1136 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 44, NO. 7, JULY 1997

Thin Oxide Thickness Extrapolation fromCapacitance–Voltage Measurements

Steven V. Walstra,Student Member, IEEE, and Chih-Tang Sah,Life Fellow, IEEE

Abstract—Five oxide-thickness extrapolation algorithms, allbased on the same model (metal gate, negligible interface traps,no quantum effects), are compared to determine their accuracy.Three sets of parameters are used: (acceptor impurity concentra-tion, oxide thickness, and temperature): (1016 cm�3, 250 A, 300K), (5 � 1017 cm�3, 250 A, 300 K), and (5� 1017 cm�3, 50 A,150 K). Demonstration examples show that a new extrapolationmethod, which includes Fermi–Dirac statistics, gives the mostaccurate results, while the widely-usedCo ' Cg (measured atthe power supply voltage) is the least accurate. The effect ofpolycrystalline silicon gate is also illustrated.

I. INTRODUCTION

CAPACITANCE–VOLTAGE (CV) characteristics havebeen used extensively in the past two decades to

extrapolate the oxide thickness of metal-oxide-semiconductor(MOS) devices [1]–[5]. Accurate oxide thickness is necessaryto measure the properties of electronic traps in the oxide andat the oxide/silicon interface, such as by the Terman method[6], [7]. Accurate oxide thickness is also needed to design highperformance MOS field-effect transistors (MOST’s) for currentand future integrated circuits because the transconductance andthe gain of MOST inverter and amplifier stages are inverselyproportional to the oxide thickness. As transistor dimensionsdecrease into the deep submicron (sub-half-micron) range,the oxide thickness must be decreased to less than 10 nmto maintain the desired MOST current-voltage characteristics[8], [9]. Furthermore, the accuracy of the operational Time-To-Failure TTF of a MOST extrapolated from acceleratedstress data is critically dependent on the gate-oxide thicknessused in the extrapolation, since the failure rate is directlyproportional to the quantum mechanical tunneling rate ofelectrons through the thin oxide potential barrier [10, 11]. Asthe oxide thickness decreases, the error in the extrapolatedTTF increases drastically because the tunneling rate dependsexponentially on the oxide thickness. For example, a 5%uncertainty in the thickness of a 50A oxide 2.5 A, orabout one atomic layer) would give 100 times uncertainty inthe extrapolated TTF at an operation voltage of 3.3 V.

This paper compares a new analytical algorithm for oxide-thickness extrapolation using metal-gate MOS capacitor(MOSC) data with four previous algorithms. The gate

Manuscript received September 27, 1996; revised February 21, 1997. Thereview of this paper was arranged by Editor D. P. Verret. This work wassupported by the Semiconductor Research Corporation (SRC) under Contracts94-SJ-145 and 95-BJ-412 with the University of Florida. The work of S. V.Walstra was supported by an SRC Fellowship.

The authors are with the Florida Solid-State Electronics Laboratory, Uni-versity of Florida, Gainesville, FL 32611-6200 USA.

Publisher Item Identifier S 0018-9383(97)04678-9.

depletion effect is illustrated in the last section for a heavilydoped polycrystalline silicon gate MOSC with a thin (50A)oxide.

II. GENERAL LOW-FREQUENCY CV EQUATION

In the theoretical investigation comparing the accuracy ofthe algorithms, the low-frequency CV (LFCV) theory will beused because it provides the “exact” analytical CV formula toenable accurate comparison without approximations other thandigitization and truncation errors from numerical computa-tion. In a practical measurement situation, high-frequency CV(HFCV) data are taken. HFCV formulae are approximationsand their errors are difficult to evaluate with confidence.Using LFCV is justified since the low-frequency capacitanceis the sum of the majority and minority carrier capacitanceswhile the high-frequency capacitance excludes the minoritycarrier capacitance. In the flat-band to majority-carrier accu-mulation range used by the algorithms, the minority carriercapacitance is relatively small, so the LFCV and the HFCVare indistinguishable. Numerical tests show negligible errorfrom neglecting the minority-carrier capacitance in the LFCVformula compared with the extrapolation errors from the fivealgorithms. A derivation of the LFCV formula is brieflyoutlined in this section.

A spatially constant dopant impurity concentration is as-sumed to be extended from the SiOSi interface deep into thesilicon substrate, i.e., constant through the voltage-dependentthickness of the surface space-charge layer. This is a goodapproximation since the surface-space charge layer is verythin in the majority-carrier accumulation range of applied gatevoltage. Interface traps are neglected because of their verylow density in production technology. In addition, the chargedensity of the interface traps is nearly independent of theapplied dc gate voltage in the majority-carrier accumulationrange used in the and extrapolation procedures, becausenearly all of the interface traps are filled with electrons (n-Sisubstrate) or holes (p-Si substrate).

One-dimensional (1-D) quantization or two-dimensional(2-D) surface energy bands will also be ignored. In thevoltage range of practical application (weak accumulation),the Si surface-potential well is not deep and the 2-D energylevel spacings are small compared with 25 meVat 300 K) so that thermal broadening would mergethe 2-D energy levels into the classical 3-D continuum.Unequivocal experimental support of this contention can befound in the temperature dependence of the lattice-scattering-limited surface channel mobility in the weak inversion range

0018–9383/97$10.00 1997 IEEE

WALSTRA AND SAH: THIN OXIDE THICKNESS EXTRAPOLATION 1137

which was shown to follow the 2-D classical distributionand its acoustical phonon scattering dependence[12]. Theoretical calculations by Ando, Fowler, and Sternat higher temperatures also suggests that quantum effectsare not as important at room temperature [13]. In addition,atomic randomness or interface roughness at the SiOSiinterface would delocalize the electron distribution from 1-Dquantization, as was first envisaged by Schrieffer in 1957[14]. A first-order quantum correction was investigated byRios and Arora [15].

The analytical model outlined here was first given by Sei-watz and Green in 1958 [16]. The capacitance–voltagecharacteristics of a MOSC are computed from two parametricequations: the voltage equation and the capacitance equation.The independent parameter is the dc surface potentialof thesemiconductor, which is the total semiconductor energy bandbending from the oxide/semiconductor (or SiOSi interface

to the ohmic contact at the back surface of thesemiconductor bulk The two equations are [17]

(1)

(2)

In (1), is the constant flat-band voltage as defined in[17]; is the static dielectric constant of silicon,

F/cm is the oxide capacitance per unit area,is the static dielectric constant of SiOF/cm is the thickness of the oxide layer;

and is the dc electric field on the semiconductor side ofthe SiO Si interface. In (2), is the capacitance of thesemiconductor surface space-charge layer. and arefunctions of The Boltzmann or fully ionized forms of(1) and (2) are well known [17]. However, the deionized [18]and the combined deionized, Fermi–Dirac [16], [19] forms, asshown below for p-type material, are infrequently used

(3)

(4)

(5)

, given in (5), is the Fermi–Dirac integral of the-thorder, as defined by Dingle [20]. is the temperature inKelvin, is the Boltzmann constant, J/K, andis the electron charge, C. is the substrateimpurity doping density and is the ground-state degeneracyfactor. , and are the normalized (by surfacepotential, Fermi level, and single-level acceptor trap energy,respectively. and are the normalized conduction and

valance band edges, and and are the conductionand valance band density of states, as defined in [17]. Allnormalized energies are referenced to the intrinsic Fermi level.

III. EXTRAPOLATION FORMULAE

The five extrapolation formulae are compared in this sec-tion. The oxide thickness is calculated from the 1-D parallel-plate capacitance formula

(6)

where is the measured area of the MOS capacitor. Notingthat in the strong majority-carrier accumulation and(low frequency) strong inversion ranges, it has been a commonmanufacturing practice to monitor an effective oxide thicknessby using measured at the operating voltage andapproximating in (6) by the measured ,because it requires only one simple measurement and isparticularly suitable for statistical analysis of large number ofsamples. This zeroth-order approximation will be denoted asthe algorithm. Equation (2) shows that thismodel assumes The other four extrapolations to bediscussed provide improved estimates offrom experimentaldata in order to give a more accurate value of

A. Maserjian–Petersson–Svensson (MPS) Metallic Algorithm

Maserjian, Petersson, and Svensson (MPS) [1] developedan oxide thickness extrapolation procedure in the asymptoticlimit of complete carrier degeneracy in order to measure3–4-nm thick MOS oxides. The MPS formula is derived from(1), (2), and the relation In the verystrong accumulation range, (3) can be approximated by theasymptotic form in the complete degenerate limit [16], whichis valid when For the p-Si MOSC, thesurface hole term dominates, thus

(7)

After some substitutions and rearrangements, we have

(8)

where

and

Thus, a straight line is obtained when the experimentaldata in strong accumulation is plotted as versus

whose intercept is Equation (8)shows that the slope of this straight line can be used tocalculate the experimental effective density-of-states orthe density-of-states effective mass which is a funda-mental parameter that is difficult to measure in degeneratesemiconductors.

1138 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 44, NO. 7, JULY 1997

B. McNutt–Sah (MS) Nondegenerate Algorithm

The Boltzmann distribution of the majority carrier concen-tration in the surface accumulation layer was assumed byMcNutt and Sah [2]. The derivative of (1) and (2) yields

(9)

The above formula contains a extension of the original solutionof McNutt and Sah [21], who neglected the surface potentialterm in (1). Equation (9) indicates that a plot of experi-mental versus will givefrom the slope. Equation (9) can also be solved quadraticallyfor as a function of and the first derivative.

C. Ricco–Olivo–Nguyen–Kuan–Ferriani(RONKF) Flatband Algorithm

Ricco et al. [4] discovered a novel property of higherderivatives of the curve at flatband, whichthey used to calculate the oxide capacitance. Their formulaeare given by

(10)

(11)

where and are the first and second derivative withrespect to Equation (11) is used first to determine thevalue at flatband from the experimentaldata. At this the and values from the least-squarefit of the data are then used in (10) to determine, which isthen used in (2) to give

Ricco et al. suggested that the effects of degeneracy, deion-ization, and surface quantization can all be neglected if theoxide thickness is extrapolated at flatband. Fermi–Dirac de-generacy from high carrier concentration, however, will notbe negligible in samples with high dopant impurity concentra-tions. More important, impurity deionization givesdistortion near the flatband [18], which introduces significanterror in the second derivative, , in (11), to be discussed inSection IV-B.

D. McNutt-Sah-Walstra (MSW) Algorithm

The McNutt-Sah algorithm can be readily extended toinclude high concentration of carriers and deionization of thedopant impurity. The two effects add a multiplicative factor

to the original MS formula, (9), as follows[22]:

(12)

where the factors and are defined in the appendixand contain the Fermi–Dirac integrals of 3/2, 1/2, and1/2orders, and the impurity deionization effect. The bracketedterm, , is the correction term due todegeneracy and deionization. As would be expected, thisterm approaches 1 in the Boltzmann low-carrier-density limitwith full ionization of dopant impurities. This formula isexact and is the baseline for comparing the oxide capacitanceextrapolation algorithms. This MSW algorithm would give the

exact value of the assumed if there were no digitizationand truncation errors in the theoretical exact dataand its derivative, Since , and are implicitfunctions of the independent parameter , (12) must besolved iteratively to give and as discussed in [22].

IV. COMPARISON OF FIVE ALGORITHMS

The preceding five oxide-thickness extrapolation algorithmswere compared: 1) , which is widely used inadvanced and future technology development and productionmonitoring with at the operating voltage; 2) the Maser-jian, Petersson, and Svensson (MPS) complete degenerateor metallic approximation; 3) the McNutt and Sah (MS)nondegenerate approximation; 4) the Ricco, Olivo, Nguyen,Kuan, and Ferriani (RONKF) flat-band approximation; and5) the McNutt, Sah, and Walstra (MSW) exact solution withFermi statistics, but in the comparisons, the full impurityionization (MSW-FI) form was used. The error in the MSW-FIalgorithm from excluding impurity deionization is shown to beinsignificant due to the much higher carrier concentration inthe accumulation range than the dopant impurity concentration.The MSW-FI is preferred since it is computationally lessexpensive and does not require knowledge of the dopantimpurity energy level.

Two oxide thicknesses (50A and 250 A), two dopantimpurity concentrations (10 cm and cm ), andtwo measurement temperatures (150 K and 300 K) will be usedto compare the accuracy of the five extrapolation algorithmsover a wide range of applied gate voltages. The CV dataare computed from the “exact” degenerate and deionizedformula, (1)–(5), listed in Section II. These data are taken tobe theexact data, whose input oxide thickness is the exactoxide thickness used for error calculation and comparison.A boron-like dopant acceptor impurity level at 0.046 eVabove the valence band edge is assumed with a ground statedegeneracy factor of 2. The excited states and spin-orbitsplit-off states are ignored. The exact CV data are generatedby numerical calculations to only seven significant figuresin both capacitance and voltage at gate voltage steps of 50mV. The seven-significant-digit limit is used to simulate themaximum experimental sensitivity in small-signal, 1 MHzCV measurements attainable in university and manufacturelaboratories. To obtain the extrapolated oxide capacitance andthe oxide thickness, versus was obtained by least-squares-fit (LSF) of ten consecutive pairs to a third-order polynomial. The derivative is evaluated fromthe analytical derivative of the polynomial at the mid pointof the ten-point voltage range, - The anddata at - is then used in four of the five algorithms(except RONKF) to extrapolate the oxide capacitance andthickness.

The RONKF flat-band method requires much finer voltagespacing because it requires second derivativein (11). Thus,a 10-nV step is used and the ten-point third-order-polynomialLSF then gives both the first and second derivatives. Theflatband gate voltage is assumed to be found when (11) issatisfied with less than 10 residual. This gives a 10 nV

WALSTRA AND SAH: THIN OXIDE THICKNESS EXTRAPOLATION 1139

Fig. 1. Oxide thickness extrapolation error atxo = 250 A,NAA = 10

16 cm�3; andT = 300 K.

Fig. 2. Oxide thickness extrapolation error atxo = 50 A,NAA = 5 � 10

17 cm�3; andT = 300 K.

accuracy in the flatband voltage defined by ,which is then used in (10) to give , and in (2) to giveIn applications, 10 nV sensitivity is difficult to attain, whichsuggests a serious application limitation.

It is important to note that we could also apply theseextrapolation formulae to experimental data, since that is theeventual goal, but then we would have no baseline thickness tomake error assessments. Each of the many non-CV measure-ment techniques for finding oxide thickness (e.g., FNT, TEM,SIMMS, XPS, Ellipsometry) have errors and nonidealities [5],[22] which would invalidate our goal of assessing the mostaccurate CV thickness extrapolation formula derived from thesame basic model (metal gate, negligible interface traps, noquantum effects).

A. Oxide Thickness and Impurity Concentration Dependencies

Figs. 1 and 2 illustrate the effects of dopant impurityconcentration and oxide thickness on the absolute percentageerror - at K for

cm 250 A) and cm 50 A), which covercurrent and future application ranges.is the oxide thicknessused in generating the exact CV data and- is the oxidethickness extrapolated from the CV data by each formula.

Fig. 1 shows the oxide-thickness error in a (10cm ,250 A) MOSC. The exact CV curve (dashed line) is alsoplotted to relate the errors to the CV curve. The MS andRONKF algorithms, which are based on the low-concentrationBoltzmann approximation, give less error (0.5%) than the

and MPS algorithms (1–10%) because the semicon-ductor is not strongly degenerate due to the medium-thickness250 A oxide, even at 5 V. The RONKF algorithm showsan error between 0.0025% and 0.12% in a very small gatevoltage range ( 10 mV) which is the very voltage rangecovered by the ten-point (1 mV step-size) LSF. Thus, eachtime the calculated flatband was within the ten points usedfor the polynomial derivative, (11) was satisfied; however, theresulting flatband estimation differs each time due to imprecisefirst and second derivatives. The range plotted is indicativeof what can be expected from the RONKF algorithm underthe assumed experimental accuracy (i.e., parameters and CVdata are all accurately known to seven significant figures).The MSW-FI extrapolation gives the best results, with errorexceeding 10 only when V. The sharpincrease in error near flatband orV in both the MSW-FI and the MPS algorithms is due toneglecting minority carriers.

Fig. 2 shows the oxide-thickness error for a higher substrateimpurity concentration with a thinner oxide cm50 A) than that of Fig. 1. The larger error in thealgorithm is due to the larger from thinner oxide, whichslows the asymptotic approach of to in (2). The errorin the MS algorithm has increased over the entire rangeof gate biases because the surface hole concentration at allbias points is greater, and hence less Boltzmann-like. Theincreased error in the RONKF algorithm is due to deionization(shown in the next section). The error in the metallic MPSalgorithm drops significantly, which is anticipated from theincreasing carrier concentration at the surface. The sharp dipin the MPS error near 4.25 V is caused by the extrapolated

value moving from an underestimate to an overestimate,thus passing through the exact value. At about1.75 Vthe MPS and MS error curves cross. This voltage pointrepresents the transition between the two asymptotic limits ofthe Fermi–Dirac distribution of 3/2 order, which approachesthe Boltzmann for , and the metallicfor , where is the Fermi energy. Because the carrierdensity in the surface accumulation space-charge layer is somuch larger than the density of the carriers trapped at theimpurities, the error of the fully ionized MSW algorithm(MSW-FI) is still less than 10 for V.

The foregoing comparison of Figs. 1 and 2 suggests thatthe MPS algorithm can give low-error oxide thickness usinga rather simple analytical solution. However, the low erroris attained only at very high electric fields that could causerapid build-up of oxide and interface trapped charges [10],[11], which would distort the CV curve and increase the error.For example, a 0.1% oxide-thickness accuracy in a 50Aoxide at cm dopant concentration using the MPSalgorithm would require V, or an oxideelectric field in excess of 7 MV/cm, which would cause arapid build-up of positive oxide charge [10, 11]. Even for 1%

1140 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 44, NO. 7, JULY 1997

Fig. 3. Effect of impurity deionization on oxide thickness extrapolation errorat xo = 50 A, NAA = 5 � 1017 cm�3; andT = 150 K.

error, the MPS algorithm would requireV or a 4.3 MV/cm oxide field, which exceeds the reliabilitydesign limit of about 4 MV/cm. One way to improve the MPSalgorithm’s applicability is to obtain the CV data at low200K) temperatures in order to better meet the assumptions madein (7). This is demonstrated in the next section.

B. Effects of Impurity Deionization

The carrier concentration and capacitance in the surfacespace charge layer are lowered at higher impurity concentra-tions and lower device temperatures because a fraction of theacceptor (boron) dopant impurity will be occupied by holesand deionized. Neglecting this impurity deionization will notsignificantly increase the error in and extrapolatedin the majority carrier accumulation range for four of thefive algorithms , MPS, MS, and MSW). However,neglecting impurity deionization greatly reduces the accuracyof algorithms that use the CV curve and its derivatives nearflatband, such as the RONKF algorithm. Fig. 3 shows theextrapolated oxide thickness errors for A,

cm eV, and K. Theseparameters are identical to those used in Fig. 2 except for thelower temperature. This and combination is selectedto give a large (62%) acceptor deionization in the quasineutralbulk region.

Comparing Figs. 2 and 3, the and MS modelsshow very little increase in error from high (300 K) to low(150 K) temperatures because their errors are already sohigh that the small effect of deionization is masked out. Onthe other hand, the MSW-FI shows a noticeable increase infractional error at 150 K, but is still less than 0.01% for

V. The basic reason for the high accuracyof the MSW-FI algorithm, which neglects deionization, is thatthe concentration of the gate-voltage-attracted holes (majoritycarriers) in the surface accumulation layer is considerablyhigher than the reduction of the hole concentration from holetrapping by the dopant impurities (deionization) in the surfaceaccumulation layer.

As the temperature is lowered from 300 K to 150 K, theMPS metallic algorithm improves from 1.5% error to 0.1%

Fig. 4. Effect of n-poly-Si-gate depletion on the accuracy of oxidethickness extrapolation algorithms withxo = 50 A, NXX (p substrate)= 5� 1017 cm�3; NGG (gate)= 1� 1020 cm�3; andT = 300 K.

error at V. The improvement is entirely dueto the increased validity of the assumptions made to expandthe Fermi–Dirac integral in (7). However, at low temperatures,quantum effects could become nonnegligible [13].

At the lower temperature, the range of RONKF solutionshas increased drastically from a tight group of 1.4% erroraround flatband to two separate clusters with 0.2% error around

V and 13% error atV. These clusters of different errors come from the slightdistortion of the CV curve which occurs when the surfacepotential passes through the single- level acceptor trap, causinga large change in the ionized acceptor concentration, whichwas noted in 1961 [18]. Thus, RONKF cannot reliably giveaccurate extrapolated values at high dopant concentrationsor low temperatures.

V. ERROR FROMPOLY-GATE DEPLETION

The analyses just presented assumed the classical metal gateemployed in the first decade of silicon integrated circuits (mid-1960’s to mid-1970’s). Since then, highly-doped thin-filmpolycrystalline silicon gates (poly-gates) have been employedfor at least two technological advantages: self-alignment [23]and impurity (hydrogen) gettering [24]. In spite of the veryhigh impurity concentration in the poly-gate, a thin space-charge layer remains at the poly-gate/oxide interface. Thisgives a capacitance in series with the oxide capacitance,which reduces the input voltage that modulates the channelconductance or the output current. This is known as thegatedepletion effect. It becomes important when the gate oxidethins down to the 25A to 100A range, or five to 20 times thethickness of the poly-gate space-charge layer5 A).

A demonstration of the gate-depletion effect on thicknessextrapolation is given in Fig. 4 using the same conditions asFig. 1, but with an n-type polysilicon gate doped to

cm The and MS methods both increasein error by about a factor of two compared to the metal-gatecase. The RONKF error has increased to about 6.1% due to thepresence of the additional space-charge layer in the polysilicongate, which further invalidates the assumptions made in theRONKF derivation. The MSW error has increased to about

WALSTRA AND SAH: THIN OXIDE THICKNESS EXTRAPOLATION 1141

3.9% at V and about 3.1% atV. The MPS solution does rather poorly (about 8.0% error) at

V, but improves significantly (about 3.0%error) at V to match the performace ofMSW near the reliability limit (oxide field 4 MV/cm) forthin oxides with polysilicon gates. We use an n-type gate on p-substrate because this ensures that the gate is in accumulationwhen the substrate is in accumulation.

If a p-type gate were used on the p-type substrate withthe same doping levels as Fig. 4, the gate would not be inaccumulation when the substrate is in accumulation at lowgate voltages, causing a distinct curvature change in the CVcharacteristics and invalidating the formulae which use theCV derivatives in accumulation. Only and RONKFcan be used in this case, both with errors in the 8–10%range.

VI. SUMMARY

A highly accurate algorithm (McNutt-Sah-Walstra) includ-ing Fermi–Dirac statistics is developed to extrapolate ox-ide capacitance and thickness for thin-oxide high dopant-impurity-concentration MOS capacitors expected in currentand future MOS circuits. A numerical example using theMSW algorithm demonstrated 0.01% theoretical error at50 A, cm , and 300 K for V

MV/cm oxide electric field), which cannot be attainedby the four previous algorithms: 1) (6.8%), 2)McNutt–Sah (1.8%), 3) Ricco et al. flatband (1.5% at flatband),and 4) Maserjian–Petersson–Svensson metallic approximation(7.0%). When polysilicon gate depletion is included, the MSWalgorithm is still superior at very low applied gate voltages(3.9% at V), while atV, the simpler MPS algorithm gives about the same results(3.0% versus MSW’s 3.1%) near the reliability limitMV/cm).

APPENDIX

The formulae needed to iterate the oxide thickness from CVdata on a p-Si MOSC are summarized in this appendix. From(3) and (4), we have

(A.1)

sign (A.2)

(A.3)

where

(A.4)

(A.5a)

(A.5b)

(A.5c)

is the effective Debye capacitance for holes.1 The, and terms assume p-type accumulation to ignore

the electrons which reduces the number of terms, but is nota necessary approximation. The term is the normalizeddensity of the space-charge layer whereis the surface charge density. The term isintegrated from at the surface to the basecontact, and proportional to The term isFor negligible deionization in p-type, the Fermi level is several

above the acceptor level, soThe general , and expressions, valid at all surface

concentrations and impurity deionization conditions, are givenin [22]. The solution, (12), is valid for the general , and

Explanations and examples for solving (12) can be found in[22]. A method for calculating the Fermi–Dirac integral andits inverse is required. Blakemore [25] has compared manyexplicit formulae for F–D integrals of 1/2 order, but betteraccuracy has been attained by implicit formulae. The three-range rational Chebyshev approximation by Cody and Thacher[26] appears to be the most accurate, as demonstrated for[27]. In our numerical solutions, the Fermi energyis firstfound from Nilsson’s full range analytical approximation to theinverse F–D integral of 1/2 order [28], which is then iteratedusing the Cody–Thacher [26] formula to reach an arbitrarilysmall error. A 10 or smaller difference in two successiveiterations was used in this paper to calculate

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1This differs from the traditional Debye length definition,q("sNV =kT )1=2, by 1=

p2 and is used to simplify the equations.

1142 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 44, NO. 7, JULY 1997

[11] C.-T. Sah,Fundamentals of Solid-State Electronics—Study Guide. Sin-gapore: World Scientific, 1993, App. B2, pp. 394–423 (in particular, seeFig. B 2.1 on p. 396).

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[18] , “Theory of the metal oxide semiconductor capacitor,” Solid-State Electron. Lab., Univ. Illinois, Urbana, Tech. Rep. 1, 1964. Awidely circulated report which is available from the Engineering Libraryof the Univ. Illinois or the Florida Solid-State Electronics Laboratory.

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[22] S. Walstra, “CV-based oxide thickness extrapolation procedure for thinoxide MOS devices,” M.S.E.E. thesis, Univ. Florida, Gainesville, Aug.,1994. (Obtainable from the Semiconductor Research Corporation asTech. Doc. T94214 and by interlibrary loan from the University ofFlorida.)

[23] L. L. Vadasz, A. S. Grove, T. A. Rowe, and G. E. Moore, “Silicon-gatetechnology,” IEEE Spectrum, vol. 6, pp. 28–35, Oct. 1969 (see alsoreferences cited).

[24] C.-T. Sah, J. Sun, and J. Tzou, “Study of atomic models of three donor-like defects in silicon metal-oxide-semiconductor structures from theirgate material and process dependencies,”J. Appl. Phys., vol. 56, no. 4,pp. 1021–1031, Aug. 1984.

[25] J. S. Blakemore, “Approximations for Fermi–Dirac integrals, especiallythe functionF1=2(�) used to describe electron density in a semicon-ductor,” Solid-State Electron., vol. 25, p. 1067, 1982.

[26] W. J. Cody and H. C. Thacher, Jr., “Rational Chebyshev approximationsfor Fermi–Dirac integrals of orders�1/2, 1/2, and 3/2,”Math. Comput.,vol. 21, p. 30, 1967.

[27] S. A. Wong, S. P. McAlister, and Z. M. Li, “A comparison of someapproximations for the Fermi–Dirac integral of order 1/2,”Solid-StateElectron., vol. 37, p. 61, 1994.

[28] N. G. Nilsson, “An accurate approximation for the generalized Einsteinrelation for degenerate semiconductors,”Phys. Stat. Sol., vol. 19, p.K75, 1973.

Steven V. Walstra (S’90) was born in Hammond,IN, in 1970. He received the B.S.E.E. (high hon-ors) and M.S.E.E. degrees from the University ofFlorida, Gainesville, in 1992 and 1994, respectively.He is currently pursuing the Ph.D. degree in elec-trical engineering at the University of Florida.

He joined the Florida Solid-State Electronics Lab-oratory, University of Florida, in 1991. He workedin the TCAD Department, Intel Corporation, SantaClara, CA, in the summers of 1995 and 1996,where he investigated intrinsic capacitance degrada-

tion and degraded circuit performance. His current research interests includeMOS device reliability, compact physical models, and thin-oxide capacitanceextraction methodology. He holds a Semiconductor Research CorporationFellowship.

Chih-Tang Sah (S’50–M’57–F’69–LF’96), for a photograph and biography,see p. 117 of the January 1997 issue of this TRANSACTIONS.