5.2 Impact of Gate Oxide Thickness Variations on HCD

The shape of the coordinate profile N_it(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 (t_ox) 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 t_ox, 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 ΔI_dlin(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 t_ox/t_n.
(a) (b)

A 5V n-type MOSFET fabricated on a standard 0.35um process with the nominal oxide thickness t_n = 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 t_n. One varied t_ox in a wide range from 0.4t_n to 1.8t_n. Afterwards, for the reference values of t_ox (t_ox/t_n 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 V_ds = 6.25V and V_gs = 2.0V). Figure 5.6b demonstrates the series of AIs calculated for the varying t_ox using the analytical formula. A direct comparison between acceleration integrals for particular thicknesses of 1.8t_n and 1.0t_n 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

**Figure 5.9:** Dependences ΔI_dlin(t) calculated for different values of t_ox.

Using acceleration integrals from Figure 5.6b, a series of reference curves ΔI_dlin was calculated for different values of t_ox, Figure 5.9. One can see that ΔI_dlin(t) substantially changes with varying t_ox. 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 t_ox is the left front of the AI. This varying front defines the difference between ΔI_dlin(t) at short stress times calculated for the devices with different t_ox. For the single-particle process, the bond-breakage events are triggered as described by the activation exponent, i.e. N_it(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 ΔI_dlin(t) curves tends to vanish at t~10⁴s. Since the parameter t_ox may fluctuate, one should consider the mean value of ΔI_dlin and its standard deviation as functions of time

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

**Figure 5.10:** (a) The mean value of I_dlin(t) vs. t calculated for different σ_d. (b) The standard deviation of ΔI_dlin(t) calculated for different σ_d.
(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 t_ox was incorporated into analytical approach to HCD modeling. It was found that only two parameters in the AI analytical formula change with t_ox. Their dependences on t_ox/t_n have been represented by fitting expressions covering the entire range of t_ox/t_n variations. This fitting allow us to avoid the time consuming Monte-Carlo simulations while studying HCD in devices with different t_ox. 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 ΔI_dlin(t) is rather sensitive to an oxide thickness change at short and moderate stress times. For longer stress times, the difference between ΔI_dlin(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 t_ox. As an example, the mean value and the standard deviation of ΔI_dlin(t) for different t_ox distributions were also calculated.