*Corresponding author. Tel.: #617-253-6907; fax 617-253-6782.

E-mail address: alsmith@mit.edu (A.L. Smith)

Physica B 273}274 (1999) 358}362

Transition metal defect behavior and Si density of states in theprocessing temperature regime

A.L. Smith!,*, S.T. Dunham", L.C. Kimerling!

!Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA"University of Washington, Box 352500, Seattle, WA 98195, USA

Abstract

In order to make predictive models of transition metal gettering during semiconductor processing, a completeunderstanding of the process variables in high temperature ranges is essential. These variables are the internal getteringsite density and capture radius, the intrinsic metal solubility, silicon doping level, the band gap, the e!ective density ofstates of the conduction and valence bands, and the transition metal defect level position in the gap. The least understoodof these parameters is the temperature dependence of the transition metal defect level position. The work of Gilles et al.and McHugo et al. demonstrates that the doping enhancement of the solubility of Fe in p-type silicon vanishes attemperatures above 10003C. They model this behavior by proposing movement at high temperature of the defect level forinterstitial Fe from within the energy gap into the valence band. We explore the available models for Si e!ective density ofstates as a function of temperature and generate a third density of states model based on 0 K ab initio band structurecalculations with the temperature-appropriate carrier occupations given by Fermi}Dirac statistics. We also consideruncertainty in E

Gin the processing temperature regime. We show that uncertainties in the Si intrinsic properties database

in the processing temperature regime can account for the available dopant-enhanced solubility data by assuming thatET

remain at a constant fraction of EG. To quantitatively model gettering processes at high temperatures, more reliable

estimates are needed for the densities of states of the conduction and valence bands, EG

and the behavior of defect levelsas temperature rises. ( 1999 Elsevier Science B.V. All rights reserved.

Keywords: Si; E!ective density of states; Fe; Defect level; Processing temperature

1. Introduction

The presence of transition metals in silicon deviceprocessing has deleterious e!ects on performance, yieldand reliability. In silicon photovoltaics, we are concernedwith the impact transition metals have on minority car-rier lifetime and thereby e$ciency. For integrated circuitapplications, we are concerned with gate oxide integrityand device parameter homogeneity across a wafer. Forthis reason, a quantitative understanding of transitionmetal equilibria and kinetics at high temperature is

needed in order to design gettering processes throughaccurate simulation. We explore the case of interstitial Fe(Fe

*), however, the method is quite general and can be

extended to other deep-level impurities in a semiconduct-ing host.

Researchers measuring Fe-dopant-induced solubilityenhancement in p-type Si have found the enhancement tobe less than they expected at temperatures &10003Cand have proposed an instability of the well-known Fe

*defect level (E

T) as these high temperatures are ap-

proached [1,2]. However, in order to infer the behaviorof Fe in the processing temperature regime, we "rst needto complete our understanding of silicon at these temper-atures.

In order to model dopant-enhanced solubility of de-fects in Si quantitatively, we need to understand the

0921-4526/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 4 7 6 - 7

temperature dependence of the various defect levels andSi parameters with precision. With regard to the defectFe, E

T, the Fermi level (E

F), and the intrinsic carrier

concentration (n*) are the controlling parameters. The Si

materials parameters, EF

and n*, in turn depend on the

e!ective density of states (DOS) of the conduction andvalence bands (N

Cand N

V, respectively) and the

semiconductor band gap (EG). In the work presented

here, we demonstrate that uncertainties in the Si intrinsicproperties database in the processing temperature regimecan account for the available dopant-enhanced solubilitydata by assuming that E

Tremain at a constant fraction

of EG.

2. Doping-enhanced solubility

Heavy p-doping increases the solubility of donor andinterstitial transition metal impurities. This e!ect isdriven by electron}hole equilibria and defect pairing toionized acceptors which we model with defect reactions[3}7]. The intrinsic solubility of Fe in Si [8] representsthe undoped reference level. The solubility in silicon isgoverned by the equilibrium between Fe from an externalsource with Fe

*donors in solid solution in the Si matrix.

In p-type silicon, increased ionization of Fe*and pairing

of Fe`*

to ionized acceptors (A~4) contributes to solubility

enhancement.The quantitative calculation for the ratio of ionized to

neutral charge states for Fe*

is given by Fermi}Diracstatistics for the defect level. The equilibrium constant forionized Fe`

*)A~

4pair formation is given by Kimerling et

al.[9]. Increased p-type doping increases dopant-en-hanced solubility by moving E

Fwith respect to the Fe

*ionization level. As more Fe

*is positively ionized, pairing

becomes more likely.

3. Si in the processing temperature regime

Our interest in Si can be categorized into three temper-ature regimes: measurement, device operation, and pro-cessing. Most of our theoretical and experimental under-standing of silicon is from very low temperatures, ap-proximately 0 } 400 K, in the measurement temperatureregime. This temperature regime overlaps reasonablywith the temperature regime of device operation, pre-dominantly between 200 and 500 K, providing accuratedata for device simulation. For the case of the muchhigher temperatures of the processing temperature re-gime, approximately 700 } 1300 K, simulations currentlyrely heavily on extrapolations from the measurementtemperature regime. In order to predict the interaction ofdefect levels with E

F, we need to understand the variation

of NC, N

V, and E

Gat high temperature. It is necessary to

extrapolate currently available values of these para-

meters well beyond the range of available measurements.In doing so, we gain an appreciation for how the uncer-tainty at processing temperatures will a!ect our calcu-lations for dopant-enhanced solubility of Fe in Si.

The available models for NC

and NV

show disagree-ment even in the measurement temperature regime. Theroutinely used ¹3@2 model found in classics such as Sze'sPhysics of Semiconductor Devices [10] is based on a para-bolic band approximation. Si, however, is known todeviate from this approximation even at low temper-atures. The valence band most strongly de"es this cat-egorization due to a lack of parabolicity in energy, an-isotropy in the constant energy contours and the e!ect ofspin}orbit coupling. At processing temperatures, theparabolic approximation is inadequate for both N

Cand

NV. A more realistic empirical "t to data up to 500 K is

provided by Green [11]. While the Green relation isa better "t to the experiment, it is only valid to 500 K andit is not valid when the Boltzmann approximation breaksdown, such as in the case of degenerately doped material.This limitation is due to the fact that density of statese!ective mass is, in general, both temperature and energydependent (see, for example Ref. [12]).

In order to generate a more physically reasonablemethod for E

Fdetermination in the processing temper-

ature regime, we determine the 0 K DOS from "rstprinciples using density functional theory within the localdensity approximation (LDA) with the Vienna ab initosimulation package (VASP) [13,14]. The LDA bandstructure-generated DOS has been shown to correlatewith experiment better than the parabolic band model[15]. VASP numerically solves the LDA Kohn}Shamequations using ultra-soft pseudopotentials [16,17] anda plane wave basis set. A cut o! energy of 300 eV wasused. The exchange and correlation functional was thatof Ceperley and Alder [18] as parameterized by Perdewand Zunger [19]. k-space sampling was performed withthe method of Monkhorst and Pack [20] using an18]18]18 grid. k-space integrations were performedusing the linear tetrahedron method including correc-tions according to BloK chl et al.[21]. We perform a rigidenergy shift to correct for E

Gwhich is well known to be

underestimated by LDA. We then use Fermi}Dirac stat-istics (F-D) with numerical integration and E

G(¹) to

determine EF(¹) as dictated by the charge neutrality

condition. Certain limitations of the calculation includeneglecting spin}orbit coupling and greater inaccuracy incalculating excited states. On the other hand, this methodcan be used in the degenerately doped regime withoutresorting to the parabolic band approximation. In Fig. 1,we compare the calculated DOS with those given bya parabolic approximation to illustrate that they are verydi!erent even in the regions near the valence and conduc-tion band edges. In fact, within &k¹ of the band ex-trema, the DOS varies approximately linearly with E asopposed to the E1@2 variation for the parabolic model.

A.L. Smith et al. / Physica B 273}274 (1999) 358}362 359

Fig. 1. Comparison of the density of states as a function ofenergy obtained ab initio to the density of states for a parabolicapproximation to the valence and conduction bands.

Fig. 2. Calculated values of EF

as a function of temperature forthree di!erent density of states approximations in the intrinsic(upper three curves) and heavily doped (bottom three curves)regimes. The dashed lines corresponds to E

Fusing the ab initio

density of states the solid lines are determined with Green'sN

Cand N

Vand the dotted lines were obtained with the

¹3@2 using the data from Sze.

Fig. 3. Comparison of calculated dopant-enhanced solubility ofFe in Si as a function of temperature using di!erent density ofstates approximations while assuming E

Tremains at a constant

fraction of EG. The doping level is N

A"1.5]1019 which corres-

ponds to the doping level of the Si investigated by McHugo et al.(solid squares denote their data points). With the exception ofthe "ne dotted line, all calculations were performed usingFermi}Dirac statistics to calculate E

F. A comparison between

the "ne dotted line (EF

obtained with the Boltzmann approxi-mation for the ¹3@2 model) and the dashed double-dot lineillustrates the error generated using the Boltzmann approxima-tion. The solid line represents the intrinsic Fe solubility.

There are additional concerns with DOS modeling atprocessing temperatures. All three of these models rely onthe assumption that the band structure is not signi"-cantly altered as temperature increases. In the processingtemperature regime, we need to consider the in#uence ofphonons on the band structure. Other factors that alsoneed consideration are thermal expansion and the in#u-ence of the energy level occupations found at high tem-perature. The method we have developed for modelingDOS as a function of temperature is a physically reason-able starting place for determining E

Fin the processing

temperature regime, but it is by no means an endingpoint.

In Fig. 2, we compare EF

for both the case of intrinsicSi and for Si with an acceptor concentration (N

A) of 1019

cm~3 using the three DOS models discussed. The esti-mate using the ab initio calculated DOS falls between thecurves calculated with N

Cand N

Vby the ¹3@2 model and

the extrapolation of Green's relations. It is important tonote that E

Fis quite sensitive to the DOS model used,

varying by more than 0.1 eV at the highest temperaturesshown.

Since EF

is not a measurable quantity, we look atestimates of dopant-enhanced solubility generated usingthe di!erent DOS models. Fig. 3 displays calculations ofFe solubility for N

A"1.5]1019 cm~3 with each DOS

model assuming ET

remains at a constant fraction of thegap. The solid line shows the intrinsic Fe solubility. Thecurve predicting the greatest dopant-enhanced solubilitywas generated using again the ¹3@2 model, however forthis case, the Boltzmann approximation is used to deter-mine E

F. The signi"cant di!erence between this curve

and that of the same DOS model with F-D integralsdetermining E

Fdemonstrates the importance of using

the appropriate carrier statistics. Due to the extremelyhigh doping level, F}D are needed.

Also in Fig. 3, we overlay the data of McHugo et al.The data is "t within error by the calculations using ourDOS method and the ¹3@2 model, with no instability ofET

required. The calculation using the Green extrapola-

tion underestimates for the data point at 10003C. The¹3@2 model when used with the Boltzmann approxima-tion overestimates the concentration at 11003C.

The last parameter we require knowledge of in theprocessing temperature regime is E

G. Again, we have no

reliable expression at these extreme temperatures. Thatmost commonly used was derived by Varshni [22] with"tting parameters valid to 750 K extracted by Alex etal.[23]. We have used this relation in the calculationsabove, but note that extrapolations of empirically basedpolynomials beyond their regime of validity are notori-ously unreliable.

360 A.L. Smith et al. / Physica B 273}274 (1999) 358}362

Fig. 4. Comparison of calculated dopant-enhanced solubility ofFe in Si for di!erent temperature dependencies of E

G. E

Twas

held a constant fraction of the gap and we use the ¹3@2 model fore!ective density of states. The dashed line was calculated usingthe Varshini relation for E

Gwith the parameters of Alex et al.

while the dotted curve calculation relies on the extrapolation ofa semi-empirical model for E

G.

Fig. 5. Comparison of calculated dopant-enhanced solubility ofFe in Si for di!erent temperature dependencies of the Fe defectlevel, E

T, in the gap. Dotted line was calculated assuming that

ET

is a constant distance from the valence band edge. For thedot-dashed curve, E

Twas held a constant fraction of the gap.

The dashed curve was obtained by assuming that ET

is a con-stant distance from the conduction band edge. The solid linedenotes intrinsic solubility.

A semi-empirical expression has been developed[24,25] and "t [26] over the 0 } 300 K temperaturerange. Extrapolations to higher temperatures are consis-tent with extrapolations of a linear "t valid up to415 K[10]. The extrapolations of these relations from415 to 750 K diverges signi"cantly from the expressionbased on Varshni's model. Nevertheless, we calculatedopant enhanced solubility comparing the two sets ofEG(¹) to observe the impact that uncertainty in E

Ghas

on our predictions. Fig. 4 shows these results where againwe assume E

Tremains at a constant fraction of the gap

and we use the ¹3@2 model for NC

and NV. Note that for

the data point at 11003C, the semi-empirical EG

calcu-lation is no longer within error.

4. Defect properties in the processing regime

In addition to the temperature dependence of funda-mental properties of the host material, quantitativeprediction of dopant-enhanced solubility for a defectrequires the temperature dependence of (i) any defectlevel in the gap and (ii) relevant parameters for reac-tions of the defect with other species in the host matrix.For the case of Fe in p-type Si, these parameters areET

and EB.

For the dopant-enhanced solubility calculations con-tained in this paper, we focus on temperatures above7003C where pairing has negligible impact on the solubil-ity enhancement. Nevertheless, to generate a completeunderstanding of the dopant-enhanced solubility at alltemperatures, the exact temperature dependence ofEB

should be determined.Motion of E

Twithin the gap will a!ect the ionization

statistics and thereby the dopant enhanced solubility ofFe in p-type Si. The defect level of a species which is very

foreign to its host matrix is expected to behave indepen-dently from the band edges.

In Fig. 5 we show calculations based on three cases ofET

behavior as temperature is increased and EG

narrows:(i) E

Tstays at a constant fraction of E

G, (ii) E

Ta "xed

energy from the valence band edge, and (iii) ET

a "xedenergy from the conduction band edge. Defect level posi-tion does a!ect the results, as expected, however, only thecase with E

Ta constant fraction of E

Gis within error of

the experimental data.

5. Conclusion

The commonly used ¹3@2 model is not accurate even inthe device operation regime and the available relations ofGreen do not extend past 500 K. We have constructeda DOS model using ab initio calculations and temper-ature-appropriate Fermi}Dirac statistics to generatea more physically motivated extrapolation of DOS intothe processing temperature regime. Nevertheless, muchremains to be explored about high temperature e!ects onthe band structure. The available data of dopant-en-hanced solubility of Fe in Si can be modeled within errorassuming E

Tremains at a constant fraction of E

Gby

either the ¹3@2 model for NC

and NV

or the ab initioDOS. There is no evidence at this time for instability ofET, however more accurate data for the Si DOS and

EG

in the processing temperature regime will shed lighton the temperature dependence of E

T.

Acknowledgements

The authors wish to thank K. Wada and M. Lipson forhelpful discussions. This work was supported by NREL

A.L. Smith et al. / Physica B 273}274 (1999) 358}362 361

under contract no. XD-2-11004-4 and by the Si-WEDS(Silicon Wafer Engineering and Defect Science) Consor-tium. One of the authors (ALS) is grateful for fellowshipsupport from the DOE Fellowship for Integrated Manu-facturing.

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