|Year : 2012 | Volume
| Issue : 3 | Page : 191-196
Aging Model for a 40 V Nch MOS, Based on an Innovative Approach
Filippo Alagi, Roberto Stella, Emanuele Viganó
Technology R&D Smart Power & High Voltage, ST Microelectronics, via Tolomeo 1, Cornaredo (Milan), Italy
|Date of Web Publication||16-Jun-2012|
Technology R&D Smart Power & High Voltage, ST Microelectronics, via Tolomeo 1, Cornaredo (Milan)
| Abstract|| |
Usually, aging models implemented in design kits are able to accurately describe parameter degradation only if kinetics during constant voltage stress is relatively simple; a methodology is proposed to extend aging simulation to a more complex behavior, and indeed implementable into a commercial software environment (in the case Eldo UDRM by Mentor Graphics). The methodology is applied to describe Ron drift of a 40 V Nch MOS transistor.
Keywords: Aging modeling, Compact modeling, First order kinetics, High-voltage MOS, Hot-carriers
|How to cite this article:|
Alagi F, Stella R, Viganó E. Aging Model for a 40 V Nch MOS, Based on an Innovative Approach. IETE J Res 2012;58:191-6
| 1. Introduction|| |
For power and high-voltage technologies, reliability concerns are becoming more and more severe: it is almost impossible to develop a device reliable on the full operating range and having competitive performances. For this reason, reliability issues must be treated at design level, thus requiring the development of aging models. Several commercial tools are available; indeed, they usually allow to accurately describe device wear out only if degradation kinetics during constant voltage (DC) stress is described by a function of the form f(t/τ(V(t)), with f independent of bias. We propose a method to extend aging simulation to a wider class of kinetics, and yet compatible with commercial tools.
| 2. Traditional Approach|| |
Most commercial aging model tools (e.g., Eldo UDRM by Mentor Graphics, RelXpert by Cadence) reproduce the simulation flow introduced by "BERT" tool, developed by Berkeley University  .
Aging simulation is divided in the following two steps:
- Stress simulation: For each device, a function "stress rate" s is defined, depending on device bias and temperature and describing instantaneous degradation rate.
Then, a periodic waveform is applied to the device. Stress rate is integrated over time giving cumulated stress S
- Post-stress simulation: Degraded model parameters value is computed as a function of cumulated stress S
The above equations are immediately derived by the parameter degradation under a constant bias stress (DC stress), provided that the kinetics is described by a monotone function of the following form (with f independent by bias):
As demonstrated in Appendix A, with this assumption degradation during a generic, non DC stress is given by:
Comparing Eq. 5 with Eq. 2, Eq. 3 we have:
Condition (4) is a severe limitation since practical cases often do not satisfy it: we propose a way to extend aging modeling to a wider class of phenomena.
We start by observing that previous results are immediately extended to the case in which DC drift is the sum of several independent components, each of which satisfies Eq. 4:
In this case, generic drift is given by:
Drift of parameter P can then be calculated by the simulator like in Eq. 5, just considering each addend in Eq. 8 like an independent degradation mechanism.
In the following sections, a physical model for Ron drift in a 40V-rated Nch MOS is presented, which natively satisfy the latter condition and an implementation in the commercial simulator "Eldo UDRM" is proposed.
| 3. 'Dispersive First Order Kinetics' Model for HV MOS Ron Drift|| |
On-state resistance drift for a 40V-rated Nch MOS has been characterized and modelized basing on "Dispersive first order kinetics" theory . Model is based on few assumptions.
Ron degradation kinetics can easily be derived by these assumptions. Integrating Eq. 10, we obtain the probability function:
- Degradation is due to the activation of defects  at Si/SiO 2 interface or in SiO 2. Activation is due to hot carriers' injection (electrons or holes) near one or more hotspots; different hotspots are assumed to be independent and their effect on Ron is plainly added.
- Ron drift is proportional to the number of activated defects. We assume that the total number of defects and their energy distribution are not changed during the stress. If we define D(Φ,t) as the defect energy distribution and p(Φ,t) as the probability that a defect of energy Φ gets activated before the time t, we have:
- Rate depends on defect activation energy ("dispersive kinetics")
- Activation rate is given by a "first order kinetics"; i.e., rate is proportional to the number of defects (with a given energy) yet not activated; the coefficient of proportionality k is the rate constant of activation reaction and depends on the instantaneous device bias:
Inserting this in Eq. 9, we obtain an expression for Ron drift during a generic stress:
If the stress signal is periodic, time integral can be computed over only one period T:
In the case of a DC stress, the time integral in Eq. 13 is trivial, and the equations reduces to:
This expression does not satisfy Eq. 4, so it could not be implemented in a simulator with the usual approach. Indeed, if we approximate the integral with a sum, we have:
Drift is the sum of independent contributions , each of which is due to defects of a given energy. A single addend is of the form "f( t/τ(V) )," having τ(V) = 1/k(ɸ,V); we are then in the case described by Eq. 7; drift model can thus be implemented in the simulator, by computing the contribution of each energy level, and then summing all addends (as in Eq. 8). Present calculation has been performed considering a single hot spot; if more are presents, their contribution is plainly added.
Range of defects energies is divided in a given number of intervals (e.g., 50). For each "energy level," a stress rate s i is defined, depending on energy and device bias; it represents the contribution to the degradation of the defects with energy in the interval; from Eq. 14 :
As usual, Eldo will compute degradation in two phases.
Let us consider a periodic stress of period T; for each energy value, stress rate s i is computed, depending on device bias and energy; stress rates are then integrated (Eq. 17), giving stress parameters S 1…S N. Since stress is periodic, integration is carried on one period:
The contribution to Ron drift of every component is calculated, and then summed giving total drift:
Thus, present results allow to extend aging modeling also to some cases in which DC degradation kinetics does not satisfy the condition of Eq. 4.
Indeed, there are some limitations. This new approach requires more computational resources than the conventional one, since it uses a "stress parameter" per each energy value, instead of a single one: this means more memory required and more time integrals to be calculated. This could be an issue for CMOS (millions of devices in a chip) but, realistically, not for HV MOS, since the instances number is much lower.
It is worth noticing that first order kinetics hypothesis is not really mandatory; the new approach of resolving kinetics as the sum of "elementary" contributions of the form f(t/τ(V)) would work also with higher order kinetics; if g is an arbitrary function of defect occupation p, activation rate may be given by
In that case, the exponential terms in Eq. 15, Eq. 19 would be replaced by different functions, but the overall simulation flow would work anyway.
| 4. Model Equations for 40v-Rated NCH MOS Ron Drift and Model Implementation|| |
The approach of previous section has been used to model Ron degradation of 40V-rated Nch drift MOS, realized in a 0.18-μm technology. To apply the method to a real case, one has to specify distribution of defects activation energy D(ɸ) and rate constant of activation reaction k(ɸ,V).
Drift is induced by hot-carriers injection in two distinct hotspots; in one of them, injected hot carriers were electrons, in the other holes. Their effect is summed:
The same model equations D(ɸ), k(ɸ,V DC ) were used for electrons and holes; only the coefficients were changed  .
Distribution of activation energy of defect was assumed to be Gaussian:
For the rate of defect activation k(ɸ,V), we adopted a modified "Lucky electron model":
Where, Θ is the "step function": Θ(x)=1 if x>0, Θ(x) otherwise.
F(V DS ,V GS ) is an "effective" electric field at the hot spot; n(V DS ,V GS ) is carriers concentration in the same point. Both F and n have been extracted by TCAD simulation for different bias values and then modeled as functions of V DS and V GS (Eq. 24 a, b).
"λ" is the carriers' mean free path; for it, we assumed a value of 7.8 × 10 -7 cm; it is to some extent conventional, but coherent with results found in literature.
Step function Θ deserves some explanation. Assuming that the hotspot has a linear size of "δ", a carrier could not receive an energy amount greater than δ times the (effective) electric field F. All the states with energy higher than Fδ will then remain inactive for any stress time. This is exactly the effect of the step function Θ(δF-ɸ) in Eq. 23: turn to 0 the activation rate of defects with activation energy over the Fδ limit. Since drift saturation is related to the activation of all available defects, a consequence is that during a DC stress, the saturation value may depend on bias: a low V DS , for example, will lead to a small value of F and so only few states will be activated. The value of δ has been adjusted to fit the experimental data; indeed, the resulting values are physically reasonable: 0.282 μm for the electrons hotspot, 0.234 μm for holes.
The carriers' concentration n and the electric field F as a function of V DS and V GS are given by eq. Eq. 24 a and b, respectively  :
Present model was implemented in the aging modeling tool "Eldo UDRM (by Mentor Graphics)"  . The energy integral of Eq. 13 was approximated with a sum of 50 elements; since degradation was described as the effect of two hotspots (Eq. 21), to each device instance were associated 100 values of "stress parameter" S, allowing a very high accuracy.
| 5. Comparison with Experimental Results|| |
Model has been extracted from stress experiments performed with constant stress voltage at different values of V DS and V GS . Measured devices had L=2 μm, total W = 100 μm, two fingers (four gates). Gate oxide thickness was 120 A. Periodically, stress was interrupted and on-resistance was measured (at V gs =5V, V ds =0.1V).
[Figure 1] and [Figure 2] show a comparison between measured percentage Ron drift and simulation results (performed with Eldo): the agreement is satisfactory. [Figure 3] (courtesy from 2) reproduces the same results as [Figure 1], distinguishing the contributions of electrons and holes, at the two hotspots.
|Figure 1: Ron drift vs time, measured with V gs -Vth = 0.25 V and different V ds values.|
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|Figure 2: Ron drift vs time, measured with V gs -Vth = 1 V and different V ds values.|
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|Figure 3: Ron drift vs time with V gs -Vth = 1 V and different V ds values; electrons and holes contributions are distinguished (courtesy of ).|
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To validate the model, a stress measurement with a periodic signal has also been performed. A trapezoidal wave was applied to the gate, switching between V low = Vth+0.8 V and V high = Vth+2 V; wave period was 100 μs, duty cycle was 10% high, 90% low; drain voltage was fixed to 44 V. As shown in [Figure 4], simulation results are accurate, demonstrating the model validity also for non-constant stress.
|Figure 4: Ron drift vs time during a stress performed with a trapezoidal wave applied to the gate (Vg1 = Vth+0.8 V, Vg2 = Vth+2 V).|
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To improve simulation speed, one may reduce the discretization intervals number in the energy integral; [Figure 5] shows Ron drift (V ds =36 V, V gs =Vth+1 V) simulated with intervals numbers N ranging from 5 to 50; it is clear that choosing N = 30 ensures a good approximation of the integral, since using more intervals gives only a slight change in the results.
|Figure 5: Ron drift vs time (V gs -Vth = 1 V, V ds = 36 V), simulated with different integration intervals for the energy.|
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| 6. Conclusions|| |
A new methodology was proposed, extending aging modeling to a wider class of phenomena and yet compatible with commercial simulators; it was applied to describe Ron drift of a 40 V NMOS, adopting a physically based "dispersive first order kinetics" model. The model was implemented in Eldo UDRM (Mentor graphics) and simulation results were compared with measures, showing a good accuracy.
| 7. Appendix A - Degradation Stimulation During a Periodic Stress|| |
Let the parameter degradation with constant stress bias be:
We assume that:
- ΔP(t) monotone
- Degradation rate is a function of stress voltage and current degradation (i.e., it does not depend on stress history):
- ΔP during DC stress is of the form:
Where, f does not depend on stress voltage.
Eq. 26 assures that given V and ΔP, the stress rate is the same during DC and non-DC stress:
By using Eq. 27 in Eq. 28, we obtain:
But, considering f -1 (ΔP) as a function of ΔP, we have:
Substituting A.6 in A.5:
And, integrating in dt, we obtain parameter drift in the generic case:
| References|| |
|1.||R H Tu, E Rosenbaum, W Y Chan, C C Li, E Minami, and K Quader, et al, "Berkeley Reliability Tools-BERT", IEEE Trans. On Computer-Aided Design of Integrated Circuits And Systems, Vol. 12, No. 10, pp. 1524-34, Oct. 1993. |
|2.||F Alagi, "A first-order kinetics ageing model for the hot-carrier stress of high-voltage MOSFETs", "Microelectronics Reliability", Vol. 51, No. 2, pp. 321-5, Feb. 2011. |
|3.||F Alagi, "DMOS FET parameter drift kinetics from microscopic modeling", Microelectronics Reliability, Vol. 50, No. 1, pp. 57-62, Jan. 2010. |
|4.||"Eldo UDRM User's Manual - Release AMS 2010.1", Mentor Graphics Corporation, pp.31-61, 2010. |
| Authors|| |
Filippo Alagi graduated in Physics at the University of Naples "Federico II" in 1987. Since then he joined STMicroelectronics, where he is still presently employed. His research interests are in the field of instability and ageing of elementary microelectronic devices.
Roberto Stella graduated in Physics at the University of Milan in 1989, with a thesis work about the modeling of the VDMOS transistor.
Since 1989 he has been working in STMicroelectronics involved in the electrical characterization and modeling of devices in smart power technologies.
Presently, he manages the group dedicated to the extraction of compact models for all BCD technologies. He published many papers about modeling and characterization of Smart Power technology devices.
Emanuele Viganó graduated in Physics at the University of "Insubria" (Como) in 2002 with a thesis work about ultrathin oxides reliability. From 2003 to 2004, he worked in Accenture Technology Solutions. In 2004, he joined STMicroelectronics where he is presently employee as a modeling engineer. His main research interests are aging simulation and its implementation in compact device models.
[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5]