A compact non-quasi-static extension of a charge-based MOS model

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A compact non-quasi-static extension of a charge-based MOS model
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  IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 48, NO. 8, AUGUST 2001 1647 A Compact Non-Quasi-Static Extension of aCharge-Based MOS Model Alain-Serge Porret, Jean-Michel Sallese, and Christian C. Enz  , Member, IEEE   Abstract— This paper presents a new and simple compactmodel for the intrinsic metal oxide semiconductor (MOS) tran-sistor, which accurately takes into account the non quasistatic(NQS) effects. This is done without any additional assumptionor simplification than those required in the derivation of theclassical description of the MOS channel charge. Moreover, themodel is valid from weak to strong inversion and nonsaturationto saturation.The theoretical results are in very good agreement with mea-sured data performed on devices of various channel length, from300 m down to 0.5 m, and in various modes of operation.  Index Terms— Charge-sheet model, compact modeling, EKVmodel, MOS transistor modeling, non-quasi-static (NQS). I. I NTRODUCTION A LTHOUGH the small-signal non quasistatic (NQS)regime of the intrinsic metal oxide semiconductor (MOS)transistor has already been investigated in several papers[1]–[10], these descriptions are often mathematically cumber- some, so that they give little insight to the circuit designers.Moreover, the operating frequency of most high frequencycircuits is forced in a range where NQS effects are almostnegligible by the capacitive load of the active devices.However, when an inductive load tunes out the capacitance atsome nodes, NQS effects might be the limiting factor. Also,the PMOS biasing current sources of RF circuits might easilysuffer from these effects because of the lower mobility of holes.Finally, low frequency circuits sometimes require extremelylong devices, like 100 m, exhibiting an NQS cutoff frequencymuch lower than 1 MHz.If the NQS regime must be checked for, a well known, butcrude, first-order approximation can be used [11], [12]. How- ever, there is no fundamental reason for banning devices oper-ating in this mode, provided that they have not to be active athigh frequency. The goal of this paper is therefore to provide asimple model to accurately predict the small signal behavior of a MOS transistor at any frequency.  A. DC Model This work is based on a charge-based description of the MOStransistor as was first proposed by D. E. Ward in [13]. Such Manuscript received August 25, 2000; revised March 19, 2001. The reviewof this paper was arranged by Editor A. Marshak.A.-S. Porret is with UKOM, Inc., San Jose, CA 95114 USA.J.-M. Sallese is with the Electronics Laboratory, Swiss Federal Institute of Technology (EPFL), Lausanne CH-1015, Switzerland.C. C. Enz is with the Swiss Center for Electronics and Microtechnology(CSEM), Neuchâtel CH-2007, Switzerland.Publisher Item Identifier S 0018-9383(01)05727-6. a description is for example used in the EKV compact model[14]. It has the advantage of being physically based and providesimple analytical formulations for long channel devices whichare valid in all modes of operation.In the context of charge-based models, the drain current iswrittenasthedifferencebetweenaforwardcomponent ,whichissolelydependentonthelocalchargedensityatthesource,andareversecomponent ,whichdependsonlyonthedraincharge.Strictlyspeaking,thispartitioningisonlyvalidwhenthemobilitycanbeassumedconstantalongthechannel,i.e.,atlowelectricalfields. However, mobility reduction due to the vertical field canbe taken into account by decreasing the mobility globally. Also,velocitysaturationcanbedealtwithbydividingthechannelintotwosections,onewithaconstantmobilityandanotherwherecar-rierstravelatsaturationvelocity[15],[16]. The states of charge inversion at the source and drain are de-scribed by the variables and , respectively, which have nodirect physical meaning and areunitless. and tend towardinweakinversionandincreaseproportionallywiththelocalcharge densitiesat thesource ordrain instrong inver-sion. The equations 1 linking these intermediate variables to thevoltages applied at the transistor terminals are then, from [14],[17], or [18] (1)where, ,and the source, drain, and gate voltages referredto the bulk;threshold voltage at ;thermodynamic voltage;transistor slope factor due to the body effect(see [11] and [14] for an expression of this parameter, or [19] if polysilicon depletion isnot negligible in the gate).With and given, the dc current becomes simply(2)with (3)whereand effective width and length of the device,including channel length modulation(CLM); 1 To avoid any unnecessary repetition, only the expressions related tothe source will be given. Use subscripts in bracket to get the expressionscorresponding to the drain0018–9383/01$10.00 ©2001 IEEE  1648 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 48, NO. 8, AUGUST 2001 Fig. 1. Small signal NQS equivalent circuit. gate oxide specific capacitance;effective average mobility, possibly takingintoaccountthehighfieldeffectspreviouslycited.Finally, the dc source, drain and gate transconductances ,and are given by(4)(5)The low frequency transcapacitances are not listed here,since they can be deducted from the general NQS expressionsor found in [9], [12], [14]. This simple and efficient formulation gives a good description of the MOS operation, from weak tostrong inversion and nonsaturation to saturation, and can beextended to introduce advanced features [14], [18]–[20].  B. NQS Small Signal Operation Keeping the same framework than for the dc equations, ithas been previously shown in [9] that a compact formulationof NQS effects can be derived without any additional assump-tion. These results, which are briefly reminded in an appendix,are at the root of the derivation presented in the next sections.However, the initial formulation is difficult to manipulate, sinceit is expressed in term of Bessel functions of fractional ordersand of complex arguments. Such functions are not available inmost programming environments, and their numerical evalua-tion tends to be slow and have a poor convergence.Therefore, the results call for a simplification in order to be-come practical. Section II proposes a simple equivalent small-signal circuit, where only two fundamental functions of fre-quency are sufficient for expressing the values of all the ele-ments.Then,byusingacarefulnormalization,SectionIIIshowsthat these functions can be represented in two simple abacus,which are valid for any transistor operating in any state. Next,in Section IV, approximate expressions are proposed. Finally,Section V compares the model with various measurements.II. E QUIVALENT  C IRCUIT By studying the relationships between the generic (trans)ad-mittances expressions given in [9], the equivalent circuit of Fig. 1 can be deducted, where the only remaining variables are TABLE IN ORMALIZATION  F ACTORS , , , and . The complete expression for thesequantities can be found in the appendix.Note that, from this point, all quantities (currents, voltages,frequencies, impedances) are normalized according to Table I.Normalized, unitless, quantities will be denoted by using lowercase letters , as opposed to “normal” variables written incapitals .Thanks to the intrinsic device symmetry, the expressions of and can be obtained by exchanging and in theformulation of and(6)(7)Therefore the complete solution of the small-signal NQS be-havior of the MOS model can be reduced to the two functionsand .III. F URTHER  F REQUENCY AND  A DMITTANCE  N ORMALIZATION A further step can be achieved by separating each of the tworemaining independent functions, and , into two factors:a low frequency term and an NQS term. At the same time, anadditional normalization of the frequency can be introduced insuch a way that the NQS term will depend only on the ratio( at the source, at the drain). Thesource factor , is close to 1 in weak inversion orlinear region. In strong inversion mode, it tends toward inforward saturation mode and toward zero in reverse saturation.Inordertoachievethissimplification,anewnormalizedchar-acteristic frequency is introduced. It corresponds to the  PORRET  et al. : COMPACT NON-QUASI-STATIC EXTENSION OF A CHARGE-BASED MOS MODEL 1649 (a) (b)(c) (d)Fig. 2. To the left: Normalized             function for      (thick gray curve) and for        or        (thin curve).  To the right:  Normalized            function for      (thick gray curve), for        (lowest thin curve) and for                           and        (other thin curves). first pole of the characteristic function of the system ( i.e.,  of thedenominator which isshared by both functions, in the ap-pendix)(8)Note that the expression of is symmetrical in and ,so that it is identical for the drain and source transconductances.With this new definition, and from the relations recalled inthe appendix, one can state(9)(10)with(11)(12)and are the normalized, low-frequency, fundamental actransconductance and capacitance.At frequencies much lower than , corresponding toquasistatic (QS) operations, the two NQS functions andtend toward unity resulting in(13)(14)It can be verified that these last results correspond exactly tothe usual small-signal QS model [12], [14]. In particular, the imaginary part of in (13) corresponds to the drain-sourcetranscapacitance.Itcomesfromthefirst-orderTaylorexpansionof in (9).The two NQS functions and are plotted inFig. 2, versus the renormalized frequency , and fordifferent values of the parameter . The two leftmostplots show that the transadmittance of the MOS device, quicklydegrades with frequency. Indeed, a phase lag slightly in excessof 45 is already observed at and a factor of 2 is lostat . Therefore, if NQS effects are to be avoided, thecondition must be enforced. Surprisingly, the functionappears to be only weakly dependent on parameter , so thatthis dependence can almost be ignored in practice.The two rightmost plots of Fig. 2 show the degradation of the fundamental instrinsic capacitance with frequency. In theNQS regime, the channel must be considered as a distributed,nonuniform, RC line. So, , which increases proportionally toat low frequencies, start to vary only with . Therefore,for and for . The exact factorof proportionality at high frequencies depends on the ratio. For (reverse saturation), the dB per decadeslope is only reached at extremely high frequencies, which arenot of practical interest and are not shown on the plots. Finally,it must be noted that the curves vary very little between ,  1650 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 48, NO. 8, AUGUST 2001 corresponding to the linear regime or the weak inversion mode, and , corresponding to complete saturation instrong inversion mode .IV. A PPROXIMATE  E XPRESSIONS  A. Linear Region or Weak Inversion ( ) The special case is important since, as noted above, itincludes both the triode regime when and, asymptoti-cally, the weak inversion mode. Moreover, theMOS behaviorinsaturation mode can be approximately described with the sameset of functions.From [9], the following simple equations can be derived:where (15)where (16)with (17)  B. Forward Saturation ( and ) For the saturation mode, from weak to strong inversion, ap-proximations which are within an accuracy of a few percentscan be derived from (9)–(12) and Fig. 2(18)(19)(20) C. General Case, Valid in any Mode of Operation As previously stated, the function can already beconsidered as independent of , even for (again, seeFig. 2). However, this is not the case for the function , thebehavior of which varies significantly when . Even so,the following approximation can be used:(21)(22)with (23)Factor is always comprised between 0 and 1.17. The approximation is done in such a way that theasymptotic behaviors, both for and , are pre-served. The accuracy is always kept better than 1% of the dctransconductance value.V. E XPERIMENTAL  R ESULTS  A. Low-Frequency Impedance Measurements In order to verify our theoretical approach, low frequencymeasurements on very long devices have first been undertaken.This kind of measurements are relatively easy to realize andquite accurate, since a precision LCR-meter can be used. Be-cause the device can be made fairly large, the extrinsic elements Fig. 3. Two-port on-wafer measurement setup.Fig. 4. Measurement of the    function (symbols), deduced from thenormalized drain-to-gate transadmittance in linear mode, measured with anHP4285 LCR-meter. The device is a   2       m PMOS transistor. The gateoverdrive voltage was set to 0.2, 0.5, 1.5, and 2.5 V. The curves correspond tothe theory. can be ignored, eliminating the need for a manual de-embed-ding. Fig. 4 is an example of such a measurement performed ona m PMOS device integrated in a 0.35 m process,and for a frequency varying between 75 kHz and 30 MHz. Itshows an excellent agreement with theory, for a broad range of overdrive gate voltages, over three decades of frequency. Theslight discrepancies appearing in the high frequency range areattributed to the extrinsic elements, mostly due to the packagingofthedevice.Notethat, sincehighgate potentialwhere applied,the effect of mobility reduction had to be taken into account. Itwas extracted from a standard dc measurement.  B. High-Frequency -Parameters Measurements In order to verify the validity of the theory for shorter devicesand at higher frequencies, test structures for on-wafer measure-ments were realized. Devices of various length between 10 and  PORRET  et al. : COMPACT NON-QUASI-STATIC EXTENSION OF A CHARGE-BASED MOS MODEL 1651 Fig. 5. Normalized gate transadmittance function, computed from the measured    parameter of the setup of Fig. 3 , with the device operating in saturationmode. 10    m NMOS device measured between 1 MHz and 1 GHz (left), 1.5    m (center) and a 0.5    m (right) PMOS devices measured between 100 MHz and10 GHz. The top plots represent magnitude, the bottom ones phase in degrees. The    axis is always the normalized frequency 2       . In each case, the gateoverdrive voltage is varied between 0 and about 1 V, corresponding to the moderate and strong inversion regions. The black curves correspond to the theory. 0.5 m were integrated on a 0.35 m process. The measure-ments were carried out by using a standard wafer prober stationand a network analyzer. This kind of -parameters measure-mentsarelessaccuratethanwithanLCR-meter,buttheyaretheonly available technique at frequencies higher than a few MHz.Note also that the measurements are even made more difficultsince a wide impedance range is required, in order to cover alarge frequency and operating point range.The two-port measurement setup is depicted in Fig. 3. Theselected de-embedding process is two-step. First, as usual[21], [22], the pad and interconnects contribution must be removed from each device under test (DUT) measurements bysubtracting the -parameters matrix of an “open” test structurefrom the -matrix of the DUTs. For the frequency range thatwas required here GHz , the series access impedancecan be assumed negligible.The second step consists in removing the effect of the ex-trinsic elements (mostly overlap and junction capacitor). Thiswas done by subtracting two -matrix of the same DUT, thefirst one with the device biased normally, and the second oneat a zero gate voltage, in order to almost achieve the flat-bandcondition under the gate.The resulting -parameters, normalized to , can be ex-pressed in terms of the fundamental transconductances(24)This relation assumes that the extrinsic capacitances vary littlewith the gate voltage. As this condition does not completelyapply to the overlap capacitances, the validity range of this kindof extraction is limited. However, if not perfect, this processstill appears to be the only simple way to capture the intrinsicbehavior of the device without adding many tuning parameters.In saturation mode and in the linear region , re-spectively, the -matrix reduces to(25)(26)Parameter in saturation, and in the linear regionhave been measured on different devices. They are plotted inFigs.5–7,respectively,andarecomparedtothetheoreticalvaluesusing the following expressions, derived from (25) and (26):(27)In each figure, the measured normalized phases and magni-tudes are compared to their theoretical value for a 10 m length
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