Solid-State Electronics Vol. 35, No. 8, pp. 1051-1057, 1992 0038-1101/92 $5.00 + 0.00 Printed in Great Britain. All rights reserved Copyright © 1992 Pergamon Press Ltd

CURRENT DEPENDENCE OF BASE-COLLECTOR CAPACITANCE OF BIPOLAR TRANSISTORS

WILLIAM Lxut and JAMES S. HARRIS Solid State Laboratory, McCuIIough Bldg No. 226, Stanford University, Stanford, CA 94305, U.S.A.

(Received 8 November 1991; in revised form 4 January 1992)

Abstract--We present analytical expressions for the base-collector capacitance of bipolar transistors in three operating conditions as the collector current density is continuously increased until the collector is fully depleted. A simple model is also presented to calculate this capacitance after base pushout occurs. The critical current densities separating each operating condition are discussed. The capacitance as a function of current density is calculated for various base-collector biases, collector thicknesses and collector dopings. The calculated results of this simple base-collector capacitance model are in close agreement with SEDAN simulation results. In addition, these results are shown to agree with published experimental work.

1. INTRODUCTION

One critical parameter in the design of high- frequency bipolar transistors and circuits is the base-collector capacitance C~. Tong and Solomon[l] demonstrated that C~ significantly affects the switch- ing time of emitter coupled logic (ecl) gates and Madihian et a/.[2] showed that this capacitance should be minimized, even at the expense of increas- ing the base resistance. For discrete transistors, C~ is also an important parameter since it inversely relates a device's maximum oscillation frequency fmax to its cutoff frequency fx[3]. Because high-performance bipolar transistors and circuits operate at high collec- tor current densities, it is important to understand the effects of high current densities on C~. In particular, in the presence of large collector currents, Cb¢ is no longer solely a function of the base-collector bias, but also a function of the current level. In this paper, we will present analytical solutions for C~ in three operating conditions which depend upon the collec- tor current density. The boundary current densities separating each operating condition will be discussed. The equations derived are physical, simple and can be calculated directly from device parameters.

This work differs from a published work[4]. The published work's formulation of Cb¢ is based on a general expression which includes a dielectric capacitance and a free-carrier capacitance[5,6]. The dielectric capacitance corresponds to the capacitance that results from the change of the carriers at the edges of the space-charge region, and the free-carrier capacitance corresponds to the capacitance that results from the change of the carriers in the volume of the space-charge region. Using this expression, the published work comprehensively considered various

tCurrently with Central Research Laboratories, Texas Instruments, Dallas, TX.

operating conditions including forward-active, saturation and quasi-saturation. It also used the expression to approximate Cb~ in the base pushout operating condition. It showed how Cb¢ varies across various base-collector biases at different current levels and demonstrated that approximating C~ solely by the dielectric capacitance leads to error, especially when the transistors are in saturation or quasi-saturation. In contrast, this work bases its derivation directly on tracing charges inside the junction. This work focuses its scope on transistors under the forward-active condition and considers explicitly how C~ varies as a function of collector current, which is of interest to designers aiming to optimize a device's f r and fmax' In addition, the variations of Cb¢ with various device parameters, such as the collector thickness, base-collector bias and collector doping are calculated. These calculations allow designers to predict device performance as the values of the device parameters are varied. In the operating condition of base pushout, a model which considers the modifications of carrier concentrations inside the current-induced base in response to a change in base--collector bias is presented. The values of C~ calculated from these simple derived equations agree with the calculations from a much more compli- cated SEDAN simulation tool which simultaneously solves the current continuity equation and the Poisson equation[7]. Moreover, the calculated values of C~ are compared and are shown to have good agreement with experimentally extracted values for a transistor[8].

2. THEORETICAL MODELS

This model is presented in terms of the I-D N-p+-n heterojunction bipolar transistor (HBT) shown in Fig. 1. This derivation is completely general,

1051

1052 WILLIAM LIU a n d JAMES S. HARRIS

X

J N

AIGaAs emil ler

X b ~ W c .i I I + ' N c Nc P + I I space charge i collector subcolleelor base = region I

X+ m P

- +

© .. © V eb V c b

Fig. 1. The HBT under consideration. Al l quantities listed are positive.

L J c

I °

however, and can be readily applied to P n-p HBTs and other types of bipolar transistors, since the base-collector junction in an HBT is a homojunction. In the presence of a collector current density J~, the base-collector depletion thickness X,, is determined by assuming the minority carrier velocity in the entire depletion region to be the saturation velocity L,~t. It is given as[9]:

/2E(Vb~+~b)( 1 _ j : ) , e ( , _ ~ ) ~ : (l)

<=v 7N7 where

J I = qv~,, N, (2)

q#,,N~(Vt~ +qS) J2 -- (3) W~ -- X,,,

is the built-in potential across the junction in equilibrium, ,u,, is the electron mobility in the collec- tor, W~ is the collector thickness and Vb~ is the base-collector bias. For typical modern microwave t r a n s i s t o r s , J2>>J¢ therefore, eqn (1) shows that X,,, increases with increasing J~ until it extends to the subcollector layer, and then the collector is fully depleted[9]. Let us first examine the case in which the depletion thickness is increasing but the collector layer is not yet fully depleted.

2.1. Partially-depleted collector condition The differential base-collector junction capaci-

tance is defined as[10]:

d i Cb¢ d V b . . . . . ,on -= q - - [n(x) or p(x)]dx. (4)

When the collector is not fully depleted, Cb~ is more easily determined from the integration of the electron concentration n(x) rather than the hole concert-

tration p(x). Figure 2 shows the variation of n(x) before and after the application of dVh~, and Cb~ is determined from eqn (4) as:

( J~) dX''XmdJ~l,sat d Vb c • Cb~= qXc--~/~V~. (5)

Equation (5) accounts for the Early effects by consid- ering the possible change in J~ upon a change of 1/~[11]. In the discussion of Early effects, it is useful to define a base-collector conductance parameter. g,,c=dJjdVb~ at a constant base-emitter bias[12]. in this case, g,c is equal Jur(xh-x~).dXp/dVhc, where X b is the metallurgical base thickness and XI~ is the depletion thickness into the p~ base (Fig. 1). By the requirement of charge neutrality, X + = ( l - - Jc/J1)" ~,n'Nc/Nb, Substituting the deriva- tives ofeqns (1) and (2) with respect to both IQ. and Jc into the expression for g,~, one obtains:

- J'-J~ f(l+qt'~'N Xb-X'~

Consequently, from eqn (5), when the collector is not fully depleted, C~ is given as:

~Osat \ [?sat

The first term in eqn (7) represents the dielectric capacitance due to the space-charge variation at the depletion edges[4]. This component can be directly calculated from Gauss' law and shown to be e/Xm if the Early effects are ignored[6]. However, inclusion of the Early effects adds a dJ~ term to the expression for dX,,. Thus, the second term ofeqn (7) originates from the additional Xm variation caused by the incremental

Base--collector capacitance 1053

electron concen trcaion profile after dVbc

N ÷ C

Jc q vsa I =

/ proffie before dVbc /

/

........................................... l .......................... 7.-..., l ~ Xm ~ ~ " ' ~

dXm

J¢

q vsa

i X rn depth into collector

Fig. 2. The electron concentration in a partially depleted collector prior to and after a small change in V~.

Early current. The third term of eqn (7) also results from Early effects, which increase the free electron concentration stored in the depletion region: An = AJJqv~at [4]. This third term has a sign opposite to that of the first two terms. The signs are different because an increment in V~ depletes charges at the edge of the junction, but increases the mobile charges inside the depletion region (Fig. 2).

2.2. Fully depleted collector condition

When J~ increases while Vb~ is held constant, X m continues to increase until the whole collector layer is fully depleted. The depletion thickness then stops abruptly at the n ÷ subcollector. The critical current density at which X~ equals W~ for a given V~ is derived from eqn (1) with some algebraic manipula- tions and is given as:

fv~+~ v~+4,}-', (8) ")'critical = ( V2 -- Vbc)" [ £ "]'2

Where I/2 is the applied base-collector bias which totally depletes We at J~ = O: V 2 = qN~. W~/(2.E) - tp. Let X + be the depletion thickness inside the subcoi- lector, and Xp, inside the base, then:

Similar to the case in which the collector is not fully depleted, the electrons are assumed to travel at a constant velocity v~t in the depleted region. Therefore, n(x) = Jc/qVsa~ across the collector and the depleted subcollector and n (x) = N¢ + at the undepleted subcol- lector. From eqn (4), the base-collector capacitance in the operating region of full collector depletion is thus

C~ = (qN + - JJqv~,t)" dX +/d V~

- (X + + Wc)/v~,,'dJddVb~.

Neglecting Jc/vs~,t in comparison with X2, one obtains:

+ Wcg"~[1 X+q-W" +~(Vb~+2e ]

_ x.+) W¢g"¢(I +--~¢ . (12) Us, at \

X,+= W e ( 7 i + I_~_~: (V~ + dp)_(qNc_Jc_~at)jLqN ¢'~-][- + _Jc]-'_V~atA 1),

X+=~IW¢(N~-qv,,,~Jc ~+X+N+] (10)

Both X + and X~ defined above are positive values. Taking the derivative of eqns (10) and (11), one obtains g,c in the case of full collector depletion:

6Usat { ( qVsat Xb--X + X_~c ) ,+ )0 +

,[ ]} - -5 1 "~- ~ ( Vb¢ -~- (~b) --1. (11)

(9)

The first term of eqn (12) is due to the variation of dV~ only, neglecting the Early effects. It is equal to E/W c, which is simply the capacitance associated with the depleted collector thickness, in series with a large capacitance associated with the narrow depletion thickness of the n ÷ subcollector. This is analogous to a pin diode. The second and third terms are analogous to the corresponding terms in eqn (7).

1054 WILLIAM LIU a n d JAMES S. HARRIS

2.3. Base pushout condition

As J~ increases past Jcr~tic,J, the slope of the electric field inside the junction decreases. The slope goes to zero and then becomes increasingly negative once Jc is greater than Jl. In order to maintain a constant voltage, Vbc + qS, across the junction, but under a restriction of having a particular slope, the magnitude of the field at the base edge of the junction continues to decrease. When the field at the base edge ap- proaches zero, a new phenomenon occurs. To prevent the field from becoming positive, the base majority carriers spill over into part of the collector adjacent to the base. This is base pushout in which the base majority carriers compensate the negative charges in part of the collector so that the electric field there remains near zero[13,14]. The high collector current density above which the base starts to push out is defined as Jchcc. It is similar to that given in Ref. [13], and is more exactly given as:

= J l ( l Vbc + q~ Jchcc + ~ ) . (13) \

Figure 3 illustrates the charge profiles in the collector during base pushout. The general features of the charge profiles shown in Fig. 3 agree with SEDAN simulation[7]. As mentioned, base pushout occurs when the base majority carriers spill over to a portion of the collector so that the electric field near the junction remains roughly zero. This portion of the collector is current-induced base and is shown as

region A in Fig. 3. Because the electric field there is roughly zero, the hole concentration nearly equals the electron concentration minus the collector doping:

p(x)=n(x)--Nc, (region A). (14)

The thickness of region A of the collector, y (Fig. 3), is determined from observing that, region B, depleted of the screening holes, is where most of the V~ drops across. From the Poisson equation and the charge neutrality requirement[l 5]:

I2.

y = We-,¢/2E "(V~+ga)\qVsat-qNc (15)

It should be noted that in the current-induced base (region A), even though the electric field is close to zero due to the base majority carrier spill over, the electric field is not identically zero. Rather, in order for the hole current in region A to be zero, there exists a small yet finite electric field inside region A, similar to that in the bulk base[16]. It is given as:

kT l dp E ( x ) - (region A). (16)

q p(x) dx '

From eqns (14) and (16), the total electron current in region A which consists of both drift and diffusion components is given as:

qD, d(pn) ( n I ~ dn Jc- P ~x -qO~ 1 + ~ ] ~ x,

(region A). (17)

mobile carrier concen.

base

charge concen.

P 0 ~ ( ns

electron

Y Z

= J c / q vsat

Nc

+ n = N

C

W C

I~ x (distance)

1 x. n

Fig. 3. The electron and hole carrier concentration profiles in the collector during the base pushout operating condition.

Base--collector capacitance 1055

Since Jc is a known quantity, the slope of the electron profile, dn/dx, can be determined from eqn (17) once n~ is known, n~ is to be determined from the conditions in region B. In region B, where the electric field is high, the collector current consists solely of a drift component with electrons travelling at v~,. Thus, the electron concentration at region B, ns, is constant:

ns = Jc/qv~t. (18)

For simplicity, nt can be taken to be n~. That is, one assumes that in the transition from region A to region B, the electron velocity monotonically increases from a low value to v~,,, and the diffusion component of the collector current gradually decreases as the electron concentration rounds off rapidly toward n~ in the vicinity of the transition. Even though this is true for Si-based devices, it is not exactly correct for GaAs. For GaAs, if one assumes that the local drift velocity depends on the magnitude of the local electric field, then electrons at low electric field could attain a velocity higher than v~,. Consequently, n~ is slightly smaller than n~, as shown in the Appendix. Nonethe- less, n~ can still be taken to be roughly equal to n~ without introducing significant errors.

During the base pushout, the incremental hole concentration integrated through the base--collector junct ion due to dV~ is simply q(n o - N~).y, where no = nl + dn/dx "y (Fig. 3) and dn/dx is determined from eqn (17). Hence, from eqn (4), C~ in the case of base pushout is:

(Wo-y ) [- n L C bc=2_~ ~_~)Lq _qN¢ +_ff y(1 n~ "~_,-]

(19)

base-collector capacitance

Cbc ( El c m 2 )

10 .6

10 .7

collector layer 10.8 partiallydepleted l l ( b a s e p ush°ut

collector layer fully depleted I

10-9

Jcritical Jchcc

1 0 - | 0 . . . . . . . . . . . ~ ' ' "~ . . . . . . . . . 10 3 10 4 10 5 10 6

collector current densi ty ( A / c m 2 )

Fig. 4. The calculated base-collector capacitance vs collector current density. The diamonds denote SEDAN simulation results. The device parameters used are: Wc = 5000 A , No=5 x 10l~cm -3, Vb¢ = 0.5 V, Xb= 1000A, Nb= l x 1019¢m -3. Material parameters used, such as

mobility, are the same as those used by SEDAN.

base-collector capacitance Cbc ( F/cm 2 )

10 .6 SEDAN simulation for Vbc= 1,5V

10 -7

10 -8

• • - Vbc=0.5 V

Vbc=l.5 V

- - - - - Vbc=4.0 V

2 ' . . . .

. • /

. , / "

/ / ' /

10-9 . . . . . . . . i . . . . . . . . I . . . . . . . . 10 3 10 4 10 5 10 6

collector current densi ty ( A~ cm 2 )

Fig. 5. The calculated base-collector capacitance vs collector current as a function of base-collector bias. The device parameters used are the same as those of Fig. 4,

except Vb~, which is varied.

3. C O M P U T A T I O N A N D A N A L Y S I S

The calculated C~ for a transistor with N c = 5 x 1016 cm -3, W c = 5000 A and V~ = 0.5 V is shown in Fig. 4. These calculated C~ in Fig. 4 can be compared with the experimentally extracted capaci- tance in Fig. 3a of Ref. [8], where the transistors had almost identical parameters to the one used in this calculation. These two figures show close resemblance in C~ throughout the entire current range. In particu- lar, in both figures, the onset of current density for which the capacitance decreases to roughly half of its low current value occurs at I x 104Acm -2. Their capacitance decrements also exhibit similar current dependence. Furthermore, the current interval in which C~ maintains a constant low value when the collector is fully depleted are roughly the same. The current density at which Cb~ dramatically increases is slightly larger than Jch~ "~ 1.2 X 105A cm :. (There- fore, the highest values o f f r andfmax of the device of Ref.[8] occur at current levels where the collector is fully depleted or just at the onset of base pushout.) It is noted that the dip at the initial stage of the base pushout predicted both by this analysis and by SEDAN is not seen in the extracted capacitance. This is because the extracted capacitance includes small but nonzero parasitic base-collector capacitance, which masks the decrement of the intrinsic base- collector capacitance. The parasitics arise from the base-collector junct ion beneath the base contacts. Once the intrinsic capacitance increases dramatically after the base pushout, the magnitude of increase in C~ in the reference is also comparable to that of the modeled result.

Figure 5 illustrates calculated results of C~ as a function of Jc under various V~ s. As V~ increases, more collector donors are ionized for a given collec- tor current, so the collector is more easily fully depleted and Jc,tical decreases. An increase in Vb~ also

1056 WILLIAM LIU and JAMES S. HARRIS

base-collector capacitance

Cbc ( F/cm 2 )

10 .6

/ / 1 0 .7 /

. . . . . . # r

• SEDAN, Wc=6OOOA - - ~ " ~ ,I t

1 oS Wc : 6oooA !~l,i' W~ = 3000

W~ : 2OOO A

, ~ , , , , / t l , , ~ , , ,~ 1 0 - 9 , , , ~ , , ~ , 1

103 104 105 106 collector current density ( A/cm 2 )

Fig. 6. The calculated base-collector capacitance vs collector current as a function of collector thickness. The device parameters used are the same as those of Fig. 4,

except We, which is varied.

the junc t ion approaches zero at higher collector currents. Ano the r consequence of having thin We is tha t dur ing base pushout , C~ is lower [equation (19)], even though the capacitance is higher pr ior to the base pushout . It is conceivable tha t for a t ransis tor with thin W~ biased at a high V~, Cb~ will not vary much from its low current value th roughou t the three operat ing condit ions.

Figure 7 shows calculated Cb~ as a function of collector doping. Because a high donor concent ra t ion prevents the injected electrons from easily compensat - ing the donor charges, J~h~c increases as Arc increases. Fur thermore , as also have been demons t ra ted in Figs 4 -6 , Fig. 7 shows that the calculated Cb~ from the simple equat ions of this analysis agree very well with S E D A N simulat ions which use complicated compute r algori thms. The agreement is remarkable even in the base pushout condit ion.

4. C O N C L U S I O N

causes the electric field inside the junc t ion to be larger. Consequent ly , larger values of Jch~ are needed to present enough mobile carriers to significantly modify the field distr ibut ion. Because Jc~it,¢,l decreases and J~h~ increases at higher Vbo the current interval in which the collector remains fully depleted is thus larger.

Figure 6 shows the var ia t ion of calculated Cb¢ with various collector thicknesses. With th inner W~, the collector layer is more easily fully depleted, rendering J~,,~c,~ to be small. In one extreme in which W~ = 2000 A, the collector layer is depleted at the given V~: wi thout any collector current . A no t he r feature of Fig. 6 is the larger values of J~h~c for devices having th inner W~. This results because for th inner W~, the magni tude of the electric field in the collector is larger so tha t the electric field at the base edge of

base-collector capacitance

ChC ( F/cm 2 )

10 6

1 0 -7

10-8

Nc=l.5 x I o l 7 /cm 3 , / /

( • is f rom S E D A N simulat ion ) Nc = 5 x t 0 l 6/cm3

10-9 , , , , , , , , I ~ , ~ , , , , , I , . . . . . . 103 104 105 106

collector current density ( A/cm 2 )

Fig. 7. The calculated base-collector capacitance vs collector current as a function of collector doping. The device parameters used are the same as those of Fig. 4,

except N~, which is varied.

A physical model for the base-col lector capaci- tance as a funct ion of collector current has been presented. The expressions for the capaci tance are analytical and can be calculated easily, faciitating the tasks of opt imizing a t ransis tor 's high-frequency preformance. The calculat ions have been verified with S E D A N are in agreement with published experimental results. The capaci tance has been investigated as a funct ion of base-col lector bias, collector thickness, and collector doping.

Acknowledgements--We thank K. Shepard and Z. Chen tot their fruitful discussions. This work was supported by DARPA and ONR through Contract N00014-K-83-0077. W. Liu was also supported by an AT&T Bell Laboratories Fellowship.

REFERENCES

1. D. D. Tang and P. M. Solomon, IEEE J. Solid-St. Circuits 14, 679 (1979).

2. M. Madihian, K. Honjo, H. Toyoshima and S. Kumashiro, IEEE Trans. Electron Devices ED-34, I419 (1987).

3. H. F. Cooke, IEEE Trans. Electron Del'ices ED-59, 1163 (1971).

4. J. J. Liou, IEEE Trans. Electron Devices ED-34, 2304 (1987).

5. J. J. Liou, F. A. Lindholm and J. S. Pard, IEEE Trans. Electron Devices ED-34, 1571 (1987).

6. F. A. Lindholm and J. J. Liou, J. appl. Phys. 63, 561 (1988).

7. Z. Yu and R. Dutton, SEDAN I I I - -A Generalized Electronic Material Device Ana(vsis Program. Inte- grated Circuits Laboratory, Stanford University (1985).

8. K. Morizuka, R. Katoh, K. Tsuda, M. Asaka, N. Iizuka and M. Obara, IEEE Electron Device Lett EDL-9, 570 (1988).

9. c .T . Kirk Jr, IRE Trans. Electron Devices9, 164 (1962). I0. S. P. Morgan and F. M. Smits, Bell Syst Tech. J. 38,

1573 (1960). 11. J. M. Early, Proc. IRE 40, 1401 (1952). 12. R. L. Pritchard, Electrical Characteristics of Transistors.

McGraw-Hill, New York (1967).

Base-collector capacitance 1057

13. D. L. Bowler and F. A. Lindholm, IEEE Trans. Electron Devices ED-20, 257 (1973).

14. R. J. Whittier and D. A. Tremere, IEEE Trans. Electron Devices ED-16, 39 (1969).

15. S. K. Ghandhi, Semiconductor Power Devices. Wiley, New York (1977).

16. R. S. Muller and T. I. Kamins, Device Electronics for Integrated Circuits, p. 202. Wiley, New York (1977).

APPENDIX

in the base pushout condition discussed in Section 2, the continuity of electron concentrations of region A and region

(a) drift i velocity

(b)

2

I I

El E 3

Vsat

Electric Field

electron COIICeI'I,

, \ : ! \ l 3 ,

\ l l F

_ _ ( ( ,

n , = J e / q v s , t

distance

Fig. A1. (a) The simplified drift velocity vs electric field typical of GaAs. (b) The magnified electron concentration

vs distance plot at the transition region C of Fig. 3.

B was briefly mentioned (Fig. 3). The continuity condition depends on whether the electron velocity monotonically increases to vsa t. For GaAs, because of the intervalley transferring mechanism of the electrons, the electron drift velocity attains a maximum value at a lower field which is larger than v~t. This effect is shown in the simplified drift velocity vs electric field diagram of Fig. Ala. As a result of the drift velocity behavior of Fig. Ala, an electron concen- tration profile in the transition region denoted as region C (Fig. 3) is as shown in Fig. AI. In both Fig Ala and b, point 3 corresponds to where the electrons travel at vsa t. The electron concentration at point 3 is thus ns=Jc/qVsa t [eqn (18)]. However, before reaching v~t at point 3, the electron velocity is higher such as that in points 1 and 2. To maintain current continuity, the electron concentration at these points must be smaller than n s. It should be noted that at point 2, there exists a negative diffusion current which partially cancels the drift current. Therefore, even though the electron velocity at point 2 is larger than at point 1, n 2 is greater than n v Since the diffusion gradient is largest at about halfway between n s and n~, we shall choose n2=(n~+nl)/2 to correspond with point 2 at which electrons attain the highest drift velocity.

From the above discussion, we utilize a linear approxi- mation for the electron profile in region C as shown in Fig. 3. Similarly, it was assumed that in the same region, the hole concentration decreases linearly from n L - N¢ to zero. With these assumptions, the net charge increases linearly from zero to n s - N¢ in a distance of z, which is determined from a Poisson equation:

E E 3 - E I z = 2q n - - ~ . (A1)

Finally, n, is determined from the current continuity requirement at point 2:

[-q qD,-]-'[- q , , = L ~ V 2 + T j LJe - ns(~ v2 - ~ ) 1 , (A2)

where V2, E3, E I are known parameters: E3= 1.5 x 10Wcm -1, E 1 = 2 x 103Vcm -1 and //2= 1.45x 107cms -I for N c = 5 × 1016 cm-317]. Using these values, one can solve for n v C~ in the base pushout operating condition is determined from eqn (19) with the substitution of n s by n 1.