An arbitrary stressed NBTI compact model for analog/mixed-signal reliability simulations

An Arbitrary Stressed NBTI Compact Model for Analog/Mixed-Signal Reliability

Simulations

Jinbo Wan and Hans G. Kerkhoff

Testable Design and Testing of Integrated Systems Group, CTIT, University of Twente

Enschede, the Netherlands

j.wan@utwente.nl, h.g.kerkhoff@utwente.nl

A compact NBTI model is presented by directly solving the reaction-diffusion (RD) equations in a simple way. The new model can handle arbitrary stress conditions without solving time-consuming equations and is hence very suitable for analog/mixed-signal NBTI simulations in SPICE-like environments. The model has been implemented in Cadence ADE with Verilog-A and also takes the stochastic effect of aging into account. The simulation speed has increased at least thousands times. The performance of the model is validated by both RD theoretical solutions as well as silicon results.

Keywords

Reliability, NBTI, RD model, Reaction-Diffusion solution, analog, mixed-signal, simulation

1. Introduction

Recent progress in advanced deep-submicron CMOS technologies make circuits faster, smaller, and operate at lower power. However, behind these benefits, the reliability becomes a major concern. The reliability issues significantly reduce the product yield and lifetime. In order to apply new CMOS technologies, especially in some highly dependable mixed-signal areas like the analog front-ends in the automotive arena [1], reliability problems need to be considered during the circuit design period. Accurate reliability simulations are in high demand to assure design success.

One of the problems for analog/mixed-signal reliability simulation is the lack of accurate aging models. This is especially true for the negative bias temperature instability (NBTI), which is probably the most critical aging effect in advanced deep-submicron CMOS technologies nowadays. The dominant degradation due to NBTI is threshold shift with time in PMOS transistors under stress. Moreover, the NBTI degradation can partly recover after the stress has been removed, which make both the measurements as well as the modelling very difficult. In fact, until now, there is still no proper solution to accurately measure the NBTI degradations [2]. The present measurement limitations block the insight into the physical cause, and hence the modelling theories for the NBTI degradation are still under debate [3], [4].

Despite much effort spent in NBTI modelling theories, there is a big gap between modelling and implementation in SPICE-like environments which are familiar to analog/mixed-signal circuit designers. In addition, the NBTI modelling publications are mainly focused on DC stresses and square-wave stresses (AC stress or dynamic NBTI in some publications), which are not typical cases in

analog/mixed-signal circuits. Now, many simulation tools apply a simplification by using the average DC stress instead of the

real stress to evaluate the NBTI degradations. This approach

will dramatically underestimate the NBTI degradation and cause a large error in long-time extrapolation [5].

In order to apply the NBTI aging simulation to analog/mixed-signal circuits, the NBTI model should be able to:

1. Handle arbitrary stress conditions, like arbitrary voltage stress waveforms and arbitrary temperature stress waveforms.

2. Be easy to implement in SPICE-like environments and embed in existing design flows.

3. Take into account the stochastic effect of aging and enable the combination with process-variation simulations.

4. Realize short simulation times, even for large circuits. Instead of proposing new NBTI theories, this paper shorts the gap between NBTI theories and SPICE-like environments. A new compact NBTI model will be introduced which satisfies the above four requirements and is suitable for analog/mixed-signal NBTI simulations in the Cadence Analog Design Environment (ADE). The new model is based on completely solving the reaction-diffusion (RD) equations [6] in a smart way. It can achieve high accuracy with only a small computational effort.

The paper is organized as follows. Section 2 briefly reviews the existing NBTI modelling theories. Section 3 demonstrates the derivation of the new compact model from the original RD theory. Section 4 evaluates the performance of the model. The model implementation is discussed in section 5 as well as simulation times and stochastic aging simulation. Section 6 provides the conclusions.

2. Brief review of existing NBTI models

Currently there are several NBTI models used in published papers. The most simple NBTI model is based on fitting the DC stress measurements in the logarithmic domain, which could be referred to as the “power model” [7]. The power model is not based on any physical meanings but is simple to extract from measurements and easy to implement in simulation software. In fact, many commercial EDA companies and silicon foundries, like Cadence and TSMC, provide NBTI simulation by means of power models. However, the power model can only handle DC stresses because it cannot model the recovering phenomenon of NBTI.

Another NBTI model is the reaction-diffusion (RD) model, which has first been discussed by Jeppson in 1977 [8]. From 2003 on, Alam [6] reviewed many old NBTI experiments and This research has been conducted within the ENIAC project ELESIS (296112)

new degr of in RD RD and prin wav How com be norm usin wav A ther limi NBT cycl mea theo a “tra “par D the only hou degr dom seco time one 3. M T relia is t mod trap circ is m inst T from RD seco 3.1. A exp clos invo diffu D inte “Si-resp into into w on-the-fly (O radation is ma nterface traps theory. Base equations an relaxation of nciple, the R veforms, like wever, it is ve mmonly used implemented mal approach ng some lin veforms, like D Although the re are many itations. For TI log-like re le dependence asurement tec ories have bee “well-based” apping/de-trap rallel diffusion Despite the d latest studies y one which urs). They sh radation, and minate in sho onds, and can e fast change

-over-six (0.1

Model detail

The new com ability simula that the ana derate stressed pping problem uits.In additio more interesti ead of heavily The classic R m other paper equations in ond part of thi

. Classic RD According to lained as the se to the Si olved during fusion, as show During the rea erface are brok -” dangling b ponsible for th o neutral H2 a o the gate-oxid OTF) measure ainly because s , which c ed on that exp nd showed the f NBTI by s RD model ca a sine wave ery difficult to mathematical in circuit si h for the sim near approxi DC and squar RD model is y papers tha example, the ecovery transi e [10]. Based chniques, a nu en proposed in model [10 pping” model n pass way” m debate of diffe [3], [13], [14 agrees with ow the RD together with ort time; it s n be grouped e NBTI behav 6) power law ls mpact NBTI ations is based alog/mixed-sig d condition a ms as compa

on, for studyin ing to investi y stressed, inst RD equations w rs, our model n a simple wa is section. D equations the RD theor generation o i/SiO2 interfa the process wn in Figure 1 action phase, ken under the bonds are rem he threshold s and diffuse aw de and eventua ements. He cla of the genera can be model planation, he e possibility t solving those an work unde

form and tria o solve those l solvers are t imulation sof mulation with imation for re waves [9].

the most acce at claim the

RD model c ient and the s d on these obs umber of new n recent years 0], Velam [11], and Kuf model [12]. ferent NBTI m ] show that th long-time me model domin h the hole tra

aturates after to accurately viour as well behaviour. model for an d on the RD t gnal circuits and suffer less

ared to heav ng the life-tim igate the long tant recovery will be review

is based on d ay. It will be

ry, the NBTI d of dangling bo ace [6]. The of NBTI b 1. some Si-H bo e electrical str maining at the shift. The H a way from the ally into the g

aims that the N ation and anne

ed by the clas proposed a s o model the s RD equation er arbitrary s angular wave RD equations too complicat ftware. Hence the RD mod particular s epted NBTI m RD model cannot explain shape of AC servations and w NBTI mode . Grasser prop mala propose fluoglu propo modelling the he RD model easurements ( nates in long apping de-trap r around thou model both s l as the long nalog/mixed-s theory. The re are workin s hole trappin vy-stressed d me of the circu g-time degrad behaviour. wed first. Diff directly solvin introduced i

degradation c onds in the re ere are two

eing reaction onds at the Si ess. The gene e interface an toms will com e Si/SiO2 inte gate poly-silico NBTI ealing ssical set of stress ns. In stress form. s, and ted to e the del is stress model, l has in the duty-d new elling posed ed a osed a ories, is the (>100 g-time pping, usand short-g-time signal eason ng in ng/de-digital uits, it dation ferent ng the in the an be egion steps n and i/SiO2 erated nd are mbine erface on. F inter T equa the n initia conc any disso Equa whic equa is coef oxid 3.2. NBT T proc degr shift form appli some resea perio brok the Und redu T num 0) [6 be ex In (3)~( Figure 1: Sch rpret interface This process i ations can be number of int al number o centration of given instant ociation rate, a ation (1) desc ch is the same ation (2)~(3), s the combina fficient, and de interface, as Our solutio TI model The reaction-ess of NBTI radations, for t. However, th m. In order to ications, one e assumptions arch in [9], od of the stre ken Si-H bond total Si-H bo er these assu uced to: The number o mber of H atom 6]. As a resul xpressed as: n fact, most R (5) to obtain t ematic descrip -trap generatio = ( − = = is referred to expressed in terface traps a f unbroken hydrogen ato t, is the and is the b cribes the gen

as the genera 2 is the co ation rate of H is the locatio s shown in Fig on for RD eq diffusion equ I and could example, in hey are very

find a solutio needs to sim s as discussed the trap gene ess till the en ds during NBT onds. Hence umptions (1) a (0, ) ∙ of interface tr ms diffused fr t, the relation = 2 RD theory-bas the threshold iption of the R on during NB ) − ( o as diffusion (1)~(3) from at any given Si-H bonds, oms at the Si oxide-field d backward com neration rate o ation rate of h oncentration H atoms, ₂ i on perpendicu gure 1. quations and quations (1)~ be used to terms of the difficult to s on which is s mplify the con d in the follow eration is slow nd of the pro TI are just a v ⁄ 0 and (2) can ∙ raps ( ) is from the Si/Si nship between ( , ) sed NBTI mod

shift. The dif

RD model use BTI stress [6]. (0, ) n. The classic [6], where instant, 0 is

(

)

ular to the sil

d the compa ~(3) describe obtain the a threshold-vol solve in a gen uitable for an nditions and m wing. Based on w from the in oduct life and very small pa 0 and ≪ be combined . equal to the iO2 interface ( n and . dels use equat fference is tha ed to (1) (2) (3) RD is s the the ce at ward e [6]. raps, ms. In l H2, usion licon act the aging ltage neral nalog make n the nitial d the art of ≪ . and (4) total ( = can (5) tions at the

various authors use different approaches to approximate the solution under DC or square-wave stress cases [8], [9], or employ complicated solvers to solve the equations in a time-consuming way to accurately evaluate the RD theory [3], [6]. Compared to other papers, we will directly solve the equation set (3)~(5) in a simple, fast way. Since in submicron CMOS technologies, the gate oxide is by far thinner than the gate poly-silicon and the H2 diffusion much faster in the gate

oxide [9], the number of H2 in the gate dioxide can be ignored

while calculating long stress-time situations. The diffusion coefficient in the poly-silicon is treated as a constant. Applying the Laplace transform for (3) to solve the differential equation, the solution for in the Laplace transform can be derived as:

( , ) = (0, ) ∙ − ∙ (6) In (6), the superscript is referred to as the Laplace transform of corresponding functions. By substitution of (6) into (5), the solution for in the s-domain can be written as:

( ) = 2 ∙ ( , ) = 2 ∙ (0, ) ∙ (7) Using (4) and (7) to eliminate and applying the inverse Laplace transform, one obtains the time-domain equation. The final result can be simplified to an equation with only physical parameters:

√ ∙ ( ) ∙ ( ) = 2 ∙ (8) In (8), the symbol ∙ denotes the inverse Laplace transform. It can be alternatively written as a convolution form in the time domain:

1 √ ∗

( )

∙ ( ) = 2

= ( ) (9) The symbol " ∗ " inside the brackets at the left side of (9) represents the convolution operator. At the right side, is proportional to the inversion-hole density, the vertical electrical field and temperature. Parameters , and

are proportional to the temperature only, as shown in (10), in which and are constant parameters. So the right side of equation (9) is a function of stress voltage, temperature, oxide thickness and unit gate capacitance. Basically, they are a function of time and can be defined as ( ).

~ − ∙

~ exp − ~ exp −

(10)

If the stress voltage and the temperature remain constant with regard to time, e. g. DC stress, ( ) will be a constant and can be written as . In this situation, (9) will degrade to the well-known closed-form solution for the number of interface traps :

( ) = ∙ . (11) In (11), is a constant and can be expressed as

= √ ∙ 3 ∙ 2 3 ∙ 5 6 = 0.94 ∙ (12) where the gamma function has been employed [15].

However, if the stress condition changes with time, it will be difficult to derive a closed-form solution. An alternative way is to modify (9) into a discrete form and find an iterative solution. Suppose the timing points are separated uniformly by ∆ seconds. The discretization of (9) can then be written as:

( + 1) − ( )

√∆ ∙ √ − ∙ ( ) = ( ) (13) In (13), parameter is the discrete time index. Rearranging (13) and using the solution of the so-called Cubic equation [16], the iterative solution for can be expressed as in Table I. The parameters and of (14) in Table I are given in (16) and (17) of Table I. ( ) which is given in (15) and (18) of Table I is a function of time, stress voltage, temperature and other process parameters.

Table I: The new compact NBTI model with uniform time step ∆ .

( ) = 0 = 0 (1)√∆ = 1 − 3+ −27−2+ 27+2 −729 + −27−2− 27+2 −729 > 1 (14) ( ) = 0.94 ∙ ( ) − exp − 2 ( ) 2 (15) = ( ) 1 √ − + 1− 1 √ − (16) = − ( )√∆ (17) = ( ) − (18) ∆ ( ) = ∙ ( ). (19)

T follo the chan thre (EO and Para para mea due whi diel C mod I, w less I com simu App with ∆ poin Tab 4. M T sinc com Sec resu 4.1. I equ show imp clos thre in p sepa the Wit dram the is b larg ( ) = 0 − 3 = ( ) The paramete ows. is stress voltage nge with dis eshold voltage OT) of the gate ( ) is the te ameters , ameters which asurements. T to NBTI can ich is the lectric permitt Compared to del requires th which can be

than the seve In order to im mpact model ulation results plying the sam h non-uniform is the specifi nt −1. The e ble I. Model evalu The compact ce the model i mparison with ond, the mod ults. . Compariso It is difficult ations except wn in expres plemented in sed-form solut eshold shifts a percentage are arated by one maximum err th the number matically and time discretiz because norma ge. Ta (1) ∆ 3+ −27− ∆ 1 ∑ rs of the equa the gate capa e applied betw crete time in e, while is e oxide. Const emperature in and activati h need to be ex The actual thre be derived fr charge of a tivity of the ga other compa hree fitting pa extracted from en parameters mplement the needs to s, which is usi me derivation m time steps ied time step expression of uation model will b is actually a so h the RD theo del will be v on with the R to get the c t with the sit ssion (11) an Matlab and tion under the are compared e shown in F e second in bo ror is about 6 of timing poi converges to zation can be ally the numb

ble II: The ne

2+ 27+ ∆ − 1 ∑ ations in Table acitance per u ween gate and

ndex . s the equivale tant is the B Kelvin at dis ion energy xtracted from eshold shift a om ( ) as single electro ate oxide. act NBTI mo arameters, , m NBTI mea in [9] and ten e model in C handle the ing non-unifor n approach, th is presented between time ( ) is the sa be evaluated i olution of the oretical soluti validated with RD theoreti closed-form so tuation of DC nd (12). Our d compared w e same DC st in Figure 2. Figure 3. The oth figures. Fi 6.7% at the in ints increasing zero. So the e neglected in ber of the tim

ew compact N 2 −729 1 ∆ e I are explain unit area. ( source, which is the unstr ent oxide thick Boltzmann con

crete time ind are model f NBTI degrad as function of s (19) in Table

on and i dels, our com

and in T asurements. Th n parameters in Cadence ADE Spectre tran rm time steps he compact m in Table II. H e point and ame as provid in two ways. RD equation ons will be m h published si cal solution olution of the C stress, whi r new model with the acc tress. The resu The relative e timing point igure 3 shows nitial timing p g, the error red error introduce the real case. ming points w NBTI model w + − 27− (21) ned as ( ) is h may ressed kness nstant dex . fitting dation f time e I, in is the mpact Table his is n [5]. E, the nsient [12]. model Here, d time ded in First, ns, the made. ilicon ns e RD ich is l was curate ulting errors ts are s that point. duces ed by . This will be Figu close Figu (DC F one squa param equa solut linea solut accu with non-unifo 2− 27+ ure 2: Comp ed-form soluti ure 3: Relativ voltage stress Figure 4 show of the RD th are-waveform meter setting ations are too

tions at two t ar approximati tion as detail urate. rm time step ∆ 2 −729 = − ( parison of ou ion (DC voltag ve error of ou sed situation). ws the compa heory based m stress. Bot and the resu difficult to be ime points pe ion. Compared led as possib ∆ . = = > ) ∆ ur iterative s age-stressed si ur iterative sol . arison betwee models in [9 th models ults match we e solved, pape er square-wav ed to [9], our m ble, while the

of our model 0 1 1 (2 (2 solution with tuation). lution in Figu en our model ] under the s using the s ell. Since the er [9] just prov ve cycle under model can give e results are 20) 22) h the ure 1 and same same e RD vides r the e the very

Figu squa 4.2. T vali [13] our vali pub also and stre simu und I stre wav also is mea wav cou stre mea para mea the limi pred dow the cycl Figu resu ure 4: Compa are-wave volt . Compariso The RD theor idated by silic ], [14], [17–1 compact mo idated by RD lished silicon o be used to su square-wave ss is shown ulation result der the square-In addition sses, the com veforms, like s o need to be v only one p asurement resu ve stress. No p ld be found. ss between th asurement res ameter settin asurements res model correc its) in each cy diction. The d wn are also ag parts which le, which a ure 5: Comp ults [19] for sq arison of our tage stress. on with the p ry for modell con results in 9] using DC a odel use the s D theoretical n data in [3], upport our co e stresses. A in Figure 5, ts agree with -wave stress. to DC stres mpact model c sine-wave and validated by si publication [5

ults for NBTI publication for Here the com he compact m sults from [5 ngs are extr sults in [5]. Fr ctly predicts t ycle, which i degradation c greeing well w do not agree are predicted parison of our quare-wave str Ou model with th published si

ing NBTI deg many publica and square-w same RD the solutions in [6], [9], [13], ompact model An example f in which th silicon meas sses and squ can handle mo d triangular-w ilicon data. U 5] which sh I degradation r NBTI under mparison und model simulati ] are shown racted from rom Figure 6, the maximum s important f changes at str with the measu e are the bott d to increas r model with ress. ur model he model in [9 ilicon results gradation has ations [3], [6] ave stresses. eory and has n section 4.1 [14], [17–19 under DC str for a square-he compact m surement data uare-wave vo ore arbitrary s wave stresses. Unfortunately, hows the si under a trian r sine wave str der triangular-ion results an in Figure 6. the model it can be seen m degradation

for circuit life ress-up and s urements. How om parts in e e faster by h the measure of our model 9] for s been ], [9], Since been , the 9] can resses -wave model a [19] oltage stress They there ilicon ngular resses -wave nd the . The and n that n (up-e-time stress-wever every y our ement Figu resul mod mism wave hot resea inter will long gene 5. Im T incor One impo aging requ The such resul strat Veri T num 1 2 3 4 5 ure 6: Compa lts [5] for trian del than mea

match in the e stress in ma research topi arch, it can b rface traps [3 tend to satura -time degrad eration (RD th mplementat The compact m rporated with problem cou ossible to sim g simulations uired aging tim

new compac h an extrapola lts in a simple egy has bee log-A, which The strategy mbers in Figure ) Transient netlist by 2) With the transistor degradatio Table II . 3) Since bot detailed d range of a is required two param to be e simulation parameter (LSE) met 4) The aged source in to model t 5) The value power fu parameter ready to b with regar arison of our ngle-wave vol asurements sh relax region any publicatio ic recently. H be explained ], [13], [14], ate after arou ation will be heory). ion in Caden model propos h transient sim uld be the s mulate severa . So the result me from the s ct model has ation because e power functi en implemen is shown in F will now b e 7: simulation is Cadence Spec stress voltag obtained from on is calculate th the transie degradation ca a few seconds, d. For each P meters in the e xtracted from n: and , s are fitted thod and store

netlist is ge series with the the threshold d e of each volta unction as sh s from stored be used in any rd to a specifie r model with ltage stress. hown. In fa

has been rep ons [10–12] an However, bas by extra hol The hole tra und thousands e dominated b ence ADE sed in this pa mulations in S simulation tim al years in b ts need to be e short-time det the possibili e the trend o ion of time. H nted in Cade Figure 7. e explained carried out on ctre. ge waveforms m Spectre, the ed using the c ent simulation alculation are , not for years PMOS transist extrapolation f m the deta as shown in using the ed in separated enerated by he gate of each degradations. age source is hown in Fig d files. Now t y kind of sim ed number of of the measurem act, this kind ported for squ nd have becom sed on the l les in additio apping/de-trap s seconds, and by interface-t aper can be e SPICE simula me. In fact oth transient extrapolated to tailed simulati ity to incorpo f the degrada Hence a simula ence ADE u using the b n a non-aged f s on each PM detailed thres compact mod n as well as done in the s, an extrapola tor, there are

function that n ailed degrada Figure 7. T least-square-e d files. adding a vol h PMOS trans calculated by gure 7 with the aged netli mulation in Spe aging years. our model ment d of uare-me a latest on to pping d the traps asily ators. it is and o the ions. orate ation ation using block fresh MOS shold del in s the time ation only need ation These error ltage sistor y the the ist is ectre

n th A t V = ⋅ Δ m n m thm A t V = ⋅ Δ thm V Δ n t A_⋅

Figure 7: NBTI simulation strategy for analog/mixed-signal

applications.

6) The stochastic effect of NBTI degradation is taken into account by randomizing the total effect interface traps under the gate area assuming a Poisson distribution [20]:

_ _ = _ _

= (∆ ) ∙

2.7 ∙ (23) 7) The aged netlist with a stochastic NBTI degradation sample is generated using the same approach as 4). Except the threshold-voltage shift is now calculated from a sampled stochastic number of interface traps.

(∆ ) =2.7 ∙ ∙ ( _ _ ) (24) The (∆ ) in expression (23) is the deterministic threshold degradation which is calculated by the model in Table I and II. The and are the transistor gate length and width. The constant 2.7 is used to take into account both the random number of interface traps and the random spatial distribution of the traps [20]. This is also the reason of the name “effect interface traps”, _ _ . The (∆ )

and ( _ _ ) in expression (24) are the stochastic

samples.

The whole process in Figure 7 has been implemented using Verilog-A in Cadence ADE. A variable “Quick_sim” is used in ADE to determine the different choices: detailed NBTI simulation to extract power-function fitting parameters, deterministic threshold aging calculation, or stochastic

Table III: The proposed compact model simulation times in

Cadence ADE vs. Ref. [12].

Circuit Num. of PMOS Stress waveform Num. of time steps Total time consumed (including Transient simulation) Opamp 11 Sine 55 0.888 seconds

(this paper) Single Test PMOS 1 Square <100 Several minutes (Ref. [12]) threshold aging calculation. The parameter sweep for “Quick_sim” in ADE can be used to run a batch process, which makes the NBTI aging simulation very easy to combine with normal process variations in Mont-Carlo simulations.

The total simulation time is in the same order of Spectre transient simulations, which is much faster as compared to normal RD solvers, e. g. several minutes for only one PMOS transistor [12]. An example run on our server with one CPU core occupation is shown in Table III.

6. Conclusions

A compact NBTI model based on directly solving the RD equations has been proposed. The model can accurately handle arbitrary stress waveforms for the up-limits and is thus very suitable for analog/mixed-signal simulations. The model has been implemented in Cadence ADE with Verilog-A, and can simulate both deterministic as well as stochastic NBTI aging effects. The simulation speed is thousand times faster as compared to other RD based models. The model is validated by both original RD theory solutions and silicon measurement results.

7. Acknowledgments

The authors want to show their appreciation to E. Maricau from the University Leuven in Belgium, for the valuable discussions on NBTI modelling.

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