PROCEEDINGS OF THE IRE Design Theory for Depletion Layer Transistors WOLFGANG W. GARTNERt, MEMBER, IRE Summary-A new class of high-frequency transistors, the De- pletion Layer Transistors (DLT), utilizes maximum attainable carrier velocities in solids by injecting electrons or holes into the high elec- tric fields that prevail in properly designed depletion layers of re- verse-biased p-n junctions. The resulting short transit times should insure operation up to microwave frequencies. The design theory is presented for a particular example of a de- pletion layer transistor, discussing its low- and high-frequency, small-signal behavior, power gain, and stability. Other conceivable structures and modes of operation for DLT's are described and the potential importance of the depletion layer principle for solid-state microwave amplification is emphasized. I. INTRODUCTION C LASSICAL transistors are limited in their fre- quency response by either or both of two factors: the carrier transit time from the emitter to the collector junction, and the rb'CT product. Various ad- vances in technology during the past few years have greatly reduced these two quantities, resulting in corre- spondingly higher operating frequencies. Further im- provements, however, have become increasingly diffi- cult, which led to the search for new principles that would obviate some of the basic limitations inherent in classical transistors. In a high-frequency transistor, the carrier transit time through the structure must be short compared to the reciprocal operating frequency, and no appreciable debunching of carrier pulses must occur during the transport process. The first requirement asks for high carrier velocity; i.e., high electric fields and, similarly, the second requirement may also be satisfied if the carriers travel in strong external fields compared to which the influence of diffusion and the carriers' own space charge is negligible. This concept forms the basic principle of a new class of high-frequency transistors known as Depletion Layer Transistors (DLT), which have recently been announced.'-4 Proper design of these devices will also keep internal RC combinations at very low values and operating frequencies up to the microwave range have been predicted. * Original manuscript received by the IRE, January 11, 1957; revised manuscript received July 1, 1957. A more detailed account of the design theory for depletion layer transistors is contained in USASEL Tech. Memo. No. M-1849; October, 1957. t U. S. Army Signal Eng. Labs., Fort Monmouth, N. J. 1 W. Gartner, "A new uhf transistor structure," paper presented at WESCON Convention, Los Angeles, Calif.; August 21-24, 1956. 2 W. Gartner, F. A. Brand, and W. G. Matthei, "Design theory and exploratory development of the depletion layer transistor," paper presented at the PGED Second Annual Technical Meeting, Wash- ington, D. C.; October, 1956. s W. Gartner, "Transistor physics," paper delivered before the Austrian Chem. and Phys. Soc., Vienna; January, 1957. 4 W. Gartner, "Transit time effects in depletion layer transistors," paper presented at the Symposium on the Role of Solid State Phe- nomena in Electric Circuits, Polytech. Inst., Brooklyn, Brooklyn, N. Y.; April, 1957. II. BASIC PRINCIPLE OF THE DLT Very high electric fields may be created in the depletion layer (DL) of a reverse biased p-n junction, and when carriers are injected directly into this region, their transit time from the emitter to the collector side of the DL is very short (see Fig. 1). In fact, in properly designed depletion layers, carriers will reach the maximum velocity they can attain in solids under any conditions. If a design around this basic principle can be found with small-signal param- eters that will give power gain, a response up to very high frequencies, possibly microwave, should be insured. In the following five sections the design theory for a typical example of a depletion layer transistor is worked out in detail. Other conceivable realizations are dis- cussed in Section VIII. III. STRUCTURAL MODELS, BASIC EQUATIONS, AND SMALL-SIGNAL SOLUTIONS The derivation of the dc anid ac current-voltage re- lationships will be carried out for the geometry and type of operation shown in Fig. 2. The analysis is one dimen- sional. It is expected to describe the essentials of opera- tion although some of the proposed structures (see Fig. 3) may differ from this simple model. Emission into the depletion la-er is assumed to occur from the "virtual emitter plane" indicated by a dashed line in Fig. 2; it may be realized physically by any kind of contact with the general characteristics described below. The current flow through the model structure is indicated in Fig. 4. The collector side has been assumed to be n type so that the active conduction current is carried by electrons which generally have a higher mobility and higher maxi- mum velocity than holes. All of the following deriva- tions are carried out for these doping conditions, and the results may be easily converted to the case of a p-type collector and holes as the active charge carriers. The symbols in Fig. 4 have the following meanings: I'n is the electron current flowing from emitter to col- lector, Ii, denotes the hole current flowing from emitter to base, Iret is the current through the reverse biased p-n junction under zero emitter current conditions. In determining the dependence of these currents on the in- put and output voltages, V1 and V2, we assume that small ac voltages, v1 and v2, are superimposed on the dc voltages, V10 and V20. The customary small-signal series expansion then yields for the emitter conduction current Ileond = Ilno + iln + ilpo + ilp Ilno + ?1n1Vi + ?7n2V2 + Ilp. + fplVl +772p 2V (1) . /n ,(t'v to ",r( 1392
Transcript
PROCEEDINGS OF THE IRE
Design Theory for Depletion Layer TransistorsWOLFGANG W. GARTNERt, MEMBER, IRE
Summary-A new class of high-frequency transistors, the De-pletion Layer Transistors (DLT), utilizes maximum attainable carriervelocities in solids by injecting electrons or holes into the high elec-tric fields that prevail in properly designed depletion layers of re-verse-biased p-n junctions. The resulting short transit times shouldinsure operation up to microwave frequencies.
The design theory is presented for a particular example of a de-pletion layer transistor, discussing its low- and high-frequency,small-signal behavior, power gain, and stability. Other conceivablestructures and modes of operation for DLT's are described and thepotential importance of the depletion layer principle for solid-statemicrowave amplification is emphasized.
I. INTRODUCTIONC LASSICAL transistors are limited in their fre-
quency response by either or both of two factors:the carrier transit time from the emitter to the
collector junction, and the rb'CT product. Various ad-vances in technology during the past few years havegreatly reduced these two quantities, resulting in corre-spondingly higher operating frequencies. Further im-provements, however, have become increasingly diffi-cult, which led to the search for new principles thatwould obviate some of the basic limitations inherent inclassical transistors. In a high-frequency transistor, thecarrier transit time through the structure must be shortcompared to the reciprocal operating frequency, andno appreciable debunching of carrier pulses must occurduring the transport process. The first requirement asksfor high carrier velocity; i.e., high electric fields and,similarly, the second requirement may also be satisfiedif the carriers travel in strong external fields comparedto which the influence of diffusion and the carriers' ownspace charge is negligible. This concept forms the basicprinciple of a new class of high-frequency transistorsknown as Depletion Layer Transistors (DLT), whichhave recently been announced.'-4 Proper design ofthese devices will also keep internal RC combinationsat very low values and operating frequencies up to themicrowave range have been predicted.
* Original manuscript received by the IRE, January 11, 1957;revised manuscript received July 1, 1957. A more detailed account ofthe design theory for depletion layer transistors is contained inUSASEL Tech. Memo. No. M-1849; October, 1957.
t U. S. Army Signal Eng. Labs., Fort Monmouth, N. J.1 W. Gartner, "A new uhf transistor structure," paper presented
at WESCON Convention, Los Angeles, Calif.; August 21-24, 1956.2 W. Gartner, F. A. Brand, and W. G. Matthei, "Design theory
and exploratory development of the depletion layer transistor," paperpresented at the PGED Second Annual Technical Meeting, Wash-ington, D. C.; October, 1956.
s W. Gartner, "Transistor physics," paper delivered before theAustrian Chem. and Phys. Soc., Vienna; January, 1957.
4 W. Gartner, "Transit time effects in depletion layer transistors,"paper presented at the Symposium on the Role of Solid State Phe-nomena in Electric Circuits, Polytech. Inst., Brooklyn, Brooklyn,N. Y.; April, 1957.
II. BASIC PRINCIPLE OF THE DLT
Very high electric fields may be created in the depletionlayer (DL) of a reverse biased p-n junction, and whencarriers are injected directly into this region, their transittime from the emitter to the collector side of the DL is veryshort (see Fig. 1). In fact, in properly designed depletionlayers, carriers will reach the maximum velocity they canattain in solids under any conditions. If a design aroundthis basic principle can be found with small-signal param-eters that will give power gain, a response up to veryhigh frequencies, possibly microwave, should be insured.
In the following five sections the design theory for atypical example of a depletion layer transistor is workedout in detail. Other conceivable realizations are dis-cussed in Section VIII.
III. STRUCTURAL MODELS, BASIC EQUATIONS,AND SMALL-SIGNAL SOLUTIONS
The derivation of the dc anid ac current-voltage re-lationships will be carried out for the geometry and typeof operation shown in Fig. 2. The analysis is one dimen-sional. It is expected to describe the essentials of opera-tion although some of the proposed structures (see Fig.3) may differ from this simple model. Emission into thedepletion la-er is assumed to occur from the "virtualemitter plane" indicated by a dashed line in Fig. 2; itmay be realized physically by any kind of contact withthe general characteristics described below. The currentflow through the model structure is indicated in Fig. 4.The collector side has been assumed to be n type so thatthe active conduction current is carried by electronswhich generally have a higher mobility and higher maxi-mum velocity than holes. All of the following deriva-tions are carried out for these doping conditions, and theresults may be easily converted to the case of a p-typecollector and holes as the active charge carriers.The symbols in Fig. 4 have the following meanings:
I'n is the electron current flowing from emitter to col-lector, Ii, denotes the hole current flowing from emitterto base, Iret is the current through the reverse biasedp-n junction under zero emitter current conditions. Indetermining the dependence of these currents on the in-put and output voltages, V1 and V2, we assume thatsmall ac voltages, v1 and v2, are superimposed on the dcvoltages, V10 and V20. The customary small-signal seriesexpansion then yields for the emitter conduction current
:0.J.P0 0@0 j DE:PLErION LAYER OF REVERSEa 0o°00 BIASEI) JUNCTION
0o0to°%0 °_ (VERY FEW CARRIERS, UNL.ESS
EMITTER r . IN,JECTED; HIGH ELECTRIC FIELDS.)CONTACT
EMITTERBARRIER
BASECONTACT
Fig. 1-Carriers injected directly into the dlepletion- layer of a re-verse biased p-n junction are swept towards tlhe collector at highvelocities.
12
Fig. 2-One-dimensional model for geometrical structure ofdepletion layer transistor.
where
7= (0li/V1)OV2-const; 7i2 = (I1oi,/aV2)V-ozxonstand the other symbols have t:he meanings indicated inthe list of symbols at the end of this paper.Under the assumption that the transi t tirne from
emitter to collector is negligible at the frequencies con-sidered, and that no recombinati.on takes place betweenemitter and collector,6 the collec-tor conduction currentis the sum of the (voltage independent) junction re-verse current, 1,o, and the injected electron current,Ilno +iln,
_I2,cond = Ico + Ilno + fnITVl + 71n2V2. (2)
The input and output conduction-i currents are thus es-sentially determined by the dependence of the emitterinjection current, 'l4ond, on the input and output vrolt-ages, V1 and V2. For a mathematical description of theinjection process and an evaluation of the -i, anothermore specific assumption about the mechanism of car-rier emission into the depletion layer must be made.The simplest possibility-supported by experimental
evidence-is that a barrier (this loose term. is used de-
6 See in this connection C.-T. Sah, R. N. Noyce, and W. Shock-ley, "Carrier generation and recombination in p-n junctionis and p-njunction characteristics," PROC. IRE, vol. 45, pp. 1228-1243, Sep-tember, 1957.
Fig. 4-Current flow through depletion layer transistor withp-type base and n-type collector.
liberately here because considerably more work is re-quired to understand the physics of such contacts) existsbetween the emitter contact and the semiconductor ma-terial immediately underneath the contact, such thatthe emitter conduction current is given by
V,,0 and v. are the dc and ac components, respectively,of the almost undisturbed6 potential in the semic:on-ductor underneath the emitter contact. This means thatthe magnitude of the injected emitter current dependssolely on the potential difference, VB, between theemitter contact and the point in the depletion regionimmediately underneath it. As an example, we rnayconsider an exponential characteristic,' I n( VB) == AO[exp (a VB) -a] from which (dal1/a VB) =aAo exp
(a VBO). The following derivations, however, do not de-pend on the specific nature of this emission function.The potential, V, = V., +v,, underneath the emitter con-
tact is a certain fraction of the output voltage,6 G. L. Pearson, W. T. Read, W. Shockley, "Probing the space
charge layer in a p-n junction," Phys. Rev., vol. 85, p. 1055; March15, 1952.
1957 J3.9 ^
+ [(aIll/aVB) + (aIl1/OVB)]VB
PROCEEDINGS OF THE IRE
Vso = Ko(X) X V20 and v8 = Kl(X, V'0) X Z2, (4)
the -nagnitude of which is determined by the impurityclistribution across the junction, the position, x, of thleemitter contact, and the applied dc bias.
In particular, the ac collector voltage feedback ratio,Kl, is equal to (i/'/ V2)v20, where 46(x) denotes the elec-trostatic potential across the depletion layer which maybe calculated if the impurity distribution and applied Ucvoltage are known. Gain considerations, which will bediscussed later, indicate that the voltage feedback ratio,KI, should be as small as possible and straightforwardcalculations on step junctions, graded junctions, prn,pvn, and p-i-n junctions show that Ki becomes smallerthe closer the emitter contact can be placed to the baseend of the depletion layer (and still be inside).
If we now substitute back into the original (1) and (2),we obtain for the conduction currents
In addition to these conduction currents, displace-ment currents will flow through the structure. Betweenany two contacts there will exist a capacitance C(V)which, in general, depends on the applied voltage,V= Vo+V, but which may be assumed to be constant ina small-signal tlheory. The permissible values for thecapacitances are determined by the desired gain andfrequency response which will be discussed later. Inview of the many conceivable geometries, the actualvalues of these capacitances shall not be calculated here,and we add a current jCCec(vi-V2) +JcoCebV1 to the emit-ter conduction current and j&C4(V2 -v1) +jcoCb,v2 to thecollector conduction current where Ce, Ceb, and Cb, arethe (ac-voltage independent) capacitances betweenemitter and collector, emitter and base, and base andcollector leads, respectively.Thus with (5) and (6) we find for the small-signal,
four-pole admittances
Y-i aIl/aVB + jCO(Ceb + Cec),
Y12 = - Kl(OIll/VB) - jWCec,
(7a)
(7b)
Y21 = - aI0/E3VB -jWC,, (7c)
Y22 = Kl(aI1/daVB) + jcw(C,b + Ce,c) (7d)
An equivalent circuit for the "intrinsic" device as char-acterized by (7) is shown in Fig. 5. The emitter and thecollector branch each contain a current source as indi-cated.The currents through the device will be somewhat
modified by the impedances of the bulk material on
both sides of the junction, rb (base series impedance),and r, (collector series impedance) and by the leakageconductance, g,, across the junction. These, however,
Cec
9 I =3i)I / Z)VB gilln= 9311fin?/BFig. 5-Equivalent circuit for "intrinsic" DLT in common
base configuration.
may easily be taken into account by simple four-poletransformations (a situation similar to the inclusion ofthe base spreading resistance in ordinary transistors).For optimum performance, all these quantities should bekept small.
Eq. (7) shows that the magnitude of the small-signal,four-pole admittance depends on the capacitances be-tween the contacts, on the nature of the emission func-tion (for which an exponential behavior was given aboveas an example), on the emitter efficiency; i.e., on thefraction of total emitter current carried by electrons,and finally on the ac voltage feedback ratio, Ki. In par-ticular, the output admittance, Y22 is by a factor Km (fornegligible capacitance effects) smaller than the inputadmittance (since Ki < 1 always), and the ratio betweenthem is the source of ac power gain in this devicewhich, without carrier multiplication, has a currentgain of not more than unity.The exact dependence of power gain on these various
quantities is analyzed in Section VII.
IV. CALCULATION OF TRANSIT TIME
To determine whether the assumption of negligibletransit time, r, from emitter to collector is justified atthe frequency of operation or whether a more elaboratetheory should be used, a calculation of the transit time
fc dxr = U
J e u(8)
must be carried out, where xe and xc are the coordinatesof the emitter and collector, respectively, and u(x) isthe carrier velocity the field dependence of which hasbeen investigated by Ryder, et al.7 At low fields thevelocity is proportional to the electric field, at higherfields it is proportional to only E"l2, and finally it reaches
7E. J. Ryder and W. Shockley, "Mobilities of electrons in highelectric fields," Phys. Rev., vol. 81, p. 139; January, 1951.
W. Shockley, "Hot electrons in germanium and Ohm's law," BellSys. Tech. J., vol. 30, p. 990; October, 1951.
E. J. Ryder, "Mobility of holes and electrons in high electricfields," Phys. Rev., vol. 90, p. 766; June, 1953.
E. M. Conwell, "High field mobility in germanium with impurityscattering dominant," Phys. Rev., vol. 90, p. 764; June, 1953, and"Mobility in high electric fields," Phys. Rev., vol. P,8 p. 1379;December, 1952.
1394 October
Gdartner: JIesign Theory for Depletion Layer Transistors
EL maximum constant (field independent') value which ise.g., approximately 6X 106 cm seconds-' for electrons ingermanium, and 8X 106 cm seconds-' for electrons insilicon at room temperature. Using the published data,7(8) may be easily integrated. From the impurity dis-tribution across the junction, the electric field is firstcalculated as a function of distance across the depletionlayer, which is then divided into regions where the car-rier velocity varies like E, E"I2 or is field-independent,and integral (8) is evaluated in each of these regions,the extension of which depends on the material used, onthe impurity distribution, and on whether the activecarriers are holes or electrons.To obtain the desired short transit times, high fields
are advantageous. They should, however, lie below thecritical voltage for avalanche breakdown8 which variesfor different materials. Utilizing controlled carriermultiplication, on the other hand, may be a desirablemode of operation for this device (see Section V).The most advantageous case is obviously the one
where the carriers have their maximum velocitythroughout the entire depletion layer. This can be ac-complished by sandwiching a lightly doped region be-tween two heavily doped regions of different conductiv-ity type and applying such a voltage that the (nearlyconstant) field throughout the middle layer lies abovethe critical field for constant carrier velocity and belowthe avalanche breakdown threshold. Under these con-ditions an electron in germanium; e.g., will traverse adistance of 10-3 cm which may be a typical value for thedepletion layer thickness in 1.7 X 10-" seconds.
V. CARRIER MULTIPLICATIONIf carrier multiplication occurs in the depletion layer,
the current amplification will exceed unity and, there-fore, this condition is of considerable interest. Here weshall analyze the case where the injected carriers aremultiplied by a factor m in the region between emitterand collector.9 This can be achieved by designing theimpurity distribution around the doping crossover insuch a way that the electric field exceeds the critical
I K. G. McKay, and K. B. McAfee, "Electron multiplication insilicon and germanium," Phys. Rev., vol. 91, p. 1079; September 1,1953.
K. G. McKay, "Avalanche breakdown in silicon," Phys. Rev., vol.94, p. 877; May 15, 1954.
P. A. Wolff, "Electron multiplication in silicon and germanium,"Phys. Rev., vol. 95, p. 1415; September 15, 1954.
S. L. Miller, "Avalanche breakdown in germanium," Phys. Rev.,vol. 99, p. 1234; August 15, 1955.
E. Groschwitz, "Zur stossionisation in silizium und germanium,"Z. Phys., vol. 143, p. 632; January, 1956.
Compare also in this connection A. G. Chynoweth and K. G.McKay, "Internal field emission in silicon p-n junctions," Phys. Rev.,vol. 106, p. 418; May 1, 1957.
9 At the AIEE-IRE Semiconductor Devices Res. Conf., PurdueUniv., Lafayette, Ind., June 25 to 27, 1956, H. Statz and R. A. Puceldiscussed a very high-frequency device which utilizes avalancheeffects in the depletion region.
The use of controlled avalanche in conventional transistors hasbeen described by S. L. Miller and J. J. Ebers, "Alloyed junctionavalanche transistors," Bell Sys. Tech. J., vol. 34, p. 883; September,1955, where additional references on carrier multiplication in semi-conductors may be found.
'fIln+ I ip
-(M-I)Iin
Fig. 6-Distribution of active conduction currents for case ofcarrier multiplication in emitter-collector region.
value for multiplication.8 The conduction current flArunder these circumstances is shown in Fig. 6 under theassumption that due to proper geometry or magneticfields'0 none or only a few of the internally createdclcarriers are captured by the emitter contact. Then theemitter current will be unchanged and the collectorconduction current is multiplied by the factor m. Thenew dc collector current is then given by
-I20 = Icom + mI1no ('9t)
and the new ac four-pole parameters are of the following0form:
yii = &I1/&VB + jW(Ceb + Cec).
Y12 = - Kl(lIa/OVB) - j@Cec.
Y21 = -m(aIl,/VB) -jCOece
Y22 = mKl(31aI/O3VB) + jCO(Ccb + Cec)
- (Ico + Il,,,)(dm/3IV2).
(iGOa)
(1.01)
(tOsc.)
(1.0d)
The last term in (lOd) describes the modulation of thedc current through the structure by variations of themultiplication ratio as a function of output voltage.This effect reduces the output impedance and shoul(d bekept low. The details of designing a localized multiplica-tion region will be the subject of a later publication.From (10) it is apparent that multiplication will in-
crease the output admittance, possibly above the inputadmittance (depending on K, and m), and the mechanisnmof power gain in this case is by current amplificationrather than through the admittance ratio.
VI. TRANSIT TIME EFFECTSOf particular interest in any high-frequency device is
its upper limit of operating frequency which the abovesimple low-frequency design theory is unable to predict.We shall, therefore, now discuss the case in which thetransit time is no longer negligible as compared to thereciprocal frequency. A phase difference between emit-ter and collector conduction currents will then occur,
10 Suggestion of F. A. Brand of Signal Corp. Eng. Labs., FortMonmouth, N. J.
0. C7 >. '7 M*;,
-q .12n =-ml in
'r'%7-0 /-N 7Jj.J.Ja,JdJ1~~~ JG Ku I, i ~-'
aLrLd several or many cycles of conduction clirreait rnabe in transit through the structure simultaneously. T]lecLdvantage of this type of operation-if it resulted inpowxer gain would evidently be that for a giveni fre-quency the width of the depletion layer couLld be largerthani for the case in which the transit tiine must be
negligible. This would result in greater ease of construc-tioni and reduction in capacitance, both of which factorswould contribute to a higher ultimate operating fre-quency for DLT's.A derivation of the small-signal admittances under
these conditions, carried out in Appendix I, leads to
Yii = gln/StW1 gin
Y12 :--WI + W2 P
gln EA gln4Y21= - --
W1+ W2 (P
W, gln . (A 1Y22= -w
Wl + W2 (P Wl + W2 (P
where
W22 e-"OW2/u 1 + jwW2/U=
EAEAU (CW2/2
WI
-~ 10'3
I-
.a 03
I-
3
a'110
(1la)( lb)
(IIc)
(lId)
(1 2a)
= 1 + g'91. (12b)Wl + W2
Terms resulting from the influence of parasitics men-
tioned in earlier sections (capacitances, series, and shuntimpedances) may be added to these characteristics ofthe "intrinsic" transistor.The influence of transit time is described by the
function (12a) while 'p is practically equal to unity inmost cases. possesses a strong frequency dependenceas shown in Fig. 7 where (e-Twi -1 +jWTr)/(Cor)2 withr=w2/u has been plotted vs Cor. As will be seen in Sec-tion VII, it causes the power gain to decrease rapidly as
soon as the frequency becomes comparable to the re-
ciprocal transit time," an effect quite analogous to thetransit time phenomena in electron tubes. Unless multi-plication or another scheme of ac conduction currentamplification is utilized, the structures must be de-signed to keep the transit time below one period of theparticular frequency.
VII. GAIN AND STABILITY OF THE DLTThe investigation of the gain question for the DLT is
complicated by the fact that the "intrinsic" DLT as
characterized by (7) is potentially unstable'2 at all ex-
"' The author owes an illuminating discussion on this subject toH. Kromer of RCA.
12 For a discussion of the problems of maximum available gainand stability in an active four-pole, see J. G. Linvill, "The relation-ship of transistor parameters to amplifier performance," paper pre-sented at the IRE-AIEE, University of Pennsylvania Conference onTransistor Circuits, Philadelphia, Pa., February 17, 1955; for furtherreferences see R. L. Pritchard, "Measurement considerations in high-frequency power gain of junction transistors," PROC. IRE, vol. 44,pp. 1050-1051; August, 1956.
w r
Fig. 7-(e-i'-i +jWr)/(wr)2 plotted as a function of 'T.
cept very high frequencies. Parasitic influences such as
mentioned in Section III will reduce the power gain andmay even make the device unconditionally stable. Thepertinent derivations are straightforward but lengthyand shall not be given here. Rather, we shall first dis-cuss the gain of the "intrinsic" DLT and then investi-gate its stability behavior. To characterize power gain,we use the ratio of the power, NL, delivered into a
matched load, YL, over the power, Ni, delivered intothe input of a transistor:
The asterisk denotes the complex conjugate. If we firstconsider the case of negligible transit time, we find from(7)
NL gln2 + w2Cec2
Ni 2Klglgln + 2W2Cec2(14)
where gi =01179 VB and g1n=OlIn/0 VB. This expressionpermits us to draw conclusions as to the design whichwill give high power gain: the emitter-collector capaci-tance, C6C, shall be as small as possible; Ki, the ac voltagefeedback ratio shall be as small as possible [at low fre-frequencies the power gain is actually given by 1/(2Kl)for unit emitter efficiency]; the ac emitter efficiencyshall be as close to one as possible so that glngl-
CCc is kept low by small area contacts and wide sepa-ration of emitter and collector; values in the range of a
few tenths of a ,utqf as they are known from microwavediodes should be realizable. Ki, as pointed out earlier,must be kept low by placing the effective emitter con-
tact13 very close to the base end of the depletion layer,at least within 1/10 of the total depletion layer thick-
13 The "effective" emitter contact may be the small forwardbiased portion of a larger contact, the rest of which is reverse biasedwith respect to the underlying depletion layer with its steep potentialgradient.
',t~ci (7s:-1 r, /,
Gdrtner: Design Theory for Depletion Layer Transistors
ness. Emitter contacts with essentially unit efficiencycan be made.2" 4 Thus for a depletion layer thickness of3 X 10-3 cm in germanium and an optimum transit time(p-i-n junction) of 5 X 10-10 second, a Ki of 0.05 (loca-tion of effective emitter contact within 1.5 X 10-4 cm
of base end of DL), an emitter-collector capacitance,CCC of 0.1 ,if, and a value of g, of 10-2 mho, one findsfrom (14) a power gain of 10 db at 1000 mc/seconds.With the advent of more refined techniques which al-low a closer control of small dimensions, an improve-ment in the maximum frequency of DLT's with negli-gible transit time may be expected. Similarly, if othersemiconductors are found which have a higher maxi-mum carrier velocity (this property has so far only beeninvestigated in germanium and silicon) higher frequencycutoff will result.
If we consider the case of controlled carrier multipli-cation, we obtain with (10)
NL m2gln2 +co2Cec2 (15)
Ni 2K,lnglgln + 2co2Cec2 - 4g,(Ico + I,no)(Om/lV2)Neglecting capacitances and the last term in the de-nominator, we find that the power gain will increase bya factor of m. In addition, because of the reduction inoutput impedance, the capacitances become less critical.Similarly, and maybe most important at the presenttime, K1; i.e., the position of the emitter contact, is no
longer the major gain-determining factor. In otherwords, the emitter contact may be anywhere in the de-pletion layer as long as only few carriers return to it(which must be avoided by proper structures or mag-netic fields'0) and a power gain of at least a value of mmay be realized as long as capacitances and the ac modu-lation of the dc current through the junction are negli-gible. With these relaxed requirements on geometricalcontrol in the structure, substantial power gains in the10,000 mc range appear feasible because it is possiblewith existing techniques to locate a contact somewhereinside a 1,u depletion layer which possesses a transit timeof the order of 2 X 10-1" seconds under optimized con-
ditions (p-i-n junction). The upper limit on the per-missible multiplication factor is imposed by the condi-tion that the space charge of the mobile carriers mustremain negligible in the depletion layer. Internal RCcombinations which also limit the frequency responseof classical transistors can be kept negligibly small byreducing the series resistance of the bulk material on
either side of the depletion layer.We have thus shown that on the basis of designs with
negligible transit time, power gain up to the microwaveregion is possible. Other potential applications of thedevice include uhf and microwave oscillators and ex-
tremely fast switches. We now want to investigate thefrequency dependence of power gain when the transit
14 W. G. Matthei and F. A. Brand, "On the injection of carriersinto a depletion layer,' J. AppZ. Phys., vol. 28, p. 513; April, 1957.
.7-
a
.6
- .5 -
3 .4 -
4,\.3-
.2
1 2 3 4 5 6 7 8 9 10 II 12_ T
Fig. 8-The function (6-i07..- 1)/(jwr) 12 plotted vs or.
time becomes comparable to one period. Substituting(11) into (13), one obtains under the assumption of(o21 and W2-.-w
NL gl 21 (e-ior - 1)/(jcor) 12
Ni (WI/W)gln2[4 - 2(sin wr)/l(r)](16)
Eq. (16) shows that even when the capacitances may beneglected, the power gain drops off with frequency as
(e-ir- 1)/(jcor) I which is shown in Fig. 8. It thus doesnot appear feasible to raise the frequency to muchabove twice the reciprocal transit time unless very highgain is available at lower frequencies which may besacrificed in this cutoff tail and still give sufficient am-plification at frequencies higher than the cutoff fre-quency.
The question of stability of a DLT may be analyzedwith the help of Fig. 9. It shows the stable and unstableregions for the external admittances, Ys and YL, of an
active four-pole, and has been derived in Appendix II.
If the external conductances are chosen to lie in the area
where the two regions which are bounded by hyperbolasand straight lines, respectively, overlap, then oscilla-tions without external feedback will result. Outside thisarea the device is stable. If the two regions do not over-
lap, which occurs for
2 Re (y,l) Re (y22) - Re (yl2y21) 1 < y12y21 (17)
the device is unconditionally stable. If we now applythese considerations to the 'intrinsic" DLT with negli-gible transit time, (7), we find that condition (17) isnever satisfied; i.e., that the two regions in Fig. 9 willalways overlap. To determine the external admittancesfor which the DLT will oscillate at a certain frequencywithout external feedback, one thus draws a diagramlike Fig. 9 for the particular frequency, and selects a pairof values for the real parts of the admittances from thedoubly shaded region, and then calculates the corre-
sponding imaginary parts from (35) and (36) in Ap-pendix II. If stable amplifier operation is desired, oneselects a pair of admittances outside the doubly shaded
Fig. 9-Stable and unstable regions for external admittancesbetween terminals of active four-pole.
region. To investigate the stability behavior over a fre-quency band, one would have to construct a three-dimensional scheme with the frequency as the third co-ordinate.The same remarks essentially hold for the multiplica-
tion case, at least as long as the last term in (lOd) isnegligible. An investigation of the stability behaviorbased on the frequency dependent four-pole admittancesof (11) leads to a complicated transcendental equationfrom which it is fairly easy to determine whether thedevice will oscillate at a given frequency. The calcula-tion of the exact cutoff frequency, however, is difficultand would exceed the scope of this discussion.With these remarks, we want to close the section on
gain and stability because this paper deals mainly withthe design theory of the device and niot its circuitry.The main object of Section VI I was to establish that thedevice described is indeed an active four-pole capable ofpower gain.
VIII. OTHER DLT DESIGNSIn this section a number of DLT-type structures are
mentioned, which appear feasible and illustrate the pre-diction that designs around the principle of carriertransport through high field depletion layers will playa similar role in very high-frequency solid-state amplifi-cation, as structures based on carrier diffusion do atlower frequencies. No mathematical analysis of theseproposed devices shall be given in this paper.
It has been shown that for a regular DLT with nocarrier multiplication the ac current amplification dropsoff rapidly when the transit time is comparable to oneperiod. Higher frequency response thus requires nar-rower depletion layers. It has also been shown that theproblem of properly locating the emitter contact in thenarrow DL to keep the ac voltage feedback small maybe circumvented by boosting the current amplificationthrough controlled carrier multiplication.
A similar principle may be applied to a structurewhere the transit time is equal to several periods of theoperating frequency. If, namely, a narrow region nearthe collector end of the depletion layer contains multi-plying fields which offset the frequency-determined re-duction in current amplification, then operation beyondthe natural cutoff frequency appears feasible. The multi-plied carriers of opposite sign must be prevented fromreturning to the emitter contact; this can be accom-plished in several different ways: one may consider theamplification of pulses where the emitter is forward bi-ased only during the duration of the pulse and reversebiased at all other times so that it will not collect carriersof the opposite sign. This device will show a certain de-lay between signal pulse and amplified pulse due to thetransit time from the emitter contact to the multiplyingregion which, in itself, may find an application. Whencw operation is desired, a magnetic field may be used todivert the returning carriers from the emitter contact.This type of operation resembles the "dissected ampli-fier" idea with the gyrating element built into the deviceclose to the amplifying region.'5
Another interesting modification of the basic DLTprinciple would be a device where the carriers are gen-erated inside the DL by modulated light or microwaveenergy, or where the multiplication factor m is modu-lated by the microwave field parallel or perpendicularto the junction. In devices like this, the transition isfound from amplification by principles at least vaguelyrelated to transistor operation and amplification utiliz-ing other solid-state phenomena.
It may also prove to be of great interest to investigatethe properties of depletion layers at low temperatureswhere the maximum carrier velocities7 and the meanfree path of the carriers increase.
IX. CONCLUSIONThe experimental phase of the development of the
device is well underway and has been partially re-ported.2'14 In particular, it has been shown that the in-jection into the depletion layer occurs as assumed in thetheory and that the low-frequency gain depends on thevoltage feedback ratio, Ki, in the predicted manner.When more complete information is available, the de-sign theory may be refined and generalized in many ob-vious ways. At that time also the noise problem may beattacked.
APPENDIX I
CALCULATION OF THE FOUR-POLE PARAMETERS OF A
DLT UNDER TRANSIT TIME CONDITIONS
In the first paper on the design theory of the proposeddevices, the effects of transit time will be analyzed un-der the simplest assumptions. The many conceivable
16 W. Shockley and W. P. Mason, "Dissected amplifiers usingnegative resistance," J. Appi. Phys., vol. 25, p. 677; May, 1954.
1398 October
1 9artner: vesign i neory jor vepterT1on -ix,yer i ruwn-si,-irs 1399
generalizations shall be carried out when more experi-mental information is available. We thus assume: 1) Allac quantities, a, may be expressed by a=ao exp (jcwt)where ao is complex; 2) only electrons are injected whichcarry the entire ac conduction current, in(x, t); 3) thevirtual emitter has no other electrical influence than theemission of carriers; 4) the carriers move with a high,field-independent, constant velocity, u, and no "de-bunching" of the carrier waveform occurs due to eitherdiffusion or the electron space charge; 5) the analysisapplies to a p-i-n structure with highly doped p and nregions. Such a junction is well simulated by a planeparallel capacitor. The width, w, of the depletion layeris then assumed independent of ac voltage.The derivation is based on the geometry of Fig. 10.
The injected electron current depends on the voltageVB= V1- VS, where V. is the potential inside the DLunderneath the emitter junction. V1 is given by
v.=f E(x, t)dx = E(-wl, t) X w, (18)-Wl
or
rW2Vs = V2 - E(x, t)dx. (19)
In the region -w1<x<0, E is independent of distance,E=E(-w1, t), because of the absence of any spacecharge, whereas in the range of O<x<w2 it is equal to
rxE(x, t) = E(-wl, t) + (1/E) fp(x, t)dx. (20)
As we are mainly interested in the linearized ac para-meters (the dc relationships are identical with those ofSection III), we separate the small ac part from the dcpart and find for (18) and (19)
VS = EAC(-W, t) X W1 (21)
and
Vs = V2 - EAC(-W1, t) X W2 + (e-jWW2/u-1)E-A W2
IlnW2+ eAco (22)
where (Au)-1Xin(x, t) = (Au)-'Xi1n(t-x/u) has beensubstituted for the ac variations of the space charge,PAC. Substituting EA from (21) into (22) thus yields
Vs = [Wl/(Wl + W2)](V2 + iln.k) (23)
where
W22 e-i-12/U-I + jWw2/u=
=EAu (WWw2/u)2
With (23) and
iln = (aIl/;aVB)VB = glnVB
(24)
(25)
b o
x =-WI
ICI.E(x t)
_ 'Vs _ -_-I X=o X= W2 X
le VIRTUAL EMITTER PLANE
Fig. 10-Geometry for the analysis of transit timeeffects in the DLT.
one finds
gln Wl glniln = -Vl - --2
9 Wi+ W2 9(26)
with
Wip = 1 + gns.
Wi + W2(27)
Since the conduction current is delayed by W2/U on itsway from emitter to collector, one obtains for the accollector electron current
According to (20), iln is the total emitter current. Thetotal collector current, however, is the sum of i2n andthe displacement current
-i2,disp = EA a9E(W21)at
,eA gln¢o= 1i0 i V+ 1±-
W+W2-1- iW2_
+ iin(l - e-ilOW21u).
wl glno--V2]Wl + W2 (P
-(29
(29)The first term in (29) is the capacitive current betweenbase and collector, the second compensates for the phaseshift in conduction current.From (26), (28), and (29) it is thus possible to calcu-
late the small-signal, four-pole admittances yik:
Yii = gln/S°
WI gln'V12 = -
Wi + W2 9O
gln . EA gln4y21 = - - -_ J
9 Wl+ W2 9
W1 gln . EA 1Y22=---1 - .
Wl + W2 (P Wl + W2 9
(30a)
(30b)
(30c)
(30d)
lY5/
PROCEEDINGS OF THE IRE
APPENDIX I I
CALCULATION OF THE CONDITIONS FOR UNCONDITIONALSTABILITY OR SELF-EXCITATION OF AN ACTIVE
FOUR-POLE
An active four-pole, the input of which is passivelyterminated by the admittance, Ys, and the output ofwhich is passively terminated by an admittance, YL,will oscillate if the condition'6
and
2s,l, = a - (a2 + b2)"'2= Re (y12y21) - Y12Y2 1 (39)
which is shown in Fig. 9. We now transform back ac-cording to (34)
Re (Ys) = s- Re (y,l)Re ( YL) = 11 - Re (y22)
(40)(41)
is satisfied.tionships:
y'm + Ye Y12
A =
Y21 Y22 + YL
Eq. (31) is equivalent to the following
Slll- s212 = a
and
Sl12 + S211 = b
(31)
rela-
(32)
(33)where
S1 = Re (y11 + Ys), S2 = Im (Yll + YS)11 = Re (y22 + YL), 12 = Im (Y22 + YL)a = Re (Y12Y21), b = Im (Y12Y21). (34)
Eqs. (32) and (33) permit us to determine the values ofYS, YL for which the four-pole will oscillate at a givenfrequency. For this purpose, we derive a solution of(32) and (33) for which s,, S2, 1,, 12 are all real andRe (Ys)>0, Re (YL)>O. We find
S2 = {b ± [b2 - 4(s,21l2 - as,lm)]112}/(211) (35)
and
12 = {b T [b2 -4(s,21l2 - aslj1)]112}/(2sj) (36)
S2 and 12 may be positive or negative, but must be realwhich leads to the condition
s1212- asil, - b2/4 < 0. (37)
Eq. (37) states that the permissible values on the s, -11plane lie within the area bounded by the two hyperbolas
2s,l, = a + (a2 + b2)1/2- Re (y12y21) + | Y12Y21 (38)
16 See, e.g., H. W. Konig, "Laufzeittheorie der Elektronenrohren,"Springer Verlag, Vienna; 1948. The discussion in the present paperis an adaptation of the methods developed by K6nig which also con.tain stability criterion of Linvill (op. cit.) in implicit form.
which shifts the coordinate axes as indicated in Fig. 9.Since Re (Ys) and Re (YL) must be positive, only thosevalues of Re (YL) and Re (Ws) will give oscillationswhich lie in the region where the positive quadrant ofthe Re ( Ys) -Re (YL) plane overlaps the area boundedby the hyperbolas (see Fig. 9). Outside this region, thefour-pole is stable. If the positive quadrant does not in-tersect the hyperbolas at all, which occurs for
2 Re (yi1) Re (y22) -Re (y12y21) < Y12Y21 |, (42)
the device is unconditionally stable.
LIST OF SYMBOLS OTHER THAN DEFINEDIN THE TEXT
A conducting cross section of transistor.EAC= ac part of electric field strength.Icom= dc reverse current under multiplication condi-
tions.'mno = dc emitter electron current.IPdc emitter hole current.ii= ac emitter electron current.ii, =ac emitter hole current.J-=V_.
N. =acceptor concentration.Nd=donor concentration.q =electron charge.w=w,+w2=total width of depletion layer.e =dielectric constant.p =electric charge density.
Pac=ac part of electric charge density.r = transit time from emitter to collector.w =circular frequency.
ACKNOWLEDGMENT
The author wishes to thank F. A. Brand and W. G.Matthei of the U. S. Army Signal Engineering Labora-tories for many discussions in connection with their ex-perimental work on the device.
