1222 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 48, NO. 6, JUNE 2001

On the Iterative Schemes to Obtain Base DopingProfiles for Reducing Base Transit Time in a Bipolar

TransistorM. Jagadesh Kumar, Senior Member, IEEE,and Vijay S. Patri

Abstract—This paper shows that base doping profiles obtainedusing any iterative scheme for reducing the base transit timein bipolar transistors for a given neutral base width must takeinto account the heavy doping effects implicitly. Comparing our re-sults with those reported earlier, we demonstrate that if the heavydoping effects are not implicitly included in the iterative scheme itwill result in a completely different base doping profile leading toan overestimation of base transit time and underestimation of baseresistance.

Index Terms—Base doping profile, base transit time, bipolartransistors.

I. INTRODUCTION

I N silicon bipolar transistors, the base transit timeis oftenthe single largest contributor to the total delay time and

determines the transistor’s high frequency performance. Thedesign of the base doping profile for reducing, by keeping theintrinsic base resistance constant, has been studied extensivelyin literature. For example, while Wijnen and Gardner [1] andSuzuki [2] compared some easily realizable base profiles,Wintertonet al. [3] suggested an interesting and not so easilyrealizable base dopant distribution using a discretised relationfor and an iterative procedure. Although the final iterativescheme in [3] includes the effect of dopant induced bandgapnarrowing and dopant dependent diffusivity, both these factorsare not considered while developing the iterative scheme in [3].However, both these factors are important in submicron narrowbase bipolar transistors in which the base doping can far exceed10 /cm .

The objective of the present paper is, therefore, to show thatif the iterative scheme used for obtaining the base profile to re-duce implicitly includes the effect of dopant induced bandgapnarrowing and dopant dependent diffusivity, the resultant basedoping distribution will be different. By comparing our resultswith that of [3], we demonstrate that the iterative scheme sug-gested in [3] underestimates the base resistance and overesti-mates the base transit time since it does not implicitly take intoaccount the heavy doping parameters.

Manuscript received March 21, 2000; revised September 18, 2000. The re-view of this paper was arranged by Editor K. O.

M. J. Kumar is with the Department of Electrical Engineering, Indian Instituteof Technology, New Delhi 110 016, India (e-mail: mamidala@ieee.org).

V. S. Patri is with DSP Group, Texas Instruments (India) Ltd., Bangalore 560017, India.

Publisher Item Identifier S 0018-9383(01)04222-8.

II. M ODELS USED IN THE ANALYSIS

A. Model for Base Transit Time

The base transit time in a n-p-n bipolar transistor is given byKroemer’s double integration relation as [4]

(1)

whereintrinsic carrier concentration;acceptor impurity concentration;minority carrier diffusion coefficient in the base;neutral base width.

By dividing the neutral base into sections of equal length,the discretised base transit time expression can be written as [3]:

(2)

where and are piecewise constant functions in theth section. To minimize with respect to , the first deriva-

tive of is set to zero, in which case (2) becomes

(3)

A fixed point iterative scheme for derived from theabove expression will converge if and only if the iteration func-tion is differentiable in the interval of interest, and the absolutevalue of its derivative is less than unity [5]. Therefore, we needto choose appropriate models for and in (3) to obtainan iterative scheme that converges.

B. Models for Heavydoping Parameters

The dependence of diffusion coefficient on doping con-centration in the th section of the base may be modeledusing an emperical fit to PISCES mobility data [6], [7], as

(4)

0018–9383/01$10.00 © 2001 IEEE

KUMAR AND PATRI: ITERATIVE SCHEMES TO OBTAIN BASE DOPING PROFILES 1223

where cm s and . Theintrinsic carrier concentration is expressed as a function of theapparent bandgap narrowing due to heavy doping effects as [8]

(5)

whereintrinsic carrier concentration in undoped silicon;dopant induced bandgap narrowing in theth sec-tion of the base;Boltzmann constant;temperature in degrees Kelvin.

The apparent bandgap narrowing may be modeled as [7], [9]

meV cm

meV cm (6)

Although there are alternative models for heavy doping pa-rameters [10], we have chosen the above simple models so thatthe iterative scheme converges. It is important to note that theabove models are simple and accurate enough to highlight whythe iterative scheme suggested in [3] does not give a correct basedoping distribution as discussed in the following sections.

III. I TERATIVE SCHEMES FOROBTAINING OPTIMUM BASE

DOPANT DISTRIBUTION

A. Both and Are Dopant Independent

While evaluating (3), Wintertonet al. [3] have assumed thatboth and are independent of and have suggestedthe following iterative scheme for for finding the optimumbase dopant distribution:

(7)

where and are given by (4)–(6).

B. Both and Are Dopant Dependent

In this case, by substituting (4)–(6) in (3), the following it-erative scheme can be obtained for the optimum base dopantdistribution

(8)

where;

;doping concentration in the th section re-sulting from the th iteration;dopant concentration resulting from the thiteration.

After constraining the maximum doping concentration to thevalue at the emitter edge of the base and the minimum dopingconcentration to the doping level at the collector edge of the base

Fig. 1. Base profiles obtained using the iterative schemes (7), (8) withN = 10 cm andN = 2 � 10 cm . The neutral basewidth is 0.1�m.

as suggested in [3], we have compared the iterative schemesgiven by (7), (8), as discussed below.

IV. RESULTS AND DISCUSSION

For a neutral base width of 0.1m, Fig. 1 shows both the basedoping profiles generated by the two iterative schemes (7), (8),using the diffusivity model of (4) and the bandgap narrowingmodel given by (6) for a peak base dopingcm at the emitter edge of the base and a minimum base doping

cm at the collector edge of the base. Thesebase doping values are particularly chosen so that bandgap nar-rowing effects are predominant in the entire base region. Al-though identical diffusivity and bandgap narrowing models areused in (7) and (8), we note in Fig. 1 that the profiles gen-erated by the two iterative schemes are not identical. In ourscheme, the nonuniform profile is more steeper than that ob-tained using Winterton’s model. Further, the uniform base re-gion on the emitter side is 18.5% of base width in the case ofWinterton’s profile while it is only 4.5% of base width in ourcase. Therefore, the reduction in base transit time using our it-erative scheme results due to both the above factors i.e., a steepnonuniform profile region and a reduced (from 18.5 to 4.5%)length of the uniform profile region on the emitter side. Thebase transit time values calculated for each of these profiles arealso indicated in Fig. 1. It is clearly evident that if the heavydoping parameters are implicitly taken into account in the it-erative scheme, the resultant base doping profile is completelydifferent leading to a lower value of and a higher base re-sistance. On the contrary, the base doping distribution obtainedfrom the iterative scheme of [3] overestimates the base transittime and underestimates the base resistance.

V. CONCLUSION

In conclusion, we have shown that base doping profiles ob-tained using any iterative scheme for reducing the base transittime in narrow base bipolar transistors for a given neutral

1224 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 48, NO. 6, JUNE 2001

base width must take into consideration the effect of dopant de-pendent diffusivity and dopant induced bandgap narrowing. Bycomparing our results with that of Wintertonet al. [3], we havedemonstrated that an iterative scheme which does not implic-itly take into account the heavy doping parameters will resultin a totally different base doping profile leading to an overesti-mation of the base transit time and underestimation of the baseresistance.

REFERENCES

[1] P. J. Van Wijnen and R. D. Gardner, “A new approach to optimizing thebase profile for high speed bipolar transistors,”IEEE Trans. ElectronDevice Lett., vol. 11, pp. 149–152, Apr. 1990.

[2] K. Suzuki, “Optimum base doping profile for minimum base transittime,” IEEE Trans. Electron Devices, vol. 38, pp. 2128–2133, Sept.1991.

[3] S. S. Winterton, S. Searles, C. J. Peters, N. G. Tarr, and D. L. Pulfrey,“Distribution of base dopant for transit time minimization in a bipolartransistor,”IEEE Trans. Electron Devices, vol. 43, pp. 170–172, Jan.1996.

[4] H. Kroemer, “Two integral relations pertaining to electron transportthrough a bipolar transistor with a nonuniform energy gap in the baseregion,”Solid State Electron., vol. 28, pp. 1101–1103, 1985.

[5] J. Stoer and R. Bulirsch,Introduction to Numerical Analysis, 2nded. New York: Springer-Verlag, 1993, p. 264.

[6] D. Burke and V. de la Torre, “An empirical fit to minority carrier mobil-ities,” IEEE Electron Device Lett., vol. EDL-5, p. 231, 1984.

[7] T. C. Lu and J. B. Kuo, “A closed form analytical BJT forward transittime model considering bandgap narrowing effects and concentrationdependent diffusion coefficients,”Solid-State Electron., vol. 35, pp.1374–1377, 1992.

[8] M. J. Kumar and K. N. Bhat, “The effects of emitter region recombina-tion and bandgap narrowing on the current gain and the collector lifetimeof high voltage bipolar transistors,”IEEE Trans. Electron Devices, vol.36, pp. 1803–1810, Sept. 1989.

[9] J. W. Slotboom and H. C. de Graaff, “Measurement of bandgapnarrowing in Si bipolar transistors,”Solid-State Electron., vol. 19, pp.857–862, 1976.

[10] D. B. M. Klassen, J. W. Slotboom, and H. C. de Graaff, “Unified ap-parent bandgap narrowing inn- andp-type silicon,”Solid-State Elec-tron., vol. 35, pp. 125–129, 1992.

M. Jagadesh Kumar (SM’99) was born in Mami-dala, Nalgonda District, Andhra Pradesh. He receivedthe M.S. and Ph.D. degrees in electrical engineeringfrom the Indian Institute of Technology, Madras.

From 1991 to 1994, he did his Post-Doctoralresearch in modeling and processing of high-speedbipolar transistors with Prof. David J. Roulston atthe University of Waterloo, Waterloo, ON, Canada.During his stay at Waterloo, he also collaboratedwith Prof. Savvas G. Chamberlain on amorphoussilicon TFTs. From July 1994 to December 1995, he

first taught at the Indian Institute of Technology, Kharagpur, and later movedto the Indian Institute of Technology, Delhi, where he was made an AssociateProfessor in the Department of Electrical Engineering in July 1997. Hisresearch interests are in VLSI device modeling and simulation, IC technology,and power semiconductor devices.

Vijay S. Patri , photograph and biography not available at the time of publica-tion.