IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-17, NO. 10, OCTOBER 1970 843

in piezoelectric semiconductors,” J . A p p l . P h y s . , vol. 38, p, [22] --, “Measurement of ultrasonic wave velocities and elastic 4295, 1967. moduli for small solid specimens at high temperatures,” .4COZcSt.

tial theorv for manv-vallev semiconductors with anisotrooic 1231 hI . Pomerantz. “Amolification of microwave ohonons in per- [12] C. Herring and E. Vogt, “Transport and deformation-poten- Soc. Amer., vol. 31, p. 287, 1959.

scattering:” Phys. Rev. , vol.‘lOl, p. 944, 1956. ~ I - L , ‘~ ~ ~~ ~~-~~ ~

manium,” Phys . Rev . Le t t . , vol. 13, p. 308, 1964. ~ ~ ,~~ . ~ ~ . ~ ~ ~ -~~ - - -

[13] A. R. Hutson, J. H. McFee, and D. L. \Vhite, “Cltrasonic [24] C. S. Smith, “Piezoresistance effect in germanium and silicon,” ~~

amplificationin CdS,” Phys. Rev. Let t . , vol. 7, p. 237, 1961. [14] F. D. Ketterer and J . E. Nelcher, “Electromechanical stream- [25] H. S. Spector, “Amplification of acoustic waves through inter-

structure instabilities.” Phvs. Fluids. vol. 12, D. 109, 1969. action with conduction electrons.” Phvs. Rev.. vol. 127. D. 1084.

~~ - P h y s . R e v . , vol. 94, p. 42, 1954.

[15] F. D,. Ketterer, “Electromechanical stream:structure instabili- ties, Massachusetts Institute of Technology, Cambridge,

[16] R. \V. Keyes, “The electronic contribution to the elastic proper-

[17] R. Klein, and \V. Rehwald, “Combined effects of sound and ties of germanium,” I B X J . Res. Develop. vol. 5, p. 266, 1961.

electric fields upon the conduction electrons i n solids, P h y s . Rev. , vol. 143, p. 479, 1966.

[18] P. G. Klemens, Physical i lcoustics, Principles and Methods, vol. 3B, \V. P. Mason, Ed. New York: Academic Press, 1965,

[19] S. T. Kowel, “Electrodynamic instabilities in the acoustoelectric p. 201.

interaction,” Ph.D. dissertation, Dept. of Elec. Eng., University of Pennsylvania, Philadelphia, 1968.

[20] J. C. McGroddy and :). I. Sa than , “.A new current instability in n-type germanium, IB.11 J . Res. Develop., vol. 11, p. 337, 1967.

[21] H. J. McSkimin, “hIeasurement of elastic constants at low temperatures by means of ultrasonic waves, J . .4ppZ. P h y s . , vol. 24, p. 988, 1953.

CSR-TR-66-16, August 1966.

.~ 1962.

[26] -, “Ultrasonic amplification in extrinsic semiconductors,’‘ P h y s . R e v . , vol. 130, p. 910, 1963.

[27]”-, “Effect of an external electric field on the velocit!, of sound in semiconductors and semimetals,” P h y s . R e v . , vol. 134, p. A507, 1964.

[28] --, “Interaction of acoustic waves and conduction electrons,” in Solid State Physics: Adoances in Research and Appl icat ions, vol. 19. New York: Academic Press. 1966. D. 291.

[29] J. A. Stratton, Electromagnetic Theoyy. fi& York: McGraw-

[30] B. Tell and G. LVeinreich, “-Acoustoelectric effect and inter- Hill, 1941, ch. 8.

valley scattering rates in antimony-doped germanium,” P h y s . Rev. , vol. 143, p. 584, 1966.

[31] G. LVeinreich, “=\coustodynamic effects in semiconductors,” P h y s . R e v . , vol. 104, p. 321, 1956.

[32] G. Lveinreich, T. $1. Sanders:,and H. G. LVhite, “Xcoustoelectric effect in n-type germanium, Phys . Rev . , vol. 114, p. 33, 1959.

[33] D. L. \f‘hite and LV. \\’ang, ‘Li\ctive CdS ultrasonic oscillator,“ P h y s . R e v . , vol. 149, p. 628, 1966.

The Saturation Characteristics of n-p-v-n Power Transistors

Abstract-The transport equations and charge-control concepts are applied in an analysis of a static conductivity-modulation mech- anism occurring in the collector region of n-p-v-n power transistors. This results in an expression for collector-emitter saturation voltage as a function of terminal currents and device parameters. An ex- pression is derived which describes the current gain characteristics of saturated epitaxial and triple-diffused devices. The analysis is also used to illustrate the relationship between the emitter metalliza- tion resistance, collector charge storage, and the turn-off crowding mechanism experienced by high-frequency saturating transistor switches.

An analysis of a time-dependent collector conductivity modula- tion process is used to derive an expression which describes the repetition frequency dependence of the collector-emitter saturation voltage of an epitaxial (or triple-diffused) transistor switching a square wave of collector current. It is concluded that frequency- dependent edge-crowding mechanisms occur only at much higher frequencies than those considered in this study.

work was supported by the Defense Research Board, Canada. Manuscript received January 9, 1970; revised April 8, 1970. This

The author is with the Department of Communications, Com-

Ottawa, Canada. munication Research Center, National Communications Laboratory,

LIST OF SYMBOLS effective collector area collector-base junction area diffusion constants for electrons and holes in the collector region angular frequency electric field strength grounded-emitter dc current gain grounded-emitter dc current gain for T/Bc, = 0 base current amplitude hole current flowing from base to collector collector current collector current density collector current density at point a. collector current density at point b. electron current density hole current density a constant which relates hole current in the high-resistivity collector region to the excess base current

Boltzmann's constant hole diffusion length collector substrate impurity density impurity density in the high-resistivity col- lector region injected carrier density in the collector region intrinsic carrier concentration equilibrium carrier densities in the collector region rated collector power dissipation net minority carrier density in the collector region a t x equals 0 and Wc total minority carrier charge stored in the collector electron charge stored collector charge per unit length of emitter stripe at point a and point b absolute temperature terminal base-collector voltage potential drop across the base-collector junc- tion collector-emitter voltage collector-emitter breakdown voltage (base

potential drop across the region of the collector defined by x c < x < bVC collector-emitter saturation voltage average collector-emitter saturation voltage during the OS time of a transistor switch width of lightly doped collector region position in the collector region the point removed from x=O a t which P = 0 x at the points a and b along the emitter stripe spatialcoordinateparalleln-ith baseand emitter stripes spatial coordinate normal to base and emitter stripes grounded-emitter dc current gain for Ir,,, > O

open)

J C b I J c a

xcal wc electron and hole mobilities a constant factor relating l i S A T ( r ) and VSAT(O) minority carrier lifetime in the heavily doped collector substrate effective minority carrier lifetime i n the lightly-doped collector region.

I , INTRODGCTIOK

H E STATIC common-emitter saturation char- acteristics of n-p-v-n transistors have been studied by a number of authors lvho have shown that the

characteristics are defined by a conductivity modulation mechanism occurring in the high-resistivity collector region 11]-[6]. Hahn [ l ] , [ 2 ] and Clark [5] have shown that the conductivity modulation mechanism is respon- sible for the tv-o-stage saturation characteristics peculiar to epitaxial and triple-diffused transistors (Fig. 1). Re-

iEEE TR,\KX,\CTIQ~'S O v ELECTRON DEVICES. OCTOBER 1970

Fig. 1. VCE-JC characteristics of an n-p-v-n transistor (triple-

curve VB'WC., -0 . V C E = 1.0 \7/cnl, IC =0.2 A/cm, I B = 5 mA4/step. diffused r)TS 410). The dashed line represents the locus of the

cently, a closed-form analysis which described the txvo- stage saturation characteristics in terms of collector current, base current, and device parameters \vas devel- oped bq- the author [6], [ 7 ] . This analysis is included in Section 11; it is used to shojv that the conductivity modulation mechanism may also define the high-current hpa characteristics of epitaxial and triple-diffused tran- sistors, and moreover, is responsible for a nonuniform turn-off mechanism experienced by high-frequency saturating switches.

In Section 111, the basic conductivity modulation analysis is extended to include time-dependent effects. A closed-form expression is developed which describes the time dependence of the saturation voltage of a de- vice switching a square wave of collector current when driven by a square wave of base current. I t is concluded that for t h e devices considered, the apparent frequency- dependence of saturation voltage should often be inter- preted in terms of the conductivity modulation mecha- nism rather than in terms of a frequency-dependent cron-ding mechanism.

I I. STATIC SATCRATION CHARACTERISTICS A . A n a l y s i s

For the purposes of this paper, a collector doping profile of the form shown in Fig. 2 is assumed (impurity density on the base side of the collector-base junction much greater than that on the collector side).' The analysis proceeds from the two transport equations and the requirement that total current be continuous:

applicable to p-n-7r-p devices. 1 .;in n-p-v-n structure is assumed, although the analysis is also

CHUDOBIAK: SATURATION OF n-p-v-n POWER TRANSISTORS 845

When n<<ivd (case b) , (1) can be simplified to

J c = q p , S d E . (10)

The regions in lvhich (8) and (10) are considered to be valid are defined by X , in Fig. 2 . The location of this breakpoint is found from a consideration of recombina- tion in the Nd region. The expression

“”1

N + c X

Fir. 2 . Collector doping profile and net minority carrier density distribution in saturated n-p-v-n transistor.

where n and 9 represent injected carrier densities and no and po represent equilibrium densities. Equations ( l ) , ( 2 ) , and (3) are applied to two distinct cases:

ni2 a ) ~t >> no = ATd, p >> po = -

b) n << 110, p <<PO. l l T d

Considering case a and assuming quasi-neutrality ( p = n) gives

dP J P = WDEP - qDP * ( 5 )

Hole current is small in the high-resistivity collector region since W, is much less than the carrier diffusion length; also, hole flolv through the NdLv+ boundary is negligible. Thus ( 5 ) can be reduced to

An expression for E follon-s from the substitution of (4) and (5) into ( 3 )

1 \ - r, .

Combining (6) and ( 7 ) gives

d p J c

dx 2 q D n -

represents the hole current from base to collector re- quired to satisfy recombination in the Nd region (a con- s tant , effective lifetime 7, is assumed).

Again n-e consider two cases:

x, 5 W’,, x, > w,. First, for x,< E’,,

With the aid of (8), x, is found to be

= i V / - 4 r c D n I B C (1 1)

I C

11-here in accordance with the sign convention of Fig. 2 , IC carries a negative sign.

The injected minority carrier density a t the collector side of the collector-base junction is then given by

dP p ( 0 ) = x,- d z

The net minority carrier density at the same point is given by

For X,> W,,

where the last term is representative of the recombina- tion in the heavily doped collector substrate. Expressing P(0) in terms of P(W,)

dP P(0) = P(W,) - w c - ax and therefore

J C X

2q D, and solving for P( Wc) , n-e have p(.) = - + P(0) . (9)

846 i E E F TR,\\7S.\CTIOI'S @ Y ELECTRON DEVICES, OCTOBER 1970

The current I B C must be represented in terminal current and known device parameters. In this paper, i t is assumed that the relationship may be de- scribed by an expression of the form

IBC = K l e +- { 20J (15)

where K is a constant less than unity. The factor K is determined by substituting (15) into (14) and setting P(wc) =ni2 /Nd . According to the present model, P ( w c ) =ni2/Nd at the breakpoint of the tu-0 stage saturation curve. I t will be shown that this approach enables one to obtain close agreement between calculated and ob- served saturation characteristics over a nide range of current levels and is, therefore, valid for purposes of device characterization for circuit application. How- ever, i t places a limit on the utility of the saturation analysis in that it precludes a prediction of device char- acteristics given doping profiles and other device design variables.

The collector-emitter saturation voltage is given by the sum of the integrals of the field expression [ ( 7 ) and ( l o ) ] and the potential drops across the forxvard-biased junctions. Therefore, for the case x, < ?Vc:

kT A-dP(0) TlsAr = VBE - -- In ___

q It % 2

KT J C X C J C ( W , - .VC) - _ In rzgD.p(O) + - q p , s d

. (16) 9

For xc 2 W,, the last term in (16) is dropped and x, in the third term is changed to W,.

Equation (16) may be simplified further since the contribution of the third term is normally less than 100 mV. This simplification will be used in the calculations given in this paper.

I t should be noted that this analysis is based on the assumption that n<<Nd in the region x c < x < W,. This assumption places a limitation on the validity of the model since at current densities of the order of qvmNd, velocity saturation occurs and further increases in cur- rent density are accompanied by increases in carrier density rather than in carrier velocity. That is, conduc- tion in the region becomes space-charge-limited and the carrier density may approach or even exceed f ! d . Ho\\-- ever, in applying the analysis to large-area power tran- sistors ( A C B > 0.05 cm*) i t has been found that velocity saturation does not normally occur a t collector current levels less than the rating specified by the manufacturer.

B . Results A set of measured VCE-IC characteristic curves for a

high-voltage triple-diffused transistor ( V / C B ~ = 200 volts, PT = 80 watts, IC = 3.5 amperes) is given in Fig. 1 (junc-

t + 135.C

O - * I L I I I I 1 1 1 .

- MEASURED --- CALCULATED Ig = 40mA

0 0 1 2 3 4 5 6 7 8 9 l O

VCE (VOLTS)

Fig. 3. Comparison of calculated and observed saturation curve: for a high-voltage triple-diffused transistor.

tion temperature = 35OC). Calculated saturation curves for the same device with junction temperatures of -40, +35, and 1 3 5 O C are given in Fig. 3. The close agreement u-ith observed characteristics may be taken as an indi- cation of the validity (and utility) of the model.

For the purposes of these calculations, the effective collector area A \\-as assumed to be equal to the area of the emitter metallization lvhile the collector lvidth Wc was determined using the procedure described in the Appendix. 'The collector impurity density Nd and mi- nority carrier lifetime 7, n-ere determined using well known techniques [ 8 ] , [ 9 ] . The temperature sensitivity of the parameter's T, , pn, and D, n-ere described by expres- sions given in the literature [ lo ] , [ l l ] . However, mea- sured values o f h F ~ o n-ere used since a valid analytical description of the temperature dependence of hFEo does not appear to be available in the literature. The calcula- tions revealed that the collector-emitter saturation voltage at the extremes of temperature is particularily sensitive to the parameters p n and hFEo and to a lesser extent, 7,. Current gain hFEo and T , both increase with temperature, thus tending to reduce the collector- emitter saturation voltage (assuming I C and Ig remain fixed while temperature is varied). However, pLn varies inversely with temperature and so contributes to an increase of saturation voltage u-ith temperature in- creases. The net effect of temperature can be described as follows: for a device operating \I-ithin the second stage of the saturation characteristics and u-ith base and collector currents maintained constant, collector- emitter saturation voltage decreases with temperature.

C. Current Gain (hF3) Fall08 at High Current Densities

The hFE characteristics of epitaxial and triple-diffused power transistors are commonly specified in data sheets as a function of collector current at a low, constant collector-emitter voltage, nominally two or three volts. I t may be seen in these plots that ~ F E falls off drastically

CHUDOBIAK: SATURATION OF n-p-v-n POWER TRANSISTORS 847

at the higher current levels. Hahn [ l ] , [ 2 ] , Larin [ 4 ] , and Chudobiak [6], [ 7 ] have suggested that the falloff of ~ , P E a t high current densities could be related to the storage of minority charge carriers in the collector re- gion. Tha t is, the devices may be operating in saturation at the higher current levels. An analysis which confirms this theory is developed in the following paragraphs by applying aspects of the saturation model described in the previous section.

The saturation voltage of a device operating lvithin the second stage of the saturation region is defined largely by the final term of (16),

The variable x, in the above equation is given by (1 1) as

\\here

Combining the above equations gives

80

70

60

P 50

40

30 \ DTS 410 VCE= 5 VOLTS

I I I I b

1.0 2 .o IC (AMP)

Fig. 4. Current gain falloff at high collector currents.

equation does predict a rapid falloff of p. 'The fact that calculated and observed results are in close agreement would appear to confirm that the p(.Tc) characteristics

1

A comparison of the calculated and observed p ( 1 C )

characteristics of a high-voltage triple-diffused transis- tor serves to confirm the validity of (17). The measured current gain characteristics for the device are sholvn in Fig. 4 n-here it may be seen that current gain decreases steadily as collector current is increased. The falloff of current gain in region A (collector-base junction reverse biased) can probably be attributed to three main fac- tors. First, the emitter efficiency falls off a t high current densities. Second, an increase of base Ividth occurs as a result of the decreasing reverse bias on the collector- hase junction. Finally, mobile charge carriers injected from the emitter tend to reduce the v-idth of the hase side of the collector-base junction depletion layers [ l 2 ] and, as IYhittier and Tremere have shonm, also con- tribute to phenomena in the collector depletion region resulting i n the formation o f a current induced base region [13] . I t is suggested that the falloff of gain i n region B is primarily due to the injection of holes from the base into the collector, and to a lesser extent, to the falloff of emitter efficiency.

The P(1,) characteristics as described by (17) are plotted in region B of Fig. 4 where i t is seen that the

are defined primarily by the injection of holes from the base into the collector rather than by the emitter effi- ciency (or other effects).

D. A Limitution of the One-Dimensional I.',y.dT Analysis

The analysis presented in Section 11-A \\-as developed utilizing the assumption that the emitter current den- sity distribution )vas uniform along the length of the emitter strips. In effect, it was assumed that the degree of saturation \vas also uniform along the length of the emitter stripe. 'This assumption enahles a one-dimen- sional approach to be applied and so considerably sim- plifies the problem. However, as \vi11 be sholvn subse- quentl?., there are instances v.here the validity of this assumption must he questioned. For this reason, it is of interest to attempt to describe analytically the actual variation of the degree of saturation along the length of an emitter stripe. Such an analysis is developed in this section by applying elements of the I'.?.4T analysis given in Section 11-A.

For purposes of discussion, it is instructive to repre- sent a saturated, large-area epitaxial (or triple-diffused)

848 JEEE TR4NSACTIONS ON ELECTRON DEVICES, VL Lun~n 1 7 , ”

Y

/EASE CONTACT /

V E R CONTACT

t

( b ) Fig. 5 . The degree of saturation as a function of the collector cur

rent density distribution; (a) coordinate system, (b) (21).

switching transistor having a comb-like structure (see Fig. S(a)) by a two-element model as shown in Fig. 6. Transistor Qa corresponds to the region of the large-area device near the emitter contact while Q b represents the region near the base contact. RE and RE correspond to the emitter and base metallization resistances.

Dc potential probing along the base stripes of satu- rated transistors has shou-n that the potential drop along the length of a base stripe is normally very much less than the terminal collector-base voltage. Therefore the terminal collector-base voltages for the tlvo ele- ments Qa and Q b are nearly equal. An expression for this potential is obtained from (16) :

For a device operating within the second stage of the saturation characteristics, the second term in (18) is small in comparison with the other terms so the equa- tion can be simplified to

Since the potential VcB is not a function of y [see Fig. S(a)], (19) may be used to describe x c as a function of y, provided J c ( y ) is known and VcBt is assumed to be a constant. Caves and Barnes [I41 have shown that the effect of the emitter metallization resistance is to pro-

Fig. 6 . i\ two-element representation of a large-area transistor having a finite metallization resistance.

duce a collector current density distribution u-hich ex- hibits a minimum in the vicinity of the base contact and which increases along the emitter stripes to a maximum near the emitter contact. Therefore, in the two-element model shown in Fig. 6, the collector current for element Qa will be sorneIvhat larger than that for Q b . By defining

and applying (19) we obtain

and therefore,

where

A ratio between the stored charge per unit length a t b and a follows from (8) and (20) :

q b

= {f - ,($ 1 - 1) tZY.

The relation between this ratio and y is plotted in Fig. 5(b) with 6 as a parameter. From this graph, i t is clear that the region near the emitter contact is much more heavily saturated than the area close to the base con- tact. For example, for values of 6 = 0.5 and y=O.9, a ratio of charge densities of 0.71 is indicated.

An interesting consequence of the nonuniform satura- tion distribution becomes apparent if one considers an n-p-v-n transistor operating as a high-frequency saturat-

CHUDOBIAK: SATURATION OF n-p-v-n POWER TRANSISTORS 849

IC t I

I VSAT VCC VCE

Fig. 7 . Load curves for the elements Q. and Q b (Fig. 6 ) during a turn-off transition.

ing switch with a significant power dissipation during the turn-off transition (such as might occur if a slightly inductive load is present). For the two-element model (Fig. 6) operating as a high-frequency switch, Qa will have a longer turn-off storage time than Q b since the storage time of an n-p-v-n device is defined primarily by the time required to remove the injected minority car- riers from the epitaxial collector layer [ l j ] . Therefore, the turn-off transition for the two elements in the VCE- I C plane may resemble that shon-n in Fig. 7. I t is evident from this illustration that Qa (i.e., the region near the emitter contact) will be preferentially heated. This con- clusion is in agreement with experimental findings previously reported [16], [17].

I t should be noted that the analysis given in this section does not apply directly to transistors having a heavily doped collector since collector conductivity modulation effects are negligible in such devices. I n fact, in the case of an alloy-like transistor, the area adjacent to the base contact is the more heavily saturated since the collector current density is least in that region. Therefore, one would expect the peak junction tempera- ture in a high-frequency saturating alloy switch to occur in the region adjacent to the base contact (rather than near the emitter contact).

111. DYSAMIC SATURATION CHARACTERISTICS A . Analysis

In the previous section it was shown that the domi- nant component of the static collector-emitter satura- tlon voltage of a n-p-v-n transistor may be described by an expression of the form

Equation (22) may be used to describe the dynamic saturation characteristics of a transistor sn-itching a square n-ave collector current if xc is represented as a function of time, A relation between x c and time follon-s from an assumption that the lightly doped

collector region may be represented by a one-lump charge-control model. The equation

dQc Qc i B C ( t ) = - + -

dt 7,

describes the flow of minority carrier charge into the collector from the base. If the assumption is made that i ~ c ( t ) is a step function, the solution to (23) is given by

Referring to Fig. 2 and applying (8) we have

Qc(t) = IBcrc(l - e - l i r c )

qAsc( t ) d p = ___ - &(I)

2 dx - qx2c(t)Ic - -___

4qDn

and so

Therefore, the time dependance of collector-emitter saturation voltage is given by the expression

+ V B E - V B C ~ . (26)

For the purpose of this paper, the term V B E - VBc t

was assumed to be a constant equal to 200 mV. The average value of saturation voltage during the

ON time of the switch may be required for some applica- tions: for example, i t may be necessary to determine the average power dissipated by the transistor switch. Assuming an O N time equal to one-half of the signal period, the average saturation voltage is obtained from (26) as follows:

Jcl.l/o J C

T +-

850

Using the substitution x = e-- t /rc , the above integral may be evaluated and VSAT found to be:

-

- J C W c 2rcJc JTF VSAT = - - - ___ q P n S d q P n y j T d T

+ VBE - VBC~. (27)

VsAT may be expressed as a linear function of J c if the substitution /3 = I c / I B is made:

+ V B E - V B C ' . (28)

For a given O N time, collector-emitter saturation voltage decays to a value which is some fraction of VSAT(0). An expression for a saturation voltage cutoff frequency may be derived if this fraction is represented as a parameter u { 0 <u< 1 1 , and if device ON time is assumed to be one-half of the signal period. From ( 2 6 ) ,

U I C T Y , J c

q P n S d qPnll'd

- --

and therefore,

Solving for the signal period T gives

[ 1

i 1 -- T = 27, In

I n'c'(l - IT)'

I i I 1D,,s,K (f - -L) J ~ F E O

or, in terms of s\\-itching frequency

I n tile follo\\,ing paragraphs it is shoxvn that ( 2 6 ) provides close agreement between theory and practice. Therefore, for the devices and frequencies considered in the paper (sxvitching frequencies of the order of 1 M H z or less), the follolving conclusions may be drawn. First,

IEEE TRANSACTIOYS OW ELECTRON DEVICES, OCTOBER 1970

DEVICE-UNDE

Fig. 8. Test circuit for monitoring the dynamic saturation voltage characteristics of a transistor switching a square wave of collector current. The delay time t~ and VI are adjusted so that the device under test enters saturation after a brief excursion into the active region of operation.

the basic conductivity modulation is valid. Second, the charging of the lightly doped collector region is ade- quately represented by a one-lump model. Third, the current I B C can, in fact, be approximated by a step func- tion. And finally, increases in saturation voltage due to a frequency-dependent crowding mechanism [18], [19], [ 2 0 ] occur at much higher frequencies than those considered in this paper.

B . Results The switching characteristics of a number of triple-

diffused and epitaxial transistors were studied using a test circuit similar to that shown in Fig. 8. Drive wave- forms to the crowbar switch and to the device under test were such that the transistor entered saturation after a very brief excursion ( ~ 0 . 0 5 ps) into the active region of operation.

Calculated and observed saturation curves for a triple-diffused high-voltage transistor switching a collector current of two amperes are given in Fig. 9, where i t can be seen that the results agree within* 10 percent. This close agreement was maintained when the base-drive amplitude \vas varied over a \vide range (5</3<30).

I t is of interest to note that the point at nhich T T B C ,

equals zero is clearly discernible on the observed n-ave- form sholvn in Fig. 9. This suggests a technique which might be used to determine the locus of the curve T'Bc, = 0. For the characteristic curves shown in Fig. 1, i t can 11e seen that there is a \vel1 defined abrupt transi- tion from the second stage of the saturation region to the active region of operation. It is evident that the curve V B C , = 0 can be approximated by a straight line. Hokvever, i t has been found that some devices having lightly doped collectors do not exhibit a well defined

CHUDOBIAK: SATURATION OF n-p-v-n POWER TRANSISTORS 851

4 I 6-

c

s 4-

u)

e

k f

'1 OJ , I b

0 O! 5 1.0 I .!5 t (us1

Fig. 9. Time dependence of saturation voltage for a high-voltage triple-diffused transistor.

t

2 N 3079 IC = 4.5 AMP. IB = 200mA

d5 I .lo I . I5

t ( r s )

(b) rig. 10. (a) VCE-IC characteristics of an epitaxial transistor

active region is poorly defined. VCE =0.5 V/cm, IC =0.2 A/cm, (2x3879) for which the transition from the saturation to the

the epitaxial transistor 2N3879. IE = 5 mA/step. (b) Time dependence of saturation voltage for

second stage (see, for example, Fig. 10(a)).z Clearly, the curve VBc, = 0 is convex with respect to the I C axis, although it is difficult to estimate the locus of the curve. The difficulty can be overcome, however, if VSAT(O) is determined using the circuit of Fig. 8, and if I C , W, and N d are known; A ( I c ) can then be determined with the aid of ( 2 2 ) . This procedure was employed to deter- mine the effective collector area a t 4.5 amperes for the device having the characteristic curves shown in Fig. lO(a). V g a ~ ( t ) was then determined using (26). The results, shown in Fig. lo@), agree well with observed values, and so tend to confirm the validity of the satura- tion model.

IV. CONCLUSIONS The transport equations and charge control concepts

have been applied in the analysis of the static and dynamic conductivity modulation nlechanisms occur- ring in the collector region of n-p-u-n (or p-n-7r-p) power transistors. Analytical expressions have been developed which describe the collector-emitter satura- tion voltage as a function of terminal currents and drive parameters. The analysis was simplified by the assumption that the hole current from the base to collector could be related to the excess base current by an experimentally determined constant factor. This simplification n-as found to be acceptable for purposes of device characterization for circuit application, although it places a limitation on the utility of the analysis i n that it precludes a prediction of device characteristics, given impurity profile and other design variables.

The basic conductivity modulation analysis has also Ileen used to describe the high-current ~ F E character- istics of n-p-v-n transistors as )yell as to illustrate the relationship betLveen the emitter metallization resis- tance, collector charge storage, and a turn-off crowding mechanism experienced by high-frequency saturating transistor sn-itches.

I t is suggested that the approach given in this paper could be extended to describe the turn-off switching time characteristics (i.e., collector charge storage time) of n-p-u-n transistors.

APPENDIX

COLLECTOR WIDTH W According to the analysis presented in this paper, the

second stage of the saturation characteristic curve terminates at the point where V B C , = O (see Fig. 1). The collector-emitter voltage along the locus of the curve V B C ~ = O is given by (16) as

For most devices, the V B C , = O curve at low current

crowding mechanism ultimately dependent on the emitter metalliza- 2 I t has been suggested that this is a result of a current-dependent

tion resistance 171.

852 IEEE TRANSACTIONS ON ELECTRO\- DEVICES, ‘;C=. TE-17, h’0. 10, OCTOBER 1970

densities ( J ~ < 5 0 A/cm2) may be approximated by a straight line. Therefore, W, may be determined from the slope of this curve, provided I V ~ , the effective col- lector area A , and VBE are known quantities (normally, VBE may be assumed to be a constant).

REFERENCES [l] L. -4. Hah!?, “The saturation characteristics of high-voltage

transistors,,, Proc. I E E E , vol. 55, pp. 1384-1388, August 1967. [2] -, .“The effect of collector re;istance upon the high current

capability of n-p-v-n transistors, IEEE Trans. Electron Devices,

[3] J . R. A. Beale and J. A. G. Slatter, The equivalent circuit of a vol. ED-16, pp. 654-656, July 1969.<!

transistor with a lightly-doped collector operating in satura- tion,” Solid-state Electron., vol. 11, pp. 241-252, 1968.

[4] F. Larin, Radiation Effects in Semiconductor Devices. Xew York: IViIey, 1968, pp. 174-177.

[5] L. E. Clark, “Characteristics of two-region saturation phe- nomena,” IEEE Trans. Electron Deoices, vol. ED-16, pp. 113- 116, January 1969.

[6] IY. J . Chudobiak, “On the static collector-emitter saturation voltage of a transistor with a lightly-doped collector,”Proc. IEEE

171 -, Ph.D. dissertation. Carleton C‘niversitv. Ottawa. On- (Letters), vol. 57, pp. 718-720, April 1969.

.~

. , tario; Canada, 1969. ’

[8] J: Hilibrand and R. D. Gold, “Determination of the impurity dlstrlbution in junction diodes from capacitance-voltage mea- surements,” RCA Reo., vol. 21, pp. 245-252, June 1960.

[9] F. Larin, Radiation Effects in Semiconductor Devices. New York: Wiley, 1968, pp, 271-273.

i ; ,~

D. 91. Evans, “The nleasure.nent of ill? temperaturedependence of the mobilit\r and eti‘ectivc: liferime o f minoritv carriers in the base region of‘silicon transistors,“ J . I3lectron. Cintr., Go], 6, pp. 204-207, March 1959, F. J. hlorin and J. 1’. hlajta. “Electrical properties of silicon containing arsenic and boron,‘’ ,Dhys. Ret., vol. 96, pp. 28-35, October 1954.

sistors. New York: \I-iley, 1966, pp , 4243 . R. D. Thornton et al., Characteristics and Limitations of Tran-

R. 1. LVhittier and D. X. Tremere. “Current gain and cutoff fre-

~~ .

quency falloff at high currents,” IEEE Tr&. Electron Devices, vol. ED-16, pp. 39-57, January 1969.

stripes in ‘comb’ structure transistors, IEEE Trans. Electron K. J . S. Cave and J. X. Barnes, “Opt,$num length of emitter

Devices (Correspondence), vol. ED-12, pp. 84-85, February 1965. F. Larin, Radiation Effects in Senziconductor Devices. New York: \Viley, 1968, pp. 80-82. W . J, Chudobiak, “On crowding effects and failure mechanisms in high power transistor switches,” Proc. IEEE (Correspon- dence), vol. 56, pp. 2176-2177, December 1968.

high-frequency transistor switches,” Microelectron. Rel., pp. 75-76, January 1970. R. B. \Yard, “A study of class-C application of power transistors at high and very high frequencies,” Stanford Electronic Lab- oratories, Stanford, Calif., Rept. 713-1, February 1963. J. Seneret, “Utilization des transistors de puissance en \’HF,” Onde Elec., vol. S L V , pp. 311-316, March 1965. S. Krishna, P. J. Kannam, and IV. Doesschate, Jr., “Some limi- tations of the power output capability of VHF transistors,” IEEE Trans. Electron Devices, 1701. ED-15, pp. 855-860, Novem- ber 1968.

- , “Fluorescence thermography and current crowding in

Anomalous Noise Behavior of the Junction-Gate Field-Eff ect Transistor at Low Temperatures

Abstract-Measurements are reported on the noise resistance and the noise conductance of the junction-gate FET in the tempera- ture range 77°K4000K. At low temperatures anomalous noise be- havior has been observed. The measurements are discussed in the light of existing theories and, when necessary, the theoretical model has been extended. The agreement is satisfactory. Generally the extra noise is caused by mobility saturation, increased free-carrier temperature, free-carrier trapping and multiplication effects in the pinched-off region. Finally, several applications are discussed in relation to the limiting noise sources.

I . INTRODUCTION

X C E P T for Ion- frequencies, the channel noise of the field-effect transistor a t room temperature is generally well understood. In well designed

modern units this is the case for frequencies higher than 10 kHz. I t has been shown by van der Ziel [ l ] and other

by the Directorate of Industrial Research, Canada.

opment Laboratories, Ottawa, Ont., Canada.

Manuscript received LIarch 12, 1970. This work was supported

The authors are with the Northern Electric Research and Devel-

authors that the limiting noise has to be attributed to the thermal fluctuations of the channel resistance. This noise source can be represented by an equivalent noise voltage source at the input of the device, which is inversely proportional to the transconductance. A t higher frequencies, the capacitive coupling between channel and gate gives rise to a noise current flowing in the gate circuit. This current is partly correlated to the original channel noise [ 2 ] .

The theory has been checked experimentally by Bruncke and van der Ziel [3] a t moderate frequencies for a large structure unit, and by the author [4] a t very high frequencies for a modern 5-pm channel device.

Besides noise due to the channel resistance, the FET will also show some noise due to the gate leakage cur- rent. Published data and manufacturers specifications are very scarce and, moreover, not up to date.

However, i t is not only at room temperature that a detailed knowledge of the limiting noise is required. Nowadays field-effect transistors find wide application