Solid-State Electronics Vol. 35, No. 11, pp. 15731577, 1992 Printed in Great Britain. All rights reserved

0038-l IO]/92 $5.00 + 0.00 Copyright 0 1992 Pergamon Press Ltd

THEORETICAL COMPARISON OF DONOR-DENSITY GRADED BASE AND UNIFORMLY DOPED BASE

IN Pnp AlGaAs/GaAs HETEROJUNCTION BIPOLAR TRANSISTORS

WILLIAM LIIJ

Central Research Laboratories, Texas Instruments, P.O. Box 655936, M/S 134, Dallas, TX 75265, U.S.A.

(Received 24 April 1992; in revisedform 18 May 1992)

Abstract-A theoretical analysis is developed to calculate both the base transit time and base sheet resistance of Pnp AlGaAs/GaAs heterojunction bipolar transistors with various base donor-density grading schemes. We demonstrate that, while for a given base transit time the donor-density graded base allow a thicker base layer than the uniform base, the graded base can potentially have higher base sheet resistance than that of the uniform base. Some care needs to be exercised in choosing a proper grading scheme to result in a minimum base sheet resistance for any given base transit time. If a proper grading scheme is used, the graded base should be used instead of the uniform base to simultaneously achieve the smallest rs and lowest R,,.

1. INTRODUCTION

Pnp AlGaAs/GaAs Heterojunction Bipolar Transis- tors (HBTs) for high frequency applications have recently received considerable interest[ l-41. In these transistors, the base transit time, ta, often dominates the overall emitter-collector transit time due to the low hole mobility in GaAs. It has been demonstrated that this transit time can be reduced by a base quasi-electric field established by grading the indium (or aluminium) composition in an InGaAs (or AlGaAs) base[3,5]. A popular alternative in estab- lishing a base electric field to reduce the base transit time is to grade the base donor density[2,6]. This investigation is concerned with the latter grading scheme and compares it to the uniform doping scheme.

If 7B were the only design criterion, then either the donor-density graded base or the uniformly doped base can be used by adjusting the base thickness appropriately to achieve the desired zg. However, in the design of Pnp HBTs, there exists another critical parameter besides 7B: the base sheet resist- ance, R,, . The base sheet resistance directly relates to both the intrinsic base resistance and the base contact resistance, and should be minimized to improve the device’s maximum oscillation frequency[7]. For a given 7,,, a donor-density graded base layer can be made thicker than a uniformly doped base layer due to the established base field in the graded base. This thicker base thus potentially lowers the base sheet resistance compared to the uniform base and is thus more desirable. However, R,, of a given base struc- ture depends on not only the base thickness, but also the doping concentration. In a donor-density graded

base, the part of the base adjacent to the collector is more lightly doped than NB max, the maximum doping concentration which is designed at the emitter edge‘of the base layer. Therefore, R,, of the graded base may actually be larger than that of a thinner base layer which is uniformly doped at Na,,,, despite the thicker base advantage enjoyed by the graded base.

In this investigation, a theoretical analysis is pre- sented to calculate the base transit times and base sheet resistances for both the uniformly doped base structures and the donor-density graded base struc- tures with various grading schemes. Since both tg and R,, are calculated and illustrated in the same figures, a decision can be reached in choosing the preferred doping scheme to result in smallest tg and lowest R,, .

These calculations demonstrate that, for a given 7g,

R,, of donor-density graded structures can either be larger or lower than that of the uniformly doped base structure, depending on the exact donor-density grad- ing scheme used. Consequently, if a proper grading scheme is used, a donor graded base should be used instead of the uniform base to simultaneously achieve the smallest 7B and lowest R,,.

2. THEORETICAL ANALYSIS

The governing equation for the base minority hole concentration, p(x), is given by[8]

J,, = w:(x) .P(x). 66) - qv&‘(x) y, (1)

where x is the distance into the base from the collector edge of the base (x = 0 at the base-collector junction, and x = W,, the base thickness, at the base-emitter junction.) In eqn (l), Jp is the minority

1573

1574 WILLIAM LIU

hole current flowing from the emitter to the collector and it is assumed to be constant throughout the entire base. c(x) is the built-in base electric field; it is to be determined shortly. pp” is the minority hole mobil- ity in the n-type base and V, is the thermal voltage. These quantities are functions of x because the donor density in the base, N,(x), varies with x to result in the base field. Because of the lack of experimental data, the minority hole mobility at a given base doping level is assumed to be the same as the majority hole mobility at the same base doping level, and is given as[9]

380

P,“(x) = (1 + 3.17 x lo-“Na(x))o.*~~ (2)

The solution of p(x) to the linear, first order differen- tial equation of eqn (1) is readily available[6,10]. After one determines an expression for p(x), the base transit time can then be given as[6]

4 WEI ZaE --

J s 0) dx

P 0

1 wB 1 X =_ s 5

i(x’) dx’dx - - V T 0 i(x) 0 P,NW>

(3)

where

c(x)=exp[l-Fdx]

= exp [S

1 d&(x) dx -v,dx . 1 (4)

Note that in Ref. [6], the base was assumed to be non-degenerate; therefore, c(x) was simply taken to be N,(x). However, since the base of HBTs under investigation is usually heavily doped, both the effects of bandgap narrowing and degeneracy of the density of states on the valence band energy

(EV) need to be addressed. From the experimental results of the bandgap narrowing due to heavy doping[ll], and the Joyce-Dixon approximation to the Fermi integral[l2], c(x) can be written as

l(x) = Ndx)exp

1.6 x 1O-8 - V,

.3&X5 (5) 1 where Nc is the density of states in the conduction band and An’s are the approximation coefficients found in Ref. [12]. The validity of the bandgap narrowing model used in this investigation[l 1] to calculate the base electric field has been discussed in Ref. [2]. Once Na(x) is known, the base transit time can then be determined from eqns (3)-(5). In addition, the base sheet resistance of the donor- density graded base is given by

1 &a =

4 I

ws (6)

P:(X). b(x) dx 0

where pr is the majority electron carrier mobility in the n-type base; it is given by[9]

7200

P’(x) = (1 + 5.51 x 10-‘7Nr,(x))o~233 ’ (7)

In the following analysis, ra and RSH for three possible kinds of donor-density grading schemes are calculated. These schemes are referred to as the linear, exponential, and complementary exponential grading, respectively

(8)

(9)

Cornplemenlnry exponentialm

Fig. 1. The donor density profiles of various donor-density grading schemes discussed in this analysis. See eqns (8)-(10). In the case of uniformly doped base (not shown here), the base is assumed to be doped

at the maximum possible base doping concentration, Ns,.

Base transit time in HBTs

I I I I I I I I I I I I I I I I,, ,

= linear grading

: exponential grading

uniform (no grading)

I I I” II II 11 8 II r 0 ‘I” ( I I

0 1 base trinsit time 7 psec )

4 5

Fig. 2. The base thickness for each grading scheme as a function of the base transit time. This calculation assumes ns min = 8 x lO”/cm and Ns_ = 5 x 10’*/cm3.

N,(x) = NB,, +NB,,-NB,,

x exp[ln(+) . y]. (lo)

N Bmax and Ne,, are the maximum and minimum base doping in the graded base, respectively, with the maximum base doping located in the emitter and to establish the base electric field in the direction to sweep the minority carriers through the base. These three donor-density grading schemes are illustrated in Fig. 1, with NBmio = 8 x 101’/crn3, and N BmaX = 5 x 10L8/cm3.

In order to fairly compare the donor-density graded and the uniform base structures, the constant base doping in the uniform base structure is taken to

beNa_, the maximum base doping in the case of the donor-density graded base structure. Consequently, from equations (3) and (6), the base transit time and the base sheet resistance for the uniform base HBTs are given as

10’

IO’

1575

Note that since we will compare the base sheet resistances at a given base transit time, the base thickness for the uniformly doped base ( Walutiorm) is thinner than that of the donor-density graded base

(wB)*

3. RESULTS AND DISCUSSION

The values of rr, and R,, for HBTs with the three grading schemes of eqns (8)-(10) are calculated. Figures 2 and 3 illustrates the calculated base thick- ness and base sheet resistance, respectively, for a given base transit time. The base doping level is assumed to vary from Netin = 8 x 10”/cm3 at the

v linear grading

1 \ - exponential grading

--c- complementary exponential grading L

- uniform (no gilding)

i

1-

0 I 2 3 4 5 base transit time ( psec )

Fig. 3. The base sheet resistance for each grading scheme as a function of the base transit time. This calculation assumes NBmin = 8 x lO”/cm and NBmu = 5 x lO’*/cm’.

1576 WILLIAM LnJ

collector end to NBmPr = 5 x 10’8/cm3 at the emitter end. (For the uniform base, the doping is assumed to be constant at Ns,,, = 5 x lO’*/cm’.) As expected, Fig. 2 demonstrates that, at a specific value of base transit time, the base thickness is thinner for the uniform base structure compared to graded base structures. Furthermore, since the exponential grad- ing results in the largest average base field among the three grading schemes, this grading scheme allows a thickest base layer when compared at a given transit time.

One usually associates a decrement of RSH with a thicker base layer. However, as demonstrated in Fig. 3, for a given base transit time, R,, is actually worst in the case of exponential grading despite the fact that such grading scheme allows the thickest base layer. This results because R,, is a function of both the base thickness and the base doping profile. As shown in Fig. 1, Ns (x) decreases rapidly from Ns max to NBmio in the exponential grading scheme. Such an exponential decrease of doping density signifi- cantly increases the base sheet resistance that can be obtained otherwise. When both factors of base dop- ing and thickness are considered, Fig. 4 shows that the complementary exponential grading gives rise to the smallest value of RSH for any given base transit time.

The observation that the complementary exponen- tial grading scheme results in the smallest value of R,, for a given transit time deserves some attention. From eqn (S), one observes that the valence band edge gradient (the base electric field), increases dra- matically as the base doping concentration becomes comparable to the conduction band density of states. Therefore, if one varies Ns(x) only slightly with x when N,(x) approaches N,, but much more dra- matically when N,(x) is <N,, then the established base field becomes relatively constant across the entire base layer, thus reducing the base transit time. Such description of Ns(x) fits closely to that of the

complementary exponential grading scheme. This grading scheme, in addition to this reduction of base transit time by a fairly constant base field, also renders the largest amount of integrated base charge because of its gradual variation of Ns(x) with x. Consequently, this grading scheme effectuates the smallest value of RSH for a given base transit time.

Figure 4 is calculated values of RSH as a function of base transit time for various grading schemes, with Ned,, = 2 x 101*/cm3, and Ns,, = 5 x 101’/cm3. These results again show that the complementary exponential grading scheme results in the smallest value of R,, for a given base transit time. More interestingly, comparison of Figs 3 and 4 demon- strates that complementary grading from 8 x 10” to 5 x 10’*/cm3 results in lower values of R,, than those from 2 x lo’* to 5 x lO’*/cm’. This is some- what surprising in that R,, should typically decrease with a high base doping. However, by lowering Ns min from 2 x lo’* to 8 x 10’7/cm3, the base electric field can be made larger and the base thickness can be made thicker for a given zs. Consequently, RSH is actually lower for the base with Netin = 8 x 10” than 2 x 10’*/cm3, despite the apparent disadvantage of having a lower doping value in the former.

4. SUMMARY

The base transit time and the base sheet resist- ance of the donor-density graded bases in various grading schemes are compared with the uniformly doped base. For a given base transit time, the complementary exponential graded base has lower values of R,, than the uniform doping base. In contrast, the exponential graded base actually can have slightly larger values of R,, than the uni- form doping base, despite the fact that it has the thickest base layer.

- linear grading

- exponential grading

0 1 2 3 4 5 base transit time ( psec )

Fig. 4. The base sheet resistance for each grading scheme as a function of the base. transit time. This calculation assumes IV,,, = 2 x lO’*/cm’ and A$_ = 5 x 10’*/cm3.

Base transit time in HBTs 1577

Acknowledgemen@-The discussions with Dr D. Hill are greatly appreciated. The help from Dr B. Bayraktaroglu, and the technical support in computation from Dr E. Ziem are also acknowledged.

REFERENCES

7. H. F. Cooke, Proc. IEEE 59, 1163 (1971). 8. R. L. Prichard. Electrical Charocterisrics of Transistors,

1. D. Hill, W. S. Lee, T. Ma and J. S. Harris Jr, Eleclron. L.-et?. 25, 993 (1989).

2. G. J. Sullivan, M. F. Chang, N. H. Sheng, R. J. Anderson, N. L. Wang, K. C. Wang, J. A. Higgins and P. M. Asbeck, IEEE Electron Device Lett. 11, 463 (1990).

pp. 330-334. McGraw-Hill, New York (1967). 9. D. Sunderland and P. D. Dapkus, IEEE Trans.

Electron Devices 34, 367 (1987). 10. See for example, H. W. Reddick, D#erentiol Equations,

DD. 58-81. John Wilev. New York (1943). 11. H. C. Casey and F: .Stem, J. o&l. Z&s. 47, 631

(1976). 3. W. Liu, D. Hill, D. Costa and J. S. Harris Jr, 12. W. B. Joyce and R. W. Dixon, Appl. Phys. Left. 31,

IEEE ZEDM, p. 669 (1990). 354 (1977).

4. D. B. Slater, P. M. Enquist, M. Y. Chen, J. A. Hutchby, A. S. Morris and R. J. Trew, IEEE Electron Device Lett. 12, 382 (1991).

5. H. Kroemer, Proc. ZEEE 70, 13 (1982). 6. J. L. Moll and I. M. Ross, Proc. IRE 44, 72

(1956).