UNIFIED MODEL FOR BIPOLAR TRANSISTORS INCLUDING THE VOLTAGE AND CURRENT DEPENDENCE OF THE BASE AND COLLECTOR RESISTANCES AS WELL AS THE BREAKDOWN LIMITS

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UNIFIED MODEL FOR BIPOLAR TRANSISTORS INCLUDING THE VOLTAGE AND CURRENT DEPENDENCE OF THE BASE AND COLLECTOR

RESISTANCES AS WELL AS THE BREAKDOWN LIMITS

F. Hébert, D. Roulston

To cite this version:

F. Hébert, D. Roulston. UNIFIED MODEL FOR BIPOLAR TRANSISTORS INCLUDING THE

VOLTAGE AND CURRENT DEPENDENCE OF THE BASE AND COLLECTOR RESISTANCES

AS WELL AS THE BREAKDOWN LIMITS. Journal de Physique Colloques, 1988, 49 (C4), pp.C4-

371-C4-374. �10.1051/jphyscol:1988477�. �jpa-00227975�

JOURNAL DE PHYSIQUE

Colloque C4, suppl6ment au n09, Tome 49, septembre 1988

UNIFIED MODEL FOR BIPOLAR TRANSISTORS INCLUDING THE VOLTAGE AND CURRENT DEPENDENCE OF THE BASE AND COLLECTOR RESISTANCES AS WELL AS THE BREAKDOWN LIMITS

F.

HEBERT*

and D. J. ROULSTON

E l e c t r i c a l E n g i n e e r i n g D e p a r t m e n t , U n i v e r s i t y o f Waterloo, Waterloo, O n t

.

^{N2L-3G1, Canada}

" ~ v a n t e k I n c . , Advanced B i p o l a r P r o d u c t s , 39201 C h e r r y S t r e e t , Newark, CA 94560, U.S.A.

~gsume'

-

Un modale pour transistor bipolaire, qui tient compte des variations des rgsistances de base et de collecteur avec la tension gmetteur-base, la tension collecteur-base et le courant collecteur, ainsi que la dgpendence en tension de la charge de base et le claquage par

avalanche, est pre/sent6. Un bon accord entre les simulations par ordinateur et exp&rimentations est obtenu.

Abstract

-

A unified bipolar transistor model, which takes into account the variation of the base and collector resistances with emitter-base voltage, collector-base voltage and collector current, as well as the voltage dependence of the base charge and the avalanche breakdown, is presented.

The agreement between computer simulations and experiments is shown to be very good.

1

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INTRODUCTION

High performance bipolar transistors typically have narrow base widths and operate at high current densities. These two effects tend to reduce the useful operating region of the transistor since punch-through and avalanche effects reduce the maximum voltage, and saturation and ohmic quasi-saturation effects increase the minimum voltage limit for high current operation. Also, the base resistance varies with operating voltage and current which affects the device performance.

Accurate and physical transistor models are required in order to properly predict the device performance. A unified model which takes into account the base punch-through and avalanche breakdowns [I], as well as the base and collector conductivity modulation is presented.

2

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ASSUMPTIONS

The modeling of the voltage and current dependence of the base and collector resistances is based on the solution of the carrier distribution within the base and collector regions.

For an NPN transistor, with the doping profile shown in Fig. la, it is assumed that when in saturation, the electrons injected from the forward biased base- emitter junction will spread into the collector region (Fig. lc). The base- collector space charge layer will then effectively disappear, and due to charge neutrality in the collector region, the hole concentration (p(x)) may be solved from the knowledge of the electron distribution (n(x)) and the epitaxial collector doping n (x) =p (x) +Nepi.

The base and collector conductivity modulation is then solved from n(x) within the transistor. Some workers have solved for the collector conductivity modulation through a solution of p(x) (holes) as a function of the base- collector voltage (Vbc) [2-51 but this neglects the effect of the base-emitter voltage (Vbe) and the fact that the solution of p(x) is difficult [5] due to the uncertainty in defining the exact point of hole injection in the collector under high level injection (HLI). An other advantage in considering n(x) is that the base conductivity modulation is solved automatically (unified model).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988477

JOURNAL DE PHYSIQUE

h

DEPLETED REGION

-

Fisure 1

-

Details of the emitter, base and epitaxial collector regions. a) typical doping profile. Carrier distribution in the normal mode (b) and in saturation (c)

.

3

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CARRIER DISTRIBUTION

For a reverse biased base-'collector junction (Fig. lb), the collector current may be computed by assuming negligible drift current:

where q is the electronic charge, Ae is the effective emitter area, pn is the average electron mobility, Vt is the thermal voltage (kT/q), Wb is the neutral base width, K is unity under low level injection (LLI) conditions and K is 2 in HLI (since Ic is made up of equal drift and diffusion components). n ( 0 ) is the injected electron density in the base at the emitter-base space chgrge layer edge (Fig. 1). It is defined as, for LLI and HLI conditions:

where n. is the intrinsic carrier concentration, Nao is the effective doping level at the emitter-base scl edge (Fig. la) and is obtained from CV

measurements or from device simulations.

In the saturation mode of operation, the base-collector junction no longer acts like a sink for the electrons, and free carriers will extend from the base into the collection region. The electron distribution may be solved from 1,:

The term in square brackets takes into account the gradual transition between LLI and HLI (K above). Nhli is the effective doping level at which HLI begins, n (Wb) is the electron carrler density at the base-collector junction equal to

(bn(wb)t~epi)

.

The electron distribution is n(x) = np(0)

-

(dn/dx) (x) where

dn/dxY I,/( [I + (n P (W t3 )/(np(Wb)+Nhli) ) I ( 4

pn

Vt) ) 4 - COLLECTOR RESISTANCE

The width of the collector region over which the conductivity is modulated by free carriers is defined as xo, as shown in Fig. lc. It is defined as:

The conductivity modulated collector resistance (Rcm) is

where A is the emitter area modified to include lateral current spreading within ?he collector region [6] and

mc

is the field dependent electron mobility [5]. Nc is the average concentration over the modulated region of width xo, compute8 using (4) above with x=(Wb+(x0/2)).

The voltage dependence of Rc is evaluated by assuming a one-sided abrupt junction

Rc = Rcm ( (Wepi'Wscl) /Wscl) + Rcext (8 1 where Wscl is computed from integration of Poisson's equation and from the knowledge of Nepi, (Wscl<Wepi) and Rcext is the current and voltage independent resistance.

5

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BASE RESISTANCE

The base resistance is inversely proportional to the base conductivity and to the base width. HLI and base widening effects will therefore reduce the base resistance. For a given increase in base carrier concentration (proportional to an increase in base conductivity) defined as Nbmr and a given increase in base width, defined as Wk, the resulting reduction in base resistance is

where Rb io is the low current intrinsic base resistance and N is the average base doping. (9) may be expanded to include the base X8ening due to saturation effects and Kirk effect, and conductivity modulation due to HLI and Early effect (variation of Qbc in Fig la). This is discussed in detail in [ 7 ] . 6

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BREAKDOWN AND CROWDING EFFECTS

The avalanche and base punch-through breakdown effects are based on the work published in [I]. Crowding effects are included by using distributed

equivalent circuits.

7

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MODEL PARAMETERS AND EXPERIMENTAL EVALUATION

Most of the model parameters may be obtained from measurements (sheet doping profile, sheet resistances, CV, etc.) or from device simulations. N may be estimated as half the collector doping (Ne i) [2]. For devices witkliightly doped bases, HLI will occur first in the bgse and the value of Nhli will tend to the average base doping.

The model has been implemented in the WATAND circuit simulator 181, and two different bipolar transistors displaying quasi-saturation effects have been considered. As shown in Figures 2 and 3, the agreement between simulations and experiment is very good. Figure 4 shows a comparison of the simulated and computed base resistance (using BIPOLE [9]). The reduction in intrinsic base resistance is properly simulated. The evaluation of the avalanche and base punch-through breakdown model has been carried out in [I].

8

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CONCLUSIONS

A physical model for the variation of the base and collector resistances of bipolar transistors with operating current, base-emitter voltage and base- collector voltage, compatible with CAD programs, is presented. The model is based on the solution of the electron distribution within the base and collector regions under LLI, HLI or saturation conditions. The model has been implemented in the WATAND circuit simulator and the agreement between simulations and experiment is very good.

C4-374 JOURNAL DE PHYSIQUE

SIMULATIONS MEASUEMENTS

EMITTER

-

COLLECTOR VOLTAGE ( V ) COLLECTOR-EMITTER VOLTAGE ( V )

Fiaure 2

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Ic.vs Vce of a poly Fiuure 3

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I, vs Vce of a PNP device.

emitter transis or. The emitter Transistor and measurements are from area is 6x41 J.I~I'.' Base current [5]. Base current values of 10, 20, values of .I, - 3 , .5, . 7 and .9 mA. and 50 JJA.

p

^{2001 I I I}

10-4 10-3 lo-* lo-'

COLLECTOR CURRENT ( A

1

W

3

-1

a

I-

Fiqure 4

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Variation of the total base resistance with current for the poly emitter trans is to^:. WATAND simulations compared to BIPOLE simulations. The distributed model is used to indicate that current crowding is not important.

+- BIPOLE SIMULATIONS I SECTION Rbb MODEL A 5 SECTIONS Rbb MODEL

REFERENCES

F. ~dbert and D.J. Roulston, IEEE T. Elec. Dev., ED-34, (1987), 2323.

L. J. Turgeon et al. Proc. IEDM, (1980), 394.

J.R.A. Beale et al. Solid-State Elect., ll, (1968), 241.

A.W. Alden et al. Solid-State Elect., 2, (1982), 723.

G.M. Kull et al. IEEE T. Elec. Dev., ED-32, (1985), 1103.

P.R. Gray and R.G. Meyer, Analysis and Desian of Analoq Intearated Circuits, Wiley, (1977), 82.

F. ~gbert and D.J. Roulston, IEEE T. Elec. Dev. to be published.

WATAND circuit Simulator, Elec. Eng. U. of Waterloo, Ont. Canada.

D.J. Roulston, Proc. CICC, (1980), 2.