*3.1. Transistor Model*

The model is an enriched variant of the classic SPICE Level 1. In particular, the transistor is represented as the series of an "intrinsic" conventional MOSFET describing the channel behavior, and a resistance for the lowly-doped N-type epitaxial region, as depicted in Figure 2.

**Figure 2.** Sketch of the transistor representation.

Let us consider the following nomenclature:


The channel region is described with the Level 1 model. If *VDSch* is lower than the overdrive voltage *VGS* − *VTH*, the DUT operates in triode mode, and the II-free drain current *IDnoII* is expressed as

$$I\_{DmolI} = K \cdot \left[ 2 \cdot (V\_{GS} - V\_{TH}) \cdot V\_{DS\text{cl}} - V\_{DS\text{cl}}^2 \right] \tag{1}$$

Conversely, if *VDSch* ≥ *VGS* − *VTH*, then the DUT is driven into pinch-off, and

$$I\_{Dmol} = K \cdot (V\_{GS} - V\_{TH})^2 \tag{2}$$

The negative temperature coefficient of *VTH* is described through the following law:

$$V\_{TH}(T) = \left[V\_{TH}(T\_0) - V\_{TH\alpha}\right] \cdot \exp\left(-a\nu\_{TH} \cdot \Delta T\right) + V\_{TH\alpha} \tag{3}$$

such an exponential model being an improvement with respect to the simple linear relationship used in [7].

The current factor *K* depends upon *T* since the electron mobility in the channel is temperature-sensitive; similar to [7,9], such a dependence is taken into account through the power relationship

$$K(T) = K(T\_0) \cdot \left(\frac{T}{T\_0}\right)^{-m(T)}\tag{4}$$

where the exponent *m*(*T*) is

$$m(T) = -a\_m + (a\_m + b\_m) \cdot \left[1 - c\_m \cdot \exp\left(-d\_m \cdot \frac{T}{T\_0}\right)\right] \tag{5}$$

II effects are accounted for as follows [7,9]. The bias- and temperature-sensitive avalanche multiplication factor *M* (≥1) is given by [12]

$$M(V\_{DS}, I\_D, T) = 1 + m\_{II} \cdot \tan\left\{ f\_l(I\_D) \cdot \frac{\pi}{2} \cdot \left[ \frac{V\_{DS} - R\_{II} \cdot I\_D}{BV\_{DS}(T)} \right]^{0I} \right\} \tag{6}$$

where *BVDS*(*T*) is the temperature-dependent drain-source breakdown voltage, expressed as

$$BV\_{DS}(T) = BV\_{DS}(T\_0) \cdot \exp(\alpha \eta \cdot \Delta T) \tag{7}$$

and *fI*(*ID*) is a nondimensional correction term to describe a potential II dependence on current (i.e., on biasing conditions), given by

$$f\_l(I\_D) = \exp(\beta\_{ll} \cdot I\_D) \tag{8}$$

Let us introduce the avalanche coefficient ξ = *M* − 1 (≥0). The II-affected drain current *ID* is evaluated as

$$I\text{ID} = I\text{D}\text{mol} + I\text{DI} = I\text{D}\text{mol} + \xi \cdot (I\_{\text{Iank}} + I\_{\text{D}\text{mol}}) \tag{9}$$

where *IDII* is the additional current component only dictated by II, and *Ileak* is a small leakage current.

The resistance*Rdrift* is expressed as the sum of (i) a bias- and temperature-dependent resistance*RJFET* to model the path composed by the accumulation and JFET regions, and (ii) a temperature-dependent resistance *Repi* for the epitaxial region beneath the JFET one [7,9]:

$$R\_{drift} \Big( V\_{GS\prime} V\_{drift\prime} T \Big) = R\_{\text{JFET}} \Big( V\_{GS\prime} V\_{drift\prime} T \Big) + R\_{\text{epi}}(T) \tag{10}$$

where

$$R\_{fET}\left(V\_{GS}, V\_{driftr}, T\right) = R\_{fET}\left(T\_0\right) \cdot \left(\frac{T}{T\_0}\right)^{m\_{R\_{fET}}} \cdot \left(\frac{1}{1 + V\_1 / V\_{drift}}\right) \cdot \left(\frac{V\_{GS}}{V\_2}\right)^{-\eta} \tag{11}$$

$$R\_{cpi}(T) = R\_{cpi}(T\_0) \cdot \left(\frac{T}{T\_0}\right)^{m\_{R\_{cpi}}}$$

*RJFET*(*T*0) being the JFET resistance at *T* = *T*0, *Vdrift* » *V*1, and *VGS* = *V*2 (*V*1 and *V*2 are fitting parameters). This formulation improves the one reported in [7] in the high-current triode region and is derived on the basis of simple arguments. First, the resistance of the accumulation region reduces with gate voltage due to the increased concentration of the attracted electrons; second, under high *Vdrift* values, the high electric field occurring in the JFET region tends to saturate the electron velocity, thus degrading the mobility.

The dynamic transistor behavior is described by improving the Level 1 capacitance models; the nonlinear nature of *CGD* and *CDS* = *CDB* is accounted for with the following expressions [9]:

$$\mathcal{C}\_{\rm GD}(V\_{\rm GD}) = \left(\mathcal{C}\_{\rm GD0} - \mathcal{C}\_{\rm GDMIN}\right) \cdot \left[1 + \frac{2}{\pi} \arctan\left(\frac{V\_{\rm GD}}{V^\*}\right)\right] \tag{12}$$

and

$$\mathbb{C}\_{\rm DS}(V\_{\rm DS}) = \frac{2}{\pi} \cdot \mathbb{C}\_{\rm DS0} \cdot \left[\frac{\pi}{2} + \arctan\left(-\frac{V\_{\rm DS}}{V^{\*\*}}\right)\right] + \mathbb{C}\_{\rm DS\rm MN} \tag{13}$$

while *CGS* was not modified [13].
