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Modeling of Defect Related Reliability Phenomena
in SiC Power-MOSFETs

4.8 Effective Single Defect Decomposition

Finding a correct and unique set of defect parameters to compute energetic barriers, c.f. (4.31), that yield charge transfer rates to reproduce the exponentially decaying recovery data obtained from electrical measurements, e.g. MSM sequences Section 2.1.3, is referred to as spectral- or Multi-Exponential Analysis (MEA). As explained in detail by Istratov et. al [244], a unique solution can only be derived with adequate Signal-to-Noise Ratio (SNR), as the signal decomposition is mathematically an ill-posed problem. Sufficient variation of experimental parameters, e.g. \( T, V_\mathrm {G} \) and a correct hypothesis (model) in combination with a reasonably accurate initial guess of the parameters are required to mitigate the numerical instabilities typically arising within MEA. In this section, a new numerical method for MEA, as imposed by the problem of finding two-state NMP defect parameters that explains experimental MSM data obtained in large area MOSFETs, is presented. Due to its inherent decomposition of parameter space, this method has been termed Effective Single Defect Decomposition (ESiD) [CSJ8].

A common assumption for the distribution of the defect parameters in previous works was a uniform [235] or linearly decaying [105] spatial defect density distribution \( N_\mathrm {T} \) with a Gaussian distribution in the energetic dimensions, which for a two-state NMP model yields the parameter tuple

(4.69) \{begin}{align} P = \left ( \overline {E}_\mathrm {T}, \sigma _{E_\mathrm {T}}, \overline {E}_\mathrm {R}, \sigma _{E_\mathrm {R}}, R, x_\mathrm {T}, N_\mathrm {T} \right ) \label {equ:defect_tuple} \{end}{align}

subject to minimize the error between computed and measured ∆Vth. The naive approach of optimization using local gradient based or iterative minimization schemes, e.g. a Nelder-Mead method [245], is simple to use, however, does not constrain the parameter space and usually requires an excellent initial guess if numerous local minima are present as is typical for highly nonlinear problems like MEA of BTI data. In case of insulators with more than one defect band of the form  (4.69), which is subject to charge capture and thus alters the observed ∆Vth, the optimization becomes even more tedious due to the extended parameter space.

Additionally, a parameter cross-correlation in the NMP model between the curvature ratio \( R \) and relaxation energy \( E_\mathrm {R} \) leads to similar energetic barriers, not resolvable by an experimental parameter variation [CSJ8]. Hence, in-line with ab-initio calculations [77], the curvature ration is fixed to \( R=1 \) and removed from the parameter space (4.69) that is optimized. As the measured ∆Vth is a result of the superposition of a large defect ensemble it can be expressed as

(4.70) \{begin}{align} \Delta V_\mathrm {th} \left ( t \right ) = \int _\Omega N \left ( \vec {p} \right ) \delta V_\mathrm {th} \left (t, \vec {p} \right ) \dd \vec {p} \label {equ:dvth_superposition} \{end}{align}

with the weight or distribution function \( N \left ( \vec {p} \right ) \) of the defect parameter vector \( \vec {p} = \left ( E_\mathrm {T}, E_\mathrm {R}, x_\mathrm {T} \right ) \) within the parameter space \( \Omega \) and the response function \( \delta V_\mathrm {th} \left (t, \vec {p} \right ) \). For the definition of the estimator on a discrete time \( t_i \) defined by the measurement input, the parameter space is discretized as \( p_j \) on a grid within a reasonable range. In our approach, a non-negative least square (NNLS) estimator is then used to infer the underlying distribution function \( N\left (\vec {p}\right ) \) from the experimental degradation \( \Delta V_\mathrm {th}\left (t,V_\mathrm {G},T\right ) \). Mathematically, this estimator can be cast as [CSJ8]:

(4.71) \{begin}{align} \vec {N} = \underset {\boldsymbol {N} \geq 0}{\text {argmin}} \norm {\boldsymbol {\delta V} \cdot \boldsymbol {N} - \Delta \vec {V}}_2^2 \{end}{align}

with the response matrix \( (\boldsymbol {\delta V})_{ij} = \delta V_\mathrm {th} (t_i,p_j) \) and the observation vector \( \Delta \vec {V}_i \) = \( \Delta V_\mathrm {th}(t_i) \). Note that a restriction to semi-positive values for the distribution function is necessary for physical reasons, since a negative density would be nonsensical. Minimization of this estimator, however, may result in a discontinuous discrete distribution function, as no restriction is given on the shape of \( \boldsymbol {N} \). Nonetheless, a physical distribution function requires smoothness, therefore a Tikhonov regularization term [246] is added to the estimator yielding [CSJ8]

(4.72) \{begin}{align} \vec {N} = \underset {\boldsymbol {N} \geq 0}{\text {arg min}} \norm {\boldsymbol {\delta V} \cdot \boldsymbol {N} - \Delta \vec {V}}_2^2 + \gamma ^2 \norm {\boldsymbol {N}}_2^2. \{end}{align}

As \( \gamma \) is a free parameter its variation for the optimum regularization has to be chosen sensibly. A too low value of \( \gamma \) results in a low error, but at the same time a large density and thus over-fitting of the problem. On the other hand, if \( \gamma \) has a too large value, the estimator is too heavily regularized. Thus, the “sweet spot" in between these two regimes has to be evaluated by the variation of \( \gamma \). In Figure 4.15 the impact of the regularization parameter \( \gamma \) can be seen with its optimum at the“corner" point of the normalized error, together with the absolute error and defect density evolution during the iteration of the estimator.

(image)

Figure 4.15: The impact of the variation of the Tikhonov parameter \( \gamma \) shows an optimum point at the corner point of the mean absolute error as exemplary shown for a Si/SiON transistor (left) (reproduced from [CSJ8]). The error increases rapidly when the density is reduced further during the iterations of the ESiD algorithm for too low densities as shown for a SiC DMOS.

The new ESiD algorithm has proven to reproduce MSM data on an established Si/SiON technology including n- and p-MOS transistors, with two different gate oxide thicknesses. The thereby extracted defect parameters are both consistent over the investigated process variations and with ab-initio calculations for suspected defect candidates in this widely studied material combination [CSJ8]. Also a consistency with parameters from a single defect characterization has been demonstrated. This renders the ESiD method the most promising approach to extract physical defect densities from MSM data obtained from SiC/SiO2 MOS devices, with its large defect densities and a multitude of potential defect candidates and therefore overlaid parameter distribution functions [CSJ8].

Its application for a comparison of SiC DMOS technologies will be shown in Section 5.1.2. For the extraction of defect parameters based on measured TAT currents, the method is only applicable when the multi-TAT term in (4.56) can be ignored as then the non-linear coupling of the system of equations for the defect occupations vanishes. With the inclusion of the multi-TAT term, a least square optimization method employing a simplex algorithm, as discussed at the beginning of this section, is used throughout this work.