SECOND-ORDER EFFECTS IN JUNCTION BIPOLAR
TRANSISTORS
Zvonko Fazarinc
Keywords: semiconductors, bipolar transistors, bipolar junction transistors, transistor models, computer analysis, Ebers-Moll model, Gummel-Poon model, Early effect, Kirk effect, C language
Abstract: The Gummel-Poon model of bipolar transistor is derived with inclusion of Early and Kirk effects.
/
Efekti drugega reda v bipolarnih tranzistorjih
Ključne besede: polprevodniki, transistorji bipolarni, BJTtransistorji bipolarni junction, modeli transistorjev, analiza računalniška, Ebers-Moll model, Gummel-Poon model, Early efekt, Kirk efekt, C jezik
Povzetek: V prispevku je izveden Gummel-Poon model bipolarnega tranzistorja, ki vključuje Earlyjev in Kirkov efekt.
1. Introduction
The first model of the bipolar junction transistor (BJT) was published in 1954 by J.L.Moll and J.J.Ebers from Bell Laboratories /1/. The model quickly became the wori^horse of the industry because it captured the state of the art of the transistor technology of that time in a most elegant way. The model was simple and sufficiently accurate to allow prediction of transistor circuit behaviour based on a few physical parameters.
The transport of charges in a semiconductor is governed by diffusion, drift and recombination. The early transistors had relatively long bases and low current gains, both resulting in a considerable recombination in the base. High level injection and other phenomena associated with drift were hardly a problem of that period. The Ebers-Moll model was based on the diffusion and recombination mechanisms only, and was as such in tune with the technological state of the 50-s and 60-s.
The semiconductor technology was making big strides in the subsequent decades and the BJT bases have shrunk down to below a micrometer. Due to this and in conjunction with constant improvement of material purity the recombination rates were suppressed to theoretical minima. At the same time the quest for ever higher current gains called for high base resistivities and the high-level injection became the limiting mechanism of current gain.
It was until 1970, that H.K.Gummel and H.C.Poon also of Bell laboratories published their Integral Charge Control model of the BJT /2/ which captured the high-level
injection mechanism as part of its physical basis. Again the model was a reflection of the state of tehnology. Gummel and Poon have correctly concluded that recombination played a minor role in physical behaviour of transistors and have ignored it in favor of including the drift mechanism in addition to the diffusion. The base current in the Ebers-Moll model was a natural consequence of recombination in the base. With no recombination considered the Gummel-Poon model requires a separate theory for the base current.
Both models, the Ebers-Moll and Gummel-Poon must treat the Early effect /3/, the Kirk effect /3/, the emitter crowding /5/ and other second-order effects as separate problems. This paper integrates the Early and the Kirk effect into the Gummel-Poon model derivation and thereby avoids the present conflict of first deriving the Gummel-Poon model under the assumption of a fixed base length, and then adding the Early and Kirk effect which both arise from a variation of the base length.
2. Definition of Model Parameters
We arbitrarily choose an NPN bipolar transistor as the basis for our analysis. The results are easily expandable to PNP structures. Extensive approximations are necessary to arrive at the basic Gummel-Poon model. A real transistor is a three- dimensional structure while the Gummel-Poon model possesses no spatial dependence at all. We start with a one-dimensional approximation of the transistor and demonstrate how the spatial dependence is integrated out. In process we make the user of the model aware of the many necesary approximations.
A! the top of Fig.1 is a schematic depiction of an NPN transistor. The polarities of externally applied biases are in agreement with normal operation while the junction biases at xi and are defined in agreement with convention.
Fig. 1:
Xi	X2
NPN Transistor Schematic
Below the transistor and in relative position to it are shown the internal concentration of electrons n and of holes p as functions of x. The concentrations shown correspond to a positive Ve and a negative Vcett. The junction at x^ is therefore back-biased.
3. Modeling of High-Level Injection
The high-level injection or the Webster effect is an integral part of the Gummel-Poon model but we will rederive it here in order to lay ground for inclusion of additional second-order effects. The Webster effect originates in the base voltage drop near the emitter junction. This directly reduces the effective bias across the junction from the full value Ve. It happens when the number of minority carriers injected into the base approaches that of the background concentration and results in a significant drop of current gain. The absence of neutrality which accompanies this phenomenon requires that the model include the electric field. We start therefore with the transport equations /6/ in terms of electron and hole flux densities Fn and Fp, respectively, which contain in addition to the diffusion term the drift term as well.


(2)
In (1) and (2) D stands for the diffusivity of respective carriers and E(x) for the electric field, q is the electronic charge, k is the Boltzmann's constant and T is the absolute temperature. Fp in the base's P-region is the majority carrier flux and as such negligible compared to Fn. We take advantage of this, set it to zero, acid find from it the electric field E(x). This field is, of course, common to both carriers and we substitute it into the first equation to obtain
^	ax p (x)
Multiply both sides by p(x) and obtain F,{x)p{x) = -Z)„

The term in brackets is recognized as the derivative of the product p(x) n(x) so the integration of the above equation between the two junctions yields
fF„(x)p(x)dx = -D„;,(x)p(x)

(3)
The product n(x) p(x) is given by the mass action law 111 everywhere except nearthe two junctions where the law of the junction applies. Therefore the following must be true
qVe/kT
n(xi) p(xi) = n iä
(4)
(5)
where n/is the intrinsic concentration of the host material. Substitute (4) and (5) into (3) and end up with the following integral equation

jF„(x)pix)cbc = -D„n}

(6)
J^l
A computer solution of (6) for any arbitrary distribution of p{x), given the other parameters, is quite feasible and may prevail in the future as a means of circuit analysis. At this time the computational effort still becomes excessive if thousands of transistors are involved even in view of the present day computer performance level. Therefore, we are forced to eliminate the last remaining spatial coordinate, i.e., xfrom our model. This we do by making the assumption of space charge neutrality which is in direct conflict with the initial goal of deriving a model subject to internal electric fields. The fact is that we cannot solve the integral in (6) for the general case by any known means. A consoling circumstance is that a very small charge unbalance is sufficient to maintain the electric fields in the region of interest and the neutrality condition can be justified on that basis. This says that sum of all charges in the region of interest must be equal to zero. In addition to mobile charges r^{x) and p(x) we
must consider the ionized innpurities which results in the comnnonly encountered neutrality condition
Nd(x) - Na(x) + p(x) - n(x) = 0	(7)
An alternative expression for the minority carrier flux density Fn(x) is in the terms of their density n(x) and velocity Vn(x)
Fn(x) = n(x) Vn(x)
A combination of (7) with the above produces
Equation (9) is the principal result of the work by Gum-mel and Poon if w is replaced by the unbiased base length wto and the effective collector junction bias Vcetf is raplaced by the applied collector-base voltage Vc. We will give this expression an interpretation later.
4. Base-Width Modulation
Parameters w and Vceif can be expressed in terms of some physical parameters illustrated in Fig.2. This depicts the electric field for our idealized structure from Fig.1 with the unbiased case shown in dashed lines.
Before we substitute (8) back into (6) we recognize that the donor density Nd in the base P-region of our NPN transistor must be zero and that in absence of recombination the minority flux density in the base Fn(x) must be independent of position xand equal to the collected flux density Fc. Consequently (6) translates into the following quadratic equation
j! i ^"v^j	xi
-^qvceff/kt _ ^qv,/kt
= 0
The solution of the quadratic yields

jN,{x)dx

■»1
■<2
2/
dx
1 - ' 1 —--2- J


•ri

The integral of dx/Vn(x) is the transit time from xi to X2 which we denote by Xn- The transit time in the base is primarily controlled by diffusion despite the electric field-presence. We will therefore use the diffusion aproxima-tion of the transit time (8)
{X2 -x,f w^
2D„
2D„
Finally we are forced to assume that the concentration of acceptors in the base Na(x) is independent of x, admittedly not a good approximation. All of this results in the highly simplified expression for the collector flux density
fc =
D„N„

The collector current density Jc is the negative of the electron flux density multiplied by q
Jc =

1-

A
4>e-ye
\ <f>e <
\
-V
^-Ibo-
leo^
'b

7
---
v
A

Fig. 2: Electric Field Inside the Transistor.
The area under the field cun/e equals the voltage across the particular junction. For the unbiased case we have in the emitter and collector junctions, respectively, the following built-in voltages
kT, ^dcNa -—In-—
and
kT, ^dcN^ — In--r—
(10)
In terms of these it is easy to extract by simple geometric considerations for similar the following expression for the diffusion width of our transistor
w
=-UoVi-yj^e -/i^/T^
(11)
leo =\/ <i>ei^/q{Na+Nae) and Ibo 'i>ce/q{Na+Nac)
Expression (11) contains the base length modulation caused by the emitter and collector voltage. The latter is, of course, the Early effect and if we substituted (11) back into (9) we would have included the Early effect in the Gummel-Poon model. But we have promised to include the Kirk effect as well so we must now focus on it before we finalize the model.
As the readers might know, the Kirk effect is also referred to as "base push-out", i.e., abase lengthening effect.
It has its origin in the resistive drop in the collector region for high currents. As the collector current rises some of the applied collector voltage l/c is expended in the collector resistance and the remaining junction voltage Vceff may be diminished to the point where no more collection of the arriving electrons takes place in the junction. The electrons start accumulating and the diffusion length of the base increases. This is illustrated in Fig.3.
From (11) we get for the length lb
lb = Ibo \/ l + {Vc-JcPcWco)/4>c
(13)
Expression (13) applies as long as lb remains real. It attains a zero value for
K-JcPcWco =-4>c or y^o
PcWco
(14)
Hence (13) applies for Jc < Jco- When Jc gets larger we have at hand the second case in Fig.3 which is easily solved for I as
/ = Wr,
JcPc
for >/,
cO-
(15)
We can now summarize this discussion by expressing the effective base width wand the effective collector bias Vceff applicable to (9) as follows
Case: Jc <Jco
W =	+ {^c ■JcPcWco)/'f>c
Vcea= -{Vc-J cPc^cc)
(16)
Fig.3:
Illustration of the Base Push-out.
Case: > Jco
W = Wt>o + Wco - {yc+4'c)ßcPc
(17)
In it we see the shape of the electric field for moderate collector current at the top and for high current on the bottom. What the two cases have in common is the area underthe field on the collector side which represents the collector voltage. One can observe the lengthening of the base region through which the electrons must diffuse as the current increases. The net effect of this is a reduced current gain but more importantly a quadratic increase of transist time, which reduces the hihg frequency cutoff of the transistor in inverse proportion. With this much physical insight we should be able to drive the effective collector junction bias Vceff and the diffusion base length w. We focus on the collector side and try to find the lengths lb or I as the case may be.
The top sketch in Fig.3 which representes a moderate current case suggests that most of the applied collector bias appears across the collector-base junction. The resistive drop in the collector region of resistivity pc is
Vceff = 0
Let us return now to expression (9), multiply and divide it by
and obtain
Who
Jc = 2/,-
^gVJkT _ ^qV^/kT
^ 1 + (18)
In (18) we have introduced the reverse saturation current density J s
Js =
qPn nj
Who Na
(19)
Jc9c(Wco-lc) = JcpcWco
with Wco defined in Fig.3. The approximation is justified for all cases of interest. The remaining collector junction bias is therefore
Vceff = -(Vc - Jc^cWco)
(12)
5. Base Current Model
To complete the model we must add the base current. For our NPN example this consists of holes flowing in the emitter-base diode and is made up of two components. One is due to carriers injected from the base into the emitter and the other has its origin in the transition
region where considerable recombination takes place. The reverse injection into the emitter is simply expressed as the diode current
^t ^ de
Ji is best determined experimentally because of ill defined parameters Nde and particularly We. The second component of the base current arises from the recombination in the junction. The Shockley-Read-Hall model/9/ relates the net recombination current J^to some more or less known parameters
dJrix)/q ax
pn-nj Tpn+T„p
Furthermore, within the junction the product of p and n must obey the law of the junction
p(x)n(x) = n^ e^^®^^
If we make the assumption that p = n in the transition region (not readily justifiable) we obtain
n = p m
Substitute this into the differential equation for Jr, integrate over the transition region width wt, and up with the following expression for the recombination component of the base current
nHe'""" -I)
Because many of the parameters in the above are not well known we lump them into a measurable parameter J2. We also recognize that for normal forward biases the exponential is much larger than 1 and the latter can be neglected. This produces for the recombination component of the base current the expression
The base current is then composed of the sum of the two componerits
(20)
6. Model Interpretation
Table 1 below lists the set of parameters used for model evaluation.
Table 1
Parameter
Value
Physical Meaning
m Na
1.45x10^°/cm®
Intrinsic Concentration Base Impurity Density
Ndc	IxlO^W	Epi Concentration
Dn	36 cm^/s	Electron Diffusivity
Wbo	1x10"" cm	Metallurgical Base
Wco	1x10"^ cm	Epi Thickness
ßmax	100	Maximum Current Gain
	0.1 Volt	Low End Bias for ß=1
All the necessary quantities can be calculated from the eight parameters given in the table. The reverse saturation current density from (19) evaluates to Js=1.2x10'^ A/crrf.The base current component Ji can be defined in terms of the maximum current gain ßmax which can be measured. Ji is then simply equal to Js/^max as is evident from Fig.4. Also evident from Fig. 4 is the fact that the second component of the base current determines the value of the emitter-base bias l/ß=r for which the current gain passes through unity. For the chosen value of 0.1 Volt we get from the intercepts of Fig.4 the
value J2 = Js e'
aO. 1/2kT
All the other quantities are compu-
ted from formulas (10), (11), (14), and (16) through (20) while the large signal current gain ß is evaluated as the ratio of Jc to Jb-
The results are plotted in Fig.4 versus the emitter-base bias Ve for l/c = 0. Intercepts of currents and of biases indicated in the graph have proven useful in determination of model parameters from measurements.
log/
A/cm^
.2	A	.6	.8
Fig. 4: Modeled Currents and Current Gain.
The collector current density and the current gain have been calculated once from (18) using (16) and (17) for w and a second time using w = wto - Ibo V^vjTvyö^
which ignores the Kirk effect. Dashed lines have been used in this latter case. A slight degradation of current gain is noticeable at high currents when Kirk effect is considered. As the collector bias is increased the degradation is still there it only occurs at higher currents. In Fig.5 the current gain is graphed for three different collector biases once with inclusion and once without the Kirk effect. The latter is plotted in dashed lines.
.?	.4	.6	.8	V,
Fig. 5: Current Gain and Base Width for Different Vc.
• The main problem caused by base push-out is not the current gain degradation but the degradation of high frequency performance. In Fig.5 the base width w is plotted for the same three collector biases as ß and the dramatic effect is inescapable. The highfrequency cutoff is inversely proportional to the square of the base width across which the carriers must diffuse and consequently the inclusion of the Kirk effect is of crucial importance where the high frequency performance is at issue. Finally in Fig.6 the common emitter characteristics of our sample transistor are shown for moderate bias conditions to ilustrate the Early effect.
Fig. 6:
2	4	6
Common Implementation in C
7. A Computer Implementation in C
C-language is gradually making inroads into scientific computing and it seems appropriate to show one implementation of our Gummel-Poon model in this language. The notation used is adhered to as closely as permitted by the language. Dots are used to mark those variables whose values may be assigned by the user.
/./Part 1: Assigned parameters
epsilon = 1e-12; q --= 1.6e-19; wbo = 1e-4; wco = 1e-3; Na = 1e16; Ndc = 1e15; ni = 1.45e10; mun = 1450;
Vc =.....(positive); Ve =......(positive);
temperature =.....(positive in Kelvin);
decimator =.....(.03) used in the example.
//Part 2: Computed Variables kT = 1.38e-23* temperature; Dn = mun*kT/q;
phic = log(Na*Ndc/(ni^ni))''kT/q;	H^c
rhoc = 1/(q*mun*Ndc);	//pc
Ibo = agrt(phic*epsilon/(q*(Na-f-Ndc);	Hko lb = lbo'sqrt(1-(-Vc/phic); Hib
Jco = (phic-fVc)/(rhoc*wco);	HJco
Js = 2*Dn*ni*ni/(wbo*Na);	HJs
J1 = Js/betamax;	//J?
J2 = Js*exp(q/(2*kT)*VbetaO);	IIJ2
niNaSquared = 2*ni*ni/(Na*Na);	//2(n/Naf
//Part 3: Tentative evaluation with approximate values
tentativeW = wbo-lb;
Vceff = - Vc;
A = exp(Ve)-exp(Vceff);
Jc = 2*Js*wbo/tentativeW*A
/(1 -Hsqrt(1 +niNaSquared*A));
Jb = Jrexp(q*Ve/kT) + J2*exp(q*Ve/(2*kT));
//Part 4: Final evaluation with fiitred feedback
if(Jc <= Jco)
{Vceff = - (Vc-Jc*rhoc*wco);
W = wbo - lbo*sqrt(1-Vceff/phic); {
else
{Vceff = 0{;
tentativeW = wbo-i-wco-(Vc+phic)/(Jc*rhoc);
w = (tentativeW-w)*decimator + w;
}
A = exp(Ve)-exp(Vceff);
Jc = 2*Js*wbo/w'A
/(1+sqrt(1+niNaSquared*A));
One may wonder why so many steps for a relatively simple set of equations. If the transistor is embedded in reactive circuitry, nothing can change abruptly and one can evaluate the model equations without any complications. But a true DC model presented here experience sudden changes of parameters. The transition from normal operation to Kirk state, for example, is very abrupt and is dependent on the collector current. The collector current, on the hand, is exponentially dependent on Vceft. We are consequently faced with a high gain feedback system of essentially infinite bandwidth. The behaviour of such systems is well known and therefore we have introduced a filter into the feedback loop. The "decimator" has a value less than unity and provides a bandwidth limitation to the system. If the Kirk effect is of no consequence Part 4 can be omitted all together resulting in dashed curves of Fig. 4 and 5.
8. Conclusion
The Gummel-Poon model has been revisited with the purpose of including the Early and Kirk effects earlier in the derivation. The result is a simpler model with fewer parameters that can be derived from the process and from simple measurements. The Kirk effect sets in abruptly and calls for softening of the response unless the external circuitry provides the cushion. If the high frequency response is not of importance, only the base-width modulation associated with the Early effect should
be retained. The current gain deterioration due to the Kirk effect can be neglected in such cases. A sample computer program cast in C-like language shows how one or both base modulation effects can be implemented in the model.
Reference
/1/ Ebers, J.J., and Moll, J.L., "Large-Signal Behaviour of Junction Transistors," Proc. IRE, 42, No.12 (December 1954), pp.1761-1772.
/2/ Gummel, H.K., and Poon, H.C., "An Integral Charge Control model of Bipolar Transistor," B.S.T.J., (May-June 1970), pp.827- 852.
/3/ Early, J.M., "Effects of Space-Charge Layer Widening in Junction Transistor," Proc. IRE, 40, (November 1952), pp.1401- 1406.
/4/ Kirk, C.T., "A Theory of Transistor Cutoff Frequency Falloff at High Current Densities," IRE Trans. Electron Devices, ED-9, No.2 (March 1962), pp.164-174.
/5/ Sze, S.M., "Semiconductor Devices", John Wiley & Sons 1986 p.126,
/6/ Muller, S.R., and Kamins, "Device Electronisc for Integrated Circuits," Second Edition, John Wiley & Sons, p.220.
/7/__pp.16-18
/8/_, p.335
/9/_, pp.226-228
prof. Zvonko Fazarinc, Ph. D Stanford University, ret. Hewlett-Packard Laboratories, ret.
880 La Mesa Drive Menio Park, CA 94025 fax (415) 854 8574
Prispelo: 05.2.93
Sprejeto: 25.2.93