1965 Correspondence

I 0' ~ ur. n v,=-5v

IO' v, =-5v

third diode, forward instead of reverse re- sistance ratio R,,/R,, is given for 3.108 dyne/ cm2 at 0.15 pA forward current.

According to these results pressure sensi- tivity of characteristics can be produced in diodes which did not show them originally and vice versa. Exposure to surrounding air can produce the effect, while it usually vanishes after exposure to ozone. Etching in boiling XaOH leads to opposing results. Wet oxygen atmosphere has relatively little influence on the effect.

It appears that these phenomena are connected with the silicon surface a t the junction. This is supported also by the fact that effects become negligible above one mA/cm2 current density in the forward direction. Pressure arises a t the junction a t the surface because the radial expansion of the molybdenum discs due to the applied axial pressure would separately be smaller than that of the silicon wafer. Considering the unit as a single elastic system, the aver- age radial or tangential pressure in the sili- con and at the junction a t the surface, -ur, is roughly

Here v denotes Poisson's ratio, E, Young's modulus, -uz, axial pressure applied by the hydraulic press, a d indexes Si and Mo cor- respond to silicon and molybdenum respec- tively. Replacing numerical values for Y

and E , -0; =0.14. -u2. If now the pressure effects on the characteristics are connected with -ur, it appears that already stresses

less than lo7 dyne/cm* can change charac- teristics noticeably.

The described effects could not be due to changes in the energy band structure and consequent changes in minority carrier density, because the stresses are very low [2]. For the same reason neither can piezoresistive changes in surface channels a t the junction account for the effects. How- ever, as the sensitivity of the characteristics to mechanical pressures is changed by am- bients and by chemical etching, it is con- ceivable that the effects might be connected with variations in the rate of surface re- combination by the mechanical stresses. Such effects might account in part also for experimental results reported previously

The diodes used in these experiments were presented by E. Lidow, International Rectifier Corp., El Segundo, Calif. and it is a pleasure to acknowledge this help.

S. KLEIN E. PIWKAS

Dept. of Elec. Eng. Israel Institute of Technology

Haifa. Israel

PI, [31.

REFERESCES [l] Rindner. W., and I. Braun. Resistance of elasti-

cally deformed shallow p n junctions. 11. J . Appl . Phys.. vol. 34, 1963, pp 195.370.

121 Wortman, J. J. , J . R. Hauser and R. M. Burger,

characteristics, J . Appl . Phys., vol 35, 1964, pp Effect of mechanical st ess on p n junction device

2122-2131. 131 Sikor&, M. E., and D. H. Bevins. PN junction

sensors. Intnnat'l Sdid-Sf& Circuits Conf, Digest of Technical Papers, pp 90-91.

623

Jointly Optimum Realizable Linear R e - and Postfilters for Systems with Samplers

Robbins, [l] and later Chang, [2] have discussed jointly optimum realizable linear pre- and postfilters to be used in connection with sampled-data links. The purpose of this letter is to point out an error in Chang's work, which has misled the authors of a t least one recent paper. [3] We use Chang's notation.

Consider the system model of Fig. 1. In this model, r ( t ) and n(t ) are realizations of the signal and noise random processes, re- spectively. L(s ) represents a desired trans- formation of the input. F(s) and G(s) are linear time-invariant filters which should be chosen so as to minimize E [ e ' ( t ) ] .

I " l t 1 SAMPLING

:,It1

Fig. 1. System model

Both authors [l] , [2] show that if F(s) is specified, the optimum G(s) is given by

[Chang's (S)].

They show in a similar manner that if G(s) is specified the optimum F(s) is given by

1 F(s) =

(c(-s)G(s)J*+(O.,,,,(s)J + . U s )

[Chang's (17)] where

[Chang's (19)].

rl condition for joint optimality of the pre- and post-sampling filters deduced by both authors [l 1, [2] is

Q(s)Q(-s) = V ( s ) U ( - s ) . [Chang's (22)].

The material following (22) in Chang's paper is based upon his assumption that this equation implies

Q(s) = U(s) [Chang's (23)]

under which Robbins's (20) and (21) be- come Chang's (25).

Actually it is required that as) and U(s) have the same poles, since both, being defined in terms of the [ ]+ operator, are required to be analytic in the right half of the s plane. However, a zero of as) may,

February 1. 1965. Manuscript received January 27, 1965; redeed

624 PROCEEDINGS O F THE IEEE June

for all Chang’s (22) tells us, occur in either U ( s ) or in U( -3). The poles of a s ) and U(s ) are those of

the joint optimization problem can be solved by treating the zero locations as undeter- mined parameters. As Robbins’ points out, one must try all possible combinations of right and left half-plane assignments of the zeros and pick the one which yields the smallest mean-squared error.

Several examples have been worked out in which

s + a W(s) = (5 + $ d ( S + $2) ’

with pl and p2 complex. In these cases the choices are

the possibility of which is considered by both authors, [l] , [2] and the additional possibility

bs + r = (s + Pl)(S + $2) ’

-bs + I U(s) =

(3 + $1) (s + $2)

is considered by Robbins but excluded by Chang. The solution procedure consists in postulating forms for as) and U(s) and then finding the unknown parameters b and r.

The ‘Robbins” solution, in which a s ) and U(s ) have zeros in different half-planes, was found to exist in each case in which W ( s ) has a zero in the right half-plane, but not otherwise. Whenever this “Robbins” solution exists, it turns out to be superior to those to which Chang limits himself, in which as)= U(s). However, no proof is known that these results will be true in general.

This work also provides a counterexam- ple to the suggestion of DeRusso [4] made in the discussion following [2] and also in [4] that the optimum prefilter is always a Wiener filter. A necessary condition for this conjecture to be true is that U ( s ) = W(s ) , but such is not the case in the examples solved thus far.

LVILLIS C. KELLOGG~ Div. of Engrg. and Appl. Phys.

Harvard University Cambridge, Mass.

REFERENCES [ l ] Robbins, H. M.. An extension of Wiener filter

theory to partly sampled systems, IRE Trans. on CircuU Theory. vol CT-6, Dec 1959, pp 362-370. (See especially footnote on p 362.)

121 Chang. S. S. L.. Optimum transmission of con-

AIEE. (Applications and Induslry). vol 79. Jan tinuous signal over a sampled data link, Trans.

1%1. pp 538-542. (See especially footnote on p 538.)

[3] Kahn, R. E., and B. Liu. Sampling represents-

Rept 7, Communications Lab., Dept. of Elec. tion and optimum reconstruction of signals. Tech

Engrg.. Princeton University. N. J.. Jun 1964. [4] De Russo. P. M.. Optimum linear filtering of sig-

n a l s prior to sampling. Trans. AIEE (Applica- t i o n s and Industry). v01 79, Jan 1%1, PP 549-555.

‘On leave from The MITRE Corp.. Bedford, Mass.

Author’s Comments2 The author wishes to thank Kellogg for

pointing out an error in his paper.’ However, the error can be corrected readily, and the corrected results confirm that of Robbins which were derived in a different way.

Because of the defining equations (8) and (19), all the poles of as) and U ( s ) are in the LHP, but nothing can be said about the zeros. Equation (23) is not valid. Using (24) only, ( 1 7 ) and (8) lead to

Similarly ( 7 ) and (19) lead to

Equations (25a) and (25b) replace (25). Equation (30) is replaced by

No other change needs to be made. SHELDON S. L. CHANG Dept of Elec. Sciences

Stony Brook, L. I., N. Y.

‘Manuscript received February 19, 1%5.

An Explanation of the “Energy” Dependence of Secondary Break- down in Transistors

In order to more effectively characterize the transistor for secondary breakdown in terms of the more conventional device parameters, a new method of thermal re- sistance measurement was developed. The new method yields a determination of the transistor thermal resistance while the de- vice is in the open base avalanche mode and while the maximum voltage and current are applied. The method of thermal resistance measurement is made by pulsing the tran- sistor in the circuit shown in Fig. 1. The transistors examined during this study were primarily silicon planar epitaxial and triple- diffused types. In addition, germanium al- loy and silicon single-diffused types were also examined.

For the purpose of describing the ther- mal measurement and results, the character- istic of a silicon planar epitaxial transistor will be used. The pulsed characteristic curve for such a device is illustrated in Fig. 2. At low values of collector current IC, the curve indicates the normal negative resist- ance characteristic leading to the collector- emitter sustaining voltage. Of prime interest to this analysis is the region of the curve

February 15.1965. Manuscript received February 1. 1965; revised

where the collector-emitter voltage again rises to a second peak V, followed by a sec- ond negative resistance region. This rise in collector-emitter voltage is attributed to a variation in resistivity of the original silicon epitaxial material, most likely a t a hot spot under the emitter region of the transistor. The curve in Fig. 2 is so drawn that any increase in the supply voltage Vcc (Fig. 1) which would carry the curve beyond the point (V8, I,) would result in the initiation of secondary breakdown.

Referring back to the assumption relat- ing to the second rise of collector-emitter voltage CCE a t a large value of collector cur- rent, it is possible to relate the second peak point of the characteristic curve (V,, I,) to a specific hot spot temperature T;,. This may be accomplished by referring to resistivity temperature curves for the starting epitaxial materials. The starting material used in the transistor under examination had a resistiv- ity of 3-5 0-cm a t 25OC. Referring to the resistivity temperature curves by Gartner,’ the peak of the resistivity temperature curve for n-type silicon occurs a t 25OOC for this material. Therefore, the hot spot tem- perature a t the point V,, I, is 25OOC. From this the transient thermal resistance et is determined to be:

For the specimen under consideration, 225OC et = -

V P x G Utilizing this technique, a series of

thermal resistance measurements as a func- tion of pulse width was made. Pulse width for the purpose of this series of measure- ments is the time the collectoremitter voltage remains a t the maximum value.

Once the thermal resistance measure- ment is obtained, it is then possible to cal- culate the “hot spot” temperature T;, prior to the onset of secondary breakdown. The spot temperature at that point is

T,. = e,(v,, ZJ + r, (2)

Therefore, this method of thermal meas- urement yields the value of transient ther- mal resistance and spot temperature prior to secondary breakdown.

Utilizing the technique of thermal meas- urement described, a series of measure- ments were made as a function of pulse width. The silicon planar transistor exam- ined had a gain bandwidth product f t = 180 Mc/s, steady-state thermal resistance 00 =30° C/W, and a thermal time constant T=20 ms. The above data were supplied by the device manufacturer.

The results of the measurements made are plotted in Fig. 3. From the plot of Fig. 3, it is evident that the transistor thermal resistance varies with pulse width, but rapidly approaches the steady-state value.

As a first approximation to peak pulse power capability of the semiconductor de-

and Application, Princeton, N. J.: Van Nostrand. 1 Gartner W. W., Transistors: Principles Design.

1960.