Characteristics of MOS Transistors

8/12/2019 Characteristics of MOS Transistors

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324

Characteristics of theMetal-Oxide-Semiconductor Transistorsc. T. SAH, MEMBER, IEEE

Summary-The theory of the MOS transistor in thegradualchannel approximation is presented with the assumption of constantsurface and bulk charge, and constant surface mobility. From thesimple theory, the complete design equations ar e derived and designcurvesare calculated. From the analysis, the equivalent circuitparameters of the device are related to the basic properties of thematerial and geometry of the device. The simple theory is thencritically compared with experimental measurements of MOStransistors with circular geometry. The comparison shows goodgeneral agreement withhe theory of the dc characteristics but dis-crepancies are found fo r the differential characteristics such asthe transconductance and the gatecapacitance. T he possible sourcesof the discrepancies a re discus sed.

I. INTRODUCTIONHE FIRST attempts t o realize active solid statedevices operating on the fieldffectrincipleappear to have been undertaken by J. E. Lilienfeld

of New York which were disclosed in a series of threepatents issued in 1930, 1932 and 1935. Anotherindependent effortwasalso undertaken at about histime by Oskar Heil in Germany which was recorded ina British patent issued in 1935.1dStudies of the fieldeffectundertaken at he Bell Telephone Laboratoriesafter the war under the direction of Shockley2 ed tothe discovery of the transistor effect by Bardeen andBrattain2b in 1948. The effect was first observed duringa series of experiments in which they ttempted omodulate the current flowing througha point contactmade on germanium by controlling the electric fieldat the germanium surface near the contact by the useof an electrode in close proximity to hecontact.2(a)This discovery led o the development of the point contacttransistors2(b)and subsequently the invention of the

Manuscript received January 31, 1964; revised March 24, 1964.The uthor was formerly with he Fairchild SemiconductorResearch and Development Laboratory. He is now with the Uni-versity of Illinois, Urbana, Ill.I(a) ,U,.S. Patent 1745175, filed October 8,1926; granted Januaryn o n

(b) U. S. Pa te nt 1877140, filed December 8, 1928; granted Sep-Ad, I Y S V .tember 13. 1932.(c) U. S. Patent 1900018, filed March 28, 1928; granted March7, 1933. Th e structures proposed in these three paten ts wererecently misinterpreted b y V. E. Bottom in Invention ofthe solid-state amplifier, Physics Today, vol. 24, pp. 24-26;February, 1964. It is pointed out by J. B. Johnson thatthese structures of Lilienfeld are indeed field effect devicesrather than bipolar transistors of Bardeenza), Brattainz (b)amplifier, to be published in P hys i cs Today .and Shockley3. See J. B. Johnson, More on the solid state(d ) British Pa tent 439457, filed March 4, 1935; granted Decem-

(a) J. Bardeen, Semiconductor research leading to the pointber 6, 1935.contact transistors, 1956 Nobel Lecture, published in(b) 9.Bardeen and W. EL Brattain, The ransistor: a semi-Science, vol. 126, pp. 105-112; 1957.

also U. S . Patent 2524035, October, 1950.conductor triode, Phys. Rev., vol. 74,p.230; July, 1948;

bipolar junction t ra ns i~ to rs ,~ll of which operate on theprinciple of minority injection rather than fieldeffect.At this time Pearson alzd Shockley had experimentedwith the fieldeffect in which modulation of the con-ductivity of a thin semiconductor film was achieved byan electric field applied perpendicular to the ilm ~ ur fa ceConsiderable smaller conductivity modulation was foundthan expected, andheynterpreted the discrepancyas due t o a high concentration of surface states at thesemiconductor surface, basedon a modelproposed byB a r d e e ~ ~he device geometry in he experiments ofPearson and Shockley is similar t o those proposed byLilienfelds and Heild which are shown im-Fig. l (a )and (b). Although these initialand the subsequent at-tempts to acheive a solid-state amplifier by modulatingthe electric field at the semiconductor surface were nottoo successful, this method of conductivity modulationhas since been employed to the study of the electronicproperties of the semiconductor surfaces,which aregenerallyknown as he fieldeffectexperiments, bynumerous workers. An alternative approach to an activesolid-state field effect device was conceived by Shockley7and later successfully built in which a reverse biased p njunction is used as the field effect electrode. The pointcontact and the bipolar junction transistors have domi-nated the development and the progress of solid-stateelectronicssince their invention. In the last five years,the semiconductor technology has been refined to a stateso that volume production of the junction gate fieldeffect transistorhas alsobecomecommercially easible.The obstacles which prevent the progress towards apractical surface controlled active fieldeffect transistorseem to lie mainly in the lack of the controllability andstability of the surface although the difficulty of ex-tremely high concentration of surface states (of theorder of 1015/cmzor one monolayer), whichexisted inPearson and Shockleys experiment14does not seem t oappear for an oxidized emiconductor surface. Thesefabrication problems are till ot completelyolved,

a W. Shockley, Th e theory of p-n junctions in semiconductorsand p - n junction transistors, B. S. T . J., vol.28, pp. 435-489,July, 1949; also U. S. Patent 2569347, September 25, 1951.W. Shockley and G. L. Pearson, [Modulation of conductanceof thin fdms of semiconductors by surfacecharges, Phus. Rev.,conductor contact, Phys. Rev., vol. 71, pp. 717-727; May, 1947.6 J. Bardeen, Surface states and rectification at a metal semi-mamum, Bell Sys. Tech. J., vol. 32,pp. 1-41; January, 1953.W. H. Brattain and J. Bardeen, Surface properties of ger-See also R.H. Kingston, Ed., Semiconductor Surface Physics,University of Pennsylvania Press, Philadelphia, Pa.; 1957.vol. 40, p. 1365-1376; November, 1952.1 W. Shockley, A unipolar field-effect transistor, PROC.RE,

V O ~ .74, pp. 232-233; July, 1948.


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1364 Sah:haracteristics of MOS TransistorsMarch 7.1933. J.E. LILIENFELD 1.900.018

DEVICE FOR CONTROLLIN0 ELECTRIC CURRENTP i l e d Marcn 28. 1928 3 ShseLs-Sbeet 1

4 5Fig. 1-The field effect transistor structures. ( a ) The metal-oxide-semiconductorransistor proposed by Lilienfeldl(G). (b)Themetal-oxide semiconductor thin film transistor roposed byHeil*(d).

however, the solutions, being hastened by he recentadvances made in the silicon transistor fabrication tech-nology using oxide surface protection techniques, seemto be just over the horizon. These technological advanceshave made it possible to fabricate and study the proper-ties of useful and relatively stable surface field effectactivedevice structures and t o study he surface physicsof the more practical surface, the silicon-silicon dioxide

In the present paper, we shall study theelementary theory and the principle of operation of thesemiconductor surface field ffect transistor in detail.Because of the device struc ture, the name metal-oxide-semiconductor transistor or MOS transistor is adoptedthroughout this paper.

11. DEVICETRUCTUREThe MOS transistor tructuresare shown in Figs.2(a)-(d). The cross-sectional viewof the circular geometry

given in Fig. 2(a ) is shown in Fig. 2(b) and the coordinateaxes are labled in Fig. 2(c). The linear geometry is shownin Fig. 2(d) where the coordinates are also shown. Thesecoordinates willbeused in he mathematical analysisof the device characteristics.

developments see G. E. Moore, Semiconduc tor ntegrated Cir-For a comprehensive review of the silicon planar technologycuits,: Chapter 5 of Principle of MicroelectronicEngineering,E. Keonjian, Ed., McGraw-Hill Book Co., Inc., New York, N. Y.;1962. C. T. Sah, A New SemiconductorTetrode, The Surface-Controlled Transistors, presented at WESCON San Fran-cisco, Calif.; August, 1961; published n PROC.RE, vol.49,pp. 1623-1634; November, 1961.(b) C. T. Sah, Effect of surface recombination and channel onp-n junctionand ransistorcharacteristics, IRE TRANS.ON ELECTRONEVICES,vol. ED-9, p p . 94-108; January,1962.

325

Fig.2-The geom etry of N-channel MOS transistors. (a) Top view.(b) Cross-sectional view. ( e ) The expanded channel region of thecircular geometry. (d) The linear structure.

These basic structures were first proposed by D. Kahngand M. M. Ata1la.l The mode of operation to bedis-cussed in this paper was conceived in Kallngs patent.

conductor Devices, filed May 31, 1960 and issued on August 27,10 U. S. Patent No. 3102230, Electrical Field Controlled Semi-1963 to D. Kahng; and U. s.Patent No. 3056888, SemiconductorTriode, filed August 17, 1960 and issued on October 2,1962 toM. M . Atalla. Some of the materials covered in these two patentswere presented at he IRE-AIEE Solid State Device ResearchConference, Pittsburgh, Pa.; June, 1960 by these authors n Silicon-Silicon Dioxide Field Induced Surface Devices.unpublished eport by H. K. J. Ihantola,DesignTheory of al This mode of operation was discussed by SA g and in anSurface-field-Effect Transistor, StanfordElectronics LaboratoryReport No. 1661-1; September, 1961.


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326 IEEERANSACTIONS ON ELECTRON DEVICES Jd?]The N regions of the structures shown in Fig. 2 areobtained by high temperature diffusion of phosphorus

impurity nto the P-type silicon substrate usingoxidemasking techniques. One of the N-type regions labeledS for source112 s electrically connected to he P-typebulk by means of a metal layer (darkarea), althoughthis nternal connection is not essential and without itthe device becomes a four terminal structure. The otherN region, labled D for drain, is usually employedas the output lead with the source as the common leadto he nput andoutput.The nput electrode is themetal electrode over the oxide labled G for gate.12For the structures shown in Figs. 2(a)-(d) which employa P-type substrate, if the surface region under the gateelectrode in he semiconductor silicon is made N-typeduring the high temperature diffusion and oxidationprocesses, then a conduction path for electrons willbepresent between the drainand the source electrodeseven at zero gate voltage. Such a structure has eIectricaIcharacteristics very similar to the unipolar junction gatefield effect tr an ~i st or .~his type of device is sometimesknown as the depletion mode tr an si ~t or ,~ ~ th ats, thecurrent pa th between the source and the drain electrodesat zero gate voltage can be reduced or the electrons inthe path can be depleted, however, it can also be operatedin the enhancement mode in which the source to draincurrent is enhanced by a positive gate voltage.Structures without the built-in channel are very usefulin digital integrated circuit app1i~ations.l~uch a struc-ture is sometimesknown as he enhancement modetransistor l 3 which appropriately describes the operationalcharacteristics of the device since it cannot be operatedin the depletion mode.The oxide under the metalgate electrode shown inFig. 2 may be formed during the phosphorus diffusionprocess. Alternatively, this oxide may be stripped offand regrown either thermally a t high temperatures oranodically at r2om temperatures. Oxide thickness of theorder of 1000 A is generally used. Thick oxide reducesthe transconductance, the gain and speed of the devicefor a given dc operating point and very thin oxide makesthe reproducibility of the devicedifficult. Themetalgate electrode andmetal electrodes which form theohmic contacts t o the N-type diffused source and drainregions are made by evaporation and photoengravingtechniques. Metals such as aluminum, silver, gold,platinum, chromium and others may be used.

unipolar junction gate field effect transistor s adopt,ed by Shockley712 The nomenclatures hereare borrowed from those for thesince the functions of t he corresponding electrodes for the two typesof field effect transi stors are identical.PRac. IRE, vol. 50, pp. 1462-1469; June, 1962.13 P. K. Weimer, The TFT-A new thin-film transistor,

14 F. M. Wanlass and C. T. Sah, Nanow att logic using field-effectmetal-oxide-semiconductor triodes, Digest of Tech. Papers,IRE International Solid State Conf., Lewis Winner, New York,N. Y.; 1963; G. E. Moore, C. T. Sah and F. Wanlass, Metal-oxide-semiconductorieldffectevices for micropowerogiccircuitry, Presented at th e 1963 Internat ional Microwatt CircuitSymposia in Europe; July, 1963.

111.SURFACENERGYAND DIAGRAMSThe operation of MOS transistors can be understoodon a quantitive and physical basis by studying the energyband diagrams under various bias conditions. In Fig.

3(a)-(f) the energy band diagrams or the variation ofthe electron potential energy along the x-axis is plottedfor a y = constant plane in the region between the Nislands for the structures shown in Fig. 2. For simplicityof preliminary discussions, the surface statesand hedifference of work functions between the metal g a belectrode and he semiconductor are neglected in hefirst four energy band diagrams. These four cases allcorrespond to thermal equilibrium, ie., no potentialand current between the source and the drain electrodes.The equilibrium condition prevails essentially for allvalues of gate voltage V,, since the silicon oxide insulat-ing layer has very high resistivity ohm-cm) a troom temperatureand extremely low leakage current.A t zero gate voltage, V , = 0, .the energy band is flatas indicated in Fig. 3(a) for the case of no surface statesand negligible metal to semiconductor work functiondifference. When a small positive voltage is applied tothe gate eIectrode V G> 0, the energy band bends down-ward as shown in Fig. 3(b). An exhaustion layer (ex-hausted of majority carriers, in this case holes) is formednear the silicon-silicon oxide interface. Further increaseof the gate voltage results in the formation of an N-typesurface layer. The on-set condition of an N-type surfaceis given by the condition of intrinsic surface which isshown in Fig. 3(c) in which the intrinsic Fermi level atthe interface E i ( x = 0 ) coincides with he Fermi levelin the bulk E , or F , which is flat and independent ofposition. Still further increase of the gate voltage re-sults in an inversion layer or an electron channel whichprovides the conducting path for electrons between thet w o N-type islands. At large positive gate voltages,the surface potential along the channel is essentiallylocked at Ec(N) , the conduction band edges of the Nislands as indicated in Fig. 3(d). This is due to the factthathe surface becomes strongly degenerate underlarge positive gate voltage and th at there is a large sourceof minority carriers (in this case electrons) from the Nsource and drain islands which can flow into the channel.Numerical caIcuIations of the exact solutions in hegradual channel approximation have verified these energylevel diagrams. (Tobe published by C. T . Sah and H. Pao.)

The conduction of the electrons in the surface channelis perhaps more clearly shown in Fig. 4, where he electronenergy surface corresponding to he conduction bandis drawn for various bias conditions. To simplify the draw-ings, the transition regions of the junctions in each of thegraphs are shown as vertical surfaces and the surface underthe metal gate electrode in theN regions are not bendedby the gate vo1tage.The equilibrium energy band contourfor zero gate anddrain to source voltage isshown inFig. 4(a) which corresponds to Fig. 3 (a). In Fig. 4(b) ,a small positive gate voltage is applied but still withV, = 0. This diagram corresponds to ha t shown in


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1964 Sah: Characteristics of M O S Transistors 327

1 . 4WThl SEMICONDUCTOROXIDS

n

V

I /4I /L,

- E

Fm p- type region-depletlon region

inversion ayer(N-typesurface channel)

Fig. 3-The surface energy band diagrams in the z-direction per-pendicular to the channel current flow direction at (a) Vc = 0,(b) small positive V,, (e) at V , for intrinsic surface, and d ) atVQ or he on-set: of electron surface channel. Surface energybands including surface states nd work function differencebetween themetaland semiconductor at (e) equilibrium and( f ) nonequilibrium.

/ Vu=O. I OP-type Si l i con ssA - 7

( 2 Z i z i T

v =v = 2 , v i o

hannel byk inched-off regionFig. 4-The equipotent ial surfaces of electrons and energy surfacesof the conduction band at various bias conditions.


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328 IEEE TRANSACTIONS ON ELECTRON DEVICES JFig. 3(b) where an exhaustion layer is created near thesurface. Further increase of the gate voltage but stil lwith the source to drain potential zero is shown in Fig.4(c). It is evident th at at thisgate voltage a surfacechannel for electrons is created since the conductionband edgebetween the source anddrain electrodes a tthe surface is now at the same level as those in the sourceand drain regions. If the gate voltage is further increased,beyond that given in Fig. 4(c), a large channel is formedand the surface is strongly degenerate N-type. Thesurface potential energy of the conduction band edge isnearly 'locked' at the position given in Fig. 4(c) due tothe strong degeneracy and the presence of two sources ofinfinite supply of electrons due t o the N' source anddrain islands. This situation is illustrated in Fig. 4(d),which corresponds to the cross-sectionalview of theenergy band shown in Fig. 3(d).If the surface states and the work function differencesbetween the metaland the semiconductor are akeninto account, the energy band diagrams shown in Figs.3(a)-(d) and Figs. 4(a)-(d) just discussed are modified.These mod5cationsare shown in Figs. 3(e) and 4(e)where the drain-to-source voltage is zero and the surfacestates are assumed to be donors and ionized and hencepositively charged. In Fig. 3(e) the dash line correspondsto the conduction band and the valence band edges ofthe insulator. In Fig. 4(e) the energy band contoursare rounded and the effect of the gate potential on thesurface energy band in the N' regions is also illustrated.These are disregarded in all other diagrams.

One importantparameter of a metal-insulator-semi-conductor sandwich is themetal semiconductor workfunction difference given by the expression+Ls= + N -x + + F + EG/2). The various quantities n his ex-pression are shown in Fig. 3(e). + M is the work functionof electrons in the metal which s the energy requiredt o remove an electron at t he Fermi surface of the metalt o infinity. x is the electron affinity in the semiconductorand is the barrier height of the vacuum level measuredfrom the bottom edge of the conduction band a t hesurface. C#Ip + E,/2 is the Fermi potential of electronsin he semiconductor measured from the bottom edgeof the conduction band. Other parameters which arenecessary to describe the electrical characteristics ofan MOS structure are hose which characterize the surfacestates and bands a t the silicon and silicon dioxide inter-face. Fora surface band, it isnecessary t o know thedensity of states n he bands as a function of energyand the total surface state concentration per unit surfacearea n order t o establish the equilibrium energy banddiagram and equipotential surfaces shown in Figs. 3(e)and 4(e). For the assumed ionized and positively chargeddonor states, the surface energy band ispulleddownas indicated in Fig. 3(e) from the flat band conditionshown in Fig. 3(a).Theamount of band bending salso dependent on the work function difference justdiscussed. It is evident from the energy band diagramshown in Fig. 3(e) that the energy band can be made

flat if +p or if the impurity concentration in the P-tybulk is increased. This can also be accomplished ifmetal work function C#IM is increased by selecting a difent metal or an alloy for the gate electrode.

The dynamic parameters of the surface statesbands are also required in order t o study the frequedependence of the electrical characteristics of the Mstructure. These are the transition probabilities of etrons and holes between the conduction or valence bastates and the surface states or the surface band statThe surface state or band parameters are onethe least known quantities in solids, specially insituation such as he present MOS structure n whthe semiconductor surface is overed with a n oxlayer which is thermally grown. In this case, the numof surface states is considerablyessin fact lethan a perfect and clean surface which has aboutdangling bonds per cm2 and approximately l O I 5 staper cm'.On the oxidized surface obtained during htemperature processes, these bonds are mostly tieduby the oxygen forming silicon-oxygen bonds and thdo not contribute to surface states. Generally for oxidizedilicon surface, the remaining surface staare donor type rather than the acceptor type from dangling bonds and have a total concentration of 1to 10 states per cm2. If this had not been the caneither transistors nd diodes nor the MOS devwould have been useful devices. This very factor pvented Pearson and Shockley from obtaining the antpated conductivity modulation by he applied surfelectric field in tlze first metal-insulator-semiconductorvice e~p erime nt.~ecause of the unavailability of the sface band and state parameters, we shall lump all surface states and denote this by a h e d charge of Qper unit surface area in this paper.The energy band diagrams are further modified ivoltage is applied between the drainand the souelectrodes. This diagram along the x axis for a smpositive drain voltage (drain junction reverse biasis shown in Fig. 3(f), which shows th at next t o the higdegenerate N-type surface channel or inversion laythere is a region which is essentially depleted of carrand labeled as the depletion region.

The energy surfaces are also sketched for three drato-source voltages and shown in Figs. 4(f)-(h). Fig. 4corresponds t o a drain voltage which is small compawith the gate voltage, and the surface channel is faiwide open along its entire length between the souand drain. I n Fig. 4(g) the drain voltage is equalthe gate voltage in excess of the turn-on gate voltaie., Vo = V G- V T where VT s determined by condition shown in Fig. 4(c). Under this condition, width of the surface channel is narrowed down t o zat the drain electrode as indicated in Fig. 4(g) and channel is said t o be pinched 08 t the drain. At pinoff, the channel current, which omes from electrflowing from the source to the drain, is essentially ctrolled by the not-pinched-offegion of the chann


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1964 Sah: Characteristics of M O X Transistors 329If the drain voltage is further increased, such as thatshown in Fig. 4(h), almost all of the additional voltageincrease, i.e., [V, - ( V , - V T ) ] ,ppears across thepinched-off region which is now extended from the drainto a point at a distance L - 4 in the channel as indicatedin the figure. The drain current is essentially constant,independent of the drain voltage, when the drain voltageis ncreased beyond th at given by Fig. 4(g). The slightincrease of the drain current from the condition in Fig.4(g) where V, = V G- V , o the condition in Fig. 4(h)where V , > V G- V r comes from the slight decreaseof the channel length of the nonpinched-offegion.This alsogives a finite, rather hanan infinite, drainresistance. This channel shortening effect is very similarto he well-known space charge layer wideningeffector Early effect ina bipolar junction transistor whichcauses a considerable collector conductance.15

The condition shown in Fig. 4(g) where the drainvoltage is equal to the excess gate voltage over the turn-on gate voltage is known as the condition of saturationsince the drain current is saturated and further ncreaseof the drain voltage results in very little increase of thedrain current.

Another important part of the energy band diagramsand surfaces is the contour of the electron and holequasi-Fermi levels or potentials which we shall discussnext. In the ast hree energy surfaces shown in Figs.4(f)-(h) where a finite drain voltage is applied, theelectron concentration in the channel isdifferent fromthe equilibrium value. In terms of an electron quasi-Fermi level F , in he Boltzmann factor, the electronconcentration may be written as

n = ni exp (F , - E,)/kT = n, exp (u - un) 1)where ni is the intrinsic carrier concentration, E i sthe intrinsic Fermi level, .u, is the normalized electronquasi-Fermi potential given by u, = - J k T and uis the normalized intrinsic Fermipotential given byu = - .E i /kT. (See also the list of symbols for the defini-tion of the notations.) Similarly, the hole concentrationin the surface region also differs from that in the bulkdue to he bending of the energy band. However, thehole quasi-Fermi level F , is essentially independent ofdistance and coincides with the equilibrium Fermi levelin the bulk. This occurs because the hole current, whichis proportional to the gradient of the hole quasi-Fermipotential, is small compared with the electron currentin thechannel since the hole current comes almost entirelyfrom the generation processes fromsurface states andbulkShockley-Read-Hall centers in thenversion and the deple-tion regions indicated in Fig. 3 f ) . Thus, the concentra-tion of holes may be written asp = ni xp (Ei F,)/kT

= ni exp up u)= ni exp up u) (2)16 J. M. Early, Effect of space-charge layer widening in junctiontransistors, PROC.RE, vol. 40, pp. 1401-1406; November, 1952.

and the variation of the hole concentration in lthe channelregioncomes mainly from the variation of the electro-static potential u x, ) in (2).

The variation of the quasi-Fermi levels for electronsmay now be considered. Sincehe electron diffusion currentoutside the channel region s small compared with thegeneration-recombination current in wide band gapmaterials such as silicon, there is a correspondingchange of the electron quasi-Fermi level with the 5 di-rection. However, since the width of the channel isusually quite small, one may assume that the electronquasi-Fermi levels essentially constant independentof the x coordinate near the surface and in the surfacechannel region as indicated in Figs. 3(f)-(h). The differ-ence between P , and F, is th e voltage difference betweenthis point in the channel and the source electrode whereF , = F,. One can readily make this conclusion by con-sidering the variation of the electron quasi-Fermi levelwith the y coordinate along the channel. At the sourceelectrode, the electron and the hole quasi-Fermi levelsconcide with the bulk Fermi level, Le. , F , = F , = E Fas indicated in Figs. 4(f)-(h). At the drain electrode,the electron and hole quasi-Fermi levels must differ bythe amount corresponding t o the drain voltage, Le.,F , - F , = -qVD. The negative signcomes from theparticular sign convention chosen for the drain voltagewhich makes the drain junction reversebiasedwhenV, > 0. It is thus evident that at ny point in thechannelwhich is adistance y from the source electrode, the voltagemeasured relative t o the source is given by - Fn- F, ) / q .Denoting the channel voltage by V(y), thenVY = pV(y)/kT = - Fm- FJ/LT

= an(y ) p = un(y) - u 0 < y < (3)In the direction other than along the surface channel,the electron quasi-Fermi level F , near the N drain

region decreases exponentially with distance away fromthe draincontact towards the bulkFermi level E,.Thisvariation of F , isalso hown in Figs. 4(f)-(h).

IV. ELEMENTARYHYSICALHEQRYIn order t o bring out some of the most important

physical parameters which control the electrical charac-teristics of the surface channel and the MOS transistor,we shall give a mathematical analysis based on a simplephysical model in his ~e ct i0 n. l~he simple physical

and recombination n p-n junction andp-n unction characteristics,l6 C. T. Sah, R. N. Noyce and W. Shockley, Carrier generationPROC.RE, vol. 45 pp. 1228-1243; September, 1957.l7 An elementary analysis due to A. Many has recently beenpublished by H. Borkenand K. Weimer in An analysis of thecharacteristics of insulated-gate thin-film transistors, RCA Re-view, vol. 24, pp. 153-165; June, 1963. This analysisextends thework of Ihnatolal-l by including th e initial charge present in thesemiconductor,although i t does not relateexplicitly the initialcharge to the material and interface properties of the insulator andsemiconductor. The analysis in this section follows essentially the

relates more explicitly the properties of the materials t o the devicesame line of attack and was quoted in parts by Sahg. It, however,characteristics tha n the work given by Borken and Weimer.


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model will be based on the following assumptions: 1) Thejunction current flowing in the drain unction is excluded,2) no electron-hole recombination and generation in thechannel from either the surface states or the SRH centers,3) the variation of thewidth of the channel is smallalong its length, 4) the substrate is nondegenerate, 5) thediffusion current n he channel snegligiblecomparedwith the current due to theelectric field along the channel,and 6) the surface states may be lumped into a fixedcharge with total charge given by Qss per unit area ofsurface.

With reference t o Fig. 2(d), it is evident th at hecurrent flowing into the draincontact I , consists oftwo parts, the current flowing into the surface channeland thecurrent flowing into the substrateover the barrierof the NP drain junction.

The surface channel current may be divided into threeparts: 1) The electron current due to the electrons flow-ing into the channel either from the drain N* islandif the drain junction is forward biased (VD< 0) or fromthe source N island if thedrain unction is eversebiased (V, > 0 ) )which corresponds t o the MOS tran-sistor mode of operation. This is the dominating channelcurrent when the MOS transistor is turned on. 2) Thehole current from the holesdue to ionized acceptorimpurity in thesubstrate.Thiscurrent s very smallcompared with the electron current when the deviceis turned on since most of the holes in the surface channelare pushed into he bulk egion by the electric fieldnormal to the urface from the large positive gate voltage.3) The electron and hole current from carrier generationand recombination from the surface states and from theShockley-Read-Hall centers inhe channel. Thesecontributionsare also quite small compared with theelectron current when the channel is turned on,whichare neglected as stated i n ( 2 ) .

Theotherpart of the draincurrent, which omesfrom the drain junction current, is excluded. It can besimplyadded to the channel current t o give the otaldrain current. For the MOS transistor mode of operation,the drain junction is reverse biased so tha t this currentwhichcomes from carrier generation in the ransitionregion, can beneglectedcompared with the electroncurrent n the channel when the device is urned on.In he off state, however, this is the residue currentwhich flows in the drain and must be taken into accountto calculate the stand-by power in switching applica-t i o n ~ . ~ ~n the surface-potential controlled tetrode modeof ope~ atio n,~( ) therain unction, which is connectedt o the emitter junction of the tetrode, is forward biased,and he drain junction current is appreciable. In thiscase, the drain unctioncurrentmay be added to thechannel current t o give the total drain current.

Following from l) , 2) and ( 5 ) ) he drain current maybe obtained by integrating the electron current densityover the cross-sectional area of the channel. Using thecoordinates shown in Fig. 2(d), this gives

330 I E E E TRANSACTIONS ON ELECTRON D E V I C E S July1, = 2 j d z , Y) dx

= -2 / unE,dx = -2 qbn(-dV/dy) dx= pZ dV/dy) 1 nndx.

The integration is taken over the width of the channex, which may be defined as the equipotential line givenby he condition u xc , y) = qVBD/kT,where VBo sthe built-in or daus ion potential of the drain junction.This l i e is shown in Figs. 4(f)-(h). I n the calculationin (4) the following relationships are used: Jnl(z,) = a, ,,ca = qp n and E = -dV/dy.

The electron mobility p, in (4) is generally ess thanthat in the bulk due to the additional surface scatteringfrom the atomic discontinuities and disorders a t hesemiconductor-oxide interface. Aneffective mobility- nthe surface space charge layer has been calculated bySchrieffer and Green, Frankl and Zemel. It is definedbY

pn = p n dz/[ n dz 5and can be used in (4).

The total electron concentration in the channel J n dzwhich appears n 5 ) may be related to the charge onthe surface states Q S Sand the charge on the metal gateelectrode CoV, using the gauss theorem of flux fromelectrostatics. If the channel width is airly uniformalong its length (assumption 3)), the flux lines are alnearly perpendicular to the silicon-silicon oxide interfaceThis is known as the gradual channel approximationwhich has beenemployedbyShockley to analyze thecharacteristics of the junction-gate fieldffect tran-sistors. I n this approximation, the gauss theorem reduceto a simple one-dimensional form given by

mQ~~ + cove = = - [ P a x 0= - q ~ m @ + p l - - n - - D - N , f N , ) d l : (6

where C, is the capacitance per unit area of the oxidelayer under the metal gate electrode which has an oxidethickness of zo and is given by

Co= Koeo / xo . 7Vo s the voltage drop across the oxide layer at a distancey from the source electrode and is given by

Vo(d = G - V(Y>. ($1charge layer, Phys. Rev., vol. 97, pp. 641-646; Februa ry, 1955.l J. R. Schrieffer, Effective carrier mobility in surface-spacein semiconductors, Phys. Rev., vol. 118, pp. 967-975; May, 1960l@R. F. Green, D. R. Frankl and J. Zemel, Surface tran sporin semiconductors, Phys. Rev., vol. 131, pp. 592-593; July, 1963R. F. Green, ,yonlocal transport and cuspidal surface mobi


8/22

1964 Sah: Characteristics of M O S Transistors 331Thus, the ota l electron charge per unitarea n thechannel, obtained from (6) using 8 ) , is given by

Q he = &sa COW, - V ) I- B (9)where Q B is the charge per unit area in he bulk defined byQB = - 9 /- N A p - A + nD - N,) d~+ q l p n d z G - 4 L m N A - p ) + y s m n d ZC (10)which shows th at the bulk charge is equal to the totaldepleted hole charge in the semiconductor due to heapplied positive gate voltage. The approximation in (10)follo-mrs from assumption (4) for a structure with P-typesubstrate.The current equation given by (4) may be written asI D= Z(dV/&/)Lco{(Qss+ & B ) / ~ o ] . V c - v(Y)) (11)using (5) and (9). This differential equation can bereadily integrated along the channel if it is assumed thatthe average surface mobility p,, the bulk charge Q B ,and the surface state charge QSs areall independentof position along the channel y. These follow directlyfrom he gradual channel approximation made in as-sumption ( 3 )which essentially states tha t u x, ) h a(.)and from assumption 6) whichassumes that Qss is aconstant. The integration over y = 0 to L gives thefollowing drain characteristics:

I D = (t cO/Lz)[(VG- V,) V - V2/2]12)where C, = C,ZL is the total capacitance of the oxidelayerunder the gate electrode and V T is the turn-onvoltage if it is positive or turn-o$ voltage i f it is negative.This threshold voltage is defined by

V T = -(Qss + Q B ) / C O . (13)It is evident from the result given in (12) that ifQ s s + B B < 0 , V , s the minimum gate voltage requiredto induce a surface channel and turn on the rain currentwhich corresponds t o the enhancement mode of operation.On the other hand, if Q Ss + Q B > 0, for example in adevice with a donor surface state of QSs > - Q B on aP-type substrate, hen an electron channel exists evenat V G = 0. In this case, V T s then the minimum gatevoltage required to tu rn off the conduction path betweenthe drain and the source.In silicon MOB transistors, usually the surface statecontribution Q s s dominates over the bulk charge Q Bin V T for a properly made device. For silicon-siliconoxide interface, the surface states are donor like with ato ta l concentration of approximately Q s s / q = 5 X1011/cm2. The turn-off voltage from this contributionfor an oxide thickness of z a = 2000 A with dielectricconstant of K O = 4 is about -5 volts using (7) for theoxide capacitance. The contribution from the bulk charge,Q B given by (lo), is of the order of +1.5 volts for P-type

bulk concentration of N A = 1015/~m314hm-em).Thus, the net turn-off voltage for this N-channel MQStransistor is V T = -5 + 1.5 = -3.5 volts which showsthat there is an N-type conducting channel induced bythe positively charged donor surface states a t zero gatevoltage which requires a negative gate voltage of -3.5volts to turn t off.

The drain characteristics given by (12) is valid onlyfor gradual channels and ceases t o be a good approxima-tion in the region of the channel where it is pinched offor nearly pinched off. As the drain voltage J7D is in-creased and the gate voltage is kept a t some value greaterthan V,, the drain current increases first linearly withV D ut the increase slows down as the drain voltage isfurther increased and the channel narrows down at thedrain as indicated in Fig. 4(f). This is evident from thefamily of drain characteristics shown in Fig. 5. At themaximum of the drain current given by (12) and shownin Fig. 5, the channel is pinched off a t the drain junc-tion as indicated in the energy surface diagram in Fig.4(g). Beyond thisdrain voltage, the gradual channelapproximation fails and (12)ceases to be valid. Thiscondition corresponds approximately t o the conditionof maximum draincurrent rom (12) andmay beob-tained from it. Setting ( a l ~ / a V , ) ~ ~0 using (12),the pinch-off condition is

V D = V D , = VQ - v, (14)which is the condition given by Fig. 4(g).Thedraincurrent at th e pinch-off condition is given by

I,, = Cii,Co/2L2)(Vo - V T ) ~ (@,Co/2L2)V:s. (15)If the drain voltage is further increased beyond the

pinch-off voltage given by (14), the pinch-off regionlengthens into the channel from the drain as illustratedin Fig. 4(h). Most of the additional voltage applied t othe drain beyond the pinch-off voltage, i.e., V , - VDs,appears across the length of the pinch-off region andresults in very ittle increase of the drain current. Aslight increase of the drain current does occur which isevident from (15) ince the length of the channel Lmust be replaced by the effective length of the channel 8.An accurate calculation of the length of the pinched-offregion is not possible in he gradual channel approxi-mation and will be covered in a future discussion. Inthe present analysis, thedraincurrent n he regionV , > V D S wll be taken as the value given by (15) andassumed to be independent of V D . his is also illustratedin Fig. 5.

Since the draincurrent or V , > V,, is essentiallyconstant or saturated to the value I D S , this current isknown as the saturationcurrent and the subscript S isused which denotes saturation. The corresponding drainvoltage a t which the channel is pinched off is known asthe saturation voltage.

The dc characteristics of the MOS transistors areplotted in Figs. 5 and 6 from (12). It is evident from thedrain characteristics, I Dversus V D hown in Fig. 5 that


9/22

332 IEEE TRANSACTIONS ON ELECTRON DEVICES J u

50

40

30

20

10

0

-10

-20 0 12D

Fig. 5-The drain characterjstics of an N-channel MOS transistor.10

Fig. 6-The transfer characteristics of an N-channel MOStransistor.the MOS transistor is very similar t o the junction-gatefield effect transistor. The dashed line in this figure sthe locus of I D S iven by (15) which separates the regionof saturation from the nonsaturated or nonpinch-off region.A small portion of the drain characteristics in the third,quadrant V D< 0 is alsoshown in Fig. 5 whichcorre-sponds to the drain junction forward biased. This doesnot include the forwardbiased drain junction currentwhich may be appreciable compared with the channelcurrent shown in Fig. 5. From this figure, it is evidenttha t he channel current is nearly proportional to hedrain voltage when the drain junction is forward biased.Thus, n his mode of operation which occurs in the

surface-potential controlled transistor tetrode^, ^' tchannel current may be represented by a drain-voltagindependent resistance which s a function of the gavoltage.

The transfer characteristics, ID versus V Gwith 8as a parameter are shown in Fig. 6. They are straightlines below saturation when the channel is not pinchoff and the gate voltage satisfies V ,- V,> VD. he intecepts of these straight lines are given by V G V T VDBetween V , - V , = 0 and V , - V , = VD,he draicurrent follows the saturation current locus given (15). The drain current is completely cut off when thgate voltage is equal tohe pinch-off voltageV , - v,= 0.A rapid experimental determination of the threshovoltage V , can be obtained from the two terminal charateristics in which the gate electrode is connected eithdirectly t o the drain or the enhancement mode Nchannel device which has V , > O and no built-in channor throughabattery V G G > \VT \ (positive side tito he drain) which has a sufficientlyhigh voltage pinch off the built-in channel for the depletion moN-channel device. In these connections, the device is the saturation regionsince V D= V , + V G G > V D sV G- V , hus, the two terminal drain current may obtained from (15) using (14) and is

I DS= LCo / 2L2 ) . V D - V G G V,12 (1which shows ha t the onsetof the drain current orresponto a drain voltage of (V , + V,,). For devices of thenhancement type, the turn-off voltage V , can bedetermined readily from a display of this characteristwithout the use of agatebattery since VT < 0. Fthe depletion type V , > 0 and a gate batteryf V G G > ( Vmust be used to determine the turn-on voltage V,. Thslight additional complication comes from the fact than N-channel MOS transistor, which has a P-type sustrate,cannot be biased witha large negative dravoltage since then the drain junction would be forwabiased and the large forward drain unction urrentwould mask off the pinch-off point.

V. SMALL SIGNALQUIVALENTIRCUITARAMETESome of the important low frequency circuit param

eters can also be obtained from the drain characteristiof (12). For example, .the drain conductance belosaturation is

g d = (aID/aV,>v. = (ACO/L~)(VG F - V D ) (1which decreases linearly with drain voltage and becomzero a t saturation, V D= V D s= V G- V T .

The transconductance isg m = ( a T D / a v G ) V ~ = @,cO/L2)VD (1

and its maximum, which occurs a t saturation, is given bg m s = @ a C O / L 2 V D S = (~wCO/L2)(VG v,)

= d2ID6,pnCO/L2. (18


10/22

1964 Xah: C haracteristicsY - Y =G T

I I I I I

83Fig. 7-The transconduct ance versus th e drain voltage charac-teristics of an N-channel MOS transistor.

2

1

I I I A 00 2 L 10 126 v - v aG TFig. 8-The transconductance versus th e gate voltage charac-teristics of an N-channel MOS transistor.

Use is made of (14) and (15) in the last two steps in thecalculation of (1%). The maximum transconductancegiven by (Ma) is also approximately the transconductanceof the devicewhen it is operated beyond the satura-tion voltage, V D> VDs.The family of transconductancecharacteristicssplotted s a function of the drainvoltage in Fig. 7 with the gate voltage as a parameterusing (18) and ( Ha ). It is evident from this figure thatbelow saturation V D < VDs theransconductancesproportional t o thedrain voltage but independent ofthe gate voltage. Beyond saturation V D > Vns thetransconductance is independent of the drain voltagebut is proportional to the gate voltage. A similar repre-sentation with the drainand the gate voltage inter-changed is shown in Fig. 8.

The voltage amplification factor of the device is= - ( a V D / a V G ) I , = vD/(vf2 VZ - VD) (l9)

of MOS Transistors 333which is identical t o the ratio gn/gd. A t the point ofsaturation V G - V T = V D the voltage amplificationfactor diverges due to g d = 0. In actual devices, due tothe channel length shortening effect beyond the satura-tion point, which was discussed in the preceding section,the drain conductance is finite in the saturation region,and hence the voltage amplification factor is also finite.Other smallsignal parameterscan alsobe obtainedfrom the model just discussed. To calculate the shortcircuit gatendrain capacitances, the expressionC = d Q / d V can be used. For the gate capacitance, thetotal chargeon themetal gate electrode at anygateand drain voltage may be obtained fromQG = K o d J L Eo(Y)Y = ( ~ o ~ o z / ~ o ) VO(Y) yd 0

= CoZ [L [VG- V(y)I dy. 20)In th e ast step of th e calculation in (20), use is made of(7) and (8). This ntegralcan be evaluated explicitlyusing the draincurrent equation 11) t o relate dy t od V and the result is

J O

The short circuit gate capacitance is thusC G = ( a Q O / a V G ) V ,

Co{l - VO2/3[2(V, - VT)- VD]' (22)= - V,/V,,)"(lD,/rD)'/3]. (224

Similarly, the short circuit drain capacitance may beobtained from the change of the carrier charge storedin he channel when thedrain voltage is varied.Thecharge stored in he channel may be obtained eitherdirectly by integration of the electron concentration inthe channel or by the use of the gauss 'theorem (6) inconjunction with the charge on the gate given by (21).Thus,&Ch = -4.z /Jn dx dy = - Q @ + c0v,where use is made of (9) and (13) for the x integral and(20) for the y integral. The short circuit drain capacitanceis thus,


11/22

334 I E E E TRANSACTIONS ON ELECTRON DEVICESThese short circuit capacitances are graphed in Fig. 9and Fig. 10 from (22) and (24) as a function of eitherthe drain or the gate voltage with the other as a param-

eter. It is evident from these figures tha t the gate capaci-tance decreases from the oxide capacitance Co belowsaturation to a constant value of 2c0/3n the saturationregion. Similarly, the drain capacitance drops from C0/2to zero in hesaturation region. The vanishing draincapacitance in the saturation regioncomes from thefact tha t when the channel is pinched off at the drain,the electron charge stored in the channel becomes inde-pendent of the drain voltage. The charge stored in thechannel Q C h is plotted from ( 22 ) in Fig. 11which showsthis constancy.

From the draincurrentequation (12) and the otalcharge on the gate electrode (21) it is alsopossible tocalculate the open circuitgatecapacitance using therelationship CGa= (aQG/aV, ) , , and the result isC,o/Ca == 1 - {v, 2 [ 3 (V , - V,) - V,].(V, - vr - V,)(aV,/aV,),,J/3IIa(v, - v2-1- VDl= 2(vG - VT)/[Z(vG- V T ) - VD]. (25)In th e last step of the calculation the voltage amplifica-tion actor rom (19) is used. The esult of the opencircuitgate apacitance given by (25) is graphed inFig. 12 as a function f the drain voltage and gate voltage

The low frequency intrinsic equivaIent circuit pararn-eters of the MOS transistor can now be derived from theterminal conductances and capacitances just calculated.The general equivalentcircuit is shown in Fig. 13 forthe channel. This is the intrinsic equivalent circuit whichdoes not include the stray capacitances, the capacitanceof the finite drain junction area which is not connectedto the channel, the lead conductances and the spreadingresistance. Thevarious capacitances and conductancesin he intrinsicequivalent circuit are readily derivedfrom the results just, obtained. Thedrain conductancegd is evidently hat given by(17)and the transconductancegm s given by (18).The etermination of the three quivalent circuitcapacitances, C,,, Cgdand Cd,, shown in Fig. 13 fromthe terminal capacitances requires slightly more addi-tional analysis. The open circuit gate capacitance givenby.(25) may be related to the equivalent circuit param-eters shown in Fig. 13 by direct calculation. Straight-forward circuit analysis of the quivalent ircuitnFig. 13 with thedrain electrode open gives the inputadmittance

using (25).

The low frequency open circuit gatecapacitance maybe obtained from (26) by letting w -+ 0.

C G O = c g s f c d ( g m g d ) / g d - (27)

DFig. 9-The short circuit gate and drain capacitance of anN-channel MOS transistor versus the drain voltage.

1.0

0.9ccCO-

0.8

0 . 1

0.5

0.4CDF

0.3

0.2

0.1

Fig. 10-The short circuit gate and drain capacitance of anN-channel MOS transistor versus the gate voltage.


12/22

1964 Sah: Characteristics of M O S Transistors35

100 I I I I I-2 0 2 8 10 0 12

Fig. ll-The total charge of the current carrier stored in thechannel of an N-channel MOS transistor.

1.0 I I I I I2 4 VG-VT 6 8 10 12Fig. 12-The open circuit gate capacitance of an N-channel MOStransistor versus the drain and gatevoltages.

Fig. 13-The small signal intrinsic equivalent circuit of anN-channel MOS transistor.

0.666

io8 . 5 00

0.500 I I I I2 v - v 8G T 10 12Fig. 14-The equivalent circuit gate tosource capacitance of anN-channel MOS transistor versus drain and gate voltages.

It is also immediately evident from the equivalent circuitin Fig. 13 that he short circuit terminal capacitancesare elated t o the equivalent circuit capacitances by

and c, = c d s - C o d . (29)Thus, the feedthrough capacitance between the gate anddrainmay be obtained by eliminating C,, from (27)and (28) and using (17)for g d and (18)for gmoThe result isc o d / c O = 1 3 V G T ) l

* 2 ( V G- V , - VD)/3[2(VG V,) - V,I2 (30)which is identical o theexpression for CDgiven by (24b).Thus, from (29) one concludes that

C d , = 0 (3 )which is also expected from the physical model.

From his simple result, one can mmediately obtainthe gate t o source capacitance C,, from (28) using (30)for C,, and (23) for C,.cu3/c0 ( 2 / 3 ) w G- vT

. [3 (vG - V T VD]/[2(VG V T - V D l 2 (32)is plotted in Fig. 14 as a function of the drain voltageor gate voltage.

The equivalent circuit capacitances in the saturationregion areobtained by setting V D= V D ,= V G- V ,in (30) and (32) and are given by

C u d a = 0 33)and

Cgss 2C0/3 . (34)The result of 33) comes from the fact thatn the satura-tion region, the channel is pinched off at the drain elec-


13/22

336 IBEE TRANSACTIONS ON ELECTRON DEVICES Julytrode so that there is no signal path between the gateand drain electrode directly except through the modula-tion of the channel width by the gate voltage in heunpinched-offegionwhichs accounted for byhetransconductance.

In small signal low frequency applications of theMOS transistors, the low frequency equivalent circuitshown in Fig. 13 is quite adequate. For high frequencyapplications, this equivalent circuit shouldbe slightlymodified by including a cutoff frequency in he trans-conductance expressiongiven by (18) in he followingform

where g is the low frequency expression given by (18).The exact calculation of the cutoff frequency a,, israther complicated and requires a small signal highfrequency analysis of the MOS transistorstructure. Asimple charge nalysis can be made to provide an adequateestimate of ugm.This is done by considering the changeof the output short circuit current in the drain when asmall pulse of charge d Q G s added to the gate electrode.The time delay of the response in the drain current isof the order

tQ = (aQQ/D)JD (36)which may be rewritten as

to = (aQ,/avc>.,/(ar~/av,)., 37)= c,/gmo = (c08 + cQd) /gW o (374

whereuse has been made of (18) for gmo and (21) and(28) for Gc. The cutoff frequency of the transconductancemay be approximated bymom A l / t m = gmo/cc = gmo / (C , s + c,3 (38)[2(vG- v, - VD]/{2[2(V, V T vDI - v: (38a)where use is made of (18), (28) and (22).The reciprocalof the transconductance cutoff frequency l / u g m= t,,is plotted in Figs. 15(a)-(b) versus drain and gate voltagesrespectively using (38) and (38a).

Another useful figure of merit for the MOS transistorin small signal high frequency applications is the mid-band gain times bandwidth product with two or moredevicescascaded. This figure of merit can be readilyobtained from the equivalent circuit shown in Fig. 13with a load conductance G and another MOS transistorstage connected t o the output terminals D and X. Thevoltage gain is

where RL = 1 / ( G + g d ) is the ota l effective load re-sistance and w L = l/RL(C,, + ed,- C u d )s the cutoff

I. 2

1.0

0.8emd/i0.6

0.4

0.2

0

I

Y = I

I I I2 4 6 V V T 0 10 12

(b)

2

u

1L

Fig. 15-The transconductance cut-off time constant of an N-channe1 MOS transistor (a) versus the drainvoltageand (b) thgate voltage.

frequency associated with the load resistance. The midband gain is obtained by setting u = 0 in (39) and i

A, = - rnoIzL (40which is normally much greater than unity, i e . , RL>>for a useful amplifier.Thus, the cutoff frequency uL = R;associated with t he load is considerably lower than thetransconductance cutoff frequency u,, a gmo. Hencethe frequency at which the voltage gain drops b y $ dbfrom the midband isgiven by w L and he gain-bandwidth product is

2a (gain-bandwidth product) = AmuL= gmo/ (C , , + CdJ - C,d) = g m o / C , , - , d ) (41= (3/2)(/&/L2)[2(VG- v T > D]* 4

This figure of merit is almost identical t o the transconductance cutoff frequency given by 38). The differ


14/22

1964 Sah : Characteristics of MOX Transistors 337enceomes from the feedthrough capacitance whichvanishes in the saturation region.In many circuit applications, the feedthrough capaci-tance is neutralized so that C,, does not appear in theabove expression and the figure of merit becomes gm0/C, ,instead of (41).In th e saturation region, the transconductance cutofffrequency and the figure of merit are identical and areobtained by using V D= V D S= VG- V T n (38a) or(41a) t o yield

w,,, = 2~ (gain-bandwidth product)= ( P n / 2 L 2 ) ( V c- V T ) (42 )= d@a/2COL2)IDS (424

where use is made of (15) for IDS.

VI. SWITCHINGIMESFor switching applications, such as n he inverter

circuit shown in Fig. 16, the switching times consist ofbothime onstant of charging and discharging thechannel andime onstant associated with the loadresistance RL and the capacitance Ci which consists ofthe drain junction capacitance and the stray capacitanceof the device header and an nd associated circuitwiring. During the turn-on transient, the channel charg-ing timeconstant dominates over RLC; while duringthe turn-off transient, RLC; dominates.

Let us first consider the turn-on transient as illustratedin Fig. 16 for a P-channel device which has no built-inchannel. The device s initially in he off state with anegative drain voltage - VDDnd zero gate voltage. Att = 0, a negative gate voltage is applied which is suffi-cient to induce a hole channel and cause holeso flow downthe drain from the source and discharge the capacitanceC; which is initially charged up to a charge of - C i V D D .If the timeconstant RLC; is large compared with thetime constant of the channel, the discharge time of C;is essentially the timeconstant of the channel sinceduring this ime the build up of charge on C; by theload current flowing through RL from the battery V,,isnegligible. The channel time constant is an intrinsicproperty of a given device and depends on the devicegeometry and material properties. It may be calculatedby assuming tha t the charge stored in the channel andthe channel currentare given by he dc value a t anyinstant of time during the turn-on ransient. The totalchannel turn-on time in this quasi-equilibrium situationis then the sum of the differential time dt required to adda differential charge dQ t o the channel by the channelcurrent I . The sum extends to the time when the channelcharge reaches the final value Q C h and the channel currentreaches the final value I,. The time dependence of I( t )is not known and for a rough approximation, it may betaken as a constant given by the final value a t the end ofthe turn-on transient, ie., I ( t ) = ID. The turn-on or

Fig. 16-Switching transients in a p-channel MOS transistorinverter circuit.

channel charging time is then

This time constant s also the transit time f the electronsacross the N-type channel for the case of constant elec-tron mobility. This is easily demonstrated by the follow-ing calculations:

P P

In the last two steps of the calculation in (43a) , use smade of (4) for d V / d y and (23) for &Ch. The total elec-tron charge in the channel given by (23 ) and (21) maybe substituted into (43 ) t o give

At saturation, the charge stored in the channel is

and the channel time constant is

The harnel time constant beyond the saturationpoint may be estimated from (46). In this region, theeffective channel length is decreased and as a result I Dis increased above IDS. Thus, from (46) it is evidenttha t the channel time constant decreases slightly as thedrain voltage is ncreasedbeyond V D s . he reciprocalof the channel time constant obtained in (46) is almostidentical t o the transconductance cutoff frequency warnsor 2~ times the gain-bandwidth product given by (42a) ,except for a numerical factor of 4.The electron transit time or the charging and dis-charging time of the channel is graphed in Figs. 17(a)-(b)as a function of the drain and gate voltage, respectively,using the expression given by (44).

Let us nowconsider a numerical example to get anindication of the contributions from various sources onthe switching time of the turn-on transient. For a P-


15/22

338 IEEE TRANSACTIONS ON ELECTRON DEVICES Ju

P I I I-2 0 2 6 a 10 12

v

(a)

I.2 i iI\ Y = 1

0 I I I I IT 10 12Fig. 17-The channel charge storage time constant versus (a) th edrain voltage and (b) gate voltage of an N-channel MOS tran-sistor. This channel timeconstant is also the electron transittime from the source to the drain.

channel MOB transistor switch, the following numericvalues are typical: channel length L = 10microns,channel width 2 = 250 microns, bulk resistivity =ohm-emscorresponding t o a donor impurity concentrtion of 4 X lo1* atoms/cxn3, average surface mobiliof holes p9 =100 cm'/volt-see, oxide thickness x. = 1000 and oxide capacitance Co = K o e o Z L / x , = 0.885 pAssuming tha t thedevice is switchedon by a gate voltagto a steady state drain current in the saturation regioof I D = IDS = 10 ma, then the saturation drain voltagcalculated using 15) is VDs = 6.9 volts. The chargintime constant of the channel is then 880 X 10-l'seeusing (46). If such a device is operated in a small signcircuit at this current level in the saturation region, thgain-bandwidth product from (41a) is 160 Me.

Let us next estimate the time constant RLC; taking load resistance of RL 1000 ohms. For this exampllet us assume that the diffused drain junction is shalloand essentially a P'N junction with a capacitance o15 pf/mm' at a reverse bias of V , = -20 v. The draijunction is assumed to have a length of 25 microns anits area is then 25 x 250 x lo-' em'. The total drainjunction capacitance is then C D = 0.094 pf. The drainlead going through the package has a capacitancethe order of 0.5 pf for a TO-5 transistor header. Thuthe main contribution of 7;s the stray circuit capactance which may be as high as 5 pf or more. Takinthis number as an estimate for Cl,, the circuit time constant is RLC; = 5000X seewhichsonsiderablygreater than the channel time constant of 880 X 10-sec. Thus, the turn-on transienthas a time constangiven essentially by the channel timeconstant of 88picosec. If the circuit time constant s considerably reduced by eliminating thetray capacitance, thethe turn-on timeconstant. is approximately given b

The relative importance of the circuit and channtime constants is reversed during the turn-off transienLet us assume that the turn-off transient starts a t somtime t, as illustrated i n Fig. 16. At t,, the gate voltagreturns t o zero and the channel is turned off with thchannel time constant of 880 picosec for the numericexample just given. However, the drain voltage transienu d t ) decreases much moreslowly towards -VDD sinthe capacitance C must be charged up to -VDD hrougRL. Thus, the turn-off timeconstant is approximateRLCl, = 5000 picosec. If the circuit time constantcomparable tohe channel time onstant, theotalswitch-off time constant would then be approximategiven by the SUM of the two time constants.

The otal switching time (turn-on plus turn-off)sthen limited by the circuit rather than the device. Thslow down by the circuit capacitance loading also occuin a complementary inverter circuit shown in Fig. 1whichemploys a complementary pair of enhancememode MOS t rans is tor~ .~~uch a circuit has essential

i t 0 = f ' C A ( R L C L ) / [ t C A + BLc;] .


16/22

1964 #ah: Characteristics of M O A Transistors 339+ DEI1

I1Fig. 18-Complementary inve rter circuit using a P-channel and anN-channel MOS transistor.zero standby power due to the power dissipation fromthe reverse biased drain junction leakage current, makingit particularly attractive in nanowatt logic circuit appli-cations. In these and other switching circuits, the princi-ple speed limitation from the stray circuit wiring capaci-tance may be largely overcome using integrated circuittechniques to limit the size of the drain junction areaand the interconnection

VII. COMPARISONF THE ELEMENTARYHEORYWITH EXPERIMENTS

In this section, the elementary physical theory ustpresented and the characteristics calculated from it arecritically compared with experimental data aken onthe circular device structure. The surprisingly goodgeneral agreement of the theory with experiments andsome of its inadequacies will be discussed in detail andit willbe the basis of an improved analysis t o be pre-sented in a subsequent paper.For an experimental test of the simple t


17/22

340 IEEE T R A N S A C T I O N S ON E L E C T R O N D E V I C E S

(d lFig. 19-The experimental characteristics of a circnlar P-channsilicon-siliconoxide MOS t,ransistor. (a ) The Gransfer charateristics. (b) The transconductance versus thegate voltag( e )The transconductance versus the dra in voltage. (d ) The shcircuit gate capacitance versus the gate voltage.


18/22

1964 Sah: Characteristics of M O S Transistors 341

(d lFig. 20-The experimental characteristics of a circular N-channelsilicon-siliconoxide MOS transistor. (a) The transfer charac-teristics. (b) The transconductance versus thegate voltage.( e ) The transconductance versus the drain voltage. (d) The shortcircuit gate capacitance versus the gate voltage.


19/22

342 IEEE TRANSACTIONS ON ELECTRON DEVICES Jcrepancy observed here is due in pa rt to the assumptionthat theeffective carrier mobility given by (5), the surfacecharge Q s s , and the bulk charge Q B given by (10) areall independent of the gate voltage or the surface electro-static potential.It is possible from the transfer characteristics given

in Figs. 19(a)-(c) t o obtain an effective carrier mobilityin the channel defined in (5). In order t o make thiscalculation from these experimental data,he oxidecapacitance over the channel surface area of the gateelectrode must be known. This can be obtained from thegeometrygiven for these devices and the otal gatecapacitance CG measured when the surface is stronglydegenerate so that it corresponds to he ota l oxidecapacitance of the gate electrode C O T o t e lor the entiregate rea A, = ZL,. The measurement of the gatecapacitance under degenerate surface condition may beobtained from the short circuit gate capacitance datashown in Fig. 19(d) where the can capacitance of 0.6 pfhas already been substrated. The tota l gate capacitancecorresponds to he maximum plateau value at largenegative or positive gate voltage and is C O T o t a l 19.6 pf.The oxide thickness may be calculated from COT=Ko~OAII/xOsing KO = 4.0 for the siliconoxide andA , = Z L , = 1.53 X 10 microns just obtained fromthe device geometry. Thus,x. = Ko~,A,/C0 otal = {5.42/[c0 .t,l(pf)])(microns)

= 0.276icrons.The gate capacitance over the active channel area is then

C = CoT(A, /A, ) = 0.55 C o T o t s l = 10.8pf48)using the value of the gate areas calculated at the be-ginning of this section from the device geometry.The effective mobility can nowbe calculated usingthe slope of the g m versus V o curve in he saturationregiongms/dV, given in Fig. 19(b) or the sloped a / d V G n Fig. 19(a)and the ideal theory givenby (15) and (19). The effective mobility isp = (2Lz /cO)(a6 /vG) tD = (L/~O)(agmS/v,)~D

= 36.4 ~ o ( a / a V G ) ; == 18.2(dgm,/dvG)V~m2/volt-sec (49)

where the oxide thickness is in microns, the drain currentin milliamperes and the transconductance per gate volt-age in micromhosper volt. For the P-channel deviceshown in Fig.19 the experimentally obtained effectivesurface mobility for holes in the channel is then p, = 115crn2/volt-sec. This is about four times less than hemobility of holes in the bulk of the N-type silicon sub-strate which s 6.0 ohm-ern and has a donor impurityconcentration ofND = 8.2 X 1014/cm3giving a bulkmobility of approximately 470 cm/volt-see.

The comparison of the short circuit input gate capatance calculated from the simple theory shown in F10 and the experimental data given in Fig. 19(d)considerably less satisfactory. It is evident from expected capacitance drop of C0/3 = 3.6pf from ideal theory which is labeled in Fig. 19(d) that the idtheory has neglected some capacitance contributionsthe short circuit gate capacitance. The discrepacomes again mainly from the contribution from dependence of the bulk charge Q B and the surface stcharge Q s s on the gate voltage which is neglected in ideal theory.A similar analysis can be made to he N-chanMOST whose experimental data are shown in Fi2O(a)-(d). The conclusions can again be made that ttheoretical quadratic dependence of the saturation drvoltage I D s on the gate voltage is well followed expermentally and similar discrepancies exist between theand experiments in the transconductance and gate capatance characteristics.

The analysis of the experimental data of these tdevices together with many other MOS transistorsthe same geometry but fabricated with differentoxithickness, bulk impurity concentration and surface stconcentration are listed in Table I. Each of the devilisted in Table I represents a typical unit of a groupabout ten devices made by a given process. All columare measured values or calculated from measured daThe column of the effective bulk charge, with a conctration NB, is calculated by assuming that the threshvoltage corresponds t o the voltage at the minimcapacitance of a MOS capacitor of the same oxide thiness and bulk concentration. This voltage denoted V,, is plotted in Fig. 21 using the wellknown Mcapacitance analysis.2o The corresponding bulk charN, (number per unit area) is obtained from Q B = qNBC o V B nd is graphed in Fig. 22. The density of the surfstates Nss is then evaluated from the calculated bcharge and the observed threshold voltage V T us

where the positive sign is for the N-channel device whthe negative sign is for the P-channel device. The dotype surface state density isshown in the last coluin Table I indicating surface state concentrationthe range of 3 to 10 X 1011/cm2.It is also interesting to note from the experimendata in Table I that the effective mobility of the carrin the surface channel is generally about 100 cm/volt-for holes and 500 cm2/volt-sec for electrons.

O D. R. Frankl, Some effects of materialparameters ondesign of surface space charge varactors, Solid State Electronvol. 2, pp. 71-76; January, 1961. Data in Fig. 21 is obtained frunpublished calculations made by A. S. Grove and C . T. Sah.


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1964 Sah: Characteristics of M O S TransistorsTABLE ISom EXPERIMENTALHARACTERISTICSF THE CIRCULAROS TRANSISTORS

343

RunNo. Substrate COT x0ResistitS c m Concentcm-3 f P19.6

0.2105.9 0.2264.00.276

28.6 0.19012.4 0.43829.0 0.2028.3 0.19215.0 0.3622.7 0.246.72 0.81

_____10.8 -12.213.2 -3.2514.2 -2.4515.7 -4.566.82 -4.3015.6 -9.0015.9 -4.88.25 -8.793.69 -23.012.5 -5.30

630417-4631010-3631010A-2631010B-2631011-16310118-6631011B-4631011C-36310143631014D-4

NPNNPPPPNN

10.56 .010.510.510.510.510.510.810.510.8

8 .2 X 10141 . 3 X 10l61 .3 x 10151.3 x 10154.6 x 10144.6 x 10144.6 x 10141.2 x 10154.6 X 10141 . 2x 106

1252323380562756211859

1.2 x1.521.61.250.890.890.89

100 9.76 X lo1 8.5 X3.132.55 4.64.12.18.4 5.3210.42 3.45.65 9.64.7

0volts

1

0.:

2-z kt

10101 I I I0 .2 .4 .6 .8 1.0x micronFig. 22-The to ta l effective bulk impurity concentration per unitarea QB = C,VB versus the oxide thickness of a silicon-siliconoxide MOS transistor.

Fig. 21-The contribution to the threshold voltage from the bulkimpurity dopingversus the oxide thickness in a silicon-siliconoxide MOS transistor.

trons in a bulk sample12 he drain current would be givenbY1, = PLoC0/L2)[a / 3 1

. [ (V,- V,) - (V , - v, - vD)3]1/2where E. is the critical electric field and is of the orderof a few hundred volts per em. Then the drain satura-Bell Sys. Tech. J.,vol. 30, pp. 990-1034; October, 1951. The analysisW. Shockley, Hot electrons in germanium and Ohms law,of the MOS transistor characteristics follows the procedure used inthe text and is similar t o th e case of the junction-gate field effecttransistor analysis made by Dacey and Ross in G. C. Dacey andpp. 1149-1189; November, 1955,who included thehot electronI. M. Ross, The field-effect transistor, Bell Sys. Tech. J. vol. 34,effect.

VIII. CONCLUSIONSFrom the critical comparison of the experimental

dataand he characteristics calculated from the ele-mentary theory, it appears that the dependence of thesurface state charge, the bulk charge, and he surfacemobility on the gate voltage or the surface potentialwell depth must be taken into account in the calculationof the differential characteristics of the device, expeciaIlyin the regionbelow saturation. In addition, since theelectric field parallel to the urface is of the order of V J L ,hot electron effect may be important. If the fieldde-pendence of the surface mobility is assumed t o be pro-portional t o (electric field)-/2 as in the case for hot elec-


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344 IEEE T R A N S A C T I O N # ON ELECTRON DEVICES Julytion urrent would have V, - VT 32 nstead of KO = the dielectric constant of the oxide( V , - V,) dependence. Thiscontradicts the experi- K , = the dielectric cont an t of the semiconductormental data which shows a parabolic dependence over k = the Boltzmann constanta wide range of current for these devices which have a L = the length of the channelrather long channel. However, for high frequency and L C = the length or width of the gate metal electrodehigh speed devices which have veryshort channel length, N , = the acceptor impurity concentration (no./cm)the hot electron effect should become important and N D = the donor impurity concentration (no./cm3)may dominate the saturation characteristics. n = the electron concentration (no./cm3)

LISTOF SYMBOLSvoltage gaingate area over the channel, A , = ZLtotal gate metal electrode area, A , = ZL,midband voltage gainshort circuit output or drain capacitancedrain o source equivalent circuit capacitance(zero)short circuit input or gate capacitanceopen circuit input or gate capacitancethe equivalent circuit gate to drain feedthroughcapacitanceCud nsaturation, i .e . , V , > VDs = VG- V,,(zero)the equivalent circuit gate to source capacitanceC,, in saturationthe capacitance corresponding to the oxide layerover the channel area A,, 6, = K O ~ O A c / ~ Othe capacitance corresponding to the oxide layerunderhe ntire gate metal electrode, C, =I C W L / X OColA, = K o ~ o /x cdiameter of thedmin island of the circulargeometryenergy level at the bottom of the conductionband edgethe equilibrium Fermi energy levelthe width of the energy gap a t a given tempera-ture involtthe intrinsic Fermi energy level, E , = --Qthe energy level at the top of the valence bandedgethe quasi-Fermi energy level of electronsthe Fermi energy level of electrons in the metalthe quasi-Femi energy level of holesthe load conductance connected between the

ni = the carrier concentration of an intrinsic speci-p = the hole concentration (no./cm3)Q B = the otal effectivebulk charge densityperunitQ C h = the total charge (coulombs) stored in the channelQCh S = QCh n saturationQ G = the otal charge (coulombs) stored on thegateQ s s = the surface state charge density perunitareaE , = the otal effective load resistance connected be-tween the drainand source electrodes, ar.=s = the inner diameter of the annular source islandT = the temperature of the sample in OKtCh = the charge storage time in the channelt C h 8 = tCh n the saturation regiont$m = the transconductance cutoff time constant, t,, =t gms = t,, in the saturation regiont L 7 = the carrier transit ime rom he source to he

men (no./cm3)area &B = COVBper unit area

electrode per unit area

l / i G 4 g d )in the circular geometry

1 G?n

drainnd source V D S =ga = theshort circuit drain conductanceg d s = gd in saturation (zero) v, =g, = the transconductance V G =gms = the transconductancen the saturation region VGG=ID = dc current flowing into the drain electrodeI D S = I D n theaturation region v, =J,, = the electron current density per unitarea along V , =

the channeldenoted by a single subscript of the electrode and the subscripts or X denotes the quantlty in the saturatl on reglon.

2% For common source connection, externd circuit voltages are v o =

drain electrode in the channelt t , in the saturationregionthe normalized electrostatic potential, u =the normalized quasi-Fermi potential of elec-trons,U,p+JkT = -F,/kTthe normalized quasi-Fermi potential of holes,the contribution to the threshold voltage f romthe bulk impurity dopingthe dc drain to source voltagethe dc potential at a distance y from the sourceelectrode measured relative to he source po-tential, V(y) = 4, - pthe normalized V(y) , (y) = qV(y)/lcT = u, u,the dc drain to source voltage at saturation,the small signal ac drain o source voltagethe dc gate to source voltagethe dc battery voltage applied between the gateand the drain electrodesthe small signal ac gate tosource voltagesthedcpotential differencebetween the sourceand the pinch-off point nearest to the source inthe channelthe de potential across the oxide a t a distancey from the source electrode

-Ei/kT = & / k T

U, p+,/kT - J k T

v,, = VG v P


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IEEERANSACTIONS ON ELECTRONEVICES 345the built-in potential across the oxide at equi- U, conductivity due t o electrons, U, = qpnnlibrium d r = the equilibrium Fermi potential, + p = -Er/qthe threshold gate voltage to urn on or turn + M = the electron work function of the metal of theoff a channel gate electrodethe width of the transition layer of the drain n the quasi-Fermi potential of electrons, += =junctionthe depth of the N + drain and source islandsthe thickness of the oxide over the channelthe distance measured perpendicular t o the oxide-semiconductor interfacethe distance from the source measured parallelt o the oxide-semiconductor interface along thedirection of the channelwidth of the device with the linear geometryshown in Fig. 2(d)permissitivity of free space, 8.85 Xfarad/cmmobility of the electrons in the channel,cm2/volt-sec.effective mobility of electrons in the channeleffective mobility of the current carriers in thechannelresistivity of the substrate in ohm-cms

, /q+ = the quasi-Fermi potential of holes, 9 = -F,/qX = the electron aEnity n he semiconductorfi = the electrostatic potential, fi = -EJqf i a = the electrostatic potential a t the oxide-semi-w = the angular frequencywarn the transconductance angular cutoff frequencyuUm8= warn n the saturation regionw L = the angular cutoff frequency due to he load

conductor interface

resistance R,.ACKNOWLEDGMENT

The author s indebted to V. H. Grinich, C. A. Bittmannand A . X Grove for many helpful suggestions on themanuscript and to 0. Leistiko and D. A . Tremere fordevice fabrication assistances. The author is also gratefult o Prof. J. Bardeen for comments and Dr. G. E. Moorefor permission t o publish this paper.

Frequency Dependenceof the Reverse-Biased Capacitanceof Gold-Doped SiliconP+N Step Junctions

Summary-The experimentally observed frequency dependencesof the reverse-biased capacitance of gold-doped silicon step junc-tions over the frequency range from 10 cps to 30 Mc are found tobe in agreement with a simple physical model which takes ntoaccount the charge condition and he charging and dischargingtime constantof the deep-gold acceptor level in the transition egionof the junction. Analysis based on the simple physical model pro-vides explicit theoretical formulas for the junction capacitance atlow- and high-frequency limits which show tha t the high-frequencycapacitance under eversebias is approximately proportional to. \ /ND - NA, and is considerably reduced below the low frequencyordc capacitance if thedonorsarenearly compensatedby thegold. The frequency effect is important for deep energy level im-purities and becomes negligible if the impurity level is at or nearthe bandedges. The presence of gold, however, has negligibleeffect on he avalanche breakdown voltage if NA < NO.

Manuscript received January 29, 1964; revised March 24, 1964.C. T. ah is with the Department of Electrical Engineering andV. G. K. Red& is with Fairchild Semiconductor Research andof Physics, University of Illinois, Urbana, Ill.Development Laboratory, Pa10 Alto, Calif.

HE REVERSE-BIASED junction capacitance ofa P'N junction has been generally observed t ofollowclosely the theoretical relationship of thedepletion approximation [l], [2] (see also Fig 1 for Ci2data)

co = /Wo = dPEND/2(VD- V). (1)In this case, the capacitance is independent of the signalfrequency of measurement. However, in highpeedsilicon diodes which contain high concentrations of goldimpurity for reducing the minority carrier lifetime andthe switching time, it has been found that the capacitanceis highly frequency dependent not only at low forwardbias where the diffusion capacitance is unimportant [2]but also at moderately large reversebias in contra-diction t o the simple theory given by 1). This experi-mental observation is demonEtrated in Fig. 1 in whichthe capacitance-voltage curves at 10 cps, 100 cps, 1000cps, 10 kc, 50 kc, 100 kc and 30 M c are shown for agold-doped and a nongold-doped P'N step junction of

Characteristics of MOS Transistors

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