Analytical base transit time of integrated bipolar transistors in quasi-saturation and hard saturation

M.M.Shahidul Hassan

Abstract: Based on the assumption of negligible recombination within the thin epitaxial collector layer of an integrated bipolar transistor switch in quasi-saturation, solutions to the collector minority carrier profile and transit time in the induced base are derived. In contrast to Dai and Yuan’s analysis (1997), the present analysis takes both the dnft and diffusion currents into account and is valid for all levels of injection. Dependence of transit time on characteristics of the epitaxial-substrate interface and recombination at the interface is studied for the transistor driven into hard saturation. At high effective surface recombination velocity, recombination at the interface cannot be neglected. The study shows that transit time increases more rapidly with collector current when the transistor operates in hard saturation and the interface is hghly reflecting.

1 introduction

In modem integrated epitaxial n+pn-n+ bipolar-transistor switches, the collector is more lightly doped than the base. The transistor with a lightly doped collector drives into quasi-saturation at low voltage and high current densities. In quasi-saturation, an injection region wdl be formed adja- cent to the metallurgical baseecollector junction. Work on the modelling of collector current at high current densities including quasi-saturation effects has been published recently [l-31. The base transit time was not studied in these papers. Base transit time is an important parameter in determining performance parameters such as the maximum frequency of operation, cut-off frequency and the noise figure of a bipolar transistor. Recently Dai and Yuan [4] have obtained an analytical equation of base transit time taking into account the high-current quasi-saturation, but drift current within the quasi-saturation region and charac- teristics of the epitaxial-substrate interface were not consid- ered in determining the minority carrier profile. The minority carrier recombination withn the collector can safely be neglected for an integrated bipolar-transistor switch with a thin collector under quasi-saturation. When the whole collector is invaded by minority carriers, the minority carrier profile and, consequently, the base transit time depend on the reflecting interface and can become sig- nificant when the interface is highly reflecting (i.e. has a low value of surface-recombination velocity).

As the present model takes into account both the diffu- sion and drift currents and also the bloclung property of low-high interface, the model predicts more accurate values of base transit time than those estimated by Dai and Yuan [4]. On the other hand, the analysis is one-dimensional and lateral base-current effects have not been considered. These

effects have an influence on collector current and transit time. A more complicated model can be adapted incorpo- rating nonideal effects.

2

2.1 Quasi-saturation At hgh collector current density and low collector-emitter voltage, an appreciable number of minority carriers are injected into the collector and an injection region is formed in the collector adjacent to the metallurgical collector-base junction of an epitaxial n+pn-n+ bipolar transistor. As a result, the injection region is electrically equivalent to a por- tion of the base region and this injection region is called the induced base region.

Taking into account the hole current in the injection region, collector current is given by

Derivation of the model equations

Jc = Jnc + Jpc (1) where Jnc and Jpc are the electron and hole current densi- ties, respectively, at any x. Jc is independent of x. In quasi- saturation, Jpc = 0 and Jc = JnC

The basic equations are

where, the symbols have their usual meanings. The collec- tor considered here is made of lightly doped n-type semi- conductor of constant doping NC bounded by a heavily doped n+ region at a distance WC.

The quasi-neutrality condition requires that p - n + N c = O f o r O < n : s W c (4)

0 IEE, 2000 IEE Proceedings online no. 2oooO201

When Jpc = 0, eqn. 3 gives the electric field within the quasi-saturation region

(5) 1 &J P dx

DOL 10.1049/1pcds:2oooO201 Paper fmt received 19th February and in revised form 10th September 1999 E = VT-- The author is With the Departnient of Electrical Engneering, King Fahd Uni- versity of Petroleum and Minerals, Dhahran 31261, Saudi Arabia where, V , = kT/q is the thermal voltage.

IEE Proc.-Circuits Devices Sysf . , Vol. 147, No. 2, April 2000 129

Using eqns. 2, 4 and 5, the electron current density withm the injection region can be shown to be

where Dni is the electron diffusion coefficient in the injec- tion region.

Integrating eqn. 6 from x = 0 to x gives

where po is the hole concentration in the collector at the edge of the collector-base junction, i.e. at x = 0.

The induced base transit time of electrons is given by 2 1 X,

( 8 ) where x1 is the width of the injection region.

Using eqn. 7 and the derivative of p(x) in eqn. 8, the inte- gration gives the following analytical expression for transit time:

+N$ In 0 -"} (9) 1 2 1 - N c

where no is the electron concentration at x = 0. If no, nl(= p1 + N,) at x = x1 and Jc are known, then the induced base transit time z, can be evaluated from eqn. 9. To determine no and Jc, the internally forward-biased col- lector-base junction voltage VcB and the emitter-base volt- age VEB must be calculated.

The value of n1 can be obtained if p(xl) is known. The value of p(xl) is always somewhat arbitrary. An assump- tion of p ( x l ) = N , may be considered, but p(xl) is obvi- ously a function of the collector current density JC [5]. Therefore, a better definition of the boundary is to adopt p(xl) in terms of collector current density JC In this work, the following condition is chosen to get a simple analytical form for p(xl):

The procedure of finding p 1 = p(xl) is similar to that pre- sented in [3] .

Using eqns. 5 and 7, the electric field is obtained as

Differentiating eqn. 11 and using eqn. 10, it can be shown that

where, E is the relative permittivity of silicon. The voltage across the injection region can be obtained

by integrating eqn. 11 from x = 0 to x = x1 and, using eqn. 7, we have

The region adjacent to the injection region is called the end region. In the end region, the excess hole concentration can be neglected. In this region, the electron current density determines both the field and electron distributions withm the end region. The voltage across this region was derived in [5] and was found to be:

For J, < J,:

For J, > J,:

Vi = 2E

where, E, is the field for whch the velocity takes the scat- tering limited value vs [6] and J, = gv,N,.

The limit of quasi-saturation mode is reached when x1 = 0. This acounts for the different current densities in

+ I c R s a t V S Wc VCE = VBE + IBRB +

pnc (2 - 1) for J , < J , (16)

When the terminal voltage between the emitter and collec- tor V,. exceeds the voltage given in eqn. 16, the device enters the active mode.

For J, > J,:

where RB is the extrinsic base resistance, and R,,, is the external series resistance in the emitter and collector. The above equations for current and voltage can be used to determine J , and the terminal voltage VcE as a function of JB. The electron concentration no and J B can be obtained if the internally forward-biased collector-base junction volt- age VcB and the emitter-base j,unction voltage VEB are known, respectively. Once Jc, no and n1 are known, it is easy to determine base transit time in the induced base from eqn. 9.

The DC collector-to-emitter voltage under quasi-satura- tion is given by

VC E = VB c 4- VC B + 113 RB + Vzn j + VI + I C R s a t (1 8 ) The emitter-base junction voltage VBE. can be obtained from Jbe(= JB) [3]. The base current JB is the hole current density Jbe injected into the emitter. As the emitter is heav- ily doped, the band-gap narrowing effect [7] needs to be considered in the calculation of VBE

2.2 Hard saturation When a transistor is driven into hard saturation, the whole collector region is invaded by minority carrier and the width of the injection region will become equal to the col- lector width W,. The structural requirement of the low- hgh junction (LHJ) at the rear contact results in a bound- ary condition whch causes the minority carrier giving the blocking property of the LHJ. The hole concentration p w at x = W, is detemined by the hole current density flowing through and it is given by

In this paper, the expression for Sefl given by Godlewski, Barona and Brandlionst [SI is used.

As the recombination within the collector is neglected, the hole current Jpc will be constant within the collector.

IEE Proc -Circuits Devices Syst., Vol. 147, No. 2, April 2000 130

The hole profie p(x) within the collector can be obtained from eqns. 1-3 and using the charge neutrality condition. The expression for p(x) can be shown to be

1.0.

5 - Jnc + m J p , -

W,i (20)

where, m = DniJDpi. The hole conentration pW at x = W, can be easily obtained from eqns. 19 and 20.

Using eqn. 8 and the derivative of eqn. 20, the induced base transit time under hard saturation can be expressed as

[(Po - 1

(21) The hole current Jpc can be obtained from JB = Jhe + Jpc [31.

The voltage across the collector is given by

(22)

(23)

The total base transit time is given by [4]

7-€! = Q J n(z)dz +rc - Jn

- WO where J,, is electron current density in the base and W, is the base width. As the base width is thin, the recombina- tion within the base can safely be neglected. Neglecting recombination within the base, zB is given by [4] as

where, n(-W,) is the electron concentration at the base side of the emitter-base junction. n(-WB) can be obtained from eqn. 5 of [9].

3 Results and discussions

In this Section, the equations derived above are used to study base transit time under different modes of operation of integrated epitaxial n + p ~ n + bipolar tranistors. The equivalent-circuit model of the transistor is similar to that shown in Fig. 1 of Dai and Yuan [4]. In the present work, recombination is neglected only when the transistor oper- ates in quasi-saturation. However, both the recombination and the bloclung property of the low-high interface are incorporated when the transistor operates in hard saura- tion, i.e. when the whole collector is invaded by minority camers. In the previous work [4], transit time was studied

IEE Proc.-Circuits Devices Syst., Vol. 147, No. 2, April 2000

when the transistor was driven in hard saturation and recombination and the blocking property of the low-high interface were not considered. The expressions for induced base transit time obtained in this work are therefore differ- ent from those obtained in [4]. The differences between the two models also lies in calculation of Jc and the internally forward-biased collector-base voltage VCB. When the transistor operates in quasi-sauration, for a given J, the emitter-base-junction voltage V& can be determined from JB [3]. Using ths value of VEB and a particular value of ter- minal voltage VcE, n,, n(- WB), nl, Vhj and V, for quasi-sat- uration are determined self consistently. For hard saturation, no, n(- W,), pw, J,,, J b s and Vinj are determined self consistently. The base transit tune can then be obtained from eqn. 24.

" 4 5 6 7 a

10 10 10 10 10 10 Set, cmls

Fig. 1 recombination velocitv Sef

Collector hole mrent density Jpc us a f ict ion of effective suface N , = 1 Y 10'6cm?, wc = 1 wi, Jc = JOOA/cm, J B = l0OAicm

The dependence of diffusion coefficients on injection [lo] and doping density [l I ] is taken into account in the present analysis. Fig. 1 shows the recombination current Jpc as a function of the effective surface recombination velocity S,,, The plot shows that, for a given collector current density JC and base current density JB, a significant fraction of the total base current J, will enter the collector at high effective surface recombination velocity Sefr and Jpc cannot be neglected even if the collector is thm.

Dependence of the reflecting property of the low-high (am') interface on induced base transit time is shown in Fig. 2. The transit time depends on the characteristics of the epitaxial-substrate interface and can become very important when this interface is highly reflecting. For given Jc and JB, the stored charge within the collector in hard saturation decreases with increase of the Sefr. Fig. 3 shows the induced base transit time as a function of collector cur- rent density Jc for a collector width of 1 . 5 ~ whde the same is plotted in Fig. 4 for a collector width of 0 . 5 ~ . The transit time increases significantly when the transistor enters hard saturation. When the collector junction is inter- nally forward biased, the charge stored withm the collector increases with the decrease of collector current and the increase of base current. For a transistor with a large base current density JB, induced base transit time also increases with collector width We The total base transit time against the collector current density JC is shown in Fig. 5. For the

131

\ I I

Se* cmls

N , = 1 1016m-3, w, = i.spm, s,,= i04cmis

Fig.2 recombination velocity S g

Dependence of d c e d base transit time T~ on effective susface-

l 2 JB= I O Ncm

I

10 20 30 0’ -2

Jc, “32 (x10 )

0

Fi . 3 Inheed h e transil time zc agabt collector current density J c for d & m t bme current ahi tks for collector wdth of1.5pn N = = 1 1 0 1 6 ~ - 3 , w,= isw, s,= 1 o 4 d s

JB= 10 Ncm

20 Ncm

40 Ncm

Fi .4 d&mt bme current densities for colgctor width of 0 . 5 ~

132

I&ed base trumit tkm t against collector nrrrent density J , for

transistor with large base current density JB, the increase of base transit time occurs for a given Jc because of the enhanced stored charge within the collector.

81 2

JB=10 A/cm

20 30 01

lo 2 -2 0

Jc, Ncm (x i0 )

Fig.5 bme nrrrent h i t i e s @a trmistor with Nc = ld6 m? and S e , = I d d s

Total base trmit time zc against collector current der@ for dijerent

Wc = 0 . 5 ~

4 Conclusions

The base transit time for the modem integrated bipolar transistor with a thm collector in quasi-saturation has been evaluated considering dnft and diffusion currents withm the collector. When the collector is invaded by minority carriers, the induced-base transit time has been studied incorporating the minority carrier bloclung property of the low-high junction. For a given base current, the maximum base transit time occurs when the transistor enters hard sat- uration. It is concluded that to keep base transit time to a minimum in hard saturation, high effective surface recom- bination velocity Seff is desired.

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IEE Proc-Circuits Devices Syst., Vol. 147, No. 2, April 2000