4-2 0 iEEE TL4NSAC’~~IQNS OX ELECTRON DEVICES, VOL. ED-17, NO. 5 , MAY 1970

REFERENCES [I] H. Kroemer, “Theory of a wide-gap enuts:er for transi~tom,”

[2 ] D. K. Jadus and D. L. Feucht, “The,,reaIization o f a GaAs-Ge Proc. IRE, vol. 45, pp. 1535-1542, November 1957.

wide band gap emitter transistor, IJ9‘h,E Trans. Elect *on Devices, vol. ED-16, pp. 102-107, January 3969.

[3] H. J. Hovel, “ZnSe-Ge heterojunctions and Reterojunct:on transistors,” Ph.D. dissertation, Carnegie-Mellon University,

[4] H. Watson, Microwave Semiconductor Devices and their Circuit Pittsburgh, Pa., 1968.

[5] G. T. Wright, “The space-charge-limited dielectric triode,” Applications. New York: McGraw-Hill, 1968, ch. 17,

[6] S. Brojdo, “characteristics cf the dielectric diode and triode at Solid-State Electronics, vol. 5, p. 117, 1962.

very high frequencies,” Sold-State Electronics, vol. 6, p. 611,

i99 D. E. Thomasand T . L. Moll. “Tunction transistor short-circuit . > current gain and phase deterhination,” Proc. IRE, vol. 46,

[lo] W. E. Beadle, K. E. Daburlos, and W. H. Eckton, Jr., “Design, pp. 1177-1184, June 1958.

fabrication, and characterization of germanium microwave transistor,” IEEE Trans. Electron Devices, vol. ED-16, pp. 125- 138, Januaryl>969.

[ill J. t e y n k e l , Drift transistor-simplified electrical characteriza- tion, Electronic and Radio Engrg., vol. 36, p. 280, 1959.

[12] R. L. Pritchard, Electrical Characteristics of Transistors. New York: McGraw-Hill, 1967.

1131 H. C. Nathanson and A. G. Tordan. “Characteristics of an ex-

.-

* * ponentially retrograded &able capacitance diode, pt. I,”

[14 J,. Shao and G. T. Wright, “Characteristics of the space-charge- Semiconductor Prod., vol. 5, p. 38, April 1962.

llmlted dielectric diode a t high frequencies,” Sold-State Elec- 1963. tronics, vol. 3, p. 291, 1961.

[7] S. Brojdo et ai., nThe heterojunction transistor and the spa-e- [15] S. R. Arnold and R. L. Pritchett, “Base-contact resistance charge-limited triode,” Brit. J . Apfil. Phys., vol. 16, p. 133, studies for microwave germanium transistors,” ch. 1 in “New 1965.

[8] D. J. Page, “A CdS-Si heterojunction transistor,” IEEE Trams. and improved transmission-type transistors,” Bell Telephone Laboratories Rept. 20, Contract DA 36-039 AMC-O2227(E),

Electron Devices, vol. ED-12, pp. 509-510, September 1965 June 30, 1965.

Multiplication in Collectcc Junctions of Silicon n-p-n and p-n-p Transistors

Abslrect-The values of 1 - l / M for both electrons and holets, where M is the multiplication factor, have been calculated in three different silicon p-n junctions. The logarithmic plot of 1 - 1/M versus the normalized voltage V/VB is well approximated by a straight line for 0.1 > 1 - 1/M>O.OOS. This range corresponds to the useful range of (YO for most bipolar transistors. An empirical expression has bee.1 obtained for the ratio VC.EO/VCBO within this range.

OR a junction transistor, the maximum collector- to-emitter breakdown occurs at the voltagc: VCEO, where the product of the low-voltage cur-

rent gain a. and the collector avalanche multiplication M is unity. In germanium, the ionization rates for hole:; ap and electrons a, are related by ap = (2.2 +0.2:\ a, [ l ] . Miller [2] noted that for germanium transistorfj

V C E O

V C B O

-= (1 - all)l’- (1:

where n varies between 3 to 6 depending on the col. lector-to-base breakdown voltage VCBO and the base re.

This work was supported by the Advanced Research Projects Agency Manuscript received August 6, 1969; revised November 17, 1969.

through the Center for Materials Research, Stanford University, Stanford, Calif.

now with Fairchild Semiconductors, Mountain View, Calif. 94040. J. L. Moll was with Stanford University, Stanford, Calif. He is

ford, Calif. 94305. J. L. Su and A. C. M. Wang are with Stanford University, Stan-

sistivity type. The relation, as given in ( l) , implies that

1 1 - - M = (2J.

V B is defined as the voltage at which M becomes in- finite. If the above equation is generalized to

1 1 -- M = F ( $ ) ,

then

V C E O ___- V C B O

- F-y1 - all).

In silicon, the ratio of an to a p varies between 2 to 30 or more as the electric field decreases [3]. The relation between avalanche multiplication and voltage cannot be approximated by (2). The values l - l /M for both electrons and holes have been calculated for three differ- ent p-n junctions. The ionization rates used were ob- tained by Ogawa [4] and are well approximated in the form suggested by Chynoweth [ 5 ] :

f f p = apoo exp (- $> = 106 exp (- ----) 2.22 x 106

E (4)

MOLL et al.: MULTIPLICATION IN COLLECTOR JUNCTIONS OF SILICON TRANSISTORS 42 1

an = anm exp (- 2)

The general expressions are

1 I s-

1 1 --

where W is the width of depletion region, and M , and M p are the electron and hole multiplication factors, respectively.

For the p-i-n junction, the electric field is constant and, as a result, the ionization rates are constant; therefore, (6) and ( 7 ) are integrable. Fig. 1 shows the resulting l-l/Mn and l-l/Mp for p-i-n junctions of various widths.

For the step and linearly graded junctions, (6) and (7) can be accurately evaluated only by numerical in- tegration; however, as a number of authors have pointed out [3], they are integrable only if the ratio between electron and hole ionization rates is constant, and if the ionization rate is a sufficiently simple func- tion of the electric field. We expect to obtain a reason- ably accurate calculation of multiplication by using y =ap/a, at the maximum electric field E, and (4) and ( 5 ) . The result for these junctions can be expressed as

4 ,

where y is the ratio a,/an a t E,, and Weff is also a func- tion of the maximum electrical field E, [SI. The values of W,ff/W are listed in Table I. For both types of junc- tions, Weff/W is a monotonically decreasing function of b/E,, where b is the ionization parameter in (4) and (5). Because b,> b,, (Wef f /W)p is less than (W,rf/W),. The approximation of (8) yields a slightly larger value for l-l/hfn compared to the exact result; the approxima- tion of (9) yields a smaller value for l-l/&fp. This is a consequence of the steeper slope of a p ( l / E ) which re- sults in the assumption of more ionization by holes and less ionization by electrons than actually occur. The differences in junction breakdown voltages, as calcu- lated by the exact numerical integration and the ap- proximate method, are only a few percent for all cases.

/ b P I I

I .o 0.8 0.6

0.4 “0

0.2

Fig. 1. Logarithmic plots of 1 -l/M% and l/Mp versus VIVB for silicon p-i-n junctions. Junction widths and breakdown voltages are: (a) 0.5 micron, 20.7 volts; (b) 3 microns, 07 volts; (c) 20

cates electron or hole multiplication. microns, 440 volts; (d) 200 microns, 3381 volts. Subscript indi-

TABLE I Weif/ W Versus b/Em

Step Junctions

0.8879 1.1099 1 5534 2.2199

4.218 3.1079

5.1059 5.9939 7.1039 7.9919

10.212 9.1019

12.2099 14.652 15.984 19.98

Weff/ w 0.4256 0.3827 0.3202 0.2591 0.2076 0.1669 0.1445

0.1112 0.1274

0.1008 0.09035 0.08181 0.06999 0.05945 0.05492 0.04467

For small M , as in the case

Linearly Graded Junctions

b / E m I Werf/W

0.7699 1 ,0779

0.6443 0.5915

2.0019 1.5399

0.4916 0.53396

3.0799 3.5419

0.4234

4.0039 0.4020

5 .OS19 0.3837

6.0059 0.3493 0.3263

7.0839 0.3045 8.0079 0.2890

12.012 9.0859 0.2736

14.014 0.2418 0.2255

16.016 0.2121 18.48 0.1985

-

of calculating VCSO where M = l/ao, the approximate result converges, as ex- pected, to those obtained by numerical integration. Figs. 2 and 3 show the approximate 1-1,” versus V/VB for several step and linearly graded. junctions, respectively, and Figs. 4 and 5 present the correspond- ing results from the numerical integration of the exact expressions (6) and ( 7 ) . The voltage VB, used for the normalization of Figs. 2 and 3 , is the breakdown voltage calculated from (6) and ( 7 ) . The approximate results and exact calculation are in good agreement for 1 - 1/M <0.1 and differ only slightly near the breakdown vol- tage.

An empirical expression reminiscent of (1) can be obtained for VCEO/ VCBO in the range for 0.1 > 1 - l /M> 0.005 which corresponds to the useful range of a. for most silicon bipolar transistors. Within this range, the plot of log (1 - 1/M) versus log ( V / VB) is well approxi- mated by a straight line:

1 - 1 = (;)n.

Equating 1 - a 0 to 1 - l /M at V = V C E o one obtains

422 IEEE TRANSACTTONS ON ELECTRON DEVICES, MAY 1970

I 1 l . l r , , , , , , , , / ~ , , , , , , , , , , , , , , , , , , , , , , ,

Fig. 2. Logarithmic plots of 1 - l / M n and l - l / M p versus V / V B 0.5- - for step junctions in silicon, as calculated from the approximation

(a) NI Crra, VB =57 volts; (b) Nr = C ~ I - ~ , V B =289 OI ' " ' " " I IO ' ' ' ' ' ' ' ' 1 100 ' ' ' ' ' l J ' l 1000 ' ' ' " ' ' IOOOC

of constant y . The impurity densitiesand breakdown voltages are: -- q-

VB =2904 volts. volts; (c) NI = 1014 cm-8, VB = 1677 volts; (d) NI = 5 X 101a cm-a, BREAKDOWN VOLTAGE, V, (volts)

Fig. 6. fn and fp versus junction breakdown voltage for various silicon junctions.

0.1 ' , I I I , , , , I I I / 1 / 1 1

IO-^ 10-2 10-1 I .o (I--!-) OR ( I - L I

Mn MP

Fig. 3. Logarithmic plots of 1 - l / M n and 1 - 1/M, versus V/ VB for linearly graded junctions, as calculated from the approxima .ion of constant y. The gradients and breakdown voltages arn: (a) a =loz3 ~ m - ~ , V B = ~ volts; (b) 102' cm-4, V ~ = 4 0 ~01;s; (c) lOlg ~ m - ~ , V ~ = 2 2 6 volts: (d) 10'6 C M - ~ , VB =3876 volts.

I.0C

Fig. 4. Logarithmic plots of 1 - l / M n and 1 - 1/M, versus V / V B for various step junctions, as calculated from (6) and (: ) . The impurity densities are those of Fig. 2.

010-3 ' I ' l i t "

I I I , , , / I 10-2 10-1 I .o

(I--!-] OR(I-') Mn MP

Fig. 5 . Logarithmic plotsof 1 - l/Mn and 1 - l / M p versus b'/ V B for various linearly graded junctions, as calculated from (6) and ( 7 ) . The gradients are those of Fig. 3.

22 t

01 ' ' ' " " ' I IO ' ' " ' ' ' I I O 0 1 " " " ' loo0 1 8 8 8 1 8 1 1 l0000 BREAKDOWN VOLTAGE, V, (volt61

Fig. 7. nn and np versus junction breakdown voltage for various silicon junctions.

where f and n must be chosen according to carrier type (f,, n, for n-p-n, and f,, n, for p-n-p), the range of breakdown voltage, and the junction type.

Figs. 6 and 7 show f,? f,, and n,, n, for a variety of p-i-n, step, and linearly graded junctions as a function of breakdown voltages. The range of f, is roughly 0.5 < fn<0.8 and f, is roughly l . O < f p < 1.02. The ideal silicon p-n-p transistor behaves almost according to Miller's empirical law, whereas the n-p-n transistor shows a drastic departure. The values of n are 4<n<20, with small values of n occurring a t lower breakdown voltages and larger values belonging to the higher breakdown voltages.

Because the variation of fn, f, and n,, np is sufficiently slow, i t is reasonable to interpolate for intermediate junctions. An additional method of interpolation for junctions that do not fall in any of the types calculated here is suggested by the successful use of an effective width. In a step or graded junction, nearly all of the ionization occurs near the high-field region of the junc-

tion within a distance of W,rf. For example, in an n+-n-pf step junction, if the width of the n region is greater than the effective width a t breakdown, the breakdown will occur at the same maximum electric field as in the ideal step junction of the same doping in the n region; if the width of the n region is less, the junction can be treated as a p-i-n junction.

CONCLUSION

‘Using (6) and (7) , and (8) and (9), respectively, the exact and approximate calculations yield essentially the

same result for 1- l/M<O.l and differ slightly near the actual breakdown voltage. An empirical relation for VcBO/VcBo in the useful range of silicon bipolar transis- tors has been obtained.

REFERENCES 111 R. A. Loran and S . M. Sze, Proc. Internatl. Conf. on Semicon-

s s (Kvoto. Japan), p. 434, 1966. ____., - ..,.. -... ., vol. 99, p. 1234, 1955.

J . -. -.1011, Physics of Semiconductors. Xew York: McGraw- Hill, 1964, ch. 11.

[4] T. Ogawa, Japan J. Appl. Phys., V O ~ . 4, p. 473, 1965. [5 ] A. G. Chynoweth, J . AppZ. Phys., vol. 31, p. 1161, 1960.

Design of an Electrooptic Light Valve Projection Display

Absfracf-A display device, e.g., projection TV, can be made by scanning, with an electron beam, a large sheet of electrooptic crystal in a light valve configuration. The design equations indicate the fol- lowing. 1) The transfer function will be reasonably linear. 2) The resolution measured at the crystal will be the order of two TV lines per crystal thickness at 15 percent sine wave response. 3) The output light spectrum can be nearly white. 4 ) Writing rates of the order of 10 MHz are feasible with conventional high-resolution CRT electron guns. 5) Large screen image brightnesses of the order of 40 to 50 fL are obtainable with xenon arc lamps and f/l collimator lenses. 6 ) Image contrast will generally be limited by the polarizer-analyzer quality. The electrooptic crystals must be several hundred milli- meters in diameter and several tenths of a millimeter thick. Obtain- ing such crystals is the most serious obstacle to device construction.

T I. INTRODUCTION

HERE ARE currently two projection television devices which work well enough to find commer- cial application: the projection kinescope and the

Eidophor light valve [l]. Unfortunately, neither de- vice possesses that combination of image brightness, cost, life, compactness, reliability, and maintenance requirements which would allow it to meet a large vari- e ty of display system applications. For example, the 7NP4 projection kinescope will produce a peak bright- ness of the order of 5 fL on a 3-meter-high display, if operated with an ultor potential of 80 000 volts, 160 watts input to the phosphor, 40 cubic feet per minute of

Manuscript received August 20, 1967; revised September 24,1969, and Dec. 11,-1969.

Rockwell, Anaheim, Calif. 92803.

Colo.

T. H. Moore is with the Autonetics Division of North American

J . D. Pace is with Ball Brothers Research Corporation, Boulder,

and Dec. 11,-1969. T. H. Moore is with the Autonetics Division of North American

J . D. Pace is with Ball Brothers Research Corporation, Boulder, Colo.

cooling air, and appropriate X-ray shielding [2]. The Eidophor uses an oil film in a schlieren optical system; distortion of the liquid by the electrostatic charge de- posited by an electron beam deflects light through the schlieren optics to the screen. The light-valve principle allows use of a very bright light source-the combina- tion of a bright source and the relatively complex op- tics may result in a somewhat brighter display than can be obtained by a projection kinescope. However, the Eidophor requires that an oil film be carried on a ro- tating mirror in a vacuum suitable for operation of an electron gun. This, in turn, requires a vacuum pump, both to maintain adequate vacuum in the presence of the oil and to permit relatively frequent replacement of electron gun cathodes.

The design equations for a device which has signifi- cant potential for projection television displays are presented below. The reader will note that the design is relatively simple and, except for the electrooptic crystal, is based on well-known techniques. Unfortunately, electrooptic crystals are not yet available in the re- quired geometry. I t is hoped that this presentation of the design equations for an electrooptic light-wave projection display will provide some incentive for the development of suitable electrooptic materials.

The Pockels effect electrooptic light valve is an elec- tronically scanned display device which creates an image by modulating a collimated beam of white light [SI. A crystal exhibiting the Pockels effect will rotate the plane of polarization of a transmit.ted light beam in proportion to the voltage across the crystal. By operat-