Incremental stress effects in transistors

,Solid-State Electronics

Pergamon Press 1967. Vol. 10, pp. 241-251.

INCREMENTAL

Printed in Great Britain

STRESS EFFECTS IN T R A N S I S T O R S * R. H . MATTSON University of Arizona L. D. YAU University of Illinois and

J. R. DuBOIS North Star Research and Development Institute

(Received 11 July 1966; in revised form 23 September 1966) A b s t r a c t - - T h i s paper reports on some studies of incremental stress effects in transistors. Theoretical and experimental work was carried out using planar transistors with the stresses applied perpendicular to the emitter-base junction. It was found that the applied incremental stress principally affects the base resistance, due to the piezo-resistive effect, and the emitter junction, due to the piezo-junction effect. It is shown that the hybrid-pi or the T-equivalent incremental transistor model can be used to predict transducer operation for varying bias conditions. Although a complete theoretical determination of the piezo-resistance effect was not practical, the magnitude of the observed piezo-junction effect was predicted using the theory developed by Wortman and his co-workers. The conclusion is that the minority carrier fluctuations induced by the stress-produced energy level changes are the major source of piezo-junction effects in transistors when uniaxial stresses are applied perpendicular to the emitter-base junction. It was found that lmiaxial stresses, applied perpendicular to the emitter-base junction, gave reproducible, predictable results. During the course of this study, it was observed that the transistor was much more sensitive to stresses applied on the line where the emitter-base junction came to the surface of the planar device, in which case the stress was applied parallel to the junction, on the junction. Although the device was more sensitive, the action was erratic and unpredictable. li~mtm6---Cet article passe en revue quelques &udes des effets de l'effort accru dans les transistors. Des travaux th~oriques et pratiques ont 6t6 fairs en employant des transistors plans ayant lcs efforts appliqu6s h la jonction base--6metteur perpendiculairement. On a trouv6 que l'effort accru appliqu6 affectait principalement la r6sistance de base ~ cause de l'effet piezo-r6sistif et la jonction d'~metteur ~ cause de l'effet piezzo-jonction. On d6montre que le module de transistor accru h circuit 6quivalent en T ou en pi-hybride peut &re employ~ pour pr6dire l'op6ration du transducteur h r~gime de polarisation variable. Malgr~ qu'une compl&e d&ermination th6orique de l'effet piezor~sistif n'&ait pas pratique, on a n6anmoins pr6dit la valeur de l'effet piezo-jonction en employant la th~orie d6velopp~e par Wortman et ses coll~gues. On conclut que les variations induites par les changements de niveaux d'6nergie issus d'efforts repr~sentent la source principale des effets piezojonction dans les transistors quand les efforts axiaux simples sont appliques perpendiculairement/l la jonction base-&netteur. On a trouv6 que les efforts axlaux simples appliqu6s perpendiculairement h la jonction base6metteur ont doun~ des rdsultats pr6dits et pouvant &re reproduits. Durant cette 6rude, on a observ6 que le transistor 6tait beaucoup pills sensible aux efforts appliqu6s sur la ligae o/~ la jonction base-6metteur apparaissait ~ la surface du dispositif plan sur la jonctlon; dans ce cas l'effort avait 6t6 appliqu~ eli parall~le/t la jonction. Malgr~ que le dispositif 6tait plus sensible, l'action &air irr~guli~re et imprdvisible. * The research was initiated by Prof. R. H. Mattson DuBois at North Star Research and Development and Mr. L. D. Yau while at the University o f Mirmesota, Institute funded by a contract from the Office of Naval and it was continued by these individuals and Dr. J . R . Research, Acoustics Programs Branch. 6 241

242

R. H. M A T T S O N ,

L. D. YAU, and J. R. D u B O I S

Zusammenfasstmg--Diese Arbeit gibt einen Bericht tiber die Auswirkungen, welche differentielle mechanische Spannungsgnderungen auf die elektrischen Eigenschaften yon Transistoren haben. Theoretische und experimer telle Untersuchungen wurden an Planartransistoren ausgeftihrt, wobei mechanisme Spannungen senkrecht zum Emitterbasistibergang aufgebracht wurden. Es stellte sich heraus, dass ~nderungen der mechanisehen Spannung hauptsiichlich den Basi-emitterwiderstand aufgrtmd des Piezowiderstandseffekts sowie den Emittertibergang beeinflussen. Das 0,-oder TErsatzschaltbild fiir den Transistor kann dazu verwendet werden, um die Veriinderung der elektrischen Eigenschaften infolge mechanischer Sparmingsiiuderungen uuter wechselnden mechanischen und elektrischen Vorbelastungen vorauszusagen. Eine vollstiindige theoretische Bestimmung des Piezowiderstandseffektes wurde zwar nicht versucht. Doch gelang die quantitative Voraussage des beobachteten Effektes der mechanischen Spannung auf den Emittertibergang unter Verwendung der Theorie yon Wortman und seinen Mitarbeitern. Daraus liisst sich schliessen, dass die Veriinderungen der Minoritiitstriigerkonzentration infolge der durch mechanische Spannung bedingten Verschiebung der Energieniveaus den Hauptbeitrag zu dem beobachteten Effekt auf den Emittertibergang yon Transistoren leistet, werm eine mechanische Sparmung senkrecht zum Emitterbasisiibergang aufgebracht wird. Einachsige Sparmungen senkrecht zum Emitterbasisiibergang erbrachten reproduzierbare und vorhersagbare Ergebnisse. Im Verlauf der vorliegenden Untersuchung wurde beobachtet, dass ein Transistor viel empfmdlicher auf mechanische Spanntmgen reagiert, welche in tier Linie wirksam sind, wo der Emitterbasisiibergang an die Oberfliiche des planaren Bauelementes trifft. In diesem Fall lag die mechanische Spannung parallel zum ~bergang. Obgleich die Transistoren gegeniiber einer solchen Beanspruchung empfindlieher waren, streuten die Ergebnisse in nicht vorhersehbarer Weise. BACKGROUND IT IS well known that stress affects the electrical characteristics of a bulk semiconductor, as well as a p - n junction diode. T h e piezo-resistance effect is attributed to the change in the resistivity of a bulk material when stressed. T h e piezo-junction effect is the name given to the change in the V - I characteristics of a p - n junction diode when a stress is applied. T h e purpose of this research was to form a theoretical foundation for incremental stress effects in transistors, and to investigate methods for optimizing the design of transistor stress transducers. This is desirable because such transducers are useful for many applications. SMITH(1) initiated a quantitative measurement of the piezo-resistance coefficients in silicon and germanium. He found that certain crystal orientations exhibit unusually large piezo-resistance coefficients. T o explain the large anisotropy of the piezo-resistance coefficients, HERRING(2) proposed a muhivalleyed energy model. By assuming ellipsoidal constant energy surfaces, he was able to account for the anisotropy of the piezo-resistance coefficients. HALL et al. (3) hydrostatically stressed germanium grown junctions and explained the observed decrease in saturation current due to stress by the stress-induced increase in the energy gap. T h e application of uniaxial stress to germanium p - n junctions (O has shown that the

saturation currents increase with applied stress for crystal orientations (111), (110), and (100). This result could not be explained using the simple energy gap theory which explained the hydrostatic situation. RINDNER and BRAUN(s) suggest that the effect in strongly inhomogeneously stressed diodes is associated with the production of generationrecombination centers by the applied mechanical stress. WORTMAN et aI. (4) show theoretically that stresses cause considerable changes in minority carrier density due to the change in the energy band structure. T h e analytical calculations of WORTMAN, et al. can explain the observed effects in germanium junctions for hydrostatic and uniaxial stresses. RINI)NER(6) has applied localized stress to shallow junctions using a steel stylus. He observed large increases in junction saturation current; this was dependent upon crystallographic orientation of the junction. EDWARDS(7) investigated some effects of localized stress on silicon planar transistors using two types of stylus, a "blunt" steel stylus and a "sharp" diamond stylus, for applying the stress to the emitter junction of the transistor. He explained his results by the change in the energy gap due to the stress applied. BULTHUIS(8) has also used the band-gap model to analyze the stress effect in transistors. TOUSSAINT and KRIEGER(°) studied a "three-layer piezo-diode" by considering three different ways of applying the

243

INCREMENTAL STRESS EFFECTS IN TRANSISTORS

stress: (a) pressure applied hydrostatically, (b) pressure applied by means of a stylus acting perpendicular to the plane of the p--n junction, and (c) pressure applied by means of a stylus acting parallel to the plane of the p--n junction. These three different cases affected the V - I characteristics of the device as shown in Fig. 1.

resistive device by its dynamic resistance. Figure 2 s h o w s a piezo-sensitive device R with a voltage

V and a stress cr applied to it. T h e current flowing through R is a function of c, and V. For the incremental model, differential changes provide: dI =--dV+--d~ 8V 8~

(1)

l o" b

jd /

o ,,

7 /

a

V

~

o

R

I (V, or)

F I G . 2. A r e s i s t a n c e

trader stress.

I f d/, d V and d~r are small signals, they can be represented by i(t), v(t), and or(t), respectively, and equation (1) can be expressed as:

i(t) = 7;- v(t) + - - ,(t). 8.

VR

FIG. 1. Idealized I-V characteristics of a reverse biased junction: (a) junction unstressed; (b) pressure applied hydrostatically; (c) pressure applied by means of a stylus acting perpendicular to the plane of the p---n junction; (d) pressure applied by means of a stylus acting parallel to the plane of the p-n junction. As a result of these various efforts, it appears that the method of applying the stress can and does affect the results. Most reported results concerning the piezo-junction effects in p--n diodes can be divided into two categories: results that can be explained using the theory of W O R T M A N , et aL (4) and results which may be explained (5) as being due to the generation of dislocations in the junction vicinity causing generation-recombination centers to be located close to the junction. T o explain the observed piezo-junction phenomenon in transistors, when applying the stress uniaxially perpendicular to junction, the theory of Wortman was found to apply. THEORY

Incremental model for a Piezo-sensitive element In a low-frequency small signal analysis, it is convenient to represent a diode or a nonlinear

(2)

T h e term 8I[8V is a dynamic conductance g, and the term (ST/&r)cr(t) is a current generated by the incremental stress cr(t). T h e incremental model representing equation (2) is shown in Fig. 3. This ~-~ a-(t)

v(t)

.,ww"

--~i(t)

ar

FIG. 3. An incremental model of a stress sensitive device at low frequeney. model will be used to account for the piezoresistance effect as well as the piezo-junction effect in transistors.

Low-frequency hybrid-~r model including the stress effects T h e hybrid-pi incremental model can represent the transistor when an incremental stress is applied to the emitter-base junction. T h e reason for applying the stress on the emitter-base junction is to take advantage of the amplification of incremental signals due to transistor action. Figure 4 shows the circuit of an N P N transistor that is biased in

244

R. H. M A T T S O N , L. D. YAU, and J. R. D u B O I S

the active region. The low-frequency incremental model of Fig. 4 is shown in Fig. 5. Figure 5 assumes that all elements of the hybrid-pi model are stress sensitive. It was experimentally observed that only the stress effects on r z and rn gave significant contribution to the output Vo(t ). This follows when

The resistance r z is an equivalent lumped base resistance, while rn is a "transformed" forward dynamic resistance of the emitter-base junction. Is~ accounts for the piezo-resistance effect o f t x and Is~ accounts for the piezo-junction effect of r n. I f the current sources Isz and ls~ are known, the

o vo(t) ~t

o-(t)

RL

Rb

lib-

,Ill

FIo. 4. Common emitter circuit for a transistor.

r~

i

r~ V r~

I~

cjm

~

----o--------

RL

Is0

o - -

Fro. 5. Hybrid-~r model of Fig. 4. the expression for Vo(t ) from Fig. 5 is used to show that Isz and Is~ are amplified by almost a factor of /3 (a.c. current gain) compared to Is, and [so. Furthermore, r u is so large that its effect is negligible in most practical cases. Figure 6 is a simplified incremental model for a mechanical-to-electrical transducer. Mechanical stress ~ t ) generates current sources Isz and Is:~. ][sx

output voltage Vo(t ) can be solved to give:

' ~-~-i-R. J Lr -T-r-~-+-RbJ kr0 + RL] t ; gm r~ = ft.

The parameters of equation (3) can be measured by standard techniques except for Isz and Inn. They were measured with the desired static stress applied to the emitter-base junction. /sn can be determined indirectly by measuring % ( 0 when R b >~ (rb+rz) or R b -+ oo. Designating the output voltage under this condition by Vob, equation (3) can be used to evaluate Is~ as:

s,,, FIG. 6. Simplified low-frequency model including stre~s effects.

(3)

(ro + Rr,~ l = Vob

-.

\r o R J fl

(4)

Knowing Is~, Isz can be obtained by choosing

245

INCREMENTAL STRESS EFFECTS IN TRANSISTORS

/ ~ voCt]

I~b Im~--]

I

When Is= and Isx are known, equation (3) can be used to predict the output voltage Vo(t) for any value of R b and Rr~.

aJe

r,

Is,

Fzo. 7. T-equlvalent circuit for Fig. 4.

another value o f R b and measuring Vo(t). Choosing R b = 0 and designating the output voltage under this condition by Vs~,

Is,=I~LVob\

]

r, j

(s)

v°''

rb

T RL

I$b

FIG. 8. C o m m o n - b a s e T - e q u i v a l e n t circuit with incremental stress.

E

T-equivalent circuit including the stress effects The T-equivalent circuit was also found to be a good model to account for the stress effects in transistors. Figure 7 shows the low-frequency incremental model for Fig.4 using the T-equivalent circuit. As in the case of the hybrid-~r model, the stress effects of r c were not significant. In Fig. 7, i e is not a terminal current, but rather a current flowing through r e. This is an important point because of the dependent current generator 0u"e that appears at the collector circuit. This shows that the stress induced incremental current must flow across the junction to cause output current. The experimental results show that the model is correct when i e is taken as the current flowing through re. Using the model of Fig. 7, and assuming r e is very large, the transducer output voltage is: ~d~ise+ [ Isbr~ ] Lrb + RbJ Vo(t ) = (6)

1 -~+

- -

re

r~ + R~

As in the hybrid-Tr model, Ise can be obtained by making R~ >>flre+r b and equation (6) can be

Icoo. Lc

[

,

I I

E 11 U Stressapparatus _

FIG. 9. Block diagram of the experimental arrangement.

M~,a,,~ 0age

246

R.H.

MATTSON,

L. D . Y A U ,

a n d J. R. D u B O I S

4 x l o -~

3x10 -~

e

I xtO - ~

10

20

Emitter

current

IE,

30

mA

FIG. 10. Evaluated Is~ as a function of emitter current for 2N1893.

rearranged to give: &~ =

(1-~)Vob

~RL

V0b = --

(7)

/~RL

where V o ~ is the output voltage when R b >~ fire + r b. Similarly, Isb can be evaluated for R b --> 0 by

emitter resistance R e very large (this may be done by using tuned circuits), and making R z + r ~ ,~ re, the current through R L will be approximately ~i e since r b is small compared to r c Even if Isb was comparable to l~e, it cannot significantly affect the output current. For the case in which R e is very large, Fig. 8 gives: 4 = Le

I~b =

- I~

(8)

and

o~Rz,

roe

4e =

where Vsb is the output voltage when R b -+ 0. For the grounded base configuration (Fig. 8), [se can be easily determined. By making the

%

=

where roe is the output voltage when R e = >~ flr~ q-r b or R e --> o0.

6 . 4xlO dynes/cm 2

_

.

.

.

.

L

2xlO -7

E v

IxlO 7

I.,-I

!J,

'

1 IO

Emitter

(I0) ocRr.

3 x l O -7

~-~

(9),

current I E , mA

FIG. 11, Evaluated Is~r vs. IE for 2N2905.

20

INCREMENTAL STRESS EFFECTS IN TRANSISTORS rOxlO -7

2n29o5:o-~ = 6.4x10 ' '~ .

' dynes/crn

247

z

I

8 x ; O -7 u~ (p

2N B93

4xi0_7 - -

I ;

o"0 = 4 . 9 X 1 0 9 d y n e s / c m

i

t 0.5

LO Bose

s

!.5 current,

2

1 2.0

2.5

mA

FIo. 12. Evaluated values of 18= as a function of base current for 2N1893 and 2N2905. of 10 9 dyn/cm 2. The experimental apparatus was

Equations (7) and (10) indicate two entirely different methods of determining Ise, and the experimental measurements showed that they are equal. This agreement is more evidence supporting the validity of the simplified model.

arranged so that a small a.c. stress could be added to the bias stress. The dynamic characteristics of the apparatus were calibrated for 30 c/s, 1.5 kc/s and 21 kc/s. A block diagram of the experimental arrangement is shown in Fig. 9. Throughout the experiment, a steel stylus of 0.003-in. dia. was applied "perpendicular" to the emitter-base junction. An incremental force of 6 . 3 x 10 a sin wt dyn was chosen. The output voltages Vo(t ) were measured across a collector resistance of one kilohm, unless it is specified otherwise. For evaluation of the model, the output voltage Vo(t ) was measured for various R~'s and

EXPERIMENTAL PROCEDURES A special stress apparatus was designed for the

purpose of applying static and dynamic stress to planar transistors. The static bias force was necessa D, to prevent the stylus from leaving the surface

of the transistor when the incremental force was applied. The transistor was biased b y a d.c. source in its active operating range with the stressproducing stylus applying a bias force of the order

RL's.

I~0.25

~0 I 4 . 9 X I0 9 d y n e s / c m 2 4wIO"2 w

E 3xlO -2 A :o

2*tO -2 I#0.05 mA

o

-6 >

IxlO -t

J

o

-IxlO"

J I

I llJIII

I

I jlllll

l

] I11111

102 Bose

I I0 ~

resistonCe

I IIIlll

1 I 111ll 104

IO s

,~,

FIO. 13. Variation of output voltage with external base resistance Rb of 2N1893.

R. H. M A T T S O N ,

248

L. D. YAU, and J. R. D u B O I S

EXPERIMENTAL RESULTS

~

Dependence of the output voltage on R b and R L Experimental results obtained from two transistors (npn and pnp) are shown. Figures 10 and 11 show the values of Isn for the 2N1893 and the 2N2905, respectively. T h e points on the curves were obtained using equation (4), as described previously. Figure 12 shows the values of Isz obtained using equation(5). Using the values ofls~ and Isz , equation (3) was used to calculatethe output voltage vo(t ) for various R~'s and RL'S. T h e smooth curves of Figs. 13-15 are drawn according to equation (3), and the points are the experimentally measured values. T h e agreement of the values obtained using equation (3) and the experimentally measured values is a strong indication that the simplified model of Fig. 6 is a good one. According to SMITH'S measurements, (:) the piezo-resistance coefficients of silicon and germanium may be positive or negative depending upon the crystal orientation and the type of stress applied. From Fig. 13, Vo(t ) is negative* for low values of R b. This implies that Isx is greater than Is= and the sign of [sz is negative in equation (3). In Fig. 15, vo(t ) is largest for the lowest value of R b. This implies that Isz and Is~ have the same sign in equation (3). Figure 13 is typical for an npn transistor when * The reference for the sign of Vo(t) was based on e(t) with compressive stress taken as positive.

; _ o o

102

I0 ~

104

Collector resistance

FIG. 14. Variation of output voltage with load resistor RL of 2N1893. the stylus is applied "perpendicular" to the emitterbase junction, while Fig. 15 is typical for a pnp transistor when stylus was applied "perpendicular" to the emitter-base junction. When the stylus was, applied "parallel" to the emitter-base junction, the

O.OE Go= 6.4xi0 9 dynes/ern z

0,05

i

-

0.04

g

o.o~

>

- :

Q. 0.02 0 0.01

Oioo

I

I i l illl

I i Illltl I01

IO z Base

10=

RL,

I !111111

iO s

I I kLIIII~'> iO*

resistance,

Fro. 15. Variation of output voltage with Rb of 2N2905.

I0 ~

249

INCREMENTAL STRESS EFFECTS IN TRANSISTORS

curves for npn and pnp transistors are similar in shape to that of Fig. 13. T h e magnitude of the output voltage, however, was about one order of magnitude larger for the same amount of emitter current I E.

Dependence of output voltage on the static stress T h e output voltage was observed to increase with the bias stress when the stylus was applied perpendicular to the emitter-base junction. I n some cases, the output voltage increased initially, then reaches a peak, and further increase in the static stress only decreased the output voltage.

for two types of transistors. Capacitors were avoided in the test circuit to avoid circuit-induced attenuation and phase shift at the frequencies of interest. T h e results are tabulated in Table 1. Although this quantitative measurement is limited to 21 kc/s, the device was tested and found to respond up to 3 Mc/s. Due to difficulty in calibrating the a.c. force at the higher frequency range, no figure is given here.

~ I / :°S /

4,i0

-~

>

N//A?

S

I

v

o

o o

tylus

Fro. 17. Junction diode under stress.

2xlO -z - - - -

COMPARISON WITH THEORY

o. 3 0 0

2xlO 9 Static

bias

4xlO 9 stress,

6xlO 9

°T

dynes/cm z

Fro. 16. Dependence of output voltage on static bias stress ao of 2N|893. A typical set of curves showing the output voltage for various static stresses is shown in Fig. 16.

It is of interest to see how the change in energy gap induced by stress theory as proposed by WORTMAN and his co-workers agrees with the experimentally measured values of Is=. T o correlate Is= with WORTMAN'Stheory, consider a simple p-n junction diode under stress as shown in Fig. 17. For an ideal diode, WOnTMAN(4) showed that only the saturation current, Is, is stress sensitive, therefore: ~I

Dependence of output voltage on frequency T h e output voltage at the three frequencies for which the apparatus was calibrated was measured

7~ ~(t)

=

OI ~I~

OIs aa

~(t).

(11)

For a localized stress applied to diode, I s can be approximated by:

Table 1. Frequency dependence of the voltage output Transistor 2N1893

2N1072

Frequency a.e. force applied Normalized (dyn) go(t) 30"0 c/s

6-3 x 103 sin w t

1 "0

1 "5 kc/s 21 "0 kc/s

1 "0 1 "1

30"0 c/s 1.5 kcls 21-0 kcls

1.0 1.0 1.2

ts = lso [1 - -As - + -As - yv(~) ] (12) At At where .[so is the unstressed saturation current, At, the total area of the junction, As the effective stressed area, and yv(a) is the ratio of minority carrier density under stress to the unstressed minority cartier density. (or here is positive for compressive stress). T h e relation between Iae and I s from the diode equation is:

Ictc = Is[exp(qv/KT)- 11

(13)

250

R.H.

M A T T S O N , L. D. YAU, and J. R. D u B O I S

combining equations (11), (12) and (13).

81

As

8~

A s +Fv A~tt] d~ "At [ 1 - -~t

- - ~( t ) =

Iae

dyv

or(t)

(14)

to a 0"003-in. dia. stylus gave an r.m.s, incremental stress or(t) of 9"8 x 10 ~ dyn c m - 2. Choosing an emitter current of 10 mA, the measured /3 was 124. These give for equation (15),

Is~= l ' 9 x 1 0 - 7 A .

Figure 18 shows a reproduction of the theoretical relationship between Yv and ~ as worked out by WORTMAN and his co-workers. Ca)

This value of Is~ has the order of magnitude that is in agreement with the experimentally measured value shown in Fig. 10.

! i ! | L

,

| i

i

io

1107

J ! ...........

....

....... I !

t ll

// // /

iloli/-,

10 a

r

109 10 '° O-, d y n e s / c m z

+- .... ,I i 10 It

10 'z

FIC. 18. Ratio of minority carrier density with stress to the unstressed minority carrier density as a function of stress in Si (after Wortman). To relate Isn to (8I/Oa)a(t), it should be noted that r~ is greater by a factor of fl compared to the dynamic resistance, re, of the emitter diode. Therefore, using I~ = Iac, we can write: 1

81

1 As

I~

/3At

1---+7~

d~"v

or(t).

(15)

At For the 2N1893, the stylus was applied perpendicular to the (111) plane. For the ratio of the stylus area to the emitter-base junction area was about ~6. For a static bias stress of 3 x 109 dyn/cm 2, Fig. 18 gives ; ¢ ~ 2 and (dyv/da)~2x10 9 dyn -x cmz. The r.m,s, value of the incremental force was {(6.3 x 10s)~/2 dyn. This force applied

A theoretical evaluation for the piezo-resistance effect Isz is complicated because r z is a "lumped" equivalent resistance of the base. Furthermore, the type of stress producing the piezo-resistance effect of r z is essentially "transverse" for the planar transistors used. This gives geometrical difficulty in the mathematical analysis if one would attempt to calculate lsx theoretically. CONCLUSIONS This study has shown the significance of the piezo-resistance effects and the piezo-junction effects in transistors. The simplified incremental model that treats only Isz and Is= is a good model in predicting the output characteristics of a piezotransistor under an incremental stress. T o maximize the stress-induced output of a transistor, Figs. 13 and 15 show that I b should be maximized. From Fig. 13, Rb should be maximized

INCREMENTAL STRESS EFFECTS IN TRANSISTORS

for the 2N1893, and from Fig. 15, R b should be minimized for maximum output for the 2N2905. From Fig. 16, the output voltage can also be increased by maximizing the static stress e0. In a practical situation, the parameters Ib, R b and eo have limitations. The noise of the transistor increases with I b and Rb, and in certain conditions* with e0. The condition R b = 0, which maximizes the output voltage for the piezo-characteristics of Fig. 15 for the 2N2905, has an advantage of low noise and more stable operation. These results are compatible with the following observations. The model appears reasonably accurate for design purposes, when stress is applied perpendicular to the emitter base junction. Maxim u m output voltage is observed at higher bias stress up to certain limits, which is compatible with the theory of Wortman et al. The deviation at very high stress could be attributed to increased piezoresistance effects. The result that increased output * Preliminary investigation shows that the noise of the transistor increases when the application of oo causes/3 to decrease. (IE was kept constant.)

251

voltage is obtained with increased R b for the npn unit (Fig. 13) and the output decreases with increased Rb for the n/m (Fig. 15) is attributable to the sign on the Isx generator due to the difference in piezo resistance coefficients. The extension of this work should be the use of this model coupled with noise theory to optimize the signal-to-noise figure rather than just maximizing the output voltage. REFERENCES 1. C. S. SMITH,Phys. Rev. 94, 42 (1954). 2. G. I-IxI~INO, Bell Syst. tech. .7. 34, 237 (1955). 3. H. HALL, J. BAnDEm,~and G. PEARSON,Phys. Rev. 84, 129 (1951). 4. J. J. WORTM~U~,J. R. HAUSERand R. J. BUaOER,Jr. appl. Phys. 38, 2122 (1964). 5. W. RI~rggR and I. BRAt2¢, 3". appl. Phys. 3'#, 1958 (1963). 6. W. RINDNV2a, .7. appl. Phys. 33, 2479 (1962). 7. R. EDW~mDS,IEEE Trans. electron Devices 11, 286 (1964). 8. K. BULTmqIS,Philips Res. Report 20, 415 (1965). 9. H. N. TOUSSAINTand F. KRIECER,Proc. IEEE 53, 752 (1965).