IEEE TRANSACTIONS ON ELECTRON DEVICES, OCTOBER 1976

base dose. Referring to Fig. 3, the second term in (2) can bc re- duced to 1 2 percent of t,, by selecting a base dose such that p, 2 2400 (Peff 2 8.6).

111. SUMMARY The electrical properties of 12L n-p-n transistor are described

with the emphasis on explaining the high intrinsic up beta and comparing this transistor to the LEC transistor. A high fl, is necessary because more than 99 percent of the input base curl-ant is lost to the extrinsic elements in a typical 12L gate. At low 1;ur- rents, t,ff can be minimized by reducing the intrinsic n-p-n llase dose.

REFERENCES S. A. Evans and Julia S. Fu, in Extended Abstracts of the Eleci’ro-

M. S. Mock, “Transport equations in heavily doped silicon, anc the chemical Society Meeting, May 1975, pp. 394-395.

current gain of a bipolar transistor,” Solid-state Electronics, ,vel.

J. Krausse, “Auger-Rekombination im Mittelgebiet Durch-Lassbe- 16, pp. 1251-1259, Nov. 1973.

Electronics, vol. 17, pp. 427-429, May 1974. lasteter Siligium-Gleichrichter Und-Thyristoren,” Solid-State

H. J. deMan, “The influence of heavy doping on the emitter efficilncy of a biDolar transistor,” IEEE Trans. Electron Deuices, vol. EC. 18, pp. 833-835, Oct. 1971. M. S. Mock. “On heavv dooine effects and the injection effickncv of silicon transistors,” k l . i7, pp. 819-824, Aug. l“974. F. M. Klaassen, “Device physics of integrated injection logic,” I1;EE Trans. Electron Deuices, vol. ED-22, pp. 145-152, Mar. 1975.

Temperature Rise in Multimesa Devices

LOWELL H. HOLWAY, JR.

Abstract-Closed form expressions are derived for the tempor- ature of multimesa diodes arranged as trimesas, pentamesas, 011 N )< N arrays.

In computing the thermal resistance of an array of IMPA7-T diodes, the “crosstalk” between the diodes is important. In tllis correspondence, a series expansion for the complete elliptic n- tegrals is used to obtain a simple closed-form expression for the temperature rise in multimesa devices. This allows a relativl?ly simple calculation of some of the data presented graphically by Frey [I].

As an example of crosstalk, Fig. 1 shows two diodes, each producing a heat flux which passes into the heat sink at the plane 2 = 0. In an approximation where the first diode can be assumed to concentrate its power q as a delta function at the origin, the Green’s function gives the temperature rise at the center of the second diode due to the presence of the first diode as

:t

where K is the thermal conductivity. If we want to include the effect of the finite area of the fi *st

diode, when the heat flux is uniform across the area of the diotie, we can use Frey’s solution [I], namely,

Diode I Diode 2

A 3 x 3 array

Trirnesa Pentameso

(b) Fig. 1. (a) Geometry for crosstalk between two diodes; (b) trimesa,

pentamesa, and N X N arrays.

are complete elliptic integrals of the first and second kind. The elliptic integrals can be expanded in a power series [2] in S, giv- ing

Now the largest value S can ever have is l/2 (when the two diodes are touching), and, even in this worst case, the error in replacing ( 2 ) by (1) is only 3.5 percent. In a multimesa device, the precise error will depend on the geometry and the number of mesas, but the percentage error in the crosstalk will always be less than it is for a two-mesa device in which the value of rola is equal to the maximum value of any two mesas in the device under consideration. Since the crosstalk is itself only a fraction of the total temperature rise, we will use (1) which will enable us to exploit the symmetry of large arrays, and obtain closed form solutions for the temperatures.

If crosstalk could be neglected, the temperature at the center of each diode would be proIK where p = QITrEn, i.e., the tem- perature is divided by the number of mesas, when ro and the total power Q is fixed. If Q and the total area of the diodes is fixed, pro/K decreases as n-1/2. When crosstalk is included, the tem- perature is increased to

Ti = - (1 + Z) Pro K

where we define the crosstalk Z as the fractional increase in temperature of one diode due to all other diodes. If I, is the dis- tance between the ith and j t h diode, then (1) gives

where the summation is taken over all diodes for,which j # i. Therefore, the crosstalk for a trimesa is

z, ro a

while the hottest diode in a pentamesa has the crosstalk Manuscript received April 12.1976; revised May 24, 1976. The author is with Raytheon Research Division, Waltham, I’/lA

02154. z = 2 f i 2 .

a

CORRESPONDENCE I195

REFERENCES [l] J. Frey: “Multimesa versus annular construction for high average

vol. ED-19, pp. 981-85, 1972. power In semiconductor devices,” ZEEE Trans. Electron Deuices,

[2] M. Abramowitz and I. A. Stegen, Eds. Handbook ofMathematica1 Functions with Formulas, Graphs, and Mathematical Tables, Na- tional Bureau of Standards Applied Mathematics Series-55, 1970. (Seep. 591, eqs. (17.3,.11 and 17.3.12). In these equations m = S2.)

0 5 10 15 20 25 30 N

Fig. 2. Normalized crosstalk for N X N arrays. Z, is fractional crosstalk for fixed ro. Fractional crosstalk for fixed total area is ZN/N In this case, radius of individual diode decreases in proportion to n-&.

If N2 diodes are arranged in an N X N array, the center diode will have the crosstalk

if N is odd. If N is even, tine four hottest diodes have the cross- talk

IN =2 ($ ( 2 +A) + 2 ( j 2 + k 2 ) - 1 / 2 ) (9) N/2 ( N D - 1

U j = O k = l

which, for a quadrimesa (N = a ) , reduces to

The numerical value for the crosstalk I N , normalized by the ratio ro/u, is plotted in Fig. 2. The curves in Frey [l, Fig. 61 can be generated conveniently from the plot of I N , by adding 1.0 to the product of ro/u times the plotted value of u/ro IN and dividing by N.

For N odd, we can show that, as N + a, IN+2 - I N + 4(ro/u) In (1 + A) so that

IN - 2 In (1 + &)(ro/u)(N - 1). (11)

As N increases, this asymptotic formula approximates either (8) or (9) with rapidly decreasing error (for N > 4, the error is always less than 3.08 percent; for N > 12, less than 0.99 percent; for N > 28, less than 0.39 percent).

It should be noted that A , the total area of the array, is nor- mally fixed, so that ro will decrease as n-1/2, and the crosstalk behaves as INn-ll2 which is also shown in Fig. 2 for comparison. In the approximation of (ll), IN = 0.995 All2 (1 - l/N)/u, so that the crosstalk produces an almost constant fraction of the tem- perature rise.

The technological advantages of multimesa arrays and a comparison of the relative advantages of multimesa arrays and annular diodes are discussed by Frey [I].

It should be noted that, in this correspondence, we have been careful to refer to the temperature rise “at the center of each diode.” In fact, the center is not always the hottest spot; the hottest spot for the two diode problem of Fig. l ( a ) can be shown to be displaced by the small distance, ro((ro/a)2 + %(ro/u)* + - .), from the center. However, the temperature at this point is only marginally higher than at the cedter, since it is multiplied by a factor (1 + + a) which, even in the worst case, Le., when the diodes are touching, amounts to a correction of only 1.6 per- cent.

ACKNOWLEDGMENT The author would like to thank J. Simpson for a helpful dis-

cussion concerning the displacement of the hot spot from the center of the diode in nonsymmetric geometries.

Correction to “Determination of the Interaction Impedance of Coupled Cavity Slow Wave Structures”

DENIS J. CONNOLLY

In the above paper1 there is an error in (3) which may be corrected by placing a parenthesis about the terms 1 - tF. That is, (3) should read:

Manuscript received May 21,1976. The author is with NASA Lewis Research Center, Cleveland, OH

D. J. Connolly, IEEE Trans. Electron Deuices, vol. ED-23, pp. 44135.

491-493, May 1976.

On Black Solar Cells or the Tetrahedral Texturing of a Silicon Surface

F. RESTREPO AND C. E. BACKUS

Abstract-High-efficiency silicon solar cells have been reported that use a surface alteration to reduce reflection. The process here reported purposely alters the cell surface with an anistropic etching in ternary mixtures of KOH, HzO, and ethyl glycol. Wafers were “sensitized” with a carbon compound to insure etching uni- formity. It is suggested that the organic compound creates S ic nucleation sites, which in turn facilitates the formation of a tet- rahedral structure on the cell surface. This structure promotes multiple interaction of the light beam between millions of pyramids per square centimeter on the surface. The surface geometry in- creases the light absorption and reduces reflectivity, thus in- creasing the cell efficiency.

I. INTRODUCTION Several efforts have been made to increase the efficiency of

silicon solar cells. Since about 7 percent of the energy incident on solar cells is reflected, an increase in the absorptivity of the silicon surface offers one mechanism for increasing efficiency. Work in the field of reflection suppression has been published as early as 1960 [l]. More recently, COMSAT Labs introduced the CNR solar cell, which has reduced reflection [2]. During the past year studies reported here have independently developed a process to produce a silicon surface for the “black solar cell.” This name is due to the blhck appearance offered by the cell when one’s vision path is normal to the cell surface. This higher ab-

Manuscript received April 10, 1976. This work was supported in part by ERDA Contract E(11-1)2590 and was performed in ASU Solid State Laboratory.

F. Restrepo was with Arizona State University, Tempe, AZ. He is now with SEMI, Inc., Phoenix, AZ 85000.

C. E. Backus is with the Arizona State University, Tempe, A2 85281.