universal, is a rather peculiar process quite different from, for example, the diffusion of arsenic in silicon. There is little about it that is understood on first principles.

·'Present address: IBM East Fishkill Facility, NY 12533.

There are two forerunners of the Fair-Tsai model. One is the model of Schwettmann and Kendall, 15-17 which pro­poses (1) the diffusion of phosphorus via E centers (vacancy­phos~hor~s ~airs) in two charge states, (2) the discharge (but not diSSOCiation) of the negative E centers at the kink concen­tratio~, and (3) the electrical compensation of the donors by negabve E centers. The other is the model of Yoshida 18-20

which also assumes the diffusion of phosphorus via the E ~enters, but. allows the E centers to dissociate as they move mto the regIOn of low phosphorus concentration. The Fair­Tsai model is essentially a combination of these two models with the addition of some ad hoc expressions for the concen~ !rations of E centers and excess vacancies. The concept of Impurity diffusion via the vacancy-impurity pair is not

14,26 d 1': new, an cannot be Jaulted per se. We shall assess the validity of the vacancy mechanism for the phosphorus diffu­sion in silicon in terms of its consistency with experimental observations. Moreover, many assumptions and suggested relations in the Fair-Tsai model can and will be assessed from a logical standpoint. The vacancy supersaturation pur­suant to E center dissociation has not been analyzed proper­ly in the Fair-Tsai model. Actual numerical calculations of vacancy transients by Yoshida, 20 and by Mathiot and Pfis­ter27 reveal that the E-center dissociation model alone will ~ot simulate the phosphorus diffusion profile; other assump­tions, such as the concentration dependence of the vacancy formation energy, will be required. Recent publications have reported disagreement between experimental results and predictions from the Fair-Tsai model. 28

-3o But there has

also been an acceptance of the Fair-Tsai model as being con­sistent with experimental results,3I,32 or simply as a basis for the explanation of related phenomena. 33

The phosphorus diffusion in silicon has remained as an important technology in silicon microelectronics. A thor­ough review of this subject is essential to the means of pro­cess modeling and is long overdue. In the present paper, we will first review some salient features of the phosphorus dif­fusion profile and ancillary phenomena associated with the phosphorus diffusion. We will then review the aforemen­tioned recent models of phosphorus diffusion in silicon. A detailed critique is presented on the Fair-Tsai model, since it

6912 J. Appl. Phys. 54 (12), December 1983 0021-8979/83/126912-11 $02.40 © 1983 American Institute of PhysiCS 6912

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is the model dominantly used in process simulation. Finally, we propose a model that is consistent with the experimental

evidence available at this time.

II. FEATURES OF PHOSPHORUS DIFFUSION IN SILICON

Phosphorus diffusion in silicon exhibits some rather un­usual characteristics that have come to be called anomalous. The present standard technique of phosphorus diffusion comprises a "deposition," in which POCI3, P20 5, or PH3 + O2 is used to react with the silicon substrate. First, note that the silicon substrate is oxidized. Secondly, the reac­tion produces "nascent" elemental phosphorus at a chemical potential far exceeding that at its solid solubility in silicon; this phosphorus then enters the silicon substrate in supersa­turation. This creates an environment in which a variety of nonequilibrium processes can occur and in tum potentially cause the anomalous behavior. Any viable model must not only be logically sound, but also be consistent with charac­teristic features. In this section we review salient features of phosphorus diffusion in silicon and comment on their prob­able causes and implications.

A. The flat region and discrepancy between atomic and electrical profiles

Phosphorus diffusion at high surface concentrations usually produces an electrical profile having a fiat zone in the surface region, i.e., a zone where the electron concentration is constant, and is substantially lower than the phosphorus atomic concentration?4-37 The electron concentration in the flat zone is a function of the diffusion temperature only, and is independent of the source, or the surface concentration once the flat zone appears. 15,16 An abundance of experimen­tal results has established29,34-36,38-43 that the precipitation of phosphorus is the principal cause of the discrepancy between the atomic and the electron concentrations. Based on the negative E-center concept, Fair and Tsai proposed a cubic relationship between the total atomic concentration and the electron concentration as

CT =n +Kn3• (2.1)

It is easily seen, however, that such a relationship will not produce a fiat zone in the electron concentration profile. Kroger pointed out24 that, instead of Eq. (1), the correct expression for the negative E-center compensation model should be

2Kn3

CT = n + 2' l-Kn

(2.2)

Equation (2) indicates the existence of an asymptotic limit of the electron concentration at n = (11K )1/2. This may ap­proximate the existence of a fiat zone. An alternative expla­nation of the flat zone is the occurrence of precipitation, discussed below.

B. Phosphorus precipitation

Phosphorus precipitates have been found where the fiat zone occurs,36.38-43.69 although the precipitates, when very

6913 J. Appl. Phys., Vol. 54, No. 12, December 1983

small, are not always readily observed by TEM. The precipi­tates have a SiP chemical composition, an orthorhombic crystalline structure, and often exhibit a rodlike morpho­logy, although a platelet morphology has also been ob­served.38-43 Abundant experimental evidence has shown that precipitation takes place during the diffusion pro­cess. 38-43 This can occur only when the phosphorus enters the silicon substrate in supersaturation.

C. Oxidation during deposition and drive-In

Diffusion of phosphorus into silicon is often carried out by an "open-tube" technique, using POCI3, P20 S' or PH3 + O2 as a diffusion source. While the detail of the phos­phorus deposition process is not known, a general picture is that the silicon substrate is oxidized, partly by POC13 (for example), and that most of the phosphorus is converted into phosphosilicate, while some of it is reduced to its elemental form. It is this reduction to elemental phosphorus that causes its fugacity to exceed that of a saturated solid solu­tion, and to lead to the precipitation discussed above. The typical rate of oxidation of the silicon substrate by POCl3 lies between the rates of oxidation in pure dry oxygen and in steam at atmospheric pressures.44 ,45 This oxidation of the silicon substrate, somewhat similar to normal thermal oxi­dation, can be expected to generate excess self-interstitials and give rise to associated phenomena as the oxidation en­hanced diffusion, etc.

D. The emitter-push effect

A peculiar phenomenon associated with phosphorus diffusion is the emitter-push effect. This is a phenomenon in which a boron base in the area underneath a phosphorus emitter is "pushed out" relative to the rest of the base area.46-52 It is clear from both theoretical arguments53 and experimental evidence54 that the emitter-push is a result of the enhancement of the local base diffusivity rather than a "vacancy wind" effect. 55 Thus, a buried boron layer would experience accelerated movements down both the front and the rear concentration gradients. That is, the push-out for a buried boron layer occurs both toward the surface and deep­er into the bulk for the upper and the lower junctions, respec­tively. From the magnitUde of the push-out in a typical case, one observes diffusivity enhancement by a factor of tens to hundreds. 18,50-52,54 The emitter-push effect can only be ex­plained by the occurrence of a supersaturation of point de­fects, vacancies, or self interstitials. Evidence of interstitial supersaturation is given in Sec. II H. However, such evi­dence may not prove the nonoccurrence of vacancy supersa­turation to its advocates.

A definitive test of whether there exists a condition of vacancy supersaturation during the phosphorus diffusion is to investigate whether an emitter-push occurs when the bo­ron base is replaced by an impurity, possibly antimony, which diffuses strictly via a vacancy mechanism. Further­more, a buried layer would provide a better experimental test vehicle than a transistor base because direct phosphorus­antimony interaction, as well as electric field effect and the

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effect of dislocation network, can be avoided. (See Note Add­ed in Proof)

E. Kinks and tails in diffusion profiles

Figure 1 shows a typical phosphorus diffusion profile, which exhibits two distinct regimes, joined at a kink. In the surface regime, the phosphorus atomic concentration profile is rather steep. The more gradual "tail" profile indicates a significantly higher diffusivity in this regime. It has been reported that the concentration at the kink is a function of diffusion temperature only, with an apparent "activation en­ergy" of 0.79 eV (Refs. 15, 16), and is independent of the phosphorus surface concentration and the diffusion time. The kink concentration does not appear to have an intrinsic significance, but is merely the concentration at which the profiles in the two regimes meet. Hence the assignment of a kink concentration with an activation energy should be re­garded as an empirical approximation for a more complex nonequilibrium situation. However, in the models of Schwettmann and Kendall I5

-17 and of Fair and Tsai,22 the

"kink concentration" is postulated to correspond to the Fer­mi level that coincides with a charge state of the E center. Since the solid solubility of phosphorus in silicon exhibits a retrograde behavior with a substantial plateau at about 3-4X 1020 atoms cm- 3 (Ref. 43) in the temperature range between 900 and 1200 °e, while the kink concentration in­creases to about 1 X 1020 atoms cm - 3 at about 1100 °e, it

2

1021 6. POCb 0.05%

o • POCb 0.27%

5

2

1020

-... E 5 u -... as - 2 0

1018

5

2

1018

5

Predep 920°C. 7 min

2 0 0.1 0.2 0.3 0.4

X, ~m]

FIG. 1. Two examples of phosphorus diffusion profiles produced with sources corresponding to two different surface chemical concentrations (adapted from Ref. 45). The solid curves are for chemical (total phosphorus) profiles and the dashed curves is for electrical profile.

6914 J. Appl. Phys., Vol. 54, No. 12, December 1963

becomes difficult to resolve the kink at this or higher tem­peratures.

In the simplest and most reasonable analysis, 14 a profile with a kink is really a composite profile of two different dif­fusing species, which convert into each other at a finite rate.

F. Concentration-dependent diffusivity and effects of surface concentration

One of the classical and the simplest concepts in nonlin­ear diffusion is the concentration dependence of diffusivity. Thus, attempts have frequently been made l9

,22 since Tan­nenbaum34 to extract a concentration dependence of diffusi­vity from the diffusion profiles by means of Boltzmann-Ma­tano analysis. But the Boltzmann transformation, on which the Matano analysis is based, requires that the diffusivity is a function only of the local concentration, and not of time, distance, surface concentration, etc. The analysis is both in­applicable and meaningless when there exists a significant (say, 10- to lOO-fold) supersaturation of point defects, which is controlled by processes occurring in the surface regime, such as precipitation, and which is a function of distance, time, and surface concentration. The indiscriminate applica­tion of the method of Boltzmann-Matano analysis has pre­viously been criticized by one of us. 14 Notwithstanding, the analysis is still widely used. 18,22 The justification given is that when the concentration of phosphorus is plotted against the Boltzmann variable xt - 112, profiles with different diffusion times but otherwise identical conditions falI closely on one another. However, our sample analyses show this to be often not true, and profile shifts of more than 20% along the (xt - 1/2) axis can be found. Further, consider the nature of the concentration dependence of diffusivity obtained from such analyses. In Fig. 2, taken from Yoshida et al., 18 we see

0 10-13

• b • III UI

N

E .2- _.-.'" E 10-"

/",'

III

. ''''''. 'u ;;:: -III PH,/SiH. 0 .--' 0

/" 0 0.36 r:: 0

/ • 0.25

'iii 10-15

::I 0 0.10 :=

0.05 i5 0.01

• 0.005

10-10

10" 10" 10" 10'·

Total Phosphorous Concentration (em-3)

FIG. 2. "Concentration dependence" of diffusivity as obtained from Boltz­mann-Matano analysis in Ref. 18. One sees a dominant influence from the surface concentration (indicated by the rightmost point on each of these curves, and corresponding to various PH 3/SiH 4 ratios). It is clearly seen that there is no unique local "concentration dependence of diffusivity."

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that the diffusivity in the tail regime is mainly determined by the surface concentration, and is quite independent of the local concentration which, as the horizontal axis, might as well be labelled the distance axis. Technically, the Boltz­mann-Matano analysis starts integration of the concentra­tion profile from the low concentration regime, without knowledge of the concentration at the surface. Physically, too, the local diffusivity cannot experience effects due to the surface concentration except through the vehicle of transient excess point defects. In the concentration regime above the kink, Fair and Tsai show22 the diffusivity to increase in pro­portion to the square of the electron concentration. As point­ed out by Fahey,59 this manner of electron concentration dependence of diffusivity is a direct consequence of Fair and Tsai's choice of a particular electron mobility-concentration relationship, used to convert resistivity data to electron con­centration. Using the most recent mobility data60

•61 for the

same analysis would have led to a linear relationship between the diffusivity and the electron concentration. A linear concentration dependence of phosphorus diffusivity has been reported by Makris and Masters62 in isoconcentra­tion diffusion, although the electron concentration they as­sumed may also be subject to error. But the near equilibrium condition in the isoconcentration diffusion is never obtained in practical diffusion; a discrepancy between their results62

and that shown in Fig. 2 should indicate a highly nonequilib­rium condition in the latter.

G. Phosphorus diffusion from polysilicon source

Recent experimental investigations63.64 have shown that when the diffusion of phosphorus from a POCl3 prede­position takes place through polysilicon layers, rather than with POCl3 depositions directly on single crystalline silicon substrates, both the profile tail and the emitter-push are sig­nificantly reduced. This is of course not unexpected. It has been pointed out by one of us 65,66 that polysilicon grain boundaries are effective sinks for both intrinsic and extrinsic point defects, as manifested in the suppression of OSF. From a related point of view, Swaminathan56 has studied the OED of phosphorus and boron buried layers, under the condition that the silicon substrate is covered by a polysilicon layer, and the thermal oxidation takes place on the polysilicon. He found that the OED is reduced as a function of the thickness of the polysilicon layer. It is plausible that this effect is also caused by the suppression of excess point defects by the poly­silicon. Furthermore, through polysilicon the phosphorus will enter the silicon substrate in reduced supersaturation, due to its more rapid precipitation in the polysilicon.

H. Direct evidence of interstitial supersaturation

An abundance of experimental evidence indicates a su­persaturation of self-interstitials, from, e,g., the formation of helical dislocations,67 the growth of extrinsic stacking faults,77.78 and the detection of the interstitial type strain diffraction contrasts,38-41.68 The origin of this interstitial su­persaturation can only be partly attributed to the thermal oxidation of the silicon substrate, for annealing at low tem­peratures in absence of an oxidizing ambient can often lead

6915 J. Appl. Phys., Vol. 54, No. 12, December 1983

to a profusion of precipitates of interstitial diffraction con­trasts, as well as the formation of extrinsic stacking faults. The work of Armigliato et al.77 using a POCl3 diffusion source showed that under their experimental conditions nu­cleation and growth of extrinsic stacking faults depend strongly on the rate at which phosphorus enters the silicon substrate but is nearly independent of the concurrent oxida­tion at the surface, Further, using a PH3-doped CVD oxide as a source Claeys et al.78 observed stacking fault growth to proceed more rapidly when diffusions were performed in a nitrogen ambient rather than under oxidizing conditions. A similar situation is found in germanium, in which the preci­pitation of phosphorus leads to the observation only of inter­stitial type diffraction contrasts.70.71 If the more mobile com­plexes are the phosphorus-vacancy pairs that agglomerate to form SiP precipitates, then vacancies would be released and become excess, contrary to direct evidence.

For completeness it should be mentioned that some in­vestigations 79,80 have reported enhanced shrinkage, rather than growth, of stacking faults in the presence of a high con­centration of phosphorus. Since these investigations have been rather limited to date we merely make the following observations rather than attempt to provide detailed expla­nations. Enhanced shrinkage (1) occurs also in presence of boron and arsenic, (2) has been observed only at high tem­peratures (> 1100 0c), (3) is dependent on whether or not the stacking fault is contained within the diffused layer,79 It is well known that the stacking fault has natural tendency to shrink because of its faulting energy (- 70 ergs cm - 2), and that the shrinkage process is characterized by a very high activation energy (about 5 e V). At high enough temperatures the shrinkage can overtake the growth even during oxida­tion, a phenomenon known as retrogrowth. 81 The rate of shrinkage is directly proportional to the silicon self-diffusi­vity.12 Thus doping enhanced shrinkage rate can be ex­plained in terms of doping enhanced self-diffusivity.

III. MODELS OF PHOSPHORUS DIFFUSION IN SILICON: A CRITIQUE

Despite the overwhelming evidence that indicates a prevalence of excess self-interstitials during the diffusion of phosphorus, the major models, namely, the Schwettmann­Kendall-Carpio (SKC) model, the Yoshida (Y) model, and the Fair-Tsai (FT) model, all hypothesize a vacancy mecha­nism of diffusion, where phosphorus diffuses in silicon as an E center, i.e., a vacancy-phosphorus pair. In this section we first discuss the general theory of the E-center diffusion mechanism as it pertains to phosphorus, to provide a back­ground for discussions of the three models,

A. The E-center diffusion mechanism and vacancy supersaturation

A premise of the E-center models is that the E center exists in a number of charge states (E -, EO, and E +) which are in different concentrations, and may diffuse at different rates. Application of the E-center diffusion concept to the case of phosphorus is understood most easily by examining

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the general diffusivity expression2,14 for the case of a vacancy

mechanism:

(1 2 r) CAv

DA = -r WA.lA --, 6 CT

(3.1)

where r, W A' and fA are, respectively, the atomic jump dis­tance, the jump frequency, and the correlation factor, and CAv and CT are the concentration of the vacancy-impurity pairs, and the total concentration of impurity A in all forms, respectively. The factor inside the parentheses is identified as the diffusivity of the E center. Equation (3.1) says simply that the diffusivity of the impurity is equal to the diffusivity of the vacancy-impurity pair, weighted by the ratio CAv/CT' For phosphorus diffusing via the E center, the term on the right­hand side ofEq. (3.1) is replaced by three terms correspond­ing to E - , E 0, and E +. Since E - is the dominant species at high phosphorus concentrations, we take E - as an example. The effective diffusivity is then determined mainly by the ratio CE - /CT actually existing, regardless of the vacancy concentration and of any nonequilibrium condition. The for­mation and dissociation of E - centers can be written as

p + + v=~+v= . (3.2)

In equilibrium, the concentration of E - centers is given by the mass action law

CE- =KECp+Cv~ , (3.3)

where we have omitted the respective activity coefficients. Analogous expressions for the neutral and the positive E centers can easily be written. Since, in the E-center mecha­nism of diffusion, only E centers can move while P + is im­mobile and is left behind, the E centers will be diffusing into a region where the concentration of P + is very low. In order to satisfy the local equilibrium relationship as expressed in Eq. (3.3), there E centers dissociate and release vacancies which can build up an excess. This vacancy excess is, in the Y and the FT models, responsible for both the enhanced phos­phorus diffusivity in the profile tail and the emitter-push effect. Two questions can be asked: (a) How much vacancy supersaturation can be built up counterbalanced by vacancy decay mechanisms, foremost among which is the rapid diffu­sion to the substrate surface. (b) If indeed a large vacancy supersaturation is created, sufficient to account for the en­hanced tail diffusivity and the emitter-push effect, why are these same phenomena not seen in arsenic diffusion which has been successfully modeled by an E-center diffusion mechanism with both CE and C v always at equilibrium con­centrations. To answer these two questions, we make an or­der-of-magnitude estimation of the possible vacancy super­saturation as follows: Assume that at x = X k , defined as the location where the kink occurs (C p = C k), E centers disso­ciate completely (actually, some dissociate right after cross­ing X k while others do so after some distance beyond x k !­Thus, the integral of all vacancies generated beyond X k IS

equal to the flux of E centers across x = X k • Since the tail portion of the phosphorus profile resembles an erfc distribu­tion, the flux of E centers across X k can be expressed as

JE=C" --. ~eff 1Tt

(3.4)

6916 J. Appl. Phys., Vol. 54, No. 12. December 1983

In a quasi-steady state approximation, J E will be equal to the vacancy flux outward across x = X k :

acv J E = -J =D --, " "ax

which can be then written as

(3.5)

To be consistent we assume the silicon self-diffusion to also proceed via a vacancy mechanism, so we may write CUD" = C,D" where C, is the concentration oflattice sites in silicon, and Ds is the silicon self-diffusivity. Thus, a quasi­steady state vacancy supersaturation is approximately given by

CkXk~Deff (t >

[iiCsD:(t) (3.6)

As an example, we take a reasonable set of parameters: Ck

= 5 X 1019 cm- 3, X k = 0.5 /-lm, Ds = 1 X 10- 16 cm2/s, Deff

= 1 X 10- 12 cm2 s -I, and (t ) = 1000 s. An estimate of the vacancy supersaturation according to Eq. (3.6) is then on the order of 10. Two points can be observed from our approxi­mate analysis: (a) Indeed a vacancy supersaturation can build up, although our analysis does not substantiate the loo-fold increase of diffusivity in the emitter-push effect ob­served by Lee and Willoughby.52.54 (b) The supersaturation is intimately tied to the effective diffusivity of the dopant.

An adjunct hypothesis in two models of the E-center mechanism of phosphorus diffusion 15-17.22 is the presence of a very high concentration of negative E centers. While this hypothesis is not essential to the diffusion mechanism per se, it is meant to explain the observation of inactive phosphorus. In one model,22 the concentration of negative E centers is proposed to even exceed the concentration of P + , and com­prises about 2/3 of the total phosphorus present. Aside from its contradiction to the experimental evidence presented in Sec. II B, this hypothesis is easily shown to be invalid from simple reasoning.

Consider first that this violates the law of charge ba­lance. If the concentration of negative E centers is higher than the concentration of positively ionized phosphorus atoms, from where do the P + V ~ pairs get their electrons? It is suggested22 that they get the electrons from the conduc­tion band, as if the conduction band was an unlimited source of electrons. Actually, the electrons in the conduction band come from two true sources: the valence band and the phos­phorus atoms. Since phosphorus doped silicon is n type, there are very few holes and the valence band is a negligible source of electrons. Therefore, the positively ionized phos­phorus atoms provide all the electrons present in both the conduction band and the negative E centers. The conserva­tion of charge (in the local charge neutrality approximation) requires that

CP

' v + Cv + 2Cu _ + n

(3.7)

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From this relationship, together with the conservation of matter

CT = Cp+ + Cp+"o + Cp+v + Cp+V~ ,

one must then have

Cp+v~ = (!)(CT - n - Cp."o

(3.8)

-Cpy -Cv -2Cu~)«!)(CT-n). (3.9)

The inequality of Eg. (3.9) says that the concentration of p + v = can account for at most one half of the inactive phos­phorus.

Moreover, the hypothesis of the dominance of CE

suffers fundamentally from the fact that, based on any rea­sonable estimate, the concentration of vacancies in silicon is simply not high enough to support these levels of C E - •

B. A different senario of the E-center diffusion mechanism

The above analysis shows only the feasibility of a va­cancy supersaturation in an E-center mechanism of diffu­sion but does not constitute a proof that this is the actual mechanism. By an analogous analysis one can show at least equally well the feasibility of an interstitialcy excess in an interstitialcy mechanism of diffusion just from the dissocia­tion of interstitialcies alone, without having to consider the simultaneous occurrence of oxidation and precipitation. Therefore, one must judge a mechanism against all available experimental observations. We have already seen in Sec. II plenty of experimental evidence against the vacancy mecha­nism. In this and the following sections we shall examine the soundness of the logic and assumptions made in developing the mechanism. Ifboth arsenic and phosphorus diffuse by an E-center mechanism, one must ask the reason for the order­of-magnitude difference in diffusivity between arsenic and phosphorus.

In our simplistic analysis of vacancy supersaturation, as well as in all the E-center models mentioned, only the disso­ciation of the E centers is considered. No thought has been given to how the E centers are formed. This leads to a one­sided question: what extent of vacancy supersaturation can occur during the phosphorus diffusion? If the formation of the E centers is also considered, the question then becomes: is there a vacancy deficit or a vacancy excess during the diffusion? There are two ways for a phosphorus atom to en­ter the silicon lattice substitutionally: (a) a silicon lattice atom can be displaced, thus generating a silicon self-intersti­tial; (b) it can consume a vacancy as the latter becomes avail­able. However, the substitutional phosphorus must then wait for another vacancy to become available to form an E center and become mobile in the lattice. Thus we see that in the vacancy mechanism, every phosphorus atom will con­sume one vacancy to enter the lattice, and an additional va­cancy to form the mobile E center. Clearly then, there is a net consumption of vacancies in the diffusion of phosphorus into the silicon lattice.

Because of a net consumption of vacancies as envi­sioned above, the surface must necessarily serve as a vacancy source, rather than a vacancy sink as suggested in the discus-

6917 J. Appl. Phys., Vol. 54, No. 12, December 1983

sion below. In order for the surface to act as a vacancy source, there must be some vacancy undersaturation, at least in the region near the surface. How severe and deeply ex­tended is the vacancy undersaturation will depend on both the efficiency of the surface as a source, and the diffusivity of vacancies in the silicon lattice. From the observations of OED and OSF we know that the silicon surface, or rather the Si-Si02 interface is not an efficient sink which allows the surface-generated self-interstitials to build up (although it is more efficient than a Si-Si3N4 interface72

). From thermody­namics, the Si-Si02 interface is necessarily also not an effi­cient interstitial or vacancy source. Thus, we see a possibility of a vacancy undersaturation even if phosphorus diffuses via an E-center mechanism.

In Sec. IV we will see, by contrast, how natural it is for the phosphorus atom to enter the silicon lattice as an intersti­tialcy, and how this process may generate excess self-inter­stitials, and possibly cause vacancy undersaturation as well.

C. The model of Schwettmann, Kendall, and Carpio (SKC)

The SKC model is formulated from three basic assump­tions: (a) Phosphorus diffuses as an E center in two charge states, E - and EO; the latter species diffuses significantly faster than the former. (b) Electrical compensation by nega­tive E centers accounts for the inactive phosphorus in the surface flat zone. (c) The kink in the profile occurs when, at a characteristic concentration, the negatively charge E center loses an electron to become a neutral E center, which then diffuses at a much higher rate, giving rise to a long and more gradual profile tail. In a sense this is a two-stream diffusion mechanism which, as pointed out in Sec. II A, is capable in principle of predicting a kinked, two-zoned profile. There are however serious problems with this model. The first con­cerns the E-center self-compensation concept which we have discussed in Sec. II A. The second problem concerns the assumption of no dissociation of the neutral E center, which apparently violates the law of mass action-unless there ex­ists an unusually large binding energy between the phos­phorus and the vacancy. No dissociation of the neutral E center also means that the profile tail consists of electrically inactive E 0 centers, in contradiction to electrical character­izations of such profile tails. Thirdly, while a fast-diffusing neutral E center may account for the kink and the tail of a phosphorus profile, it is not able to account for the emitter­push effect which occurs at a place, as in a buried layer, rather far away [e.g., 10 /-lm (Ref. 54)] from any phosphorus species. Finally, we note that the model, which assumes a system in equilibrium, cannot account for the polysilicon effect (Sec. II G).

D. The model of Yoshida (Y) and the model of Mathiot and Pfister (MP)

The model of Yoshida et al. 18.19 is based on the concept of the dissociation of the E centers, essentially as described in Sec. III A; but the vacancy excess resulting from the disso­ciation of E centers is calculated numerically. We have al­ready discussed in Sec. III B that in considering only the

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dissociation, rather than both the dissociation and the for­mation of the E-center, one is surely led (or misled) to the unwarranted conclusion of a vacancy supersaturation. This fundamental issue aside, this model is not able to produce the characteristic kink and the flat zone in the profile. In a later model, Yoshida 20 invoked the additional assumption that the enthalpy of formation of the neutral vacancy decreases quadratically with the phosphorus concentration as follows:

Hf(Cp ) = H~/(1 + BC;), (3.10)

where B is an adjustable parameter. In this way the equilibri­um vacancy concentration in the surface region is made to increase extremely rapidly with the phosphorus concentra­tion, by almost a factor of 1000 above and beyond that made possible by electronic charging. This in turn brings about a much flatter profile in the surface region. Additional expedi­encies are made, that the equilibrium concentration of neg a­tive vacancies, under an intrinsic condition at a given tem­perature (say 900°C), is made to vary by a factor of 10 for fitting different diffusion profiles, and that the parameter B is given a very strong temperature dependence. The Math­iot-Pfister (MP) model27 is essentially similar to the Y mod­el, except that, to similate the surface flat zone, a formula for concentration dependence of the vacancy formation enth­alpy is postulated from a concept of percolation cluster, 75 instead of Eq. (3.10).

Serious problems with the Y and the MP models, in addition to those mentioned earlier, are: (a) The equilibrium vacancy concentration as a function of phosphorus concen­tration according to Eq. (3.10) or the MP model is inconsis­tent with the isoconcentration diffusion results of Makris and Masters,62 which show the concentration of neutral va­cancies to be independent of phosphorus concentration. In principle at least, an isoconcentration diffusion takes place under conditions nearest equilibrium, and should yield reli­able information concerning equilibrium parameters. (b) Ac­cording to the results of most investigators,34-45 it is the elec­tron concentration profile, and not the total phosphorus concentration profile, that shows a nearly flat surface region. Instead, both the Y and MP models assume the total phos­phorus concentration profile to be nearly flat in the surface region.

E. The model of Fair and Tsai (FT)

The Fair-Tsai model is essentially a combination of the Y model (E-center dissociation) and the SKC model (the dis­charge of E - at the kink, and the electrical compensation by negatively charged E centers to account for the inactive phosphorus). Hence our criticisms in Secs. III B-III D con­cerning the E-center mechanism of phosphorus diffusion in general, and the Y and the SKC models in specifics, also apply to the FT model. In addition, the FT model employs a number of assumptions in order to obtain "steady-state va­cancy supersaturation" without having to actually solve the diffusion equations as in Y (Refs. 18-20) and MP. 27,75 This is achieved by a series of mass-action expressions, despite that the vacancy supersaturation is a nonequilibrium pheno­menon dependent on the rates of vacancy generation and

6918 J. Appl. Phys., Vol. 54, No. 12, December 1983

decay. These assumptions are seriously flawed, and we will in this section comment only on the logic of these assump­tions peculiar to the FT model.

(1) The FT model assumes phosphorus to exist at high concentrations predominantly as negatively charged E centers. To account for the inactive phosphorus, Fair and Tsai concluded22 that about 2/3 of the phosphorus is paired with doubly negative vacancies (their conclusion No.2). This violates the law of charge balance, as discussed in Sec. IlIA. If the concentration of negative E centers is higher than the concentration of positively ionized phosphorus atoms, holes would have to be the majority carriers on account of charge balance. This contradicts the known fact that phosphorus doped silicon is n type.

(2) Fair and Tsai22 have defined an "intrinsic concentra­tion of vacancy-phosphorus pairs" [their Eq. (13) and what follows], and used this quantity to obtain the concentration of vacancy-phosphorus pairs at any finite concentration of phosphorus by scaling with the ratio (n/nj )3. The definition of an "intrinsic concentration of vacancy-phosphorus pairs" seems to have no physical meaning; for intrinsic silicon, the concentration of phosphorus is theoretically zero. Hence the intrinsic concentration of vacancy-phosphorus pairs must also be zero. Interestingly, Fair has also similarly defined "intrinsic concentrations" of vacancy-arsenic pairs vAs and of vacancy-double arsenic pairs vAs2, for which he has pro­posed the following relationships 74:

C~As = 4.3X 1015 exp[ - (0.69 eV)/kTJ cm- 3; (3.11)

C:'AS, = 1.7x 1018 exp[ - (1.43 eV)/kTJ cm-3 .(3.12)

Since the equilibrium concentration of vAs - pairs must fol­low mass-action law as

C:'As =KC> CAs' , (3.13)

one must ask what is the intrinsic concentration of arsenic C:"'s' ? Ditto for C~s,.

(3) We now come to the most serious flaw in the Ff model: The steady-state vacancy supersaturation is obtained in this model by means of a series of mass-action equilibrium relations, in contrast to the conventional balancing of the rate of vacancy generation with the flux of vacancy out-dif­fusion [see Eq. (3.5) and Sec. III A]. It is proposed that p +v = pairs from the surface flat zone diffuse down the con­centration gradient and then start to dissociate when the Fermi level drops below Ec - 0.11 eV; the liberated vacan­cies then flood the bulk.

A number of mass-action relations are then used, start­ing with the proposition that the concentration of p +v =

pairs at the surface can be expressed as a ratio to the "intrin­sic concentration of p + v = pairs," the ratio being deter­mined by the surface electron concentration n, through the relationship [their Eq. (13)]

(3.14)

(We have already commented on the lack of justification in defining an intrinsic concentration of vacancy-phosphorus

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pairs in Sec. III E 2.) The concentration of singly negative vacancies at the kink is then related to the concentration of pv- pairs at the surface through an equilibrium mass action [their Eq. (16)], as

Cv- = (ni)2 Cpv - , C'v- ne C~v-

(3.15)

where ne is the electron concentration at the kink. Since mass action is a local phenomenon, it is difficult to under­stand why it should involve the electron concentration at the kink with the concentration of pv- pairs at the surface. To obtain the "total concentration of generated vacancies," they sum different vacancy ratios [their Eq. (17)]. The sum­mation of absolute values is meaningful, not the summation of ratios.

While in a homogeneous system one may speak of the "total concentration of generated v- vacancies," in an inho­mogeneous system with an impurity profile whose depth is about 1-2/1000 ofthe thickness of the substrate, the "total concentration of generated v- vacancies" is meaningless un­less integrated over the region where these v- vacancies are generated, and then divided by the thickness of the substrate throughout which these vacancies are present.

Finally, the "total concentration of generated v- va­cancies" are equated with the "steady-state concentration of v - vacancies referenced top + v ~ and p + v - pair dissociation events." However, the excess vacancies so generated will de­cay by escaping to the substrate surface. One can estimate the lifetime of generated excess vacancies based on their dif­fusion to the surface from where they are generated, at Xk' to be approximately = xUDv' or on the order of 0.0 1-1 s. Tak­ing a diffusion time of the order of 103 s, one sees that no more than 10- 3 of the generated vacancies can stay in the substrate as a steady-state concentration.

(4) Since everything in the FT model is expressed in terms of equilibrium mass action relations, as culminated in their "steady-state vacancy concentration" [their Eq. (17)], clearly the model is incapable of explaining the effect of poly­silicon source, and the lateral and the depth factors in the emitter-push effect.

From the above discussion one can conclude that the relationships developed in this model do not have a genuine physical basis. It must then follow that the apparent good agreement with experiments is due to fitting of the data with enough parameters. We should like to emphasize that, as with any empirical model, a semblance with actual diffusion profiles can be obtained with sufficient adjustable param­eters, but extension of the model to other conditions fre­quently give poor results. We now summarize the experi­mental findings which is in contradiction to the FT model: (1) The kink and the emitter-push effect is greatly reduced for phosphorus diffusions done through a polysilicon lay­er63

,64; (2) electrically inactive phosphorus present at high concentrations is due to precipitation28 ,29,42-44; (3) there is no functional relationship between inactive and active phos­phorus concentrations28

,29; and (4) under the same experi­mental conditions in which the terms of the model were de­rived, preexisting oxidation stacking faults grow rather than

6919 J. Appl. Phys., Vol. 54, No. 12, December 1983

shrink.78 As a consequence of point (3), Eq. (2.1), which re­lates the total to the electrically active phosphorus concen­trations, is not meaningful. This observation is made here in view of the fact that Eq. (2.1) has been utilized by different investigators in related studies. Among others, Ho and Plummer33 have used it in their study of impurity enhanced oxidation, and Lecrosnier et al.32 have utilized it in their investigation of the gettering of heavy metals by phosphorus layers.

F. Comparison of phosphorus and arsenic diffusion

It is of interest to compare the diffusion of phosphorus with the diffusion of arsenic in silicon since arsenic diffusion has been modeled more successfully employing the E-center mechanism as outlined in Sec. III A. The diffusivity of ar­senic has been calcul,ated using the general expression of dif­fusion, Eq. (3.1), with the ratio of CAs vieT taken at its equi­librium value specified by the law of mass action. That is, in contrast to the phosphorus models, arsenic diffusion appears to be described very well by a model in which E centers dissociate instantaneously as they migrate from regions of higher to lower arsenic concentrations and no vacancy ex­cess is built up. An interesting question can now be posed: why does arsenic not have an enhanced tail as does phos­phorus? Why does it not bring about an emitter-push effect? In other words, why is there no vacancy excess in the case of arsenic? A plausible answer may be that, because the diffusi­vity of the vacancy is so high, a significant vacancy excess can exist only when the rate of vacancy generation is also very high. which can happen only when the flux of E centers is very large. in other words, when the diffusivity of the im­purity is very high. The dependence of attainable vacancy supersaturation on the effective diffusivity of the dopant is given in Eq. (3.6) of our approximate analysis. The fact that arsenic diffuses much more slowly than phosphorus may be invoked to explain the reduction in vacancy supersaturation, but does not explain why it should diffuse much more slowly via the same E-center mechanism. One way out is to assume that arsenic does not diffuse via the E center in silicon. This, however, will bring forth many difficulties. An alternative is to assume that arsenic and phosphorus diffuse in silicon through different mechanisms. We hold the latter viewpoint and show in the next section that phosphorus diffusion can be explained most completely and consistently by a dual mechanism of vacancy and interstitialcy diffusion6 in which the interstitialcy component plays a major role.

IV. OUR PROPOSED MODEL

Phosphorus diffusion in silicon is a good example of things in the real world: it is not simple and ideal. In the present critique we shall outline a model of phosphorus dif­fusion in silicon that is consistent with all the experimental observations mentioned in Sec. II; we will deal with the quantitative aspects of the model in a future paper. The es­sence of the model can be summarized in the following points:

(1) Diffusion from supersaturated sources: Just as chemical reactions in explosives can produce gas pressures

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far exceeding the equilibrium pressure of a system, the chemical reactions of doping sources such as P 205' POCI3,

and PH3 + O2 with both oxygen and silicon can produce elemental phosphorus at a fugacity far exceeding that of a saturated solid solution. The phosphorus then diffuses into silicon in supersaturation. There is a fairly prevalent miscon­ception, seen particularly commonly in reports dealing with annealing of implanted silicon, that the fraction of atoms above the solubility limit cannot diffuse. The truth is, indi­vidual atoms execute random walk without knowing the sol­ubility limit; otherwise phenomena such as precipitation could not take place. The fraction of atoms above solubility cannot move only when they are frozen at a sufficiently low temperature, or when they have agglomerated into a second phase-the precipitate phase. We believe that under typical conditions, phosphorus diffuses in the surface region of the silicon substrate under supersaturation, and is accompanied by a concurrent precipitation. This precipitation accounts for the fraction of phosphorus that is inactive as observed experimentally. The phase rule requires that, with the coex­istence of the precipitate phase, the concentration of either the dissolved phosphorus atoms or positive ions should be a constant; this accounts for the occurrence of a surface flat zone in the electron profile. There is direct experimental evi­dence of the formation of phosphorus precipitates during rather than after the diffusion (Sec. II B). The effect of dop­ing through a polysilicon can be explained as a suppression of phosphorus supersaturation, due to an enhanced precipi­tation at the polysilicon grain boundaries.

(2) Phosphorus diffuses in silicon via a dual mechanism, i.e., a mixture of vacancy and interstitialey mechanisms, as proposed earlier by one of us6 in the explanation of OED. (More extensive studies of OED are given in Refs. 7-11, 73.) This dual mechanism also offers an easy explanation of why the effective diffusivity of phosphorus (and of boron as well) is more than one order of magnitude higher than that of arsenic. It is further assumed that, although the equilibrium ratio of the interstitialey concentration to the substitutional concentration is about 1 or slightly less, the rate of conver­sion from an interstitialey configuration to a substitutional configuration is small. This, as can be shown, leads to a kinked, two-zone concentration profile. The conversion from the interstitialey to the substitutional configuration is the second source of excess self-interstitials during the diffu­sion of phosphorus in silicon.

Provided phosphorus can exist in silicon as an intersti­tialey, a phosphorus atom would quite naturally choose to enter the silicon lattice as an interstitialey rather than as a substitutional. To enter the lattice as an interstitialey, all it has to do is to get into an adjacent lattice site by sharing it with the host atom already there. (See Fig. 3.) The phos­phorus interstitialey so formed immediately becomes mo­bile, until it converts into a substitutional by ejecting a sili­con self-interstitial. In contrast, to enter the silicon lattice as a substitutional, the phosphorus atom must wait for a va­cancy to come along; it still cannot move until another va­cancy comes along to pair with it as an E center. (See discus­sion in Sec. III B.)

6920 J. Appl. Phys., Vol. 54, No. 12, December 1983

on

Legend

IX> Phosphorous Atom

• Silicon Atom

7

(a)

I-

(b)

®e Phosphorous Interstitialcy

•• Self-Interstltialcy

o Vacancy

®-O E-Center

FIG. 3. Conceptual picture of (a) the generation of a silicon self-interstitial and (b) the consumption of two vacancies as a phosphorus atom enters the silicon lattice substitutionally.

(3) Supersaturation of self-interstitials: The existence of a significant supersaturation of self-interstitials or vacancies is necessary to explain the emitter-push effect. We favor a supersaturation of self-interstitials in order to be consistent with experimental evidence. Moreover, as discussed in Sec. III B, diffusion of phosphorus via an interstitialey mecha­nism will surely lead to a generation, and hence a supersatur­ation of self-interstitials; in contrast, diffusion of phosphorus via a vacancy mechanism results in a net consumption rather than a generation of vacancies, and may well lead to an un­dersaturation rather than a supersaturation of vacancies. There are at least four possible sources of interstitial genera­tion: The precipitation of interstitia ley phosphorus, the con­current thermal oxidation of silicon, (Sec. II C) the displace­ment of lattice atoms when phosphorus atoms enter the silicon substrate (Fig. 3), and the conversion of an intersti­tialey phosphorus to a substitutional phosphorus. There are two ways in which the conversion may be accomplished; they may be expressed as follows:

and

k,

S+Pj~Ps +1, k' ,

k' 2

(4.1)

(4.2)

In the above expressions, S, V, I, Ps ' and P j denote, respec-

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tively, the silicon lattice atom, the vacancy, the self-intersti­tialcy, the substitutional, and the interstitialcy phosphorus, respectively. The first reaction will produce excess self-inter­stitials and the second reaction will lead to a vacancy under­saturation.

It should be noted that, while an interstitialcy supersa­turation is required for the emitter-push effect, it is not re­quired for the explanation of the kinked two-zone diffusion profile, which comes as a consequence of a two-stream mechanism, provided, as we noted earlier, that the intersti­tialcy to substitutional conversion is sufficiently slow.

A complete description of the phosphorus diffusion then comprises the following set of nonlinear simultaneous equations: [We let the 5 symbols in Eqs. (4.1) and (4.2) to also stand for their respective concentrations]

aps = ~ (D aps) at ax s ax

- PPTs + (k) + k2V) Pj - (k;I + k~) Ps ; (4.3)

aPj = ~ (D apj) at ax I ax

-PPTj +(k; I+k~)Ps -(k) +k2V)Pj; (4.4)

aI

at

a21 D[ - + kIP. - k )' P I ax2 I s

- k 3(IV -I*V*) +PPTj ;

av =Dv a2 v at ax2

(4.5)

+k~Ps -k)PjV-k3(IV-I*V*)+PPTs · (4.6)

An expression for S is not needed since it is nearly constant (at 5.5 X 1022 cm -3). The terms PPTj and PPTs denote local rates of precipitation by the agglomeration of phosphorus interstitialcies and substitutionals, respectively. The kinetics of precipitation is a very complex problem, and requires knowledge about the density of precipitates as well as the kinetics of nucleation; the importance of the role of nuclea­tion diminishes as time becomes longer. If the precipitation takes place during cooling or during annealing, PPTj and PPTs may be obtained from Ham's theory of precipitation, 82

the applicability of which has been shown in the case of the precipitation of oxygen. 72,83,84 But the precipitation of phos­phorus has been shown42 to be concomitant with the diffu­sion of phosphorus into the silicon lattice; this is also expect­ed from our model of a supersaturation of phosphorus forced into the silicon lattice during the deposition process. We be­lieve that this condition is better approximated by a quasis­teady state diffusion-limited growth of precipitates. Then it can be shown that, if the precipitates are assumed to be spherical, the rate of depletion of interstitialcy phosphorus by precipitation is given by

PPTj = 2rrNv::'2 [2Dj (Pj - P r)] 3/2t ) /2 , (4.7)

where N is the density of precipitates and v m is the molecular volume of SiP. If the precipitates are spheroidal, the result will differ only slightly by a geometrical factor. One can sim­ilarly write an expression for PPTs .

6921 J. Appl. Phys .• Vol. 54. No. 12. December 1983

The last term in Eq. (4.5) represents the generation of a self-interstitial when an interstitialcy phosphorus atom joins a SiP precipitate. The last term in Eq. (4.6) represents the generation of a vacancy when a substitutional phosphorus atom joins a precipitate; the substitutional phosphorus atom can move only when paired with a vacancy, which is released when the phosphorus atom precipitates out.

v. CONCLUSION

In this paper we have reviewed various phenomena as­sociated with the diffusion of phosphorus in silicon. In parti­cular, we have shown that the majority of the assumptions and expressions given in the literature (i.e., the SKC, Y, and Ff models) are neither self-consistent nor in agreement with many physically observed effects and data. Moreover, first order calculations based on the correct approximations show that the E-center mechanism cannot quantitatively produce or sustain enhanced diffusion of phosphorus or of other coupled species. We have shown that only a model of mixed vacancy and interstitialcy mechanisms can be consis­tent with all experimental observations.

Note Added in Proof We have recently completed this experiment. The results show that under a phosphorus "emitter" the diffusion of a phosphorus buried layer is en­hanced and the diffusion of an antimony buried layer is re­tarded [Po Fahey, R. W. Dutton, and S. M. Hu (to be pub­lished)]. Similar results have been independently obtained recently by R. M. Harris and D. A. Antoniadis (to be pub­lished). This indicates concurrent interstitial supersatura­tion and vacancy undersaturation during the diffusion of phosphorus into silicon.

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