Solid-State Electronics 76 (2012) 112–115

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Solid-State Electronics

journal homepage: www.elsevier .com/locate /sse

Letter

A formula for the central potential’s maximum magnitude in arbitrarily dopedsymmetric double-gate MOSFETs

Francisco J. García-Sánchez ⇑, Adelmo Ortiz-CondeSolid State Electronics Laboratory, Simón Bolívar University, Caracas 1080, Venezuela

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 January 2012Received in revised form 28 April 2012Accepted 1 May 2012Available online 29 June 2012

The review of this paper was arranged byProf. S. Cristoloveanu

Keywords:Double Gate MOSFETSymmetric DG MOSFETDoped-body DG MOSFETUndoped-body DG MOSFET

0038-1101/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.sse.2012.05.001

⇑ Corresponding author.E-mail addresses: fgarcia@usb.ve, ortizc@usb.ve (F

We propose here a simple approximate, yet accurate, formula to easily calculate the maximum magni-tude of the electric potential at the channel width’s midpoint of symmetric Double Gate (DG) MOSFETswith any arbitrary body doping concentration. According to it this maximum magnitude depends only onthe body’s thickness and doping concentration, apart from its usual dependence on quasi-Fermi levelsplitting. The new formula allows to easily determine the critical body thickness value above whichany arbitrarily doped symmetric DG MOSFET can be satisfactorily modeled using a conventional bulkMOSFET model. This critical value depends only on body doping concentration, The proposed approxi-mate formula also constitutes a useful means to quickly calculate initial values in iterative numerical cal-culations of doped symmetric DG MOSFET models.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Double Gate (DG) MOSFETs with undoped bodies, sometimesinadequately referred to as ‘‘intrinsic channel’’, are especiallyappealing for scaling MOSFETs down to nanometer sizes [1].Numerous physics-based compact models for these devices havebeen already devised [2–10]. Most of these models generally startby assuming that the semiconductor body that sustains the chan-nel has zero doping concentration, i.e., it is assumed as intrinsic,in spite of the presence of unavoidable unintentional doping. Theuse of this approximation, as well as the assumption of only onecarrier type, facilitates deriving simple analytical solutions fromPoisson’s equation [11,12]. Rigorous surface-potential solutionsfor undoped-body symmetric DG MOSFETs, considering both typesof carriers, have been also proposed based on Legendre’s incom-plete elliptic integral of the first kind [13].

However, real and supposedly undoped body devices are not inreality intrinsic because they contain unintentional residual impu-rities, in concentrations frequently up to 1014 cm�3 or even higher,depending on the fabrication process used. Therefore, the commonand helpful assumption of zero body doping concentration is ques-tionable [14–27]. An approximate methodology based on succes-sive point iteration was presented in 2008 by Liu et al., tocalculate the potentials of doped symmetric DG MOSFETs [28].

ll rights reserved.

.J. García-Sánchez).

In what follows we address the particular issue of determiningthe maximum magnitude that the electric potential can attain atthe body thickness midpoint, commonly referred to as the centralpotential, in arbitrarily doped-body symmetric Double Gate MOS-FETs, including unintentional small concentrations. The descrip-tion presented here does not consider any possible short channeleffects since it is based on a one-dimensional solution across thebody thickness, 0 < x < tSi. It is classical in that it does not deal withquantum effects which start to become significant at body thick-ness below �10 nm [29,30] and assumes non-degenerate statistics.

2. The central potential’s maximum magnitude

The position dependent electrostatic potential, w(x), across thechannel thickness can be exactly obtained in general from anumerical solution of the symmetric boundary value problem de-fined by Poisson’s equation with given conditions at both surfaces(x = ±tSi/2) and at the center of symmetry located at the channelthickness midpoint (x = 0). The adjective ‘‘symmetric’’ is used here,as is usually done, to indicate structural and electrical symmetry,i.e., both gate materials have the same work function, they are con-nected together, and both insulators are of the same material andhave equal thickness.

The analytical 1-D solution of w(x) is well known for the case ofa hypothetically intrinsic body device [10]. That solution whenevaluated at x = 0 and at x = tSi/2, relates the surface potential, wS,to the central potential, w0, at any position along the channel

F.J. García-Sánchez, A. Ortiz-Conde / Solid-State Electronics 76 (2012) 112–115 113

length. Such a description readily yields a simple relationship be-tween the maximum magnitude attainable by the central potentialand the device’s body half-thickness, tSi/2 [6–9]:

tSi

2¼ p

ffiffiffiffiffiffiffiffiffiffiffiffieSiv th

2qni

rexp �w0 max � V

2v th

� �: ð1Þ

where eSi is the Silicon permittivity, vth = q/kBT is the thermal volt-age, w0max, is the maximum magnitude of w0, V represents the qua-si-Fermi level splitting along the channel length, and the rest of thesymbols have their usual meaning. Solving (1), the central potentialmaximum magnitude in hypothetically intrinsic body symmetricDG MOSFETs is given by the simple formula:

w0 max ¼ V � 2v th ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2qni

p2eSiv th

stSi

2

" #: ð2Þ

This maximum value is very useful for modeling purposes ingeneral, and especially for iterative calculations. Unfortunately, tothe best of our knowledge, a corresponding simple formula forarbitrarily doped-body symmetric DG MOSFETs is not yetavailable.

In order to express the channel potential as a function of chan-nel depth for arbitrarily doped-body devices, we numerically solvethe 1-D symmetric boundary value problem [31], and calculate thevalues of w0max versus tSi/2. The resulting exact values, at V = 0,numerically calculated for several body-doping concentrationsincluding zero (intrinsic), are presented in semi-logarithmic scaleas the plots with square symbols in Fig. 1. The first implicationfrom these plots confirms that the value of w0max depends exclu-sively on two body parameters: its doping concentration and itsthickness.

Note that in the case of the hypothetically intrinsic body, the in-versely logarithmic dependence of w0max on body half-thickness,tSi/2, as described by (2) and indicated by the open triangular sym-bols in Fig. 1, correctly portrays the numerically calculated w0max,up to very large values of half-thickness (tSi/2 < 105 nm). That is be-cause the formula incorrectly predicts that w0max = 0 at a finite

Fig. 1. Maximum central potential magnitude, w0max, as a function of Silicon bodythickness, for several body-doping concentrations (including the hypotheticalintrinsic case).

value of tSi/2 � 7 � 104 nm, instead of correctly indicating thatw0max ? 0 asymptotically as tSi/2 ?1. However, (1) and (2) aremore than adequate for any reasonably practical value of tSi/2.We have extended this figure’s horizontal scale to an unrealisticvalue of 100 lm merely to stress the point that the maximum mag-nitude of the central potential in hypothetically intrinsic bodysymmetric DG MOSFETs becomes negligible only for extremelylarge body thickness.

A more significant observation that can be recognized fromFig. 1 is that all doped-body cases exhibit similar (parallel)straight-line dependence (inversely logarithmic dependence ontSi/2) as that of the hypothetically intrinsic case, if their body thick-ness is sufficiently small. It is also evident that the extent of thebody thickness range where such an inversely logarithmic depen-dence holds, grows as the doping concentration decreases, in thelimit reaching infinity as the concentration goes to zero.

Alternatively, we could say, by observing Fig. 1, that there is avalue of body half-thickness tSi/2 around which w0max abruptlydrops to zero, and that such a critical value becomes smaller asthe body-doping concentration increases. Any arbitrarily dopedbody symmetric DG MOSFET with a body thickness above this va-lue, which we shall refer to as the critical body thickness, will ceaseto be fully depleted and electrostatically behave almost as (twice)infinite body thickness devices (partially depleted bulk MOSFETs),because the maximum magnitude of its central potentialw0max ? 0. For instance, according to Fig. 1, a device with a bodydoping concentration of 1018 cm�3 begins to be partially depletedand behave as a bulk device when its body half-thickness tSi/2 islarger than about 30 nm. Thus, this device could be described bya conventional bulk MOSFET model, because for all practical pur-poses its w0 is in essence pinned at zero for all values of appliedgate voltage.

Therefore, it follows that it would be useful to be able to de-scribe the previous numerically calculated behavior of w0max forarbitrarily doped devices, as portrayed in Fig. 1, by means of a sim-ple and explicit formula, much like the well known one used forhypothetically intrinsic devices (2). For that purpose, we proposea semi-empiric approximation that describes well the half-thick-ness, tSi/2, as a function of w0max in arbitrarily doped-body sym-metric DG MOSFETs. The approximation is constructed by joiningthe two asymptotic behaviors exhibited by the numerically calcu-lated plots of Fig. 1, using a joining function of the typeln(1 + exp(x)) that retains the phenomenological meaning of theasymptotes. Assuming a non-degenerately doped n-channel de-vice, without loss of generality the proposed formula is:

tSi

2� p

ffiffiffiffiffiffiffiffiffiffiffieSiv t

2qp0

rln 1þ p0

ni

� �3

exp �3w0 max � V

2v th

� �" #( )1=3

: ð3Þ

where the thermal equilibrium hole density p0 is related to the con-centration of ionized acceptors by the well known relation:

p0 ¼ N�A =2þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðN�A =2Þ2 þ n2

i

q: ð4Þ

where N�A is the uniform concentration of ionized impurities in thesemiconductor body.

The result of using (3) is presented by the continuous curvesshown in Fig. 1. A very good match is observed to the numericallycalculated exact values for any doping concentration, including thelimiting case of the hypothetically intrinsic body, as analyticallycalculated with (1). Note that in that particular case (p0 = ni) (3) re-duces to (1), since the two summands of the logarithm’s argumentin (3) obey the following inequality:

limp0!ni

p0

ni

� �3

exp �3w0 max � V

2v th

� �" #<< 1: ð5Þ

114 F.J. García-Sánchez, A. Ortiz-Conde / Solid-State Electronics 76 (2012) 112–115

Therefore the logarithm may be well approximated as:

ln 1þ exp �3w0 max � V

2v th

� �� �� �1=3

� exp �w0 max � V2v th

� �: ð6Þ

Substituting (6) into (3), with p0 = ni , reduces it to the knownformula of w0max for the hypothetically intrinsic body device ex-pressed by (1).

Finally, solving (3) for w0max, yields the value of the central po-tential maximum magnitude in any arbitrarily doped-body sym-metric DG MOSFETs as a simple explicit approximate formula:

w0 max � V � 2v th lnni

p0exp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2qp0

p2esv th

stSi

2

!3

� 1

24

35

1=38><>:

9>=>;: ð7Þ

It is also easy to prove that (7) becomes (2) when the dopingconcentration vanishes (p0 ? ni).

It is important to keep in mind that the present analysis is basedon a 1D model for the case of long channels. In short channel MOS-FETs, the source/drain depletion regions make the vertical deple-tion region deeper and thus would alter the present results. Athorough discussion of the dependence of full depletion onset onchannel length considering 1D and 2D models is presented in [32].

3. Analysis and discussion

As mentioned before, Fig. 1 indicates that if the body thicknessof a symmetric DG MOSFET, with a given body doping concentra-tion, is larger than a certain critical value, tSiCrit , the maximummagnitude of its channel central potential becomes w0max � 0.Therefore its central potential necessarily will be w0 � 0 at any va-lue of VGS. This indicates that the device is not fully depleted andbehaves just as a (two) conventional bulk MOSFET(s) and that itcould be modeled as such.

An easy way to estimate this critical body thickness is by simplyletting w0max ? 0 in Eq. (3):

tSiCrit

2¼ lim

w0 max!0

tSi

2� p

ffiffiffiffiffiffiffiffiffiffiffieSiv t

2qp0

rln 1þ p0

ni

� �3

exp3V

2v th

� �" #( )1=3

: ð8Þ

It is worth pointing out that rigorously speaking, and judging bythe exact numerical solutions, w0max does not become zero at ex-actly the value of tSiCrit indicated by (8), but at slightly larger values.This small underestimation of tSiCrit by (8) becomes clearly notice-able considering the limiting case of the hypothetical intrinsicbody, as we mention before. If we were to let w0max ? 0 in Eq. (1):

tSiCrit

2¼ lim

w0 max!0

tSi

2� p

ffiffiffiffiffiffiffiffiffiffiffiffieSiv th

2qni

rexp

V2v th

� �intrinsic case; ð9Þ

we would get according to (9) a value of tSiCrit/2 � 7 � 104 nm (atV = 0), a very large value indeed but obviously wrong, since a hypo-thetical intrinsic body DG MOSFET thickness must be theoreticallyinfinite in order for its w0max to become strictly zero. Notwithstand-ing the above, the value given by (8) still constitutes a convenientway to establish tSiCrit for any practical purposes.

Fig. 1 also indicates that when tSi < tSiCrit, the central potentialmaximum magnitude w0max is well described by the asymptotes:

w0 max ¼ V � 2v th ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2qn2

i

p2eSiv thp0

stSi

2

" #; for tSi < tSiCrit ; ð10Þ

which correspond to the parallel straight-line segments at tSi < tSiCrit

values shown in Fig. 1 for different doping concentrations.As a further comparison between unintentionally doped (un-

doped) and hypothetical intrinsic body cases, let us subtract (10)from (2) using (4). The result is:

w0 max Dop � w0 max Int ¼ v th lnN�A2niþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN�A2ni

� �2

þ 1

s24

35; ð11Þ

which as expected corresponds to the difference between the equi-librium Fermi levels of the doped and hypothetically intrinsic cases.To illustrate the difference, consider a real ‘‘undoped’’ symmetricDG MOSFET, with a typical unintentional doping concentration of�1014 cm�3, and body half-thickness, e.g., tSi/2 = 100 nm, which isshorter than its critical value (tSiCrit/2 � 2.8 � 103 nm at V = 0). Itsw0max = 0.58 V, as given by (10), is higher than that of its hypothet-ically intrinsic body counterpart w0max = 0.35 V, as given by (2), by adifference of 0.23 V, as given by (11).

Consequently, when describing the central potential of real ‘‘un-doped’’ symmetric DG MOSFETs, their unintentional doping mustbe taken into consideration. Using the central potential maximummagnitude expression for the hypothetical intrinsic body case, gi-ven by (2), would undoubtedly result in a wrong result, for any gi-ven body thickness, that will be: smaller than its true value by anamount given by (11) if its tSi < tSiCrit, and larger by an amount givenby (2) if its tSi > tSiCrit , because its true value would be w0max � 0 Vin this second case.

4. Conclusions

We have presented a novel simple approximate formula for reli-ably calculating the maximum magnitude of the central potential,w0max, of symmetric DG MOSFETs with arbitrary body doping con-centration, including the unintentionally doped and the hypothet-ically intrinsic cases.

As expected, the proposed expression is only a function of dop-ing concentration and semiconductor body thickness. It allowsdetermining the size of body thickness above which the central po-tential of any arbitrarily doped symmetric DG MOSFET stays prac-tically at zero for any gate bias. Above such critical body thicknessthe device ceases to be fully depleted and exhibits partially de-pleted electrostatic behavior similar to that of a (two) conventionalbulk MOSFET(s). This happens, for example, at body thickness lar-ger than tSi/2 � 35 nm in practical devices with �1018 cm�3 bodydoping concentration. On the other hand, a hypothetically intrinsicbody device would behave as a partially depleted (w0 � 0 at anyVGS) only if its body were extremely thick, at least of the order ofhundreds of micrometers.

The proposed novel formula offers a valuable tool for analyzingand modeling the behavior of the electrostatic potential across thebody thickness of arbitrarily doped symmetric DG MOSFETs. Be-cause it provides a quick means to estimate w0max, it ought to beuseful for iterative numerical calculations of the potential. The for-mula may be used also for arbitrarily doped body Cylindrical Sur-rounding Gate MOSFETs, since it can be easily modified for thosestructures by straightforward geometrical transformation.

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