292 IEEE TRANSACTIONS ON ELECTRON DEVICES July frequency for which the length a is a quarter-wave or

Thus, the output power can be written independently of the diode dimensions m

The quantity in brackets is proportional to u8c-1’z for abrupt (Esaki) junctions and is of the order of magnitude of 10 mw for germanium. The quantity (WJW - w / w ~ ) is equal to unity when W/W? = 0.618. Thus, (11) indicates that the tunnel-diode oscil- lator structure shown in Fig. ](a) should give about 10 mw of power output at 60 per cent of the resistive cutoff frequency.

MINIMUM BIAS RESISTANCE

In the derivation of ( I l ) , i t was msumed that the bias resistor

was equal to zero. In a practical case, RB must have a finite value large enough to avoid excessive heating in the vicinity of the diode. If the resistor is assumed to be convection cooled from the outside, the transverse heat conductivity is

K = - k M , 2bw t

where ICM is the thermal conductivity of the resistor metal. The power dissipated in the bias resistor is related to the transverse temperature rise AT and the bim voltage by

Thus, if the allowed temperature rise is specified, the dimensional ratio 2b/t has the lower bound

E>-, VB t - (15) $2 AT

where the ratio of thermal to electrical conductivity of the bias resistor i s given by

= 2.44 x 10-’T volts2/”K CJH

the Wiedemann-Pranz ratio. For germanium at room temperature with an allowed tem- perature rise of 1O”K, (15) requires that 2b/t 2 11.7.

Analysis of the distributed oscillator cir- cuit2 indicates that a finite bias resistor will reduce the power output to one half of its zero resistance value when

where

Thus, if we define a frequency W b , a t which the finite bias resistor has reduced the power output by one half

where

A plot of w ~ / w ~ vs A is presented in Fig. 2. When A = 2, W b is about 78 per cent of w?. We have then the requirement

A c

Fig. 2-Ratio of bias resistor cutoff frequency to resiative cutoff frequency vs

The ratio 2b/t is limited by (15). The ratio 6/2b must be large compared with unity, as indicated in (3), so (4)-(6) are valid. Finally, the ratio must be large compared with unity from (1). Thus, the fist problem in designing a quarter-wave oscillator for the maximum power output indicated in (11) will be to choose a metal for the bias resistor with a conductivity greater than that of the semiconductor, as required by (20).

If we assume that the bias resistor is made of silver (uM = 6.15 X IO7 mhos/m), 6/b = 5, 2b/t = 11.7 and uQc = 2 X lo6 mhos/m, the finite value of the bias resistor will begin to limit the output power a t frequencies greater than about

w 5 z lOg/c. (2 1) A. c. SCOTT

Dept. Elec. Engrg. University of Wisconsin

Madison, Wis.

Some Comments on “Webster’s Equation””

Those readers, who are familiar with transistor design from everyday professional practice will most certainly have been con- fronted with what has become known as Webster’s equation in transistor literature. This rather fundamental equation relates common emitter current gain to emitter current 1~ in the following way:

where

It was first derived by Webster.1.2 As presented here, i t holds for a P-N-P tran- sistor and the meaning of the symbols is the following:

s = surface recombination velocity, w = average base width, A , = area of ring-shaped surface region surrounding the emitter dot, D, = diffusion coefficient of holes, A = emitter-junction area, U b = base region conductivity, u6 = emit- ter region conductivity, Le = diffusion length of electrons in the emitter region, L b = diffusion length of holes in the base region, pe = mobility of electrons.

1 W. M. Webster “On the variation of junction- * Received March 22, 1963.

transistor current-akplification factor with emitter current,‘’ PROC. IRE, vol. 42, pp. 914-920; June, 1954.

41.3-419; June, 1’958. 2 F. J. Biondi Transistor Technology, 701. 2 , p.

1965

Equation (1) gives Lyca in terms of the variable z, which in turn is directly pro- portional to the emitter current I E [see (1 a)]. Thus, a t least theoretically, one should be able to trace the function a,b(IE) to find out its main features and to see how changes in parameters (material and geometrical) included in (1) influence these features. Unfortunately, however, functions g(z) (called ‘Ifield-factor” by the author) and f(z) (called “fall-of-factor” by the author) are not simple, so that the above-men- tioned discussion of (1) in terms of varia- tions of its various parameters inevitably involves point-by-point computation of the whole curve every time. One outstanding feature of the a,b(IE)function is a m a x i m u m in current gain found a t a certain emitter current I,+ It would be of interest for the transistor designer to have a simple method of determining position, absolute value and curvature of this maximum which would equally allow him to make a qualitative and quantitative estimate as to what happens to the maximum when parameters change. Knowledge about the maximum means in this case a good deal of informa- tion on the whole curve itself. A method permitting simple analysis of the properties of the m a x i m u m in ( 1 ) shall now be pre- sented.

We start with rewriting Webster’s equa- tion (1) in the following form:

Correspondence 293 creasing ( E + V ) . Increasing S would mean either increasing s (the surface recombination velocity) or decreasing A , the emitter area. Increasing s is in contra- diction to the requirement of a high current gain and A is limited by the current carrying capacity of the device. Generally, therefore, E is decreased in order to shift the maxi- mum to higher values of emitter current. Decreasing E means increasing emitter eficiency, which may be achieved by doping the emitter region more heavily (using In-Ga or In-Al-emitters instead of pure In-emitters). Low-level ( e . g . , microwatt- level) transistors must have the Lycb maxi- mum at low current levels. This can be achieved by decreasing the ratio S:(E + V) . Here again, increasing ( E + V ) is senseless. Decreasing S is the correct answer. This can be achieved by lowering s, which in turn is limited by the efficiency of the surface‘ treatments known in the art. Frequently the emitter area A is increased for decreasing S.

Substitution of (13) into (6) leads, after some algebra, to

aCb(max.>

where S has been substituted for the surface term, E for the emitter eficiency term and V for the volume recombination term.

The jield factor, g(z) , and the fall-of- factor, f ( z ) , are not independent of each other, as can be shown by their definitions given b y W e b ~ t e r . ~

f ( x ) = 1 + ’. N d (4)

where p , = hole densit.y in the base region at the emitter junction, N a = donor con- centration in the base region.

Combining ( 3 ) and (4) leads to

Subst,itution of (5) into (2) gives

The condition for a , b to have a maximum can now be written as follows:

The function f ( z ) can be plotted, if one takes into account (4) and the relationship between z and p,/iL’, given by Webster.4

pp. 418419. s lb id . , Appendix I, p . 418, and Appendix 11,

4 I b i d . , Appendix 11, p. 419.

By combining (4) and (8) one gets

2f - 2 - l n f = x . (9) Fig. 1 is a plot of f ( z ) obtained from (9). One can deduce immediately from Fig. 1 that f ( z ) is a monotonously increasing function of z, so it may be concluded that

- + 0. df ax

our condition for Olcb to have a maximum reduces, therefore, to

Differentiation of (6) with respect to f and substitution into (11) leads to the following solutions for f:

Sincef(z) must be greater than unity for all (positive) values of z (see plot in Fig. l), we may drop the negative sign in (12) and write

From (13) and from the f(z)-plot of Fig. 1 the value of hs can be found readily for any set of S, E and V . IE(max) may then be evaluated from (la).

Equation (13) shows that a maximum for Dlcb can only exist if S 2 ( E + V ) , becausef 2 1 for any value of z ! It further shows that the position of the maximum depends on fhe ratio S:(E f V ) only. The maximum will be shifted to high current levels 1, for a high ratio S:(E f V). This is most important for power transistors, where reasonable current gain a t high cur- rent levels is a requirement. It could be achieved by either increasing S or de-

= 2[X1/’ + (E + V)1’2]-2. (14) Equation (14) describes a,b(max) in terms of the designable quantities S, E and V.

In order to obtain some information con- cerning the shape of the m a x i m u m (e.g., whether it is $at or sharp) we must calculate the second derivative of aCb with respect to z, which is inversely proportional to the radius of curvature of the ~ l , b ( z ) curve. By differentiating Dlcb(z) twice we obtain

Again, the operator d / d z can be expressed as

Fig. 1-Graphical plot of the “fall off factor.” f(z>, as given by Webster.

294 IEEE TRANSACTIONS ON ELECTRON DEVICES July Combining (15) with (16) we get

At the maximum aacb/df = 0, as required by (11). Thus, (17) reduces to

for f = fma,. (18) df/dz can be calculated from (9) t o give

a2a,,b/ajz is obtained by differentiating (6) twice with respect to f, so that from (I$), (19) and (13), the latter being the expression

for fmaa, the second derivative of LY,~ with respect to z at the maximum can be cal- culated. We omit here the rather lengthy calculation which is, by the way, straight- forward, without any additional assump- tions. The result is

A

We have a sharp maxinzum, if the second derivative of Olcb is large and a flat one, if it is small. Equation (20) bells us that a large value of S flattens the maximum, whereas a large value of ( E + V ) gives rise to a sharp maximum. It can also be concluded from (20), that the curvature has a stronger dependence on S (5/2 -power!) than on ( E + V ) (3/2 -power!).

To summarize, it is claimed that the foregoing analysis of “Webster’s equation” may provide a great deal of facilities in the design of a desired current-gain vs emitter current behavior for transistors. It is hoped, that the examples stated above are helpful in visualizing these possibilities. The equa- tions derived in this paper, which all follow directly from Webster’s equation, could be a valuable tool for the design engineer to play with when he has to make his choice of geometrical and material parameters. They are of a simple structure, easily analizable and may save the engineer lengthy computations for every variable he wants to investigate the influence of. The results presented here might stimulate wider use of “Webster’s equation” in design work.

0. H. JAKiTS Semiconductor Division

Ebauches S.A. ?Jeuchdtel/Switzerland

Contributors

David L. Bobroff (M ’46) was born in

July 17, 1917. He Milwaukee, Wis., on

received the B.S. de- gree in electrical en- gineering from the University of Wis- consin, Madison, in 1939, and the E.E. degree from the Mas-

sachusetts Institute of Technology, Cam- bridge, in 1955. He has also completed three years of graduate study in physics and mathematics a t the University of Wis- consin.

From 1941 to 1949, he worked in acous- tics, hydrodynamics and electromagnetics at the Naval Ordnance Laboratory, Wash- ington, D. C. From 1949 to 1955, he was associated with the Electrical Engineering Department of M.I.T. as a student In- structor, and Research Worker. Since 1955 he has been a staff member of the Research Division of the Raytheon Company, Wal- tham, Mass. From 1953 to 1961 he was engaged in research on low-noise micro- wave amplifiers and oscillators, and milli- meter generators. Since late 1961 he has been working in the field of optical masers.

Mr. Bobroff is a member of the American Physical Society.

.:. Joseph Juifu Chang (M ‘58) was born in Szechwan, China, on December 1, 1928. He received the B.S. degree, magna cum laude, from the Tai- wan College of Engi- neering, Tainan, Tai- wan, China, in 1953.

petitive basis, a Li Foundation fellowship He won, on a com-

and began his graduate work in 1956 a t Purdue University, Lafayette, Ind., where he received the M.S.E.E. degree in 1957, and the Ph.D. degree in 1961 with thesis work in the field of semiconductors.

From 1957 to 1959, he was with Reming- ton Rand Univac, Philadelphia, Pa., par- ticipating in project lightning. Since March, 1961, Dr. Chang has been a member of the technical staff at Bell Telephone Labo- ratories, Murray Hill, N. J., where he worked on varactor diodes and concentra- tion-dependent diffusion. At present, he is

engaged in the development of integrated devices.

Dr. Chang is a member of the American Physical Society, the American Association for the Advancement of Science, and Sigma Xi. .:.

Tsung-Shan Chen (A ’48-M ’55) was born in Hopei, China, on February 2, 1913. He received the B.S. degree from Tang- s h a n C h i a o - T u n g University in Hopei Province, China, in 1933, and the M.S. degree from Purdue

University, Lafayette, Ind., in 1934. He attended the Massachusetts Institute of Technology, Cambridge, and received the M.S. degree in 1935 and the D.Sc. degree in 1938, both in electrical engineering.

He was Professor of Electrical Engineer- ing at the National Central University, China, until 1947. He joined RCA in 1951, and is now with the Microwave Tube De- partment, Electron Tube Division, Harri- son, N. J.