5.2 Impact of Gate Oxide Thickness Variations on HCD

The shape of the coordinate profile Nit(x) is defined by the device architecture. As such, the topological features such as the doping profiles, oxide thickness, gate extension, etc. can strongly affect the device life-time under hot-carrier stress. At the same time, these architectural peculiarities vary from device to device even in the case when these devices are fabricated within the same technological cycle [1]. As a result, it is expected that such fluctuations can impact HCD. In this Section of the work, the developed analytical approach to hot-carrier degradation modeling described in the previous Section is employed in order to analyze the impact of the oxide thickness (tox) variation on hot-carrier degradation.

Figure 5.6: The AI profiles calculated with (a) the calibrated TCAD model and with (b) the analytical model for different oxide thicknesses (except tox, the device topology is identical).
(a) (b)

A "brut-force" approach to analyze the impact of device architecture variations on the behaviour of the transistor degraded during hot-carrier stress may be performed in the following manner. At the initial step a set of topologically identical devices differing only in the value of the fluctuating parameter is generated. Then, for each particular device (with the unique value of the varying parameters) the entire described procedure is applied. The results are then binned into histograms and statistically processed, weighting the statistical distribution of the fluctuating quantity.

Due to the stochastic nature of the MC approach, such a computational procedure, however, would lead to an enormous huge computational burden. And even more dramatic, if we vary this geometrical parameter with a reasonably small step, we should pay especial attention to the calculation accuracy. Otherwise, no prominent difference between results calculated for fairly similar (but still not identical) devices will be obtained. To summarize, all these circumstance make this "brut-force" strategy practically unrealistic.

Figure 5.7: Comparison between AIs calculated with the TCAD model and the analytical expression for two different oxide thicknesses.

Therefore, an analytical approach is used, which is calibrated using the results obtained with the TCAD version of the HCD model. We vary an architectural parameter (in this work the oxide thickness) and for some reference values we launch the entire routine of the TCAD model to obtain the exact dependences ΔIdlin(t). Then, we interpolate parameters in the analytical formula for the acceleration integral in order to cover the entire range of the fluctuating parameter.

Figure 5.8: Dependences of parameters (a) A1 and (b) β on the relative oxide thickness tox/tn.
(a) (b)

A 5V n-type MOSFET fabricated on a standard 0.35um process with the nominal oxide thickness tn = 14.83nm is used. That is, the model was calibrated to describe the linear drain current change under hot-carrier stress for the device with oxide thickness tn. One varied tox in a wide range from 0.4tn to 1.8tn. Afterwards, for the reference values of tox (tox/tn was 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, and 1.8) the complete simulation scheme (including the Monte-Carlo step) is performed and the AI profiles are calculated, see Figure 5.6a (stress voltages were Vds = 6.25V and Vgs = 2.0V). Figure 5.6b demonstrates the series of AIs calculated for the varying tox using the analytical formula. A direct comparison between acceleration integrals for particular thicknesses of 1.8tn and 1.0tn is presented in Figure 5.7. From Figure 5.6 it can be concluded that the slope of the right front of AI (represented by fragments I3, I5 see Figure 5.2b) remains the same (slight deviations in the shape of I3, I5 and the ledge I4 boundaries are related to the statistical noise within the Monte-Carlo approach). Therefore, the only varying parameters arethe height of I1 (parameter A1) and the slope of I2 (parameter β). Their dependences are shown in Figure 5.8 and represented by fitting formulae

(5.10)
This expression is further used for statistical analysis of ΔIdlin in the case of a fluctuating tox.
Figure 5.9: Dependences ΔIdlin(t) calculated for different values of tox.

Using acceleration integrals from Figure 5.6b, a series of reference curves ΔIdlin was calculated for different values of tox, Figure 5.9. One can see that ΔIdlin(t) substantially changes with varying tox. This change - especially pronounced for moderate stress times, i.e. less than 103s - tends to saturate at 104s. For all gate oxide thicknesses the peak of the acceleration integral is pronounced at the same position and the fragment varying with tox is the left front of the AI. This varying front defines the difference between ΔIdlin(t) at short stress times calculated for the devices with different tox. For the single-particle process, the bond-breakage events are triggered as described by the activation exponent, i.e. Nit(x,t) = N0(1–exp[-I(x)νt]), where ν is the attempt rate. Therefore, at large stress times all the bonds in the vicinity of the AI peak are predominately broken, independently of the particular oxide thickness. As a result, the difference between ΔIdlin(t) curves tends to vanish at t~104s. Since the parameter tox may fluctuate, one should consider the mean value of ΔIdlin and its standard deviation as functions of time

(5.11)

where D is the distribution of the oxide thickness and σd the standard deviation. As an example, these dependences were evaluated for situations when tox obeys Gaussian and uniform distributions. In the first case the standard deviation of the oxide thickness was σd = 5%, 10%, 20%, and 30% of tn. In the second case it is assumed that tox is homogeneously distributed in the interval of [tox–3σd; tox+3σd] and σd has the same values as in the first case (in the case of a uniform distribution with such a span, the standard deviation, differs from σd). The results are presented in Figure 5.10. Notwithstanding the fact that ΔIdlin(t) is very sensitive to tox variations (Figure 5.9) the mean value ⟨ΔIdlin⟩(t) substantially differs from the nominal one (i.e. calculated for the fixed value tox = tn) only in the case of a wide uniform distribution, see Figure 5.10 demonstrating rather low ΔIdlin(t) dispersions at all stress times also confirms this trend.

Figure 5.10: (a) The mean value of Idlin(t) vs. t calculated for different σd. (b) The standard deviation of ΔIdlin(t) calculated for different σd.
(a) (b)

Using the TCAD version of our physics-based model of hot-carrier degradation a series of carrier acceleration integrals for devices with identical architecture but with different oxide thicknesses was calculated. Then, the information describing how the acceleration integral changes with tox was incorporated into analytical approach to HCD modeling. It was found that only two parameters in the AI analytical formula change with tox. Their dependences on tox/tn have been represented by fitting expressions covering the entire range of tox/tn variations. This fitting allow us to avoid the time consuming Monte-Carlo simulations while studying HCD in devices with different tox. Moreover, following the suggested approach it is possible to analyze the impact of oxide thickness variations on the linear drain current change vs. time during hot-carrier stress. The calculations have demonstrated that ΔIdlin(t) is rather sensitive to an oxide thickness change at short and moderate stress times. For longer stress times, the difference between ΔIdlin(t) obtained for different oxide thicknesses tends to vanish. This is because at long times all Si-H bonds in the vicinity of the acceleration integral peak are predominately broken independently of tox. As an example, the mean value and the standard deviation of ΔIdlin(t) for different tox distributions were also calculated.



I. Starkov: Comprehensive Physical Modeling of Hot-Carrier Induced Degradation