Statistical mechanics based model for negative bias temperature instability induceddegradationSufi Zafar Citation: Journal of Applied Physics 97, 103709 (2005); doi: 10.1063/1.1889226 View online: http://dx.doi.org/10.1063/1.1889226 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/97/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in General framework about defect creation at the Si ∕ Si O 2 interface J. Appl. Phys. 105, 114513 (2009); 10.1063/1.3133096 Physical mechanisms of negative-bias temperature instability Appl. Phys. Lett. 86, 142103 (2005); 10.1063/1.1897075 Study of negative-bias temperature-instability-induced defects using first-principle approach Appl. Phys. Lett. 83, 3063 (2003); 10.1063/1.1614415 Bonding constraints and defect formation at interfaces between crystalline silicon and advanced single layer andcomposite gate dielectrics Appl. Phys. Lett. 74, 2005 (1999); 10.1063/1.123728 Electrical and physical characterization of deuterium sinter on submicron devices Appl. Phys. Lett. 72, 1721 (1998); 10.1063/1.121163

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Statistical mechanics based model for negative bias temperature instabilityinduced degradation

Sufi Zafara!

Semiconductor Research and Development Center, Research Division, T. J. Watson Research Center, IBM,Yorktown Heights, New York 10598

sReceived 17 September 2004; accepted 16 February 2005; published online 4 May 2005d

A physics based model for negative bias temperature instabilitysNBTId induced degradation isproposed. Like previous models, this model attributes NBTI to depassivation of Si–H bonds at theSiO2/Si interface. The two distinguishing features of the proposed model are:sid statisticalmechanics is applied to calculate the decrease in interfacial Si–H density as a function of stressingconditions, andsii d hydrogen diffusion in the oxide is assumed to be dispersive and the diffusingspecies is identified with the positively charged hydrogen ionsHi

+d. The model assumes that as Hi+

diffuses away from the interface into the oxide, interfacial and bulk traps are created. Based on thesemodel assumptions, an equation for the threshold voltage shiftsDVtd is derived as a function ofstressing time, oxide field, temperature, oxide thickness, and initial Si–H density at the interface.The model predicts thatDVt would increase with a power law dependence at earlier stressing timesand would saturate at longer times. The power law increase at earlier stress times is attributed todispersive hydrogen diffusion andDVt saturation at longer stress times is ascribed to occur when allbonded hydrogensSi–Hd has been removed from the interface. These and other model predictionsare verified using published NBTI data from various research groups for p-channel field effecttransistorsspFETsd. In addition, the model is shown to be compatible with NBTI data for HfO2

pFETs. ©2005 American Institute of Physics. fDOI: 10.1063/1.1889226g

I. INTRODUCTION

Over the last few decades, the continuous miniaturiza-tion of devices in integrated circuits has led to an exceptionalenhancement in device performance and densities. As thefield effect transistors are scaled down with a concomitantdecrease in silicon oxide film thickness, the reliability ofthese oxide layers becomes an increasingly important issue.1

A major reliability issue for aggressively scaled devices withoxide thickness of,20 Å is the negative bias temperatureinstability sNBTId.1,2 NBTI manifests itself as a shift in thethreshold voltage and a decrease in the inversion layer mo-bility under negative bias stressing at elevated temperatures.These instabilities can lead to circuit failures, and are there-fore of both technological and scientific interest.

For the last several years, NBTI has been studied inp-channel field effect transistorsspFETsd where the degrada-tion is most dominant. Various experimental studies showthat positive charges are created both at the SiO2/Si interfaceand in the oxide under negative bias stressing, thereby caus-ing the threshold voltagesVtd to shift and the channel mo-bility to decrease.1–5 There are several models for NBTI in-duced degradation.2–7 One of the most widely acceptedmodels is the reaction-diffusion model2,3,6 that attributesNBTI induced degradation to depassivation of Si–H bonds atthe oxide/Si interface and the diffusion of hydrogen relatedspecies away from the interface. The strength of this modelis that it predicts the threshold voltage shiftsDVtd to increasewith a power law dependence on time. However, the model

does not predict the saturation as observed after prolongedstressing nor does it derive the dependence ofDVt on stressoxide field from first principles.

In this article, a model for NBTI is derived from firstprinciples. Like the reaction-diffusion model, this model alsoattributes the creation of positive charges to the depassiva-tion of Si–H bonds at the Si/SiO2 interface. In this model,depassivation is considered as a two step process: desorptionof bonded hydrogen from the interface and the diffusion ofdesorbed interstitial hydrogen away from the interface.Bonded and interstitial hydrogen at the interface is assumedto be at thermal equilibrium at all times and statistical me-chanics is applied to calculate the density of desorbed inter-stitial hydrogen atoms as a function of stressing conditionsand time. Interstitial hydrogen is assumed to be mainly posi-tively chargedsHi

+d with a diffusion constant that decreaseswith time sdispersive diffusiond. As Hi

+ diffuses away fromthe interface into the oxide, fast interfacial and slow oxidetraps are assumed to be created. Interfacial and bulk oxidetraps are assumed to have neutral and charged states depend-ing on their electron occupancy. Unoccupied interfacial andoxide traps are identified with the observed positive chargesat the interface and in the oxide, respectively. Based on theideas outlined above, an equation forDVt is derived as afunction of stressing time, oxide field, temperature, disper-sive hydrogen diffusion parameter, and the initial Si–H den-sity at the oxide/Si interface. The model equation is verifiedusing previously published data at various stress conditionsfor pFETs with SiO2 and SiON gate dielectrics.5,8–10 Thecalculated results are also compared with the NBTI inducedadElectronic mail: szafar@us.ibm.com

JOURNAL OF APPLIED PHYSICS97, 103709s2005d

0021-8979/2005/97~10!/103709/9/$22.50 © 2005 American Institute of Physics97, 103709-1

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interfacial trap density increase as measured by chargepumping experiments.5 In addition, the proposed model isshown to be compatible with HfO2 pFETs.

The article is organized as follows. The proposed modelis discussed in Sec. II, calculation details are given in Sec.III, and the model is verified by comparing calculated resultswith measurements in Sec. IV. In Sec. V, the model is shownto be compatible with NBTI data for oxide/HfO2 pFETs. Theimplications of the model on observed recovery and acstressing effects on NBTI are discussed in Sec. VI. Finally,the conclusions are discussed in Sec. VII.

II. PROPOSED MODEL

In this section, a model for NBTI induced degradation atconstant stress bias is proposed. Based on statistical mechan-ics principles and the dispersive diffusion of protons, anequation for the threshold voltage shiftsDVtd is derived as afunction of stressing conditions. The model equation hasthree fitting parameters defined in terms of physical quanti-ties such as stressing electric field, temperature, oxide thick-ness, initial interfacial Si–H bond density and dispersion inhydrogen diffusion. In this article, a simplified notation con-vention is used where the same notation denotes a defect andits corresponding density. For convenience, symbols used inthe model are not only discussed in the text but are alsosummarized in Table I.

A. Interfacial and bulk oxide positive charge densities

Like previous models, this model identifies the de-passivation of Si–H bonds at the oxide/Si interface as themechanism for NBTI induced positively charged defects atthe interface and in the oxide. The model assumes that asSi–H bonds break, positively charged atomic hydrogensHi

+dis created; the justification for this assumption will be dis-cussed later. As Hi

+ diffuses away from the interface, newinterfacial traps are assumed to be created at the level«int asshown in Fig. 1. The interfacial traps are assumed to bedangling bonds with neutral and charged states: neutral when

the «int trap level is occupied by an electron and positivelycharged when the trap level is empty. The dangling bondtraps are assumed to be fast, i.e., traps remain in equilibriumwith the Fermi level at the oxide/Si interface as the gatevoltagesVgd is rapidly varied during sensing. Since unoccu-pied dangling bond traps are positively charged, observedinterfacial positive charges are identified with empty interfa-cial traps. Using Fermi statistics to calculate the probabilitysPintd for a «int level to be empty, the increase in interfacialpositive charge densitysDQintd can be written as

DQint = q · Pint · DHsstd,

Pint =1

1 + 2 ·e−s«int−«Fermi,Vtd/kT and s1d

«Fermi,Vt= sEgap− dbulkd.

DHsstd is the bonded hydrogen density removed fromthe interface and is equal to the total interfacial trap densityincrease after stressing timet. The energy levels in Eq.s1dare measured with respect to the silicon valence band edge.«Fermi,Vt

is the Fermi level position at the interface whenVg=Vt, Egap is the silicon band gap, anddbulk is the Fermilevel position in bulk silicon as shown in Fig. 1. T is thedevice temperature in Kelvin, k is the Boltzmann constant,andq=1.6310−19coulomb.

We will now discuss the bulk oxide traps that are as-sumed to be created as Hi

+ diffuses away from the interfaceinto the oxide. Hi

+ ions are assumed to interact with the ox-ide, thereby creating new oxide traps at the level«ox; themicroscopic identity of these oxide traps is not known. Likethe interfacial traps, these oxide traps are also assumed tohave neutral and charged states depending on the occupancyof «ox: neutral when filled with an electron and positivelycharged when empty. The oxide traps are assumed to beslow, i.e., they are not able to follow the rapid variation inthe gate voltage during sensing and would therefore reflectthe occupancy that occurs during stressing. Since unoccupiedoxide traps are assumed to be positively charged, the ob-served oxide positive charges are identified with these emptyslow oxide traps. Using Fermi statistics, the probability Pox

for an oxide trap level«ox to be unoccupied is obtained andthe increase in positive charge density in the oxidesDQoxdcan be written as

TABLE I. Summary of notations used in the model

Vt 5 threshold voltage;Vg 5 applied voltage at the gate;DVtstd 5 threshold voltage shift after stressing time t;DHsstd 5 Si–H density decrease at the interface at stressing time t;«int 5 trap level at the oxide/Si interface;«ox 5 trap level in the oxide;dbulk 5 Fermi level position above valence band edge in bulk Si;Eox 5 applied stress field in the oxide at the oxide/Si interface;Hi 5 neutral atomic hydrogen occupying interstitial site;Hi

+ 5 proton occupying interstitial site;Ni 5 interstitial site density at the interface;EH 5 energy for converting bonded hydrogen at the interface into Hi;Doo 5 diffusion constant without dispersion for Hi

+;H•s0d 5 Si–H bond density before stressing;dox 5 equivalent oxide thicknesssEOTd;kox 5 permittivity of SiO2; andT 5 device temperature during stressing

FIG. 1. Energy level diagram showing band bending of conduction bandsc.b.d and valence bandsv.b.d in silicon substrate at negative stress voltages.The term«F is the Fermi level;«int and«ox are the proposed levels associatedwith interfacial and bulk oxide traps, respectively;dbulk is the position ofFermi level in the bulk silicon as measured from valence band edge.

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DQox = q · Pox · DHsstd,

Pox =1

1 + 2 ·e−s«ox−«Fermi,Eoxd/kT and s2d

«Fermi,Eox= sdbulk − gEox

2/3d.

«Fermi,Eoxis the Fermi level position at the silicon inter-

face at Eox, where Eox is the oxide electric field at the siliconinterface during stressing. As shown in Fig. 1 and discussedlater in Sec. III, the quantitysgEox

2/3d is the decrease in theelectronic energy due to band bending in the substrate withgas a constant. As in the case of Eq.s1d, the energy levels inEq. s2d are also measured with respect to the silicon valenceband edge. In summary, the model proposes that the de-passivation of Si–H bonds at the SiO2/Si interface createsfast interfacial traps and slow oxide traps. The occupancyprobability Pint of an interfacial trap is independent of stressfield Eox, whereas the occupancy probability Pox of slow ox-ide trap depends on Eox. The empty states of interfacial andoxide traps are identified with the observed interfacial andoxide positive charges, respectively.

Since the threshold voltage shift is proportional to thecharge densities at the interface and in the oxide,DVt can bewritten as

DVtstd = sDQint + DQoxd · sdox/koxd, s3d

whereDQint andDQox are given by Eqs.s1d ands2d, respec-tively, kox is the permittivity of SiO2, and dox is the equiva-lent oxide thickness. To keep the fitting parameters to a mini-mum without obscuring the main physical ideas, the centroidfor oxide charges is approximated to be at the interface at alltimes.

B. Derivation of DHs„t…

As stated earlier,DHsstd is the Si–H density decrease atthe SiO2/Si interface after stress time t and is equal to theinterfacial de-passivated site density increase. To derive anequation forDHsstd, we make three key assumptions.sidDesorption of bonded hydrogen from the interface is a ther-mally driven process during NBTI stressing. This impliesthat bonded and interstitial hydrogen at the oxide/Si inter-face are in thermal equilibrium at all times.sii d Once inter-stitial hydrogensHid is formed, it converts into a positivelycharged state at negative stress voltages. This assumption isbased on interstitial hydrogensHid studies in silicon.11 Thesestudies show that Hi in silicon introduces a donor level in thesilicon band gap and the charged state of Hi can be neutral orpositive depending on the electron occupancy of this level.Hi at the oxide/silicon interface is assumed to introduce asimilar donor level in the silicon band gap. Since the Fermilevel is near the valence band edge at negative bias stressconditions, the donor level would be unoccupied and inter-stitial hydrogen would be in positively charged statesHi

+d. Inother words, the Hi density ,Hi

+ density at negative biasstress voltages. This assumption is further supported by re-cent studies that show Hi

+ is the only stable state near theSiO2/Si interface.12,13 siii d Hi

+ diffusion in the oxide is as-

sumed to be dispersive, i.e., DH=D00t−s1−bd, where D00 is the

diffusion constant without dispersion, andb is a measure ofdispersion with its value decreasing from 1 to 0 with increas-ing dispersion. Dispersive diffusion is thought to occur whenthe diffusion activation energy is not single valued but has acontinuous distribution over a width. This assumption is sup-ported by experimental studies that show protonssHi

+d havedispersive diffusion in oxides14 and other amorphous mate-rials such as hydrogenated amorphous silicon.15 Based onthese three assumptions, an equation forDHsstd is derived asfollows.

Since de-passivated sites are assumed to be created dueto desorption and diffusion of protons away from the inter-face under the influence of an applied stress field, the rate ofincrease of the de-passivated site densitysHsd is equal to theionic current due to protons and can be written as

dHs/dt = Hi+ · mH · Eox = Hi

+ · sDH/kTd · Eox, s4d

where Eox is the stressing oxide field and Hi+ is the proton

density at the interface;mH is proton mobility and is equal tosDH/kTd from the Nerst–Einstein equation, where DH is theproton diffusion constant: DH=D00t

b−1. Integration of Eq.s4dwould provide an equation forDHs provided the interstitialhydrogen density is known. The interstitial hydrogen densityis estimated as described below.

As stated earlier, bonded and interstitial hydrogen densi-ties at the interface are assumed to be in thermal equilibrium.This implies that statistical mechanics can be applied to cal-culate the Hi density by writing the partition functions ofhydrogen binding sites at the interface and interstitial sites.We will first discuss the hydrogen binding site at the inter-face. A hydrogen binding site is proposed to have two hydro-gen occupancy states: an occupied or hydrogenated state thatis identified with Si–H and an empty state that is identifiedwith a dangling bond. The proposed hydrogen level diagramis shown in Fig. 2; this hydrogen energy level diagram isanalogous to an electronic energy level diagram. EH is theenergy level associated with bonded hydrogen,m is the hy-drogen chemical potential, and Ei is the energy level associ-ated with interstitial hydrogen; Ei is set as the reference levelsEi =0 eVd and other levels are measured with respect to it.Based on the hydrogen energy level diagram, the partitionfunction ZH for hydrogen binding site at the interface can bewritten as

FIG. 2. Proposed hydrogen energy level diagram. Ei is the energy level of apositively charged atomic hydrogen occupying an interstitial site at theoxide/Si interface; this level is set as the reference level and all other hy-drogen levels are measured with respect to it, i.e., Ei =0 eV. EH is the energylevel associated with bonded hydrogen at the SiO2/Si interface,m is thehydrogen chemical potential.

103709-3 S. Zafar J. Appl. Phys. 97, 103709 ~2005!

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ZH = 1 +e−sEH−md/kT. s5ad

Using the partition function, the de-passivated site densityHs can be written as

Hs = Nt/ZH, s5bd

where Nt is the total density of hydrogen binding sites at theoxide/Si interface and is constant for a given sample.

Similarly, an interstitial site partition functionsZid canalso be written based on the hydrogen energy level diagramshown in Fig. 2

Zi = 1 +e−sEi−md/kT = 1 +e+m/kT. s6ad

Using Zi, the interstitial hydrogen density Hi can be writtenas

Hi = Nie−sEi−md/kT/Zi < Nie

+m/kT, s6bd

where Ni is the total density of interstitial sites at the inter-face. Substitutingm from Eq. s6bd into Eq. s5bd, we obtainthe following equation:

Hi = Ni ·eEH/kT ·SNt − Hs

Hs

D . s7d

Since Hi <Hi+, Eq. s7d can be substituted for Hi

+ into Eq.s4d and we obtain

dHs

dt= SNt − Hs

Hs

D ·eEH/kT · Ni · D00 ·SEox

kTD · t−s1−bd. s8d

By integrating Eq.s8d and assumingDHsstd!Nt, the fol-lowing equation forDHsstd is obtained:

DHsstd = H•s0d · s1 − e−st/tdbd,

s9d

t = SNi ·eeH/kT · D00

kT · bD−1/b

· Eox−1/b,

where H•s0d is the initial Si–H bond density at the interfaceat stress time t=0 s It may be noted that since de-passivationof the interface is assumed to create interfacial traps, Eq.s9dpredicts the interfacial trap density increase as a function ofstressing time, voltage, and temperature. This equation willbe verified in Sec. IV B using charge pumping experimentalresults for NBTI. Substituting Eq.s9d for DHs into Eq. s3d,the model equation for the threshold voltage shiftsDVtstdd isderived as

DVtstd = DVmax· s1 − e−st/tdbd, s10d

where,DVmax,t, andb are the three model parameters andare defined as follows.b is a measure of dispersion in hy-drogen diffusion and its value decreases from 1 to 0 as dis-persion increases. Dispersive diffusion is thought to occurwhen the diffusion activation energy is not single valued buthas a continuous distribution over a widths that depends onthe disorder in the amorphous material. Dispersion increasesas the ratios /kT increases fors.kT, and dispersion is zerofor s&kT. Based on the observed dispersive behavior ofhydrogen diffusion,15 b is predicted to increase with increas-ing temperature and be independent of stress oxide field. TheparameterDVmax is the saturating value ofDVt that would

occur after prolonged stressing.DVmax is the maximum shiftthat would occur when all the interfacial Si–H bonds havebeen de-passivated andDVmax is defined as

DVmax= sDQintmax+ DQox

maxd · sdox/kd

with,

DQintmax=

q · H•s0d1 + 2e−seint−Egap+dbulkd/kT s11d

DQoxmax=

q · H•s0d

1 + 2e−seox−dbulk+gEox2/3d/kT

,

where, DQintmax and DQox

max are the maximum interfacialand bulk positive charge densities, respectively, and are ob-tained by substitutingDHsstd=H•s0d into Eqs.s1d and s2d.From the above equation,DVmax is predicted to increase withstress oxide fieldsEoxd due to the contributions of slow oxidetraps. The model parametert is defined in Eq.s9d. The termt is the time whenDVt reaches 63% ofDVmax and is there-fore a measure of the NBTI induced degradation rates.t ispredicted to decrease with a power law dependence on Eox

with 1/b as exponent:t~Eox−1/b. It may be noted that the

model parameterst andb are related through Eq.s9d.In summary,DVt is attributed to the de-passivation of

Si–H bonds at the oxide/Si interface mediated by dispersiveHi

+ diffusion, thereby creating fast interfacial traps and slowoxide traps. Statistical mechanics is applied to calculate thede-passivated site density increase at the interface as a func-tion of stress conditions. This increase in depassivation ofSi–H bonds is equated with a concomitant increase in inter-facial and oxide traps. Both the interfacial and oxide trapsare assumed to have neutral and positively charged statesdepending on the occupancy of their respective levels andthe occupancy probabilities are estimated using the Fermistatistics. Based on these principles, model Eqs.s9d–s11d forNBTI induced degradation are derived. These model equa-tions will be verified using published NBTI data for pFETs atvarious stress conditions. The model is also shown to becompatible with NBTI data for SiON/HfO2 pFETs withtungsten gates.

III. CALCULATION DETAILS

Before discussing the calculated results in the next sec-tion, the details of the calculations are described. The stressoxide field is Eox=Vox/dox where Vox is the voltage drop inthe oxide and dox is the equivalent oxide thickness. Vox canbe obtained by numerically solving the equation

Vg = Vox + csi + cpoly + Vfb, s12d

where Vg is the applied gate voltage.csi is the voltage dropdue to band bending in the silicon substrate at inversion, andcpoly is the voltage drop due to depletion in polysilicon gate.The csi and cpoly are estimated from Refs. 16,17.csi=g ·sVox/doxd2/3, where g=1.61·sqq /msi

1/2d2/3 withmsi as the effective mass in the silicon substrate formotion perpendicular to the interface. cpoly

=skox2/2q kpolyNpoly

2d ·sVox/doxd2, where kox and kpoly arethe permittivities of oxide and polysilicon, respectively, and

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Npoly is the dopant density in p+ polysilicon gate. Vfb is thedifference between the Fermi level in the gate and siliconsubstrate: Vfb=s«Fermi,gate−dbulkd, where «Fermi,gate is theFermi level in the gate as measured with respect to the sili-con valence band edge. In the present calculations, Npoly=131020cm−3, «Fermi,gate ,0 eV for p+ polysilicon gates and«Fermi,gate,0.56 eV18 for tungsten gates. In Ref. 8, the dopantdensity in the silicon substrate is,231017cm−3 and is as-sumed to be the same in references where dopant densitieswere not specified. Thedbulk values corresponding to a dop-ant density,231017cm−3 are estimated by taking into ac-count the temperature dependence of silicon band gapsEgdand the Fermi level:19 dbulk=0.98 eV and Eg=1.124 eV at25 °C, dbulk=0.86 eV and Eg=1.098 eV at 125 °C anddbulk

=0.85 eV and Eg=1.080 eV at 175 °C.

IV. MODEL VERIFICATION

To verify the model Eqs.s9d–s11d, calculated results arecompared with published NBTI data for pFETs with oxidesof different thicknesses and gate materials. Measured NBTIinduced degradation data discussed in this section were ob-tained by applying a constant stress voltage that was periodi-cally and briefly interrupted to measure the shift in thresholdvoltagesDVtd or interfacial trap density increase.

A. Predicted dependence of DVt on stress time

Equations10d predicts the dependence ofDVt on stresstime with b, DVmax, and t as model parameters.b is pre-dicted to be independent of stress field and increase withincreasing temperature while the predicted dependence oft

and DVmax is given by Eqs.s9d and s11d, respectively. Toverify the model, we demonstrate that Eq.s10d not only pro-vides good fits to the data but the values ofb, DVmax, andtestimated from the fits are consistent with model predictions.

Figure 3sad shows the comparison between measuredand calculatedDVt as a function of stressing time for variousstress voltages at 125 °C for SiON/polysilicon pFETs withSiON equivalent oxide thicknesssEOTd of 16 Å. Symbolsdenote the measured data from Ref. 8 and solid lines are thecalculated results. Each measured curve is fitted using modelEq. s10d with b ,t, andDVmax as fitting parameters. The cal-culated curves show thatDVt increases with a power lawdependence at the initial stages of stressing and saturates atlonger times. The model provides good fits to the data overseveral decades of stress time at various stress voltages. Totest the model predictions forb ,t, and DVmax the depen-dence of these extracted parameters is analyzed as a functionof stress oxide field. From the fits in Fig. 3sad, b is indepen-dent of stress voltage and is equal to 0.255; this observedstress voltage independence is consistent with the predictiondiscussed in Sec. II B. The model Eq.s9d predicts thattwould decrease with a power law dependence on the stressoxide field with an exponent equal to 1/b. To verify thesepredictions, the dependence oft on stress oxide fieldsEoxd isexamined as shown in Fig. 3sbd. Symbols denotet valuesobtained from the fits in Fig. 3sad, whereas the solid line isthe power law fit with 3.8 as the exponent. We now comparethis measured exponent value with predicted value. Sincefrom Fig. 3sad the measuredb=0.255, the predicted expo-nent values=1/bd is 3.9 in good agreement with the mea-sured value as shown in Fig. 6; Fig. 6 summarizes the com-

TABLE II. Summary of parameter values estimated from fits shown in Figs. 3scd, 4scd, 5scd, and 9scd. H•s0d isthe initial Si–H density at the oxide/Si interface,«int and«ox are the trap levels associated with the interfacialand oxide traps created due to depassivation;«int and«ox are measured with respect to the silicon valence bandedge at the interface.

pFET T s°Cd H•s0dscm−2d «intseVd «oxseVd

SiONs16 Åd /poly 125 2.631012 0.29 −0.19SiO2s33 Åd /poly 175 1.531012 0.24 −0.16SiON s17 Åd/W 125 1.431012 0.17 −0.12SiON s17 Åd/W 25 1.231012 0.22 −0.09SiON/HfO2/W 125 1.831012 0.25 −0.05

FIG. 3. Comparison between predicted and measured results for SiONsEOT=16 Åd pFETs with polycrystalline silicon gates.sad Dependence ofDVt on stresstime at various stress voltages and 125 °C. Symbols are measured data points from Ref. 8. Solid lines are fits using model Eq.s10d with b ,t, andDVmax asfitting parameters; fits show thatb=0.255 and is independent of stress voltage.sbd Dependence of model parametert on stress oxide field; symbols aretvalues obtained from fits in Fig. 3sad and solid line is the power law fit with an exponent of 3.8.scd Dependence of model parameterDVmax on stress oxidefield; symbols areDVmax values estimated from fits in Fig. 3sad and solid line is the fit using model Eq.s11d with H•s0d ,«ox, and«in as fitting parameters;parameter estimates are listed in Table II.

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parison between measured and predicted exponent values fora variety of pFETs and will be discussed later. Hence, theobserved dependence oft is consistent with the model pre-diction of t~Eox

−1/b. Figure 3scd shows the dependence ofDVmax on the stress oxide field. Symbols denoteDVmax val-ues estimated from Fig. 3sad, and the solid line is the fit usingEq. s11d with «ox,«int and H•s0d as fitting parameters. Asshown in Fig. 3scd, Eq. s11d not only provides a good fit tothe DVmax data but also yields feasible estimates of«ox,«int

and H•s0d as listed in Table II. Hence, the model predictionfor DVmax is verified.

A similar analysis was performed for NBTI measure-ments for SiONsEOT,17 Åd pFETs with tungsten gates asreported in Ref. 8. Figure 4sad shows the comparison be-tween calculated and measuredDVt dependence on stresstime at 125 °C for various stress voltages. Symbols are themeasurements and solid lines are fits using model Eq.s10dwith b ,t andDVmax as fitting parameters. As in the case forpFETs with polysilicon gates, the model provides good fits tothe data over several decades of stressing time at variousstress voltages.b ,t, and DVmax are extracted from the fitsand compared with model predictions. From the fits in Fig.4sad, the extracted value ofb=0.25 and is independent ofstress oxide field as predicted. Figure 4sbd shows thatt de-creases with a power law dependence on Eox with 4.9 as theexponent; this extracted exponent value is within 22% of thepredicted value of 1/b=4 as shown in Fig. 6. Hence, theobservedt dependence is consistent with the model Eq.s9d

prediction. Figure 4scd shows the dependence ofDVmax onEox; symbols denoteDVmax values estimated from Fig. 4sadand the solid line is the fit obtained using Eq.s11d with«ox,«int and H•s0d as fitting parameters. Once again, Eq.s11dnot only provides a good fit to theDVmax data but also yieldsfeasible estimates of«ox,«int and H•s0d as listed in Table II.Hence, the model prediction forDVmax is verified. SimilarNBTI measurements were also performed at 25 °C, and theanalysis results are summarized in Figs. 4sbd, 4scd, and 6 andin Table II, consistent with model predictions.

To illustrate the applicability of the model to a wideroxide thickness range, NBTI data from Ref. 9 are analyzedand compared with the calculated results. In Ref. 9, the de-pendence ofDVt on stress time was measured at variousstress voltages and 175 °C for 33-Å-thick SiO2 pFETs withp+ polysilicon gates. Figure 5sad shows the comparison be-tween measured and calculated curves at various stress volt-ages. As shown in the figure, the calculated curves are ingood agreement with the measurements and the model pa-rametersb ,t, and DVmax are extracted from the fits. Onceagain, theb value obtained from the fits is independent ofstress voltage and consistent with the model predictionb=0.78. Figure 5sbd shows that the model parametert de-creases with a power law dependence with increasing Eox

with 1.2 as exponent; this exponent value is consistent withthe predicted value of 1/b=1.3 as also compared in Fig. 6.Hence,t dependence is consistent with the model predic-tions. Figure 5scd shows the dependence ofDVmax on Eox;

FIG. 4. Comparison between predicted and measured results for SiONsEOT=17 Åd pFETs with tungsten gates.sad Dependence ofDVt on stress time as afunction of stress voltage at a constant temperature of 125 °C, symbols are measured data points from Ref. 8 and solid lines are fits using model Eq.s10d withb ,t, andDVmax as fitting parameters. Fits show thatb=0.25 and is independent of stress voltage.sbd Dependence of model parametert on stress oxide fieldsEoxd at 125 and 25 °C; symbols are values obtained from fits shown in Fig. 4sad; solid lines are the power law fits with exponent values shown in the figure.scd Dependence of model parameterDVmax on stress oxide field; symbols areDVmax values obtained from fits in Fig. 4sad; solid lines are fits using model Eq.s11d with H•s0d ,«ox, and«in as fitting parameters; parameter estimates are listed in Table II.

FIG. 5. Comparison between predicted and measured results for SiO2 s33 Åd pFETs with polycrystalline silicon gates.sad Dependence of threshold voltageshift sDVtd on stress time as a function of stress voltage at a constant temperature of 175 °C. Symbols are measured data points from Ref. 9. Solid lines arefits using model Eq.s10d with b ,t, andDVmax as fitting parameters; fits show thatb=0.78 and is independent of stress voltage.sbd Dependence of modelparametert on stress oxide field; symbols aret values estimated from Fig. 5sad and solid line is the power law fit with an exponent=1.2.scd Dependence ofmodel parameterDVmax on stress oxide field; symbols areDVmax values estimated from Fig. 5sad and solid line is the fit using model Eq.s11d with H•s0d ,«ox

and«in as fitting parameters; parameter estimates are listed in Table II.

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the symbols denoteDVmax values estimated from Fig. 5sad,and the solid line is the fit obtained using Eq.s11d with«ox,«int and H•s0d as fitting parameters. As listed in Table II,H•s0d, «ox and «in values obtained from the fit are feasible.Hence, the model prediction forDVmax is verified.

Before concluding the above discussion, we will discussFig. 6 which summarizes the comparison between the mea-sured and predicted power law exponent as defined in Eq.s9dfor a variety of pFETs. As discussed earlier, Eq.s9d st~Eox

−1/bd predicts the power law exponent fort to be theinverse of parameterb. In Fig. 6, the dependence of thepower law exponent onb is shown. The solid line is thepredicted valuess=1/bd and symbols are measured exponentvalues obtained from Figs. 3sbd, 4sbd, 5sbd, and 9sbd withcorresponding measuredb values obtained from Figs. 3sad,4sad, 5sad, and 9sad. As shown in Fig. 6, the predicted expo-nent values are in good agreement with measured values,thus verifying the model Eq.s9d.

In summary, the model Eq.s10d is shown to providegood fits to theDVt dependence on stress time data overseveral decades at various stress conditions for pFETs withdifferent oxide thickness and gate materials. The dependenceof the extracted parametersb ,t, andDVmax on stress oxidefields is consistent with model predictions and estimates ofphysical parameters such as interfacial and oxide trap levels

and initial Si–H bond density are feasiblesTable IId, thuslending support to the validity of the proposed model.

B. Predicted dependence of interfacial trap densityincrease on stressing

As discussed in Sec. II B,DHsstd is the bonded hydro-gen density removed from the interface after stress timet andis given by Eq.s9d. As bonded hydrogen is removed from theinterface, new interfacial traps are created. Hence, Eq.s9dwould predict the dependence of interfacial trap density in-creasesDNitd as a function of stressing time, stress oxidefield, and temperature. Equations9d predicts thatDNit wouldincrease with a power law dependence on stress time in theinitial stages of stressing and would saturate to a maximumvalue of H•s0d, where H•s0d is the Si–H bond density at theSiO2/Si interface before stressing att=0 s. Since the H•s0dvalue is a property of the interface and processing condi-tions, the saturating value forDNit is predicted to be inde-pendent of stress oxide field Eox and temperature. To verifythe model predictions forDNit, the NBTI induced increase ininterface trap density as reported in Ref. 5 is compared withcalculated results. In Ref. 5, the increase in interfacial trapdensity was measured using the charge pumping method as afunction of stress time at various constant stress voltages andtemperatures; measurements were made on SiONsEOT=20 Åd pFETs with p+ polysilicon gates. Figure 7sad com-pares the calculated and measured dependence of the inter-facial trap density increase on stress time as a function ofstress voltage at a constant temperature of 125 °C. Symbolsare the measured values and solid lines are fits using Eq.s9dwith H•s0d ,t and b as fitting parameters. As shown in Fig.7sad, the calculated curves provide good fits to the data andsaturate to the same maximum value of 631011cm−2 inde-pendent of stress voltage, consistent with model prediction.Figure 7sbd compares calculated and measured interfacialtrap density increase on stress time as a function of stressingtemperature at a constant stress voltage of −0.25 V. Onceagain, symbols are measured values and solid lines are fitsusing Eq.s9d with H•s0d ,t, andb as fitting parameters. Thecalculated curves are in good agreement with measured

FIG. 6. Comparison between the measured and predicted power law expo-nent values for a variety of pFETs; the power law exponent is defined in Eq.s9d st~Eox

−1/bd. The solid line denotes the predicted exponent valuess=1/bd and symbols are measured exponent values obtained from the powerlaw fits in Figs. 3sbd, 5sbd, 6sbd, and 9sbd with the corresponding measuredb values obtained from Figs. 3sad, 5sad, 6sad, and 9sad.

FIG. 7. Comparison between predicted and measured results for SiONs20 Åd pFETs with polycrystalline silicon gates; measured results are from Ref. 5 andwere obtained by charge pumping method.sad Dependence of interfacial trap density increase on stress time at different stress voltages and 125 °C. Solid linesare fits using model Eq.s9d with b ,t, and H•s0d as fitting parameters.sbd Dependence of interfacial trap density increase on stress time as a function of stresstemperatures at a constant stress voltage of −2.5 V; solid lines are fits using model Eq.s9d with b ,t, and H•s0d as fitting parameters; estimated values of thefitting parameters are listed in Table III.

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curves and the estimated values of H•s0d ,t, andb obtainedfrom the fits are summarized in Table III. As shown in TableIII, H •s0d is independent of temperature whilet decreasesandb increases with temperature, consistent with model pre-dictions. In summary, the NBTI induced increase in the in-terfacial trap densities as measured by the charge pumpingmethod is consistent with calculated results. However, addi-tional charge pumping experiments over longer stressingtime periods are required to unambiguously verify the modelprediction forDNit saturation.

C. Predicted dependence of DVt on deuteriumpassivation

From Eqs.s9d and s10d, DVt is predicted to be propor-tional to the hydrogen diffusion constantsDdoo for t!t.Since the diffusion constant is inversely proportional to thesquare root of the mass of the diffusing species, it is pre-dicted thatDVt would be 1.4 times smaller for deuteriumannealed devices as compared to hydrogen annealed devices.This prediction is verified by the experimental result reportedin Ref. 10. In Ref. 10, 38-Å-thick SiO2 pFETs with polycrys-talline silicon gates were subjected to postmetal anneals inH2 and D2 ambient, andDVt versus stress time curves weremeasured at 150 °C and −2.9 V for 5000 s. TheDVt curvefor D2 annealed pFETs were observed to be 1.5 times smallerin comparison to that for H2 annealed devices as shown inFig. 8. Hence, the experimental results reported in Ref. 10are in quantitative agreement with the predicted 1.4 factordecrease for D2 annealed devices.

V. APPLICATION OF THE MODEL TO HfO 2 pFETS

In Ref. 8, the NBTI induced dependence ofDVt onstress time was measured at various stress voltages and125 °C for SiON/HfO2 pFETs with tungsten gates; theequivalent oxide thicknesssEOTd of the SiON/HfO2 stack is,16 Å. Using the data reported in this reference, the pro-posed model is shown to be compatible with SiON/HfO2

pFETs. Figure 9sad shows the comparison between the cal-culated and measured dependence ofDVt on stress timecurves at various stress voltages. Each measured curve isfitted using model Eq.s10d with b ,t, and DVmax as fittingparameters. The calculated results show thatDVt increaseswith a power law dependence at the initial stages of stressingand saturate at longer times. As shown in the figure, themodel provides good fits to data over several decades ofstress time at various stress voltages. To test the model pre-dictions for b ,t, and DVmax, the dependence of these ex-tracted parameters is analyzed as a function of the stressoxide field at the SiON/Si interface. As shown in Fig. 9sad,b=0.3 is independent of stress voltage, consistent with theprediction forb. Figure 9sbd shows the dependence oft onEox. Symbols denotet estimated from fits in Fig. 9sad. Thesolid line is the power law fit with exponent value of 3.8.This measured exponent value is consistent with the pre-dicted value of 1/b=3.3 as compared in Fig. 6. Hence, theobserved dependence oft on Eox is consistent with themodel prediction. Figure 9scd shows the dependence ofDVmax on Eox; symbols denoteDVmax values estimated fromFig. 9sad and the solid line is the fit obtained using Eq.s11dwith «ox,«int and H•s0d as fitting parameters. From the fit,parameter estimates are obtained and are listed in Table II.

TABLE III. Summary of parameter values estimated from the fits shown inFig. 7sbd.

°C H•s0dscm−2d b t ssd

200 731011 0.310 1045175 731011 0.301 6850150 731011 0.299 22 883125 731011 0.300 126 000100 731011 0.295 850 00075 731011 0.285 4.3106

50 731011 0.247 83107FIG. 8. Dependence ofDVt on stress time for hydrogensH2d and deuteriumsD2d annealed pFETs; symbols are data from Ref. 10. Deuterium annealeddevice shows lowerDVt by 1.5 factor in comparison to hydrogen annealeddevice, in quantitative agreement with the predicted decrease of 1.4 factor.

FIG. 9. Illustration of the applicability of the model to SiON/HfO2 pFET with stack EOT,16 Å and tungsten gates.sad Dependence ofDVt on stress timeat various stress voltages and 125 °C; symbols are measured data points from Ref. 8 and solid lines are fits using model Eq.s10d with b ,t, andDVmax as fittingparameters; fits show thatb=0.30 and is independent of stress voltage.sbd Dependence of model parametert on the stress oxide field; symbols aret valuesestimated from fits in Fig. 9sad and solid line is the power law fit with an exponent =3.8.scd Dependence of model parameterDVmax on stress oxide field.Symbols areDVmax values estimated from fits in Fig. 9sad and the solid line is the fit using model Eq.s11d with H•s0d ,«ox, and«in as fitting parameters;parameter estimates are listed in Table II.

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These estimated values are feasible, thereby verifying themodel prediction forDVmax. In summary, the model Eq.s9d–s11d are compatible with NBTI data for SiON/HfO2/WpFETs.

VI. MODEL IMPLICATIONS FOR NBTI RECOVERYAND AC STRESSING

Experimental studies show that NBTI induced thresholdvoltage shift recovers when the stress bias is removed andthe extent of recovery depends on dc stressing conditionssuch as temperature.5,20 In this model, the slow oxide trapswould contribute towards recovery. As the negative stressfield is reduced, the Fermi level at the interface would moveto a new position. Since the equilibrium density of positivelycharged oxide traps decreases with decreasing gate voltagefEq. s2dg, the positively charged oxide trap density woulddecrease until a new equilibrium density corresponding toVg=0 V is reached after several seconds. Hence, the recov-ery is attributed to the change in the charge state of the slowoxide traps. The proposed oxide traps would also impact thethreshold shift due to ac stressing. Since the oxide traps areassumed to be slow, they would not be able to reach equilib-rium during the on cycle of ac stress and would start torecover during the off cycle. Consequently, the contributionsof the slow oxide traps would become increasingly smallwith increasing frequency of ac stress, consistent with mea-sured results.5

VII. CONCLUSIONS

In conclusion, a model for NBTI induced degradation inpFETs is proposed based on physics principles. Using theprinciples of statistical mechanics and dispersive hydrogendiffusion, equations for NBTI induced degradation as a func-tion of stress time, oxide field and temperature, and oxidethickness are derived. The model equations are verified by

comparing measured results at various stress conditions for avariety of pFETs with model predictions. The model is alsoshown to be compatible with NBTI induced degradation datafor HFO2 pFETs.

ACKNOWLEDGMENTS

The author would like to thank J. Stathis, A. Callegari, T.H. Ning, M. Frank, W. Haensch, and R. Jammy for discus-sions and comments. The author would also like to thank G.Singco for help with data analysis programs.

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