*3.2. Parameter Extraction Procedure*

The threshold voltage *VTH* and the current factor *K* were extracted in a very wide *TB* range spanning from 300 to 500 K by resorting to the traditional *quadratic extrapolation method* [14,15] applied to *ID*–*VGS* transfer characteristics measured under isothermal (pulsed) conditions at *VDS* = 20 V with the curve tracer mentioned in Section 2. Then the parameters in (3), (4), and (5) were calibrated so as to ensure the best matching between experimental data and the following relations:

$$V\_{TH}(T\_B) = \left[V\_{TH}(T\_0) - V\_{TH\alpha}\right] \cdot \exp\left[-a\_{V\_{TH}} \cdot (T\_B - T\_0)\right] + V\_{TH\alpha} \tag{14}$$

$$K(T\_B) = K(T\_0) \cdot \left(\frac{T\_B}{T\_0}\right)^{-m(T\_B)}\tag{15}$$

with

$$m(T\_B) = -a\_{\rm m} + (a\_{\rm m} + b\_{\rm m}) \cdot \left[1 - c\_{\rm m} \cdot \exp\left(-d\_{\rm m} \cdot \frac{T\_B}{T\_0}\right)\right] \tag{16}$$

The comparison between the measured *VTH* and the optimized (14) is shown in Figure 3. It can be inferred that the DUT exhibits (i) a high *VTH*(*T*0) (≈6.4 V) and (ii) a high negative temperature coefficient of *VTH*(*T*) at low/medium *TB* compared to similarly-rated Si power MOSFETs. Both findings were attributed to the high density of SiC/SiO2 interface traps (quantum states originating from the thermal oxidation of the SiC surface); more specifically, (i) electrons are captured by traps and do not contribute to channel formation, thereby leading to a high threshold voltage *VTH*(*T*0), and (ii) *VTH* markedly reduces with temperature due to the concurrent effect of more broken bounds that release electrons, and of the emission of inversion electrons from the traps, the latter effect being almost absent in the Si counterparts [16–18]. It must be underlined that the strong negative temperature coefficient in turn contributes to a significant positive temperature coefficient of *ID*, which may exacerbate the ET feedback [19,20].

**Figure 3.** Threshold voltage *VTH* vs. baseplate temperature *TB*: comparison between experimental data (red circles) and model (14) (solid blue line) with optimized parameters.

The comparison between the measured *K* and model (15) and (16) with tuned parameters is shown in Figure 4. The temperature sensitivity of *K* is only related to that of the channel electron mobility μ*<sup>n</sup>*, which is due to the interplay between (i) the Coulomb scattering with the filled (charged) interface traps, leading to a positive temperature coefficient induced by the trap discharging (release of electrons) with increasing temperature, and (ii) the acoustic-phonon scattering favoring a negative coefficient, where (i) prevails at low temperatures and (ii) dominates at high temperatures [18,21–23]. This behavior is accurately described by the *m* model (16).

**Figure 4.** Current factor *K* against baseplate temperature *TB*: comparison between experimental data (red circles) and model (15) and (16) (solid blue line) with optimized parameters.

The accuracy of the parameter calibration for the *VTH* and *K* models is witnessed by the comparison reported in Figure 5 between (2) and the *ID*–*VGS* transfer characteristics measured under isothermal conditions at *VDS* = 20 V and various *TB* values in a current range wherein the DUT operates in pinch-off (note that *ID* ≈ *IDnoII*). An inspection of the curves reveals the considerable positive temperature coefficient of *ID* within a wide range of currents.

**Figure 5.** *ID*–*VGS* transfer characteristics of the DUT at *VDS* = 20 V and *TB* = 303, 348, 423, and 473 K: comparison between experimental data (red circles) and model (2) with calibrated parameters for the *VTH* and *K* formulations (solid blue lines).

The parameters of the drift resistance *Rdrift* in (10), (11) were optimized by comparing the *ID*–*VDS* output characteristics at various *VGS* and different *TB* values in triode region with the model (1) applied to the scheme in Figure 2. Figure 6 reveals the accuracy of the extraction at *TB* = 303 K.

**Figure 6.** *ID*–*VDS* output characteristics of the DUT at *VGS* = 10, 12.5, 15, 17.5, 20 V and *TB* = 303 K: comparison between experimental data (red circles) and the model described in Section 2, where *Rdrift* is given by (10), (11) with tuned parameters (solid blue lines).

The parameters of the II model given by (6) with (7) and (8) were tailored on the basis of experimental data determined under UIS conditions, and 2D numerical simulations of the *TB*-dependent *ID*–*VDS* avalanche curve at *VGS* = 0 V of an individual semi-cell of the DUT carried out with TCAD Sentaurus [24] (it must be remarked that the term "cell" here does *not* correspond to the discretization adopted in this work and identified by *N*, but it is one of the tens of thousands of identical blocks in which the effective active area of the MOSFET can be partitioned).

The parameters used in the expression of the capacitances *CGD* (12) and *CDS* (13) were calibrated to match the experimental gate and drain waveforms during an ILS turn-off and turn-on transient at different supply voltages.

All the optimized parameter values are reported in Table 1.


**Table 1.** Optimized parameter values used in the transistor model.
