0038-I101/87 $3.00+ 0.00 Copyright0 1987PergamonJournals Ltd
Solid-State Electronics Vol. 30, No. 1, PP. 771-772,1987 Printedin Great Britain.All rightsreserved
A. VAN DER ZIEL’ and T. G. M. KLEINPENNING~ ‘Electrical Engineering Department, University of Minnesota, Minneapolis, MN 55455, U.S.A. and 2Electrical Engineering Department, Eindhoven University of Technology, Eindhoven, The Netherlands (Received 13 December
1985; in revisedform
11 August 1986)
Abstract-The diffusion theory of h.f. bipolar transistor response usually makes two basic assumptions, one at the emitter junction and one at the collector junction. The validity of these assumptions is here investigated and corrections are proposed.
theorem, ii is equal to the time average of i(x),
The earliest theory of the h.f. bipolar transistor response was a diffusion theory and the chief interest was in the h.f. input and output admittances and in the transfer admittance[l]. With the development of digital circuits, however, the emphasis shifted to the time delays in the various parts of the transistor; this led to alternate approaches that will not be discussed here. In the diffusion approach to n +-p-n transistors two assumptions are often made: 1. The h.f. electron concentration on the emitter side of the base region is assumed to follow the h.f. base voltage ubeexp (jwt) instantaneously. 2. The transit time zd, through the base-collector space charge regions is assumed to be negligible in comparison with the diffusion time zdr of the carriers through the base region. These assumptions are well satisfied for low-speed transistors with a low value off,, but may not always be valid for high speed transistors with fr’s of 10GHz. We now discuss the corrections that then have to be made to the theory. The effect of the transit time td, through the base-collector space charge region is taken into account as follows. Let i, be the a.c. convection current entering the space charge region, then the convection current i(x) in the space charge region at x may be written as i(x)
exp[ -jwz (x)],
time from the base side of the collector space charge region to the point x in that region. Since the carriers move through the space charge region with a limiting drift velocity u, of about lO’cm/s we have t(x) = x/u,. To calculate the current i: in the external lead we can now apply Ramo’s theorem[f]. According to this where T(X) is the transit
i: = -
exp( -jwz) dz
In first approximation ii has the well-known phase shift -wT,/~, but ultimately the amplitude of ii is reduced by the factor: sin wr,/2 wtdr/2
Here td, can be expressed as follows. If the carriers move with the limiting velocity o, and wsc is the length of the base-collector space charge region, =dr =
We thus see that there is only a phase shift effect if wz,/2<< 1, whereas in addition Isin(wTd,/2)/ (07~,/2) 1 is much smaller than unity if (wT~,/~) > 1. This will be the case for microwave transistors operating around 10 GHz. As an example take N, = 10z3/m3 in the base, Nd = 10z2/m3in the collector and (V, - Vdr/)= 16V. In that case TV,= 1.54 x IO-” s and wr,/2 = 0.48 at 10 GHz, so that the factor sin(wt,+/2)/(wrd,/2) is still very close to unity and only the phase shift is important. But if we had chosen Nd= 102’/m3 in the collector, in order to reduce the feedback capacitance C,, still further, TV, would be 3 times larger, and in that case sin(wtd,/2)/(wzd,/2) N 0.69 at 10 GHz. As far as the first assumption is concerned, it is easily corrected on a formal basis by writing: n, (0) = n,,ejw’f(jw,),
where f (jwt) is an unknown function and T is the transit time through the emitter-base space charge region. A good estimate of z and f (jwr) has not been made so far, since the carrier flow is due to diffusion 771
A. VAN DER ZIEL and T. G. M. KLEINPENNING
in the presence of a retarding field. Qualitatively one would expect the following: (a) for WT<<1 one would expect 1f (jwz) I= 1 and there will be a small phase delay C/J,corresponding to a time delay 7be= C#J/W, where 7be is proportional to the transit time 7, but not equal to it. (b) for w7 2 1 one would expect lf(jw7)l < 1 in addition to the phase delay C#J= 07~~. It is not expected that case (b) will be easily reached, but it would be interesting to evaluate 7 and f(jw7) theoretically. CONCLUSION
The diffusion theory of h.f. transistor response can become inaccurate for microwave transistors at the highest frequencies, because of transit time effects in the base-emitter and in the base collector space
charge regions. Both effects are easily corrected on a formal basis. The first effect is probably small but needs evaluation. The second effect can be more significant, especially for microwave transistors at large collector back bias and low collector doping and is evaluated easily. For transistors operating in its useful frequency range f /fr < 1, both effects are insignificant if fT< 1 GHz.
REFERENCES 1. W. Shockley, Electrons and Holes in Semiconductors, D. Van Nostrand Co, Princeton, NJ (1952); A. van der Ziel, Solid State Physical Electronics, jrd Edn. PrenticeHall, Enalewood Cliffs. NY (1976). 2. S. i. Sze, Physics of S~mico~ductb Devices, 2nd Edn. Wiley, New York (1981). 3. S. Ramo, Proc., IRE, 27, 584 (1939); A. van der Ziel, Noise, p. 125, equations (6), (6a). Prentice-Hall, Englewood Cliffs, NY (1954).