1188 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 48, NO. 6, JUNE 2001

Hole Quantization Effects and Threshold VoltageShift in pMOSFET—Assessed by Improved

One-Band Effective Mass ApproximationY. T. Hou and Ming-Fu Li, Senior Member, IEEE

Abstract—Threshold voltage ( ) shift due to quantummechanical (QM) effects in pMOSFET is investigated basedon a six-band effective mass approximation (EMA). Due to thevalence band mixing, both subband energies and density of states(DOS) show remarkable difference from those derived fromtraditional one-band EMA using the bulk Si effective masses. Incomparison with the experimental results, it is found that shiftin pMOSFET is significantly overestimated by the traditionalone-band EMA, however it corresponds with our six-band EMAcalculation. Based on the numerical results of our six-band EMA,new effective masses are determined empirically and their electricfield dependence is also evaluated. Using these new effective massesinstead of the bulk effective masses, one-band EMA still displayeffectiveness in describing hole quantization and shift in anempirical manner. A set of constant energy quantizaiton/DOSeffective masses(0 29 1 14 0 22 0 75 0 24 0 66 0) for thefirst three subbands, neglecting their electric field dependence,is proposed for the modeling of QM effects in pMOSFET in thisimproved version of the one-band EMA formula. Computingtime is minimized and results can be obtained with sufficientaccuracy and correspond well with reported experimental data,thus the improved one-band EMA formula provide a firm groundin routine device simulation for deep submicron MOS devices.

Index Terms—CMOSFETs, inversion quantization, quantummechanical effects, semiconductor device modeling, thresholdvoltage.

I. INTRODUCTION

T HE two-dimensional (2-D) nature of carriers in MOSFETshas been known for a long time [1], [2], but it plays a more

prominent role only recently when the scaling of MOSFETenters into deep submicron era. Due to the higher substratedoping concentration and thinner gate oxide in modern devices,the electric field near the Si/SiOinterface is strong enougheven at the threshold region, so that the quantum mechanical(QM) effects becomes noticeable. In this situation, the classicalthree-dimensional (3-D) treatment becomes inadequate andcarriers must be treated as quantum 2-D. The quantization ofcarriers will lead to a substantial change of the band structureand density of states (DOS) [1]–[3], and as a consequence, a

Manuscript received September 27, 2000; revised December 8, 2000. Thiswork was supported by Singapore NSTB/EMT/TP/00/001,2 and the NationalUniversity of Singapore RP3 982 754 Grants. The review of this paper was ar-ranged by Editor C.-Y. Lu.

The authors are with SI Nano Device Laboratory, Department of Electricaland Computer Engineering, National University of Singapore, Singapore119260 (e-mail: elelimf@nus.edu.sg).

Publisher Item Identifier S 0018-9383(01)04219-8.

threshold voltage shift in MOSFET. This phenomenonhas been experimentally observed both in nMOSFETs andpMOSFETs [4]–[6]. Since is a key parameter affecting thedevice performance, a physical insight into the QM effectsand an accurate modeling of the shift will be of greatimportance. For nMOSFET, the electron quantization as wellas the shift is well understood in one-band effective massapproximation (EMA) [4], [5], [7]–[9]. However, hole quanti-zation in pMOSFET has not been studied in as much detail aselectrons due to the complexity of the valance band structureof Si [10]–[15]. Previously, one-band EMA was also appliedto hole quantization, using constant effective masses derivedfrom bulk Si (brief as traditional one-band EMA) to calculatethe hole subband energies and shift in pMOSFET [3], [10],[13]. Such a procedure omits band mixing effect and the resultsobtained are not reliable as shown later. A few studies abandonthe use of the EMA and used the pseudopotential full-bandself-consistent formalism, accounting for the complicatedvalence band structure [11], [12]. Although these full-bandself-consistent treatments may achieve the reliable resultsin principle, they are computational prohibitive for routinedevice simulations. We have developed a simple but reliablemodel based on six-band effective mass equation for holequantization [16]. Instead of the self-consistent method, thepotential in the inversion layer is approximated by a periodictriangular well (zigzag) structure. An accurate modeling of2-D hole DOS and the shifts in pMOSFET by this modelis reported. As a further simplification, a new set of empiricalhole effective masses suitable for this hole quantization effectunder electric field is also derived from the numerical results.With band-mixing effects included in these empirical holeeffective masses, we have proven that shift in pMOSFETcan still be described properly under the framework of theone-band EMA (brief as improved one-band EMA). Comparedto the traditional one-band EMA, the improved one-band EMAshows better agreement with the experiments.

II. THRESHOLDVOLTAGE SHIFT DUE TO QM EFFECTS—ONE BAND EMA

Although the actual potential profile within the inversionlayer can be determined by solving self-consistently the cou-pled Schrödinger and Poisson equations [1], such a treatmentis tedious and is not always necessary. It has been shown that,triangular well is a quite good approximation of the potential in

0018–9383/01$10.00 © 2001 IEEE

HOU AND LI: HOLE QUANTIZATION EFFECTS AND THRESHOLD VOLTAGE SHIFT 1189

weak inversion layer [3] and its validity in threshold region isalso justified [9]. In triangular well approximation, the potentialwell is written as

(1)

where is the surface electric field in Si substrate andisthe axes perpendicular to the Si/SiOinterface. The devices areusually fabricated on (001) Si and thus we take here.Moreover, we take the top of the valence band at the substratesurface as the energy reference, and for convenience the holeenergies are all expressed as positive values.

Under the depletion approximation, surface potential, andsurface electric field of the substrate with doping havethe relation [17]

(2)

Here, we neglect the contribution from the inversion charge be-cause it is much less than the depletion charge at threshold re-gion (less than 1%). The 3-D inversion charge density atcanbe given by

(3)

where is the effective DOS at the top of the valence band.The classical definition of is the gate voltage when surface

potential , with [17]. The2-D is determined by the gate voltage to populate the 2-Dinversion layer to the same inversion charge density as the 3-Done, namely, [10]. In the 2-D QM description, ifthe subband dispersion and DOS are determined, theinversion charge density is

(4)

At threshold region, can be simplified to Maxwell-Boltz-mann statistical function [9]. By equating in (4) to

in (3), we obtain , the surface potential at2-D . Finally, the shift due to QM effect can be expressedas [10]

(5)

Under traditional one-band EMA, as widely used for elec-trons, the derivation of shift occurs in a simple and straight-forward manner. In triangular well approximation, the energiesof subband minima can be expressed analytically as [1], [18]

(6)

where is the energy quantization effective mass of the bulkSi. The 2-D DOS is independent of energy, where

and ( for valence band of Si) are the DOS effectivemass of the bulk Si and degenerate factor of the ith subband.Then, (4) has an analytical form as

(7)

For complicated subband structure, (4) can only be evaluatednumerically.

III. SUBBAND STRUCTURE AND DENSITY OF STATES BY

SIX-BAND EMA

The above traditional one-band EMA has also been employedto calculate the hole subband energies as well as theshift inpMOSFET [10], [13]. However, coupling of the valence bandscasts doubts on the validity of this approximation [11], [12].To account for band mixing effect, we have developed a modelfor hole quantization based on the multiband effective massmethod. Six bulk bands are included in our model, they areheavy, light and spin-orbit split-off holes and their spin degen-erate bands. The spin-orbit split-off energy for Si is only 44 meV[19] and thus the coupling of the split-off band to other bands isnot negligible. From the effective-mass theory, the Hamil-tonian can be obtained by the unperturbed bulk band Hamil-tonian adding a diagonal electric potential energy term[19]. When a periodic triangle well potential (zigzag) with pe-riod is used to replace the single triangle well as describedin (1), the derivation of eigenenergies follows a similar treat-ment in III-V multiple quantum wells [20] with the eigenfunc-tions expanded by plane waves.is selected to be sufficientlylarge so that the coupling between the neighboring wells is neg-ligible and sufficiently small. The calculation of hole subbandenergies are efficient. Details of the model will be presentedelsewhere [16]. The difference of the subband minimum ener-gies at classical threshold region for varying substrate dopingbetween traditional one-band EMA and our multiband modelis given in Fig. 1(a) as a relative deviation in percentage. Theelectric fields versus are determined from (2). In traditionalone-band EMA, the energies are determined by (6). The energyquantization and DOS effective masses are deducedfrom bulk Si valence bands [3], [10], [13]. Their values are sum-marized in Table I. The energies for the first lowest subband

obtained from two methods are fully consistent. Wehave found that this subband is purely heavy hole at [12],[16]. It is not surprising from the symmetric consideration of theHamiltonian, where nondiagonal items for heavy hole all vanishat . For and subbands, band mixing occurseven at [12], [16], which results in the inaccuracy ofthe traditional one-band EMA in the determination of subbandenergies, as shown in Fig. 1(a). With the increase of substratedoping (surface electric field increases correspondingly),the relative deviation also increases. It is also found that the de-viations of and subbands are in opposite signs.For , there exists band mixing for all subbands. Due to thestrong band mixing effect, the dispersion of the hole subband isfar from parabolic and also becomes dependent of electric filed.

1190 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 48, NO. 6, JUNE 2001

TABLE ISUMMARY OF ENERGY QUANTIZATION AND

DOS EFFECTIVE MASSES INUNIT OF FREE ELECTRON MASSm . BULK SiVALUES ARE FROM[3]. THE FIELD DEPENDENTEMPIRICAL VALUES ARE

OBTAINED BY FITTING TO THE NUMERICAL DATA OF SIX-BAND EMACALCULATION AT DIFFERENTFIELD F (IN UNITS OF MV/cm)

In Fig. 1(b), we plot the DOS at threshold voltage for a typicalsubstrate doping density cm . In traditionalone-band EMA, the DOS profile is the well-known step-likefunction, shown as dashed line in Fig. 1(b). The difference isvery clear on the DOS obtained by our six-band EMA. The DOSprofile is deviated from the step-like function because the dis-persion is no longer parabolic. Furthermore, the DOS from ourmodel is overall much larger than that obtained from the tradi-tional one-band EMA.

In order to obtain a greater insight of the DOS characteristics,we show in Fig. 2 the separate DOS profiles of the first threesubbands at for cm cmand cm . The subband DOS profiles are all deviatedfrom the step-like shape of 2-D carriers with parabolic disper-sion. The first subband shows a much slower slopeas the DOS rises from the subband minimum. At higher ener-gies, the DOS approaches an almost constant value. AlthoughDOS profiles of other two subbands remain a rapid increasenear the subband minima, they show very different behavior athigher energies. The DOS profile of subband forms twoquasi-plateaus while two peaks appear near the subband min-imum of the subband. These peaks are caused by thecamel back structures [21] near on the subband dispersioncurve [16]. From Fig. 2, the DOS profiles of hole subbands aregenerally irregular, especially near the subband minima. Sincethe extent of band mixing depends on the electric field, the DOSprofile also becomes electric field dependent. We will not dealwith other subbands of higher order because the first three sub-bands in total have accounted for almost all the inverted holecharges in our case. Even near threshold region, occupation ofother higher subbands can be negligible especially at higher

, where QM effects are noticeable and is of interest in ourdiscussion.

IV. THRESHOLDVOLTAGE SHIFT IN pMOSFET

Using the DOS profiles and subband energies obtained, thethreshold voltage shift at various substrate doping can be de-termined numerically from (3)–(5), the results for 14-nm oxide

Fig. 1. (a) Relative deviation of subband energies of first three subbands atthreshold voltage derived by the traditional one-band EMA from those of oursix-band EMA, as a function of substrate dopingN . (b) DOS for substratedoping5 � 10 cm at threshold voltage. Solid and dashed lines are fromour six-band EMA model and from traditional one-band EMA, respectively.

Fig. 2. DOS of first three lowest subbands at threshold voltageby our multiband EMA model for three typical substrate dopingN = 1 � 10 cm ; 5 � 10 cm and 1 � 10 cm . �E isthe energy referenced from the subband edge.

HOU AND LI: HOLE QUANTIZATION EFFECTS AND THRESHOLD VOLTAGE SHIFT 1191

Fig. 3. (a) Threshold voltage shifts�V due to quantum effects at varioussubstrate dopingN with oxide thickness of 14 nm. Solid triangles anddashed line are from traditional one-band EMA, using self-consistent [10]and triangular well approximation, respectively. Solid circles are from thepseudopotential full-band self-consistent method [11], and the solid line is ourresult with six-band EMA model. (b)� (�V ) [defined by (8)] as a functionof substrate dopingN .

is shown in Fig. 3(a). The same curve under the traditionalone-band EMA is also shown as dashed lines. For compar-ison, we also presented the reported data from traditionalone-band EMA calculation [10] and from pseudopotentialfull-band self-consistent formalism [11]. The result of oursix-band model agrees well with the full-band pseudopotentialcalculation. This demonstrates that our simplified modelremains accurate in the modeling of shift due to hole QMeffects. However, the traditional one-band EMA, either byself-consistent treatment or in triangular well approximation,both overestimates the shift in pMOSFET, provided that theeffective masses from bulk Si are used. It can be ascribed to itsunderestimation of the DOS as indicated in Fig. 1(b) and theinaccurate description of subband energy levels, as shown inFig. 1(a). If we define as

(8)

Fig. 3(b) shows that is increased when the dopingconcentration increases.

V. EMPIRICAL EFFECTIVE MASS—IMPROVED ONE-BAND EMA

As noted above, the DOS profiles of hole subbands aredeviated from the simple step-like function of traditionalone-band EMA. Its shape is not regular especially near theband minimum, where the carriers are mostly populated.Moreover, the detailed profile of DOS also depends on the

Fig. 4. Surface electric fieldF dependence of (a) empirical energyquantization effective massm , and (b) empirical DOS effective massm ,determined from the numerical results of six-band EMA calculations (solidsymbols). The solid lines are the fitting by polynomials up to second order ofF with fitting parameters summarized in Table I.

surface electric fields. Therefore, the hole subband structurecannot be adequately described by a simple analytical model(parabolic, etc.) and numerical method is necessary for accuratecharacterization. However, in some cases, we may not needto know the physical details of the hole subband structure butonly of some macroscopic electrical parameters, such as,are interested. In these cases, an improved one-band EMA as-suming a parabolic dispersion is valuable. In such a treatment,(6) and (7) are still valid, however the and are notderived from the bulk Si, but from the numerical results of thesix-band EMA calculation in an empirical way. The resultsare plotted in Fig. 4 as a function of the surface electric field,instead of substrate doping , for general purpose. Attentionis first paid to empirical energy quantization effective massfor the first three subbands, as shown in Fig. 4(a). They aredetermined by (6) inversely when the numerical energies ofthe subband minima have been obtained by six-band effectivemass equations. For , the empirical energy quantizationeffective mass is and is independent of because ofits purity of heavy hole as discussed before. For and

subbands, the empirical effective masses display anelectric field dependent behavior due to the field dependenceof band mixing. The empirical effective mass value ofsubband increases as increases. Near , this bandis mainly light hole (bulk mass ). As electric fieldincreases, more and more split-off hole (bulk mass ) ismixed into this band, manifesting itself as an increase of theempirical effective mass. On the other hand, the decrease ofthe empirical effective mass of subband with increasingelectric field corresponds to the increased mixing of light holeinto this primarily split-off hole subband near .

1192 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 48, NO. 6, JUNE 2001

For subband DOS, we introduce the empirical DOS effectivemass which can be determined from (4) and (7). The inver-sion charge density occupying subbandis

(9)

Here, is the subband minimum and is the relative en-ergy referenced from the subband edge. Combined with (7), theempirical DOS effective mass is

(10)

The obtained empirical DOS effective masses are shown inFig. 4(b). These effective mass values for the and

subbands are weakly dependent of in the range ofour interests up to 1 MV/cm. This can also be clearly seenfrom Fig. 2, where the DOS profiles are nearly the same fordifferent substrate doping concentration. For subband,the dependence (and therefore dependence) of DOSis stronger as indicated in Fig. 2(c). We have fitted the elec-tric field dependence of these empirical effective masses inpolynomial form up to the second order of . The obtainedparameters are all summarized in Table I. In general, the holequantization behavior can be described empirically in thisimproved one-band EMA using these electric field dependenteffective masses. However, it must be noted that (7) is basedon the Maxwell-Boltzmann statistical function and it is onlyapplicable when the inversion is not very strong. Here theempirical DOS masse depends only on the electric field, andis independent of the Fermi level. In strong inversion case,Fermi-Dirac distribution should be used and the empiricalDOS masses will also depend on the Fermi level. Moreover,in strong inversion case, the inversion charge also affects thesurface electric field and surface potential.

Although some of these empirical effective mass values areelectric field dependent, this dependence can be neglected in thefirst order approximation. The reason is that at room tempera-ture most of inversion holes are occupied on the subband(over 70%) and the contribution of and subbandsis relatively small. From Fig. 4, both the energy quantizationand DOS masses of subband have a weak depen-dence. This leads us to propose a set of constant empirical ef-fective mass values to describe the shift. They are found tobe for the three subbands.In Fig. 5, we show the shifts calculated from these constantempirical effective masses under one-band EMA. Comparingto the numerical results, it is verified that the shifts can beobtained by this field independent empirical effective massesmethod with sufficient accuracy.

Finally, comparison with the experimentally observedshift is given in Fig. 6. The experimental data for 23 nmand 15 nm oxides are from [6]. Our calculation of shiftsin pMOSFET by the improved one-band EMA using fieldindependent empirical effective masses is in well agreement

Fig. 5. Threshold voltage shifts�V at various substrate dopingN . Thenumerical results by six-band EMA calculations are shown as solid circles whilesolid line is by improved one-band EMA using our new constant empiricaleffective masses listed in Table I.

Fig. 6. Experimental and calculated threshold voltage shifts as a function ofsubstrate doping. Solid circles and triangles are experimental data for 23- and15-nm oxide [6]. Solid and dashed lines are their respective calculation usingour improved one-band EMA with constant empirical effective masses listed inTable I.

with the experiments. This demonstrated that our improvedone-band EMA method will be valuable for the modeling ofhole quantization and shifts in pMOSFETs. Our methodhas also been used to calculate the inversion layer capacitanceand hole direct tunneling current giving rise to results thatcorresponds well with the experiments. The results will bepublished elsewhere.

VI. CONCLUSIONS

In conclusion, the hole quantization is studied by a six-bandEMA, which takes into account the band mixing effect. Theresults demonstrate that the band mixing cannot be neglected.The traditional one-band EMA using bulk effective massescannot describe the hole quantization accurately. It overallunderestimates the 2-D hole DOS and overestimatesshiftin pMOSFET. The shift obtained by our six-band effectivemass model in zigzag well approximation is found to remainsufficiently accurate compared to the reported pesudopotentialfull-band self-consistent calculation with more efficiencycomputationally. Based on this, an improved one-band EMAis developed in two hierarchies. In the first hierarchy, the

HOU AND LI: HOLE QUANTIZATION EFFECTS AND THRESHOLD VOLTAGE SHIFT 1193

empirical effective masses are derived from the numericalresults of the six-band EMA and are electric field dependent.In the second hierarchy, further simplifications are made suchthat the electric field dependencies of the empirical effectivemasses are neglected. Hole quantization and shift canbe described adequately by the improved one-band EMA ineither first or second hierarchy. The methods developed inthis work with different approximations, due to their accuracyand computational efficiency, provide a firm ground in routinedevice simulation for deep submicron MOS devices.

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Yong-Tian Hou received the B.S. and M.S. degreesin physics from Peking University, Beijing, China, in1990 and 1993, respectively. Currently, he is pursuingthe Ph.D. degree in electrical engineering at NationalUniversity of Singapore.

In 1993, he joined Tianma Microelectronics Co.Ltd., Shenzhen, China. From 1998 to 1999, he wasa Research Engineer in the Department of Electricaland Computer Engineering, National University ofSingapore. His general research interests include QMeffects, tunneling current, and reliability physics of

ultrathin gate oxide in deep submicron MOSFETs.

Ming-Fu Li (M’91–SM’99) graduated from theDepartment of Physics, Fudan University, Shanghai,China, in 1960.

He joined the Department of Applied Physics,University of Science and Technology of China(USTC). In 1978, he joined the Graduate School fac-ulty, Chinese Academy of Sciences, Beijing, wherehe became a Professor in 1986. He has also served asAdjunct Professor at the Institute of Semiconductors,Chinese Academy of Science, Fudan University,and USTC, Hefei.He was a Visiting Scholar at Case

Western Reserve University, Cleveland, OH, in 1979, University of lllinoisat Urbana-Champaign from 1979 to 1981, and was a Visiting Scientist atthe University of California at Berkeley and Lawrence Berkeley NationalLaboratories from 1986 to 1987, 1990 to 1991, and 1993. He joined theDepartment of Electrical Engineering, National University of Singapore, in1991, and became a Professor in 1996. His current research interests are in theareas of reliability physics in deep submicron CMOS devices, analog IC design,and wide energy gap group III nitride. He has published over 150 researchpapers and two books, includingModern Semiconductor Quantum Physics(Singapore: World Scientific, 1994). He has served on several internationalprogram committees and advisory committees in international semiconductorconferences in China, Japan, Canada, Germany, and Singapore.