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Explicit analytical expressions for intrinsic base resistance and cutoff frequency of bipolar transistors biased at high injection
Solid-State Electronics Vol. 34, No. 10, pp. 1113-1117, 1991 Printed in Great Britain. All rights reserved
0038-1101/91 $3.00+0.00 Copyright © 1991 Pergamon Press plc
EXPLICIT ANALYTICAL EXPRESSIONS FOR INTRINSIC BASE RESISTANCE AND CUTOFF FREQUENCY OF BIPOLAR TRANSISTORS BIASED AT HIGH INJECTION T. C. Lu and J. B. K u o Rm 526, Department of Electrical Engineering, National Taiwan University, Roosevelt Rd No. 1, Sec. 4, Taipei, 107 Taiwan (Received 30 October 1990; in rev&edform 13 March 1991) Abstract--Present analytical expressions for intrinsic base resistance and cutoff frequency of bipolar transistors based on an independent charge/current model do not provide accurate results at high injection. In this paper, explicit analytical expressions for intrinsic base resistance and cutoff frequency using an improved charge/current model incorporating base widening at high injection for bipolar transistors are presented. Compared to fully numerical simulation results, the improved model provides much more accurate expressions for intrinsic base resistance and cutoff frequency of bipolar transistors at high injection.
!. INTRODUCTION
The intrinsic delay of a bipolar transistor is determined by the base resistance r b and cutoff frequency fr[1-7], which are strongly influenced by the excess charges in the base region at high injection. The DeGraaff-Kloosterman formulas[8/ of independent charge and current for modeling bipolar transistors can characterize bias-dependent r b and fv as compared to the Ebers-Moll and G u m m e l - P o o n models[9,10/. However, the Degraaff-Kloosterman formulas cannot model bias-dependent effects at high injection accurately as a result of inclusion of base widening effects at boundary conditions only and attempting to smooth the transition between high and low current regions. In this paper, an improved charge/current model including base widening effects at high injection for modeling r b and J~ of a bipolar transistor is presented. 2. THE I M P R O V E D C H A R G E / C U R R E N T M O D E L FOR TRANSISTOR AT H I G H INJECTION
In the DeGraaff-Kloosterman formulas[8/, the electron current density at high injection in the bipolar transistor, with doping profile as shown in Fig. la, is: Jn = 2qD, [n(0) -- n(Wb) ], I/Vb
where q is the electron charge, Dn is the average electron diffusion coefficient over the entire base region, Wb is the base width, n(0) is the electron concentration at the base edge of base-emitter junction and n(Wb) is the electron concentration at the base edge of base-collector junction. However, the
DeGraaff-Kloosterman model considers base widening effects only at n(Wb), which is correlated to I , as indicated by curves (1) and (2) in Fig. la. In fact, the most important term Wb~fris not considered. Consequently, rh and fT derived from the model are not accurate at high injection. Now, including base widening at high injection, the electron current density equation is modified as[1 I]: Jn = 2.qDn [n(0) -- n(Weeft)] ,
(1)
1/t/beff
where Wbefr is the effective base width considering base widening at high injection, and n(0) is derived from n(0)[n(0) + NB(0) ] = n~ exp(qVBEkT ), where NB(0 ) is the base doping concentration at the base edge of the base-emitter junction. Without base widening, n(Wb¢fr) = n(Wb) ~ O. With base widening, n (Wbofr) = N c which affects the I r only as pointed out in DeGraalTs paper, where N c is the doping concentration in the epicollector region. As base widening exists, the effective base width Wbe~ is calculated using the following equation[l 2-14]: W~e~= Wb + WomB.
(2a)
For a high field case where the bipolar transistor is operated at a large VcB, the current-induced base width for the J~ > J0 case is: Wcm= W c ( I _ "k/Jc/J°--qvsNc~qvs No/'
(2b)
where J0 = qvs(N¢ + 2eVcB/qWZc), vs is the saturation velocity, ~ is the permittivity of silicon and Wc is the epitaxial collector width.
1113
1114
T C. Lu and J. B. Kuo
(a) 1021 e9 'E 1 0 2 °
(1)----
/I
1018
"~',... ~ . _ ~ .............................(2)
(J~ >>qv~N~= r) for
the high field
[~4qD'n(O) W~-b-b-~~ c F (~bb-~ W~) (Jo--
Ir=5]A
(Ic=5.9x10"4)
rSqDnn(O)(Jo__- r)W~
+L
/
----<.'>..-":.:...j
e- 1017 0 0
.&
1x1041
(2) ......... Electron concentration
1019
t=,
Doping profile Electron concentration
phenomenon exists case:
+
~
-- r)2
,
(4a)
1016
10 15
I I • Wbeff (1) Woelf (2)
0 Wb
0
X
I I I I I I I I 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Depth (i~m)
(b)
1 (Jo - r)W~ Qr=IA¢ qn(O)(Wb+ We)+4 nn
I
1.8
1[
2(Wb+
W¢) qn (0) (J0
-- r ) W~
D,
E +
(Jo--r)2 W:]L2~
~
-j
j,
(4b)
Q~ and lr can be obtained similarly, but they have little effect. Therefore, they have been neglected when considering the total excess charge in the base. 3. C O M P A R I S O N S WITH SIMULATION RESULTS
C Fig. 1. (a) The doping profile and the electron distributions (1), (2) of a vertical bipolar transistor biased at high injection with V,~ = 2 V. (b) The cross section of the bipolar device.
Based on equations (1) and (2), the forward and reverse currents and charges of the improved model are listed below:
2qD~ If = ~ Aon(0),
The figures-of-merit of the improved model have been assessed using a fully numerical simulation tool[15]. In order to simplify the analysis, we have concentrated on the intrinsic region of the bipolar device as shown in Fig. lb. Figure 2 shows the total excess charge in base region vs collector current for the bipolar device biased at VcB = 2 V. The total excess charge in base region calculated by fully numerical simulation is defined as the increase in the total electrons from the base-emitter junction to the collector contact when the biasing of the transistor changes from low injection to the present condition. As shown in Fig. 2, the total excess charge in base region based on the DeGraaff-Kloosterman formulas has been underestimated due to the absence of the base widening model and as a result of attempting to smooth the transition between low and high 3.0 --
(3a) o
2qDn
(3b)
~ 8 o
1 Qf = ~qAe Wb¢~n(O),
Qr
I = 5qAe Whegn(W,e~),
(3c) (3d)
where I r and/~ are forward and reverse currents, Qr and Qr are forward and reverse charges. Solving equations (2) and (3), an improved model with independent charge and current at high injection can be obtained for the situation when the base widening
Vcb.2V
2.5
l/
//--=-Pisces Our model "-~-ff data
/
2.0 1.5
e-
o
o ca
1.0
0 I--
/ /
1 -5
,
,
, ,,..a
De Graaff's model / 7 " . ~ ..... I
10-4
10-3
10-2
I ¢ (amp) Fig. 2. The total excess charge in the base region vs collector current for the bipolar device biased at V~b= 2 V.
Base resistance and cutoff frequency of bipolar transistors current regions. On the other hand, the improved charge/current model does show a much closer fit to the fully numerical results, which leads to much more accurate r b and fT.
3.1. Base resistance The base resistance of a bipolar device is composed of extrinsic and intrinsic portions. The extrinsic base resistance is dominated by the parasitic resistance in the region near the base contact. On the other hand, the intrinsic base resistance is subject to the biasing condition of the device. In order to simplify the analysis and to emphasize the bias dependence of the base resistance, we have focused on the intrinsic region of the bipolar device as shown in Fig. 1b in the following analysis. Based on the DeGraaff-Kloosterman independent charge/current model, the intrinsic base resistance has been modeled by Yuan et al.[16] and Satake and Hamasaki[17] using base widening and conductivity modulation as two independent parameters. In fact, these two parameters are dependent. As a result, the accuracy of their base resistance expressions is limited. Although an improved method to analyze base resistance more accurately has been described by Jo and Burk[18], it is too complicated for modeling purposes. Here, combining the improved charge/ current model including base widening with the model presented by Hauser[19], a more accurate analytical expression suitable for circuit simulation at high injection is now described. The base resistance considering only the emitter current crowding effect is[17,19]: Ps WEW//tan Z -- Z'], r b - WB--~-E\ Z-ta--~-Z J
(5)
! lOB ~
-
-
qp(x )l~p
1115
where Ge = S~'E NE(X) dx, NE(X) is the doping concentration in the emitter region and nie is the intrinsic carrier concentration including bandgap narrowing effect[20,21], n~e=n~exp(AEge/kT ) where AEse= CIlnNE/No, for NE>N0; C j = l . 8 7 x 10-3eV, N O= 7 x l017 cm 3. Based on eqns (4), (7-9), a more accurate base resistance expression without using base widening and conductivity modulation as independent parameters has been obtained. Figure 3 shows the r b vs collector current for the bipolar transistor biased at VCB= 2 V using fully numerical simulation, Satake's method[17] based on the DeGraaff-Kloosterman model and the improved model. For fully numerical simulation, the base resistance is extracted by directly measuring the voltage drop between the base electrode and the center of the intrinsic device area[22]. In Satake's method[17], the base resistance is calculated using base widening and conductivity modulation as two independent factors (fPo,fcM). As shown in Fig. 3, from Satake's method, the base resistance is quite different from the fully numerical simulation result at a high current due to the underestimate of the excess charge in the base. On the other hand, the base resistance expression using the improved charge/current model shows a much better agreement with the fully numerical result.
3.2. Cutoff frequency Cutoff frequency (fT) is an important parameter in describing the speed performance of a bipolar transistor. At high injection, the cutoff frequency falls off due to the spread of the neutral base layer into the collector region[23,24]. Derived from Gummel's formula[25], the currently available analytical model of fT is expressed as the total delays in the base, emitter and collector regions[26]:
Z is the solution for:
I~ W~w Z tan Z = WBLe qPB 2 k ~ '
f T I - - 2n zf+
(6)
where WEW is the emitter stripe width, L E is the emitter stripe length, #p is the hole mobility and p(x) is the hole concentration in the base region. From (5) and (6), r b can be rearranged as: rb =
WEW //tan Z - - Z'] ~w~. \ Z tan: Z ,} q#P Jo p(x) dxL E
_
Z tan Z -
WEW (tan Z -- Z'], q~pQ,BL~E\-~ta~nZ--Z-- j
Itp Q "BLE 2k T '
(7) (8)
where Q~ is the excess charge in the base (Qf) divided by A,:
=q-~n2~[exp(qVb~/kT)- 1],
(9)
gm
gm
RcCjc +
~
"
where ~r is the minority carrier base transit time, Cje is the base-to-emitter junction capacitance, Cjc is the base-to-collector junction capacitance, gm is the transconductance, Rc is the equivalent resistance in the neutral collector region, Wis the base-to-collector depletion width and v~, is the electron saturation velocity. This fT equation is not explicit in showing the base widening effects for the bipolar transistor biased at high injection, where the fall-off in fT is substantial. In fact, an explicit analytical form of the fT equation in terms of collector current at high injection can be heuristic. Here, an analytical formula of f r explicitly in terms of base widening effects for the transistor biased at high injection has been created using the improved charge/current model. Solving equations (4a) and (4b), an analytical expression of the cutoff frequency at high injection
1116
T. C. Lu and J. B. K u o
1600 -
1011
I~
= 1,oo
!/
Vcb =2V
\
Iv m 1000 to
o,oo
~ B a s e d on \ Satake's method
&~\ \ Pisces
600
¢' (/} ¢~
400
\
200
I I Ix
rn
k\=
'~
1010
o
Our model
"x,
De Graaff's model
10 9 O
..,
\,~
',...,
o
\ ~ ""~'--+.a.a, I
0
IO-4
Vcb =2V
%.
Pisc~ data
"~
data
.•L
Io-3
108
1o-2
0.1
I
I
I
Our m o d e l I
0.5
1.0
1.5
2,0
has been derived using the improved charge/current model:
where (2 is the total charge in the entire device. At high injection for the high field case, dQ/dI can be approximated by dQr/dI:
Fig. 4. The cutoff frequency f~ vs collector current for the bipolar transistor biased at V,b = 2 V. 5. C O N C L U S I O N
In this paper, explicit analytical expressions for intrinsic base resistance and cutoff frequency using an improved charge/current model incorporating base widening at high injection for bipolar transistors are presented. Compared to fully numerical simulation results, the improved model provides much more accurate analytical expressions for intrinsic base re-
½(Wb+ W J LF2 - ( J ° - r ) W ~ ( 2 q n ( O ) ( W h + mc
I 3.0
I c ( a m p ) x l 0 -3
I e (amp)
Fig. 3. The base resistance r h vs collector current for the bipolar transistor biased at V~b= 2 V.
dQ
1 25
2
Figure 4 shows the J~. vs collector current for the bipolar transistor biased at Vc-~ = 2 V using a fully numerical simulation, the DeGraaff Kloosterman model and the improved model. The cutoff frequency based on fully numerical simulation is calculated by taking the derivative of the integration of the total electrons over the entire device area with respect to the biasing current[22]. As compared to the D e G r a a f f Kloosterman model result, the improved charge/ current model curve demonstrates a much better agreement with the fully numerical result owing to the inclusion of the excess charge in the base region.
4. D I S C U S S I O N
For the r~ and fT consideration, only the Qr term is important. In fact, the improved model can bring in more influential results for the situations where the Qr term is not negligible. For example, substrate-current-related device behavior can be analyzed much more closely by the improved model. In addition, the precision of the Qr equation can be enhanced by evolving the n(Wbe~r) term, which is not exactly Nc (greater than No) at high injection, as shown in Fig. !, owing to the fact that the electrons are traveling at a saturated velocity for the high field case.
W~)W~(J°-r) ~
(J°-r)2W~
W
4
~2] .
'
sistance and cutoff frequency of bipolar transistors at high injection.
Acknowledgements--The authors would like to thank Professor R. W. Dutton for helping us obtain the license to use the PISCES program. The work is supported under R.O.C. National Science Council Contract No. 79-0404-E002-47.
REFERENCES
1. D. D. Tang and P. M. Solomon, Bipolar transistor design for optimized power-delay logic circuits. IEEE J. Solid-State Circuits SC-14, 679~84 (1979). 2. M. H. White and M. O. Thurston, Characterization of microwave transistors. Solid-St. Electron. 13, 523 542 (1970). 3. K. G. Ashar, The method of estimating delay in switching circuits and the figure of merit of a switching transistor. IEEE Trans. Electron Devices 497 506 (1964). 4. M. Y. Ghannam, R. P. Mertens and R. J. Van Overstraeten, An analytical model for the determination of the transient response of CML and ECL Gates. IEEE Trans. Electron Devices 191-201 (1990). 5. M. Nanba, T. Shiba, T. Nakamura and T. Toyabe, An analytical and experimental investigation of the cutofffrequency fr of high-speed bipolar transistors. IEEE Trans. Electron Devices 1021 1028 (1988). 6. J. D. Cresler, D. D. Tang, K. A. Jenkins, G. P. Li and E. S. Yang, On the low-temperature static and dynamic
Base resistance and cutoff frequency of bipolar transistors
7.
8.
9. 10. I 1.
12. 13. 14. 15. 16.
properties of high-performance silicon bipolar transistor 1EEE Trans. Electron Devices 1489-1502 (1989). Y. W. Chen, K. H. Lou and J. B. Kuo, Twodimensional analysis of a BiNMOS transistor operating at 77 K. Digest 1990 V L S I Process and Device Workshop, Tokyo (1990). H. C. DeGraaff and W. J. Kloosterman, New formulation of the current and charge relations in bipolar transistor modeling for CACD purposes. IEEE Trans. Electron Devices 2415-2420 (1985). J. J. Ebers and J. L. Moll, Large-signal behavior of junction transistors. Proc. I R E 42, 1761-1772 (1954). H. K. Gummel and H. C. Poon, An integral charge control model of bipolar transistors. Bell Syst. Tech. J. 49, 827-852 (1970). T. C. Lu and J. B. Kuo, An improved charge/current model incorporating base-widening at high injection for bipolar transistors. Being reviewed for publication (1991). P. Antognetti and G. Massobrio, Semiconductor Device Modeling with SPICE. McGraw-Hill, New York (1988). G. Rey, F. Dupuy and J. P. Bailbe, A unified approach to the base widening mechanisms in bipolar transistors. Solid St. Electron. 863-866 (1975). S. M. Sze, Physics o f Semiconductor Devices. Wiley, New York (1969). M. R. Pinto, C. S. Rafferty and R. W. Dutton, PISCES 2: Poisson and continuity equation solver. Tech. Rept. Stanford Univ. (1984). J.-S. Yuan, J. J. Liou and W. R. Eisenstadt, A physicsbased current-dependent base resistance model for advanced bipolar transistors. IEEE Trans. Electron Devices 1055 1062 (1988).
1117
17. H. Satake and T. Hamasaki, Low-temperature (77 K) BJT model with temperature dependences on the injected condition and base resistance. IEEE Trans. Electron Devices 1688-1697 (1990). 18. M. Jo and D. E. Burk, An intrinsic base resistance model for low and high currents. IEEE Trans. Electron Devices 202-209 (1990). 19. J. R. Hauser, The effects ofdistributed base potential on emitter-current injection density and effective base resistance for stripe transistor geometries. IEEE Trans. Electron Devices 238-242 (1964). 20. J. del Alamo, S. E. Swirhun and R. M. Swanson, Measuring and modeling minority carrier transport in heavily doped silicon. Solid-St. Electron 28, 47 54 (1985). 21. S. E. Swirhun, Y.-H. Kwark and R. M. Swanson, Measurement of electron lifetime, electron mobility and bandgap narrowing in heavily doped p-type silicon. I E D M Tech. Dig., p. 24-27 (1986). 22. J. B Kuo, T. S. Yang, R. W. Dutton and B. A. Wooley, Two-dimensional transient analysis of a collector-up ECL inverter. IEEE Trans. Comput.-,4id. Des. 1038-1045 (1989). 23. C. T. Kirk Jr, A theory of transistor cutoff frequency (fr) falloffat high current densities. I R E Trans. Electron Devices ED-9, 164-174 (1962). 24. R. J. Whittier and D. A. Tremere, Current gain and cutoff frequency falloff at high currents. IEEE Trans. Electron Devices ED-16, 39-57 (1969). 25. H. C. Poon, H. K. Gummel and D. L. Scharfetter, High injection in epitaxial transistors. IEEE Trans. Electron Devices. May, 455458 (1969). 26. E. S. Yang, Microelectronic Devices. McGraw-Hill, New York (1988).
0038-1101/91 $3.00+0.00 Copyright © 1991 Pergamon Press plc
EXPLICIT ANALYTICAL EXPRESSIONS FOR INTRINSIC BASE RESISTANCE AND CUTOFF FREQUENCY OF BIPOLAR TRANSISTORS BIASED AT HIGH INJECTION T. C. Lu and J. B. K u o Rm 526, Department of Electrical Engineering, National Taiwan University, Roosevelt Rd No. 1, Sec. 4, Taipei, 107 Taiwan (Received 30 October 1990; in rev&edform 13 March 1991) Abstract--Present analytical expressions for intrinsic base resistance and cutoff frequency of bipolar transistors based on an independent charge/current model do not provide accurate results at high injection. In this paper, explicit analytical expressions for intrinsic base resistance and cutoff frequency using an improved charge/current model incorporating base widening at high injection for bipolar transistors are presented. Compared to fully numerical simulation results, the improved model provides much more accurate expressions for intrinsic base resistance and cutoff frequency of bipolar transistors at high injection.
!. INTRODUCTION
The intrinsic delay of a bipolar transistor is determined by the base resistance r b and cutoff frequency fr[1-7], which are strongly influenced by the excess charges in the base region at high injection. The DeGraaff-Kloosterman formulas[8/ of independent charge and current for modeling bipolar transistors can characterize bias-dependent r b and fv as compared to the Ebers-Moll and G u m m e l - P o o n models[9,10/. However, the Degraaff-Kloosterman formulas cannot model bias-dependent effects at high injection accurately as a result of inclusion of base widening effects at boundary conditions only and attempting to smooth the transition between high and low current regions. In this paper, an improved charge/current model including base widening effects at high injection for modeling r b and J~ of a bipolar transistor is presented. 2. THE I M P R O V E D C H A R G E / C U R R E N T M O D E L FOR TRANSISTOR AT H I G H INJECTION
In the DeGraaff-Kloosterman formulas[8/, the electron current density at high injection in the bipolar transistor, with doping profile as shown in Fig. la, is: Jn = 2qD, [n(0) -- n(Wb) ], I/Vb
where q is the electron charge, Dn is the average electron diffusion coefficient over the entire base region, Wb is the base width, n(0) is the electron concentration at the base edge of base-emitter junction and n(Wb) is the electron concentration at the base edge of base-collector junction. However, the
DeGraaff-Kloosterman model considers base widening effects only at n(Wb), which is correlated to I , as indicated by curves (1) and (2) in Fig. la. In fact, the most important term Wb~fris not considered. Consequently, rh and fT derived from the model are not accurate at high injection. Now, including base widening at high injection, the electron current density equation is modified as[1 I]: Jn = 2.qDn [n(0) -- n(Weeft)] ,
(1)
1/t/beff
where Wbefr is the effective base width considering base widening at high injection, and n(0) is derived from n(0)[n(0) + NB(0) ] = n~ exp(qVBEkT ), where NB(0 ) is the base doping concentration at the base edge of the base-emitter junction. Without base widening, n(Wb¢fr) = n(Wb) ~ O. With base widening, n (Wbofr) = N c which affects the I r only as pointed out in DeGraalTs paper, where N c is the doping concentration in the epicollector region. As base widening exists, the effective base width Wbe~ is calculated using the following equation[l 2-14]: W~e~= Wb + WomB.
(2a)
For a high field case where the bipolar transistor is operated at a large VcB, the current-induced base width for the J~ > J0 case is: Wcm= W c ( I _ "k/Jc/J°--qvsNc~qvs No/'
(2b)
where J0 = qvs(N¢ + 2eVcB/qWZc), vs is the saturation velocity, ~ is the permittivity of silicon and Wc is the epitaxial collector width.
1113
1114
T C. Lu and J. B. Kuo
(a) 1021 e9 'E 1 0 2 °
(1)----
/I
1018
"~',... ~ . _ ~ .............................(2)
(J~ >>qv~N~= r) for
the high field
[~4qD'n(O) W~-b-b-~~ c F (~bb-~ W~) (Jo--
Ir=5]A
(Ic=5.9x10"4)
rSqDnn(O)(Jo__- r)W~
+L
/
----<.'>..-":.:...j
e- 1017 0 0
.&
1x1041
(2) ......... Electron concentration
1019
t=,
Doping profile Electron concentration
phenomenon exists case:
+
~
-- r)2
,
(4a)
1016
10 15
I I • Wbeff (1) Woelf (2)
0 Wb
0
X
I I I I I I I I 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Depth (i~m)
(b)
1 (Jo - r)W~ Qr=IA¢ qn(O)(Wb+ We)+4 nn
I
1.8
1[
2(Wb+
W¢) qn (0) (J0
-- r ) W~
D,
E +
(Jo--r)2 W:]L2~
~
-j
j,
(4b)
Q~ and lr can be obtained similarly, but they have little effect. Therefore, they have been neglected when considering the total excess charge in the base. 3. C O M P A R I S O N S WITH SIMULATION RESULTS
C Fig. 1. (a) The doping profile and the electron distributions (1), (2) of a vertical bipolar transistor biased at high injection with V,~ = 2 V. (b) The cross section of the bipolar device.
Based on equations (1) and (2), the forward and reverse currents and charges of the improved model are listed below:
2qD~ If = ~ Aon(0),
The figures-of-merit of the improved model have been assessed using a fully numerical simulation tool[15]. In order to simplify the analysis, we have concentrated on the intrinsic region of the bipolar device as shown in Fig. lb. Figure 2 shows the total excess charge in base region vs collector current for the bipolar device biased at VcB = 2 V. The total excess charge in base region calculated by fully numerical simulation is defined as the increase in the total electrons from the base-emitter junction to the collector contact when the biasing of the transistor changes from low injection to the present condition. As shown in Fig. 2, the total excess charge in base region based on the DeGraaff-Kloosterman formulas has been underestimated due to the absence of the base widening model and as a result of attempting to smooth the transition between low and high 3.0 --
(3a) o
2qDn
(3b)
~ 8 o
1 Qf = ~qAe Wb¢~n(O),
Qr
I = 5qAe Whegn(W,e~),
(3c) (3d)
where I r and/~ are forward and reverse currents, Qr and Qr are forward and reverse charges. Solving equations (2) and (3), an improved model with independent charge and current at high injection can be obtained for the situation when the base widening
Vcb.2V
2.5
l/
//--=-Pisces Our model "-~-ff data
/
2.0 1.5
e-
o
o ca
1.0
0 I--
/ /
1 -5
,
,
, ,,..a
De Graaff's model / 7 " . ~ ..... I
10-4
10-3
10-2
I ¢ (amp) Fig. 2. The total excess charge in the base region vs collector current for the bipolar device biased at V~b= 2 V.
Base resistance and cutoff frequency of bipolar transistors current regions. On the other hand, the improved charge/current model does show a much closer fit to the fully numerical results, which leads to much more accurate r b and fT.
3.1. Base resistance The base resistance of a bipolar device is composed of extrinsic and intrinsic portions. The extrinsic base resistance is dominated by the parasitic resistance in the region near the base contact. On the other hand, the intrinsic base resistance is subject to the biasing condition of the device. In order to simplify the analysis and to emphasize the bias dependence of the base resistance, we have focused on the intrinsic region of the bipolar device as shown in Fig. 1b in the following analysis. Based on the DeGraaff-Kloosterman independent charge/current model, the intrinsic base resistance has been modeled by Yuan et al.[16] and Satake and Hamasaki[17] using base widening and conductivity modulation as two independent parameters. In fact, these two parameters are dependent. As a result, the accuracy of their base resistance expressions is limited. Although an improved method to analyze base resistance more accurately has been described by Jo and Burk[18], it is too complicated for modeling purposes. Here, combining the improved charge/ current model including base widening with the model presented by Hauser[19], a more accurate analytical expression suitable for circuit simulation at high injection is now described. The base resistance considering only the emitter current crowding effect is[17,19]: Ps WEW//tan Z -- Z'], r b - WB--~-E\ Z-ta--~-Z J
(5)
! lOB ~
-
-
qp(x )l~p
1115
where Ge = S~'E NE(X) dx, NE(X) is the doping concentration in the emitter region and nie is the intrinsic carrier concentration including bandgap narrowing effect[20,21], n~e=n~exp(AEge/kT ) where AEse= CIlnNE/No, for NE>N0; C j = l . 8 7 x 10-3eV, N O= 7 x l017 cm 3. Based on eqns (4), (7-9), a more accurate base resistance expression without using base widening and conductivity modulation as independent parameters has been obtained. Figure 3 shows the r b vs collector current for the bipolar transistor biased at VCB= 2 V using fully numerical simulation, Satake's method[17] based on the DeGraaff-Kloosterman model and the improved model. For fully numerical simulation, the base resistance is extracted by directly measuring the voltage drop between the base electrode and the center of the intrinsic device area[22]. In Satake's method[17], the base resistance is calculated using base widening and conductivity modulation as two independent factors (fPo,fcM). As shown in Fig. 3, from Satake's method, the base resistance is quite different from the fully numerical simulation result at a high current due to the underestimate of the excess charge in the base. On the other hand, the base resistance expression using the improved charge/current model shows a much better agreement with the fully numerical result.
3.2. Cutoff frequency Cutoff frequency (fT) is an important parameter in describing the speed performance of a bipolar transistor. At high injection, the cutoff frequency falls off due to the spread of the neutral base layer into the collector region[23,24]. Derived from Gummel's formula[25], the currently available analytical model of fT is expressed as the total delays in the base, emitter and collector regions[26]:
Z is the solution for:
I~ W~w Z tan Z = WBLe qPB 2 k ~ '
f T I - - 2n zf+
(6)
where WEW is the emitter stripe width, L E is the emitter stripe length, #p is the hole mobility and p(x) is the hole concentration in the base region. From (5) and (6), r b can be rearranged as: rb =
WEW //tan Z - - Z'] ~w~. \ Z tan: Z ,} q#P Jo p(x) dxL E
_
Z tan Z -
WEW (tan Z -- Z'], q~pQ,BL~E\-~ta~nZ--Z-- j
Itp Q "BLE 2k T '
(7) (8)
where Q~ is the excess charge in the base (Qf) divided by A,:
=q-~n2~[exp(qVb~/kT)- 1],
(9)
gm
gm
RcCjc +
~
"
where ~r is the minority carrier base transit time, Cje is the base-to-emitter junction capacitance, Cjc is the base-to-collector junction capacitance, gm is the transconductance, Rc is the equivalent resistance in the neutral collector region, Wis the base-to-collector depletion width and v~, is the electron saturation velocity. This fT equation is not explicit in showing the base widening effects for the bipolar transistor biased at high injection, where the fall-off in fT is substantial. In fact, an explicit analytical form of the fT equation in terms of collector current at high injection can be heuristic. Here, an analytical formula of f r explicitly in terms of base widening effects for the transistor biased at high injection has been created using the improved charge/current model. Solving equations (4a) and (4b), an analytical expression of the cutoff frequency at high injection
1116
T. C. Lu and J. B. K u o
1600 -
1011
I~
= 1,oo
!/
Vcb =2V
\
Iv m 1000 to
o,oo
~ B a s e d on \ Satake's method
&~\ \ Pisces
600
¢' (/} ¢~
400
\
200
I I Ix
rn
k\=
'~
1010
o
Our model
"x,
De Graaff's model
10 9 O
..,
\,~
',...,
o
\ ~ ""~'--+.a.a, I
0
IO-4
Vcb =2V
%.
Pisc~ data
"~
data
.•L
Io-3
108
1o-2
0.1
I
I
I
Our m o d e l I
0.5
1.0
1.5
2,0
has been derived using the improved charge/current model:
where (2 is the total charge in the entire device. At high injection for the high field case, dQ/dI can be approximated by dQr/dI:
Fig. 4. The cutoff frequency f~ vs collector current for the bipolar transistor biased at V,b = 2 V. 5. C O N C L U S I O N
In this paper, explicit analytical expressions for intrinsic base resistance and cutoff frequency using an improved charge/current model incorporating base widening at high injection for bipolar transistors are presented. Compared to fully numerical simulation results, the improved model provides much more accurate analytical expressions for intrinsic base re-
½(Wb+ W J LF2 - ( J ° - r ) W ~ ( 2 q n ( O ) ( W h + mc
I 3.0
I c ( a m p ) x l 0 -3
I e (amp)
Fig. 3. The base resistance r h vs collector current for the bipolar transistor biased at V~b= 2 V.
dQ
1 25
2
Figure 4 shows the J~. vs collector current for the bipolar transistor biased at Vc-~ = 2 V using a fully numerical simulation, the DeGraaff Kloosterman model and the improved model. The cutoff frequency based on fully numerical simulation is calculated by taking the derivative of the integration of the total electrons over the entire device area with respect to the biasing current[22]. As compared to the D e G r a a f f Kloosterman model result, the improved charge/ current model curve demonstrates a much better agreement with the fully numerical result owing to the inclusion of the excess charge in the base region.
4. D I S C U S S I O N
For the r~ and fT consideration, only the Qr term is important. In fact, the improved model can bring in more influential results for the situations where the Qr term is not negligible. For example, substrate-current-related device behavior can be analyzed much more closely by the improved model. In addition, the precision of the Qr equation can be enhanced by evolving the n(Wbe~r) term, which is not exactly Nc (greater than No) at high injection, as shown in Fig. !, owing to the fact that the electrons are traveling at a saturated velocity for the high field case.
W~)W~(J°-r) ~
(J°-r)2W~
W
4
~2] .
'
sistance and cutoff frequency of bipolar transistors at high injection.
Acknowledgements--The authors would like to thank Professor R. W. Dutton for helping us obtain the license to use the PISCES program. The work is supported under R.O.C. National Science Council Contract No. 79-0404-E002-47.
REFERENCES
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