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Microelectronics Reliability 46 (2006) 731–742 www.elsevier.com/locate/microrel

Understanding threshold voltage in undoped-body MOSFETs: An appraisal of various criteria F.J. Garcı´a Sa´nchez *, A. Ortiz-Conde, J. Muci Solid State Electronics Laboratory, Simo´n Bolı´var University, Apartado 89000, Caracas 1080, Venezuela Received 17 May 2005; received in revised form 5 July 2005 Available online 19 September 2005

Abstract Undoped-body MOSFETs are currently becoming increasingly important and the value of threshold voltage is often used to assess the reliability of fabricated devices. However there exists a disparity of threshold voltage criteria proposed for these novel devices. The concept of threshold voltage in undoped-body MOSFETs is examined and various existing criteria are analyzed and compared in an eﬀort to clarify the ambiguity of the meaning of threshold and understand its dependence on technological parameters in these devices. Phenomenological considerations are also presented to shed light on the behavior of the sub-threshold slope with changing semiconductor body thickness. 2005 Elsevier Ltd. All rights reserved.

1. Introduction The continuing miniaturization of MOSFETs is having considerable impact on their models and model parameters. The inherent randomness of dopant impurity locations within a very small channel, not only turns questionable the concept of impurity density itself, but also gives rise in real life to signiﬁcant ﬂuctuations of the MOSFET characteristics. To alleviate this serious problem, it has been proposed to do away altogether with doping the channel region and instead use an undoped (or lightly doped) body to sustain the channel. The idea of an undoped body, sometimes referred to as ‘‘intrinsic channel’’, is expected to become progressively more ubiquitous in the coming years as miniaturization further advances.

*

Corresponding author. Fax: +58 212 9064025. E-mail addresses: [email protected] (F.J. Garcı´a Sa´nchez), [email protected] (A. Ortiz-Conde), [email protected] (J. Muci).

Threshold voltage has been traditionally the most important parameter to model MOSFETs as a consequence of the prevalence of threshold voltage-based regional models. However, today the interest in knowing its exact value for circuit simulation purposes is diminishing with the increasing advent of non-threshold voltage-based global models, where this parameter is absent. Nonetheless, the concept of threshold still remains signiﬁcant especially from the point of view of the device designer. The eﬀect that the variability of technological fabrication parameters has on its value is often used as a measure to assess device reliability. The dependence of threshold voltage on gate oxide thickness and semiconductor ﬁlm thickness variations is of particular importance. The absence of dopant atoms in the undoped-body MOSFET channel turns worthless the concept of threshold voltage commonly used in the literature for doped body devices [1] based on the condition of surface potential pinning at a critical band-bending, approximately equal to twice the Fermi potential. Consequently, a

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variety of new criteria have been recently proposed in the literature for undoped-body MOSFETs. Some criteria use the strong conduction region of the transfer characteristics, others portray the sub-threshold behavior, yet others are based on transition points between weak and strong conduction. Because of this variety, reported values end up being dependent upon the particular criterion used in each case. Still, at present there seems to be no general consensus as to how to deﬁne the threshold voltage of undoped-body MOSFETs. It seems therefore pertinent to reconsider the very concept of what is meant by threshold in undoped devices and to compare the several schemes that have been put forward. In what follows we analyze the concept and compare existing deﬁnitions of threshold voltage for undoped-body MOSFETs. The related behavior of the sub-threshold slope factor as a function of semiconductor body thickness is also examined. We have chosen to purposely avoid the use of the terms ‘‘depletion’’ and ‘‘inversion’’ which are improper for these devices because of the intrinsic nature of the body where the channel is formed.

2. Undoped MOSFET basic model An ideal long channel symmetric undoped-body double gate (DG) MOSFET structure will be considered for generalityÕs sake, considering inﬁnite semiconductor ﬁlm thickness devices to be a limiting case. Undoped here means truly intrinsic. It does not pretend to include residual doping as may exist in some real so called ‘‘intrinsic’’ devices. An ideal symmetric DG MOSFET means that the two gates have the same work function, both are separated from the semiconductor by the same oxide and oxide thicknesses, and the same bias is applied to both of them. The direction across the channel will be referred to as x, with x = 0 deﬁned as the midpoint of the semiconductor ﬁlm body. It will be assumed that the quasi-Fermi level is constant across the body thickness (x direction), so that current ﬂows only along the channel length (y direction). The energy levels will be referenced to the electron quasi-Fermi level of the n+ source. No second order eﬀects, such as gate current, quantum conﬁnement, short channel, drain induced barrier lowering, carrier velocity, etc, will be considered, since the intention here is to examine only the fundamental concepts upon which basic threshold voltage deﬁnitions are built. An n-channel device will be used for discussion without loss of generality. Neglecting the contribution of holes and assuming w 1/b, where b = q/kT is the inverse of the thermal voltage, the gate voltage may be written using the Poisson–Boltzmann formalism as [2–5]: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 2kTni eS bðwS V Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ V GF ¼ wS þ e 2 1 ebðw0 wS Þ ; ð1Þ C ox

where VGF = VGS VFB, with VGS being the gate-tosource voltage and VFB the ﬂat-band voltage, V is the voltage along the channel, which is equal to zero at the source end and to VDS at the drain end, wS is the surface potential at either side of the semiconductor ﬁlm body (x = ±tSi/2), w0 is the potential extremum at the middle of the semiconductor ﬁlm body (x = 0), at which point the electric ﬁeld becomes zero, ni is the intrinsic carrier concentration, eSi is the semiconductor permittivity, Cox = eox/tox, is the gate oxide capacitance per unit area, eox is the oxide dielectric permittivity, tox is the oxide thickness, and k and T are BoltzmannÕs constant and absolute temperature, respectively. The surface potential, wS, and middle of ﬁlm potential extremum, w0, may be related by [3–5] bðw0 wS Þ ¼ cosðfÞ; ð2Þ exp 2 where the argument f is given by sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ q2 ni bðw V Þ tSi f¼ e 0 ; 2kT eS 2

ð3Þ

with values of f restricted to 0 6 f < p/2. Eq. (2) is valid for semiconductor ﬁlm thickness, tSi, of up to about twice the intrinsic Debye length, or close to 100 lm in silicon. For greater tSi, including the inﬁnitely thick case, the potential extremum at the middle of the semiconductor ﬁlm body will be assumed to be w0 = 0. For brevityÕs sake, we will refer to VGF = VGS VFB from here on as simply the ‘‘gate voltage’’. Substituting (2) into (1) yields the following gate voltage equation: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2kTni eSi bðwS V Þ V GF ¼ wS þ e 2 sinðfÞ. ð4Þ C ox The surface potential may be obtained solving (4) in terms of the terminal voltages [6], resulting in an expression of the form: wS ¼ V GF V ox ; where the oxide voltage, Vox, is given by " # rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ bðV GF V Þ 2 btox kTni eSi V ox ¼ W . sinðfÞe 2 b eox 2

ð5Þ

ð6Þ

In the above equation ‘‘W’’ represents a short hand notation for Lambert W functionÕs principal branch [7]. Notice that the solution in (6) is also valid for the inﬁnitely thick MOS device, in which case the semiconductor body thickness dependent term sin(f) ! 1 as f ! p/2 [8]. Fig. 1 presents sin(f) as a function of gate voltage, VGF, for diﬀerent values of silicon ﬁlm thickness, tSi. The curves reach a value of one at increasingly higher gate voltages, as tSi decreases. Values of sin(f) < 1 are a manifestation of the coupling between both gates. We may conclude that the DG device presents a

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and w0 both approximately follow VGF in the subthreshold region until they start to depart from VGF. Note that w0 stays close to wS, following VGF, only up to a point where it quickly saturates, while wS continues to raise. The saturation value of w0, w0 max, is a function of tSi, which is found by equating the argument given by (3) to its maximum possible value of f = p/2, that is, 2 1 2p eSi kT w0 max ¼ V þ ln . ð7Þ b q2 ni tSi

Fig. 1. Value of sin(f) as a function of gate voltage for several silicon ﬁlm thicknesses.

behavior similar to that of the inﬁnitely thick device in the range of gate voltage where sin(f) ! 1. Surface potential, wS, and middle of the semiconductor ﬁlm body potential extremum, w0, as functions of VGF are shown in Fig. 2 for diﬀerent values of tSi. They are calculated by solving (2) and (5) with a given oxide thickness, tox. We observe that, at the given tox, wS is practically independent of tSi for the range of 10 nm < tSi < 1, and equal to the surface potential of the inﬁnitely thick device. We also see that at ﬁrst wS

The vertical separation within the supra-threshold region between wS and w0 depends on tSi, in such a manner that w0 ! wS when tSi is very thin, and w0 ! 0 when tSi is very thick (tSi ! 1), as hinted by the evolution of the w0 curves in Fig. 2. The relevance of the middle of ﬁlm potential extremum saturation value, w0 max, might not be totally apparent from the observation of wS in Fig. 2, for the range of 10 nm < tSi < 1, where wS is practically independent of tSi and equal to that of the inﬁnitely thick device. However w0 max is of paramount importance to understand the behavior of thinner devices. As was already mentioned, w0 max increases with decreasing tSi. In very thin devices the value of w0 max may even surpass that of wS of the inﬁnitely thick device. When this happens, w0 forces its corresponding wS to be above that of the inﬁnitely thick device. The value of tSi where this starts to happen depends on the value of wS of the inﬁnitely thick device, which in turn depends on tox. Fig. 2 shows how wS for thickness tSi = 1 nm and 10 nm rises above that of the inﬁnitely thick device (for tox = 2 nm). This is better illustrated in the expanded view at the bottom of Fig. 2. The importance of w0 max will become even more evident when we analyze the behavior of the charge in the sub-threshold region. The eﬀect of varying gate oxide thickness, tox, is illustrated in Fig. 3. It indicates that changing tox only significantly aﬀects the supra-threshold behavior but not so the sub-threshold behavior. The evolution of the three surface potential curves shown in Fig. 3 already suggests a threshold shift towards lower gate voltage values when tox increases, since it indicates that the curves for larger tox depart from the wS VGF condition at lower VGF. This behavior will be discussed later. Mobile charge induced in the channel, which represents the total net charge in the semiconductor, may be easily computed with the use of Gauss Law at the surface together with the surface potential expression, since there is no ﬁxed depletion charge because of the undoped nature of the body: Q ¼ 2C ox V ox .

Fig. 2. Gate voltage dependence of the potential extremum, w0, at the middle of the ﬁlm (dashed lines) and surface potential, wS, (solid lines) for several values of ﬁlm body thickness.

ð8Þ

The factor of two in (8) accounts for the contribution of both symmetric halves of the semiconductor ﬁlm body. Substituting (6) into (8) yields the expression for the total mobile charge:

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Fig. 5. Mobile charge as a function of gate voltage for several semiconductor ﬁlm thicknesses. Fig. 3. Surface potential as a function of gate voltage for three oxide thicknesses.

" # rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4C ox btox kTni eSi bðV V Þ W e GF Q¼ sinðfÞ . b eox 2

ð9Þ

The eﬀect of tox on the mobile charge is illustrated in Fig. 4. It again indicates that changing tox only signiﬁcantly aﬀects the supra-threshold behavior. The three semilogarithmic curves shown in Fig. 4 also suggest, as does Fig. 3, the same threshold shift towards lower values when tox increases. Mobile charge is again presented in Fig. 5 at one value of tox for varying ﬁlm body thickness, tSi. We ﬁrst observe that in the supra-threshold region the curves are independent of tSi, except in the cases of very thin devices, tSi < 10 nm as shown. Secondly, we notice that in the sub-threshold region there is a vertical displacement of the curves with increasing tSi, which is consistent with the fact that sub-threshold volume charge is directly proportional to tSi. Fig. 5 also reveals that, as a consequence

Fig. 4. Mobile charge per unit area induced in the channel as a function of gate voltage for three oxide thicknesses.

of the increase in charge with increasing tSi, there is a horizontal shift of the whole sub-threshold characteristics towards lower values of gate voltage as tSi increases. Volume charge dominates in undoped DG MOSFETs when gate voltage begins to increase until w0 approaches w0 max, making sin(f) to depart from zero (see Fig. 1). Above this point volume charge ceases to increase signiﬁcantly and any further increase of gate voltage beyond this point will result mostly in increasing surface charge. Therefore, from that point on in the sub-threshold region sin(f) ! 1 and the charge of the DG MOSFET starts to approximately follow the same behavior as that of an inﬁnitely thick device. This behavior of the charge gives rise to the possibility of a change in the sub-threshold slope, which is clearly visible in Fig. 5 for the thicker devices. The double nature of the sub-threshold slope factor, S, of thick devices is presented for clarity in Fig. 6 for the case of a tSi = 1 lm device. The slope transition point from S = 60 mV/dec to S = 120 mV/dec occurs when wS approaches the value of w0 max, since below this point the slope is governed mainly by volume charge, and mainly by surface charge above it. It is important to emphasize that this slope change is only visible whenever the semiconductor ﬁlm is suﬃciently thick so that w0 max remains below wS of the inﬁnitely thick device. If the semiconductor ﬁlm thickness is so thin that w0 max surpasses the wS of the inﬁnitely thick device (for example tSi = 1 and 10 nm for tox = 2 nm), then the charge does not start to follow the sin(f) ! 1 behavior until reaching a higher gate voltage already within the supra-threshold region. That is, the S = 120 mV/dec sub-threshold region disappears. Thus, only one sub-threshold slope would be observable in such very thin devices, as presented in Fig. 5 for the two thinner devices shown (tSi = 1 and 10 nm). It seems therefore inappropriate to speak in general of a single sub-threshold slope in undoped DG MOSFETs,

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Fig. 6. Sub-threshold slopes for two semiconductor ﬁlm thicknesses. The intersection of the two straight lines deﬁnes the sub-threshold slope transition point.

because, in principle, there are two sub-threshold regions potentially present. Of course, in very thin DG devices only a single sub-threshold slope may be indeed evident throughout the entire sub-threshold region, as already mentioned. To aid in understanding sub-threshold slope behavior, S is presented in Fig. 7 as a function of VGF for various tSi, including the inﬁnitely thick device case. The transitions from a slope value of 60 mV/ dec to a value of 120 mV/dec are clearly visible for the intermediate tSi devices. The observed evolution of these two sub-threshold regionsÕ transition point with varying tSi posses additional fundamental questions as to how and where to deﬁne threshold in undoped devices. At this point it is perhaps pertinent to recall that we are ignoring quantum eﬀects in this classical analysis. Carrier quantum conﬁnement is very signiﬁcant in extremely thin DG devices (tSi < 10 nm) and should be taken

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into account when studying them. However, we have been considering such thin devices here without taking into account quantum conﬁnement with the aim of visualizing the evolution of the classical equations. A simple way to at least qualitatively consider carrier conﬁnement is to use a thickness correction resulting in a reduced eﬀective tSi and an increased eﬀective tox. Allowing for these corrections will cause the condition w0 max > wS of the inﬁnitely thick device to occur at a lower gate voltage than that predicted by the classical equations. The dual sub-threshold behavior might need not be considered in practical thin DG devices, since it is only exhibited by relatively thick devices. However, knowing how it comes about allows an understanding of its evolution from thin to thicker devices, and to grasp the nature of the sub-threshold characteristics shift as the semiconductor ﬁlm thickness changes, which aﬀects the threshold voltage.

3. Characterizing threshold voltage Threshold is frequently conceptualized as the condition where the conduction band of the undoped-body is bent close to the conduction band of the n+ source, at which point the surface potential is said to be ‘‘pinned’’ and no longer follows the gate voltage, as it does in the sub-threshold region [4]. This condition happens in undoped devices when the surface potential becomes approximately equal in magnitude to half the semiconductor band-gap energy (0.56 V for Si). This is of course only a crude estimate, since it is not clear exactly at what point the surface potential is to be assumed relatively independent of gate voltage. Obviously, this rough approximate criterion of threshold voltage is not a useful measure for variability assessment since it is a constant independent of technological parameters. Therefore more precise and parameter dependent criteria are needed as tools for reliability studies. 3.1. Linearly extrapolated threshold voltage

Fig. 7. Subthreshold slope as a function of gate voltage for several semiconductor ﬁlm thicknesses. Dotted horizontal line indicates S = 90 mV/dec.

A linear extrapolation approach has been traditionally used to determine threshold voltage in conventionally doped devices. It is deﬁned from the supra-threshold transfer characteristics as the value of gate voltage required to make the linearly extrapolated drain current (or mobile charge) go to zero, at vanishingly small drainto-source voltage. In undoped devices it could be equally deﬁned from the Vox vs VGF characteristics in the suprathreshold region, since this is equivalent to extrapolating the Q vs VGF characteristics, because the mobile charge is the total charge by virtue of the absence of ﬁxed depletion charge. We will ﬁrst consider an inﬁnitely thick MOSFET for simplicity, since the linearly extrapolated threshold

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voltage is obtained from the supra-threshold region, which is mostly independent of tSi, as we saw in Fig. 5. Letting sin(f) = 1 in (6) the oxide voltage at V = 0 is rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 2 btox kTni eSi bV V ox ¼ W e GF . ð10Þ eox 2 b Since the Lambert W function is fully diﬀerentiable, we may take the derivative of oxide voltage with respect to gate voltage: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 W bteoxox kTn2i eSi ebV GF b dV ox ð11Þ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ . dV GF 1 þ 2 W btox kTni eSi ebV GF eox 2 b Now the linearly extrapolated threshold voltage, VTe, for inﬁnitely thick MOSFETs may be described as V Te ¼ V GFe

V ox ; dV ox =dV GF jV GFe

ð12Þ

where VGFe is the point where the tangent of the Vox VGF curve is extrapolated to zero, as shown in Fig. 8(a). Substituting (11) into (12) we get, " rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# 2 btox kTni eSi bV 1þW ð13Þ V Te ¼ V GFe e GFe . b eox 2 It is obvious from (13) that this expression does not produce a unique value of threshold voltage, because it depends on the particular choice of VGFe. In practice the extrapolation point is usually chosen as the point of maximum slope in the drain current–gate voltage transfer characteristics. However, considering (10) we may easily conclude that the slope of Vox in the ideal device, and hence the slope of Q and IDS, do not reach a maximum but continuously increase with increasing gate voltage (see Fig. 8(b)). The fact that Q does not become linear as gate voltages increases can be also understood by ﬁnding its asymptote at inﬁnite gate voltage. This asymptote at V = 0 is given by ( 4C ox lnðuÞ lnðuÞ½lnðuÞ 2 Q¼ u lnðuÞ þ þ b u 2u2 ) lnðuÞ½9 lnðuÞ þ 2 lnðuÞ2 þ 6 þ þ ; ð14Þ 3u6 where bV GF u¼ þ ln 2

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 2eSi qbni . 2C ox

ð15Þ

In real devices the transconductance usually does reach a maximum since it starts to decrease mainly because of mobility and parasitic source and drain resistances eﬀects. It is therefore understandable that the linearly extrapolated threshold voltage may not be deﬁned independently of second order eﬀects on the drain

Fig. 8. Surface potential and oxide voltage (a), with oblique dotted lines corresponding to VGF and VGF/2. Their ﬁrst (b) and second (c) derivatives with respect to gate voltage. Vertical dashed lines indicate threshold voltages: VTe by linear extrapolation from Vox = wS (a), VT1/2 at dVox/dVGF = 1/2 (b), and VTsd at the maximum of the second derivative (c).

current, unless of course some speciﬁc criterion is a priori established for unambiguously choosing the extrapolating point VGFe in the ideal device. An adequate choice could be the point at which the surface potential wS and the oxide voltage Vox become equal. A threshold voltage criterion has been proposed by Taur et al. [5] for undoped symmetric DG MOSFETs based on linearly extrapolating their supra-threshold transfer characteristics to zero current. The proposed expression has the following form [5]: V Tt ¼ V 0 þ d. where VTt is the threshold voltage and sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 2 2 2eSi kT ; V 0 ¼ ln b tSi q2 ni

ð16Þ

ð17Þ

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3.3. A phenomenological view of threshold

and

2 qðV g V 0 Þ ; d ¼ ln b 4rkT

ð18Þ

where Vg is a gate voltage point in the characteristics to extrapolate from, and r¼

eSi tox . eox tSi

ð19Þ

Substituting back into (16) produces the ﬁnal threshold voltage expression for undoped symmetric DG MOSFETs: sﬃﬃﬃﬃﬃﬃﬃﬃﬃ# " 2 eox ðV g V 0 Þ 2kT V Tt ¼ ln . ð20Þ b 2kTtox eSi ni The above equation indicates, as any linear extrapolation criterion, that the value of threshold voltage will be dependent upon the point chosen to linearly extrapolate from. It also seems to indicate that the threshold voltage is independent of silicon ﬁlm thickness. However, a closer look reveals that it is in fact weakly dependent on tSi through the V0 term present in (20). Wong and Shi [9] propose to determine threshold voltage in inﬁnitely thick devices from the quasi-linear dependence of mobile charge on VGF in the supra-threshold region of the transfer characteristics. Quasi-linearity is said to be attained when the diﬀerential mobile charge capacitance becomes ten times the oxide capacitance. The resulting threshold voltage for undoped-body inﬁnitely thick MOSFETs is kT kT e2ox V Tws ¼ 2ðlnð10Þ 1Þ þ ln . ð21Þ q ni eSi q2 t2ox 3.2. Maximum of the second derivative A common way to extract the threshold voltage is to ﬁnd the value of gate voltage where the derivative of the transconductance is at a maximum [10]. According to this criterion, the threshold voltage of the ideal device is the gate voltage corresponding to the maximum of the second derivative of the carrier charge, or of the oxide voltage for that matter. Fig. 8(c) presents the second derivatives of Vox and wS of inﬁnitely thick MOSFETs showing the coincidence of their maximum and minimum, respectively. The point so deﬁned corresponds to the condition where the rates of change of Vox and wS with VGF are the steepest. Equating the third derivative of Vox given by (10) to zero and solving for gate voltage, an expression for the threshold voltage for inﬁnitely thick MOSFETs at the maximum of the second derivative, VTsd, may be written as kT kT e2ox 1 lnð2Þ þ ln V Tsd ¼ . ð22Þ q ni eSi q2 t2ox

Let us now look into a diﬀerent approach to deﬁne threshold based on the phenomenological idea that threshold represents a crossover from one type of surface potential behavior to another. Threshold voltage is then deﬁned as the gate voltage where this crossover occurs while going from one behavioral region to another: the region where wS VGF to the region where wS is almost pinned. According to (5) for any MOSFET we may write dwS dV ox þ ¼ 1. dV GF dV GF

ð23Þ

Based on the above, we propose that threshold voltage could be deﬁned as the value of VGF where the rates of change of wS and of Vox with respect to VGF become equal. The reason for this is can be seen in Fig. 8(b) and (b). Before threshold, as VGF increases, charge increases only slightly and thus the rate of change of Vox remains close to zero. After threshold the charge begins to rise steadily and the rate of change of Vox approaches one. Likewise, before threshold as VGF increases, most of the applied VGF is taken up by wS (semiconductor band bending) which closely follows VGF. Thus the rate of change of wS remains close to one. After threshold wS becomes relatively pinned as it becomes largely independent of VGF. Thus the rate of change of wS approaches zero. The behavioral crossover point occurs when the increasing rate of change of Vox and the decreasing rate of change of wS with VGF are both equal (=1/2). This should be compared to the already mentioned maximum of the second derivative criterion, by which threshold is the condition where the rates of change of Vox and wS with VGF are the steepest. Comparison of Fig. 8(b) and (c) indicates that the threshold voltage values obtained from these two deﬁnitions do not coincide. This is because neither the Vox characteristics nor consequently its derivative are symmetrical, as can be seen in Fig. 8(a) and (b). Equating the ﬁrst derivative of Vox with respect to VGF, given by (11), to 1/2 yields the threshold voltage for inﬁnitely thick MOSFETs kT kT e2ox 2 þ lnð2Þ þ ln V T1=2 ¼ . ð24Þ q ni eSi q2 t2ox 3.4. Other threshold voltage deﬁnitions An asymptotic criterion was proposed in 2003 for understanding the oxide thickness dependence of threshold in inﬁnitely thick devices. It is based on the intersection of the sub- and supra-threshold surface potential asymptotes [8]. The expression resulting from this intersection for inﬁnitely thick MOSFETs is given by

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V Ta

0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 b kTn2i eSi A; ¼ 2bW 1 @ C0

ð25Þ

where W1 is the short-hand notation for the 1 branch of the Lambert function [7]. Another threshold voltage criterion for undoped symmetric DG MOSFETs dependent on both silicon body and gate oxide thickness has been proposed by Sallese et al. [11]. It is based on deﬁning a charge qint such that 1 q V Ts ¼ ln int ; ð26Þ b 2 where qint ¼

qbni tSi . 4C 0

ð27Þ

This results in the following expression for the threshold voltage of undoped symmetric DG MOSFETs: 1 4eox V Ts ¼ ln . ð28Þ b qbni tSi tox There are still other methods to determine threshold voltage, such as those based on various current and transconductance ratios. Recently the use of the transconductance-to-current ratio has been proposed [11,12]. A method to extract threshold voltage is based on the fact that gm/ID deviates from its maximum value at weak inversion by a ﬁxed factor that depends only on the level of inversion [12]. Because gm/ID is equivalent to the inverse of sub-threshold slope factor, the method is conceptually analogous to the sub-threshold slope factor transition point criterion at a given value, which will be discussed in Section 4.1.

about 100 mV) as ﬁlm thickness decreases (from 200 nm). This interpretation should be approached with caution, since the shift, which corresponds to volume charge and occurs while wS < w0 max, can not always be attributed to threshold since there are cases where there is a double sub-threshold slope present, as was earlier mentioned. The fact is that only the sub-threshold Q VGF characteristics with sub-threshold slope factor, S = 60 mV/dec are shifted by changing tSi, as seen in Fig. 5. As was earlier explained, that shift only aﬀects the threshold point if the semiconductor ﬁlm is so thin that w0 max becomes larger than wS of the corresponding inﬁnitely thick device. Only in those cases, a shift of the sub-threshold characteristics towards larger gate voltages with decreasing tSi will increase the threshold voltage point. 4.1. Sub-threshold slope factor The value of the sub-threshold slope factor, S, may be used as an indication of the transition point between the two sub-threshold regions, that is, from DG-like (S = 60 mV/dec) type of behavior to inﬁnitely thick-like (S = 120 mV/dec) type of behavior. One way to represent this transition is by the point where the sub-threshold slope factor, S, is at a mid value of 90 mV/dec. This transition point is indicated in Fig. 6 by a dotted horizontal line. Notice that in the thinner devices (below 10 nm for tox = 2 nm) there is not an intermediate sub-threshold region with S = 120 mV/dec and the characteristics go directly from sub-threshold with S = 60 mV/dec to supra-threshold. This means that in these cases the sub-threshold transition point value is larger than the threshold voltage of the corresponding inﬁnitely thick device, and thus, it itself assumes the role of threshold voltage in these thin body DG MOSFETs.

4. Shift of the sub-threshold characteristics 4.2. Drain current components A gate voltage shift of the sub-threshold characteristics is observed in DG MOSFETs when the semiconductor ﬁlm thickness is changed. Wong and Shi [13] have proposed that in addition to the threshold voltage obtained from the supra-threshold region a second threshold voltage be introduced for DG devices deﬁned within the sub-threshold region. This ‘‘sub-threshold’’ threshold voltage is introduced to account for the gate voltage displacement of the sub-threshold characteristics towards lower values as tSi changes, such as described earlier and depicted in Fig. 5. To that end, a point is deﬁned in the sub-threshold region at an oxide ﬁeld of 3 · 103 V cm1. Shifts in such a threshold voltage are obtained by subtracting the corresponding gate voltage for tSi = 200 nm from the gate voltage for a given tSi < 200 nm. According to this deﬁnition, this ‘‘subthreshold’’ threshold voltage is said to increase (by

The transition from DG-like behavior to inﬁnitely thick-like type of behavior may also be determined also from the drain current. The drain current as a function of the terminal voltages of a symmetric undoped-body DG MOSFET for all bias conditions can be written as [14,15]: W 1 2 2 ID ¼ l 2C 0 V GF ðwSL wS0 Þ ðwSL wS0 Þ L 2

kT þ 4 C 0 ðwSL wS0 Þ þ tSi kTni ebðw0L V DS Þ ebw00 q ð29Þ where l is the mobility, W the channel width, L the channel length, and the second subscripts ‘‘L’’ and ‘‘0’’ indicate the potentials at the drain and source ends

F.J. Garcı´a Sa´nchez et al. / Microelectronics Reliability 46 (2006) 731–742

Fig. 9. Drain current as deﬁned by (9.1) and its two components IDw and IDs.

y = L and y = 0, respectively. This equation is equivalent to that presented in [5], although the derivation and appearance are diﬀerent. Other approximate charge-based expressions for the current that have been proposed may be related to this formulation [15]. At low VGF the ﬁrst term in (29) vanishes. The third term is negative, but together with the second term gives

W ð30Þ I Dw l tSi kTni ebw00 ebðw0L V DS Þ . L This sum of the second and third terms given in (30) represents the weak conduction region component of the current. The remaining component that dominates at high VGF in the strong conduction region is I Ds ¼ I D I Dw .

ð31Þ

Fig. 9 presents the normalized (divided by lW/L) total drain current and the two components IDw and IDs, separately plotted vs gate voltage in order to visualize their relative importance. Since the total current is approximated by IDs in the strong conduction region and by IDw in the weak conduction region, the location of the intersection on the gate voltage axis of these two current components, shown in Fig. 9, approximately corresponds to the already mentioned transition from the S = 60 mV/dec sub-threshold region to the S = 120 mV/dec sub-threshold region, as will be shown later in Fig. 11.

5. Discussion Threshold voltage should be understood as the crossover into the strong conduction region where the surface potential is said to be practically pinned. Fig. 10 presents a comparative view of the variation of ﬁve such threshold voltage criteria with gate oxide thickness for inﬁ-

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Fig. 10. Oxide thickness dependence of various threshold voltage criteria for undoped inﬁnitely thick MOSFET: asymptotic potential intersection VTa, linear extrapolation VTe, VT1/2 at dVox/dVGF = 1/2, and VTsd at maximum of d2 V ox =dV 2GF .

nitely thick devices. Although the magnitudes are diﬀerent, depending on the particular deﬁnition, all of them decrease logarithmically with increasing gate oxide thickness. It is important to point out that this behavior is opposite to that exhibited by highly doped-body (>1017 cm3) MOSFETs, where the threshold voltage increases with increasing gate oxide thickness due to the presence of ﬁxed depletion charge. Between these two types of behavior, that is, threshold voltage decreasing or increasing with tox, there is a doping concentration range (around 1015 cm3) where the threshold voltage remains fairly constant for varying gate oxide thickness. This observation is in complete agreement with a doping concentration dependence study of threshold behavior recently published [9]. The relative invariance of threshold voltage with tox within a certain doping level might be of technological interest and have practical implications [16]. The asymptotic threshold voltage criterion, VTa, in (25) based on the intersection of the sub- and suprathreshold surface potential asymptotes produces the highest values. It can be demonstrated that the value produced by this criterion corresponds to half the value of gate voltage at which the surface potential wS and the oxide voltage Vox become equal, indicated at the upper right corner of Fig. 8(a). The use of this criterion led to predict for the ﬁrst time that, contrary to conventionally doped devices, the threshold voltage of undoped MOSFETs should decrease as the oxide thickness increases [8]. The same type of behavior was later conﬁrmed for undoped DG MOSFETs [9]. The linearly extrapolated threshold voltage, VTe, criterion gives the second highest values, when the extrapolation point is chosen at the point where the surface potential wS and the oxide voltage Vox become equal.

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Table 1 Threshold voltage (V) of undoped inﬁnitely thick MOSFETs according to several criteria and linearly extrapolated from MEDICI simulated current tox (nm)

VTa

VTe

VT1/2

VTSW

VTsd

MEDICI

1 10

0.638 0.500

0.579 0.477

0.553 0.433

0.551 0.431

0.491 0.371

0.560 0.455

The criterion at the value of VGF where the rates of change of wS and of Vox with respect to VGF become = 1/2, VT1/2, produces results very close to those of VTws. However, whereas VTe may be easily measured from the drain current–gate voltage characteristics, VT1/2 may not be measured directly. The maximum of the second derivative, VTsd, which may be easily measured, exhibits similar behavior but lower values. VT1/2 may be obtained from VTsd just by the addition of a constant. Regardless of the particular criteria, all indicate that threshold voltage decreases logarithmically with increasing gate oxide thickness. Table 1 presents the threshold voltage of inﬁnitely thick MOSFETs with two oxide thicknesses as predicted by several criteria and linearly extrapolated from MEDICI simulations from [9]. Threshold voltage of thick devices is independent of semiconductor ﬁlm thickness, and practically equal to that of the corresponding inﬁnitely thick device, because the transition from one sub-threshold region (S = 60 mV/dec) to the other (S = 120 mV/dec) occurs below threshold and does not aﬀect it. But if the device is suﬃciently thin the threshold voltage may increase with decreasing semiconductor ﬁlm thickness. This is because the transition point from one sub-threshold region to the other begins to determine threshold whenever the gate voltage at this point becomes larger than the inﬁnitely thick device threshold voltage. We may therefore distinguish three distinct conduction regions in a symmetric undoped-body DG MOSFETs: two sub-threshold weak conduction regions with sub-threshold slope factors of S = 60 and 120 mV/dec, and one supra-threshold strong conduction region. In thick devices the three regions are clearly visible in the transfer characteristics. The two sub-threshold slope regions have been reported before in thick devices. For example, a sub-threshold slope change in a 200 nm thick device is apparent in Fig. 2 of [13]. Another more obvious observation of the two sub-threshold slopes has been recently reported [11]. The transconductance to current ratio presented (Fig. 4 [11]), which is equivalent to the inverse of the sub-threshold slope factor, clearly shows the presence of two sub-threshold slopes in the thicker devices (tSi = 200 nm and 1 lm). The sub-threshold region with slope factor of S = 120 mV/dec disappears in thin devices, as presented in Fig. 5, and there is a direct transition from the S = 60 mV/dec to the supra-threshold region.

Fig. 11. Semiconductor ﬁlm thickness dependence of various criteria for undoped symmetric DG MOSFETs: threshold voltage at the second derivative maximum (circles); subthreshold slope transition point, deﬁned by S = 90 mV/dec (triangles); and intersection of drain current components (squares). Also shown for comparison is the saturation value of the middle of ﬁlm potential extremum (dotted line).

Fig. 11 presents the sub-threshold transition point variation with silicon ﬁlm thickness. It includes V90, the transition deﬁned at the midpoint of 90 mV/dec between the two sub-threshold regions, and the transition deﬁned at the intersection of the two current components. The maximum value of the potential extremum at the middle of the body, w0 max, is also included for reference. Additionally the ﬁgure presents the threshold voltage deﬁned by the maximum of the second derivative, VTsd. The sub-threshold slope transition point, deﬁned either by V90 or by the current components intersection, decreases towards zero gate voltage as the semiconductor ﬁlm thickness increases towards inﬁnity. The variation of w0 max with tSi, as given by (7), is responsible for this behavior of the sub-threshold slope transition point which occurs slightly below w0 max, as shown in Fig. 11. Fig. 12 presents a closer look at the semiconductor ﬁlm thickness dependence of the maximum of the second derivative threshold criterion for three oxide thicknesses in undoped symmetric DG MOSFETs. It shows a dip between the decreasing values for thin devices to the almost constant values for the thicker devices. This dip is hardly noticeable in Fig. 11 because of the scale and the few data points shown there. Fig. 12 illustrates that the fact that threshold voltage decreases with increasing gate oxide thickness is also correctly portrayed by VTsd for both thick and very thin undoped DG MOSFETs. As semiconductor ﬁlm thickness is reduced to very small values, the sub-threshold region with sub-threshold slope factor of S = 120 mV/dec vanishes when the sub-threshold transition point surpasses the value of

F.J. Garcı´a Sa´nchez et al. / Microelectronics Reliability 46 (2006) 731–742

Fig. 12. Semiconductor ﬁlm thickness dependence of second derivative maximum threshold voltage criterion for three oxide thicknesses in undoped symmetric DG MOSFETs.

the corresponding bulk device threshold voltage (see Figs. 5 and 7). Then, the transition point itself deﬁnes the threshold voltage of these very thin devices, which increases with decreasing semiconductor ﬁlm thickness. We therefore conclude that the sub-threshold slope transition point may adequately represent threshold only for very thin DG devices, and that it would predict a wrong value if used to represent the threshold voltage of thick devices. On the other hand, the value measured at the maximum of the second derivative (of oxide voltage, mobile charge, or drain current), VTsd, adequately portrays the threshold behavior of both thin and thick devices. It stays fairly constant with tSi, for a wide range of tSi, but it is forced to follow the sub-threshold transition point at very small tSi. A threshold voltage measured at the maximum of the second derivative of the current therefore seems to be an adequate tool to examine the behavior of the threshold voltage with respect to semiconductor ﬁlm and oxide thickness for both thick and very thin undoped DG MOSFETs.

6. Conclusions We have presented a critical appraisal of several existing threshold voltage criteria for undoped-body MOSFETs, both symmetric DG and inﬁnitely thick devices, and have introduced some new phenomenological concepts. Threshold voltage decreases with increasing gate oxide thickness and is largely independent of semiconductor ﬁlm thickness, unless the device is very thin, in which case the threshold voltage increases with decreasing semiconductor ﬁlm thickness. Furthermore, we have shown that in theory there are two sub-threshold weak conduction regions present in undoped DG MOSFETs, with theoretical sub-threshold slope factors of S = 60 and 120 mV/dec. The value of

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VGF where the transition between these two sub-threshold regions occurs decreases for increasingly thicker devices. In very thin devices the sub-threshold slope transition point itself deﬁnes the threshold voltage, whereas in thick devices it does not adequately portray the threshold voltage. Threshold voltage criteria based on the sub-threshold slope transition point alone can not represent the threshold behavior of undoped devices since these criteria wrongly indicate that the threshold voltage value goes to zero as the semiconductor ﬁlm thickness increases to inﬁnity. Conversely, tSi-independent criteria, as those derived only from the supra-threshold region, fail to portray the sub-threshold slope transition point which determines threshold in very thin devices. Finally, it is proposed that the threshold voltage criterion based on the maximum of the second derivative of the charge, and consequently the measurable position of the drain current second derivative, is a very suitable way to portray the threshold voltage behavior of both thin and thick devices as a function of semiconductor ﬁlm and oxide thicknesses for reliability assessment purposes.

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