*2.1. Fundamentals on Schottky Barriers*

When metal and semiconductor are put in intimate contact, a Schottky barrier forms, whose height (*φB*) is the most significant electrical parameter associated to this system. According to the well-known Schottky–Mott rule [10,11], the Schottky barrier height ideally depends only on the metal work function (*WM*) and semiconductor electron affinity (*χS*) [12], as expressed by:

$$
\phi\_B = \mathcal{W}\_M - \chi\_S. \tag{1}
$$

Hence, considering the work function of the typical metals used for SBDs (in the range 4–5.5 eV [13]) and the electron affinity of 4H-SiC (*χSiC* = 3.2 eV [14]), Schottky barrier height values between 1.0 and 2.3 eV are expected, i.e., much higher than in silicon. However, in real cases, the Schottky barrier height does not simply obey to the Schottky–Mott rule, but may depend on other factors, like the surface preparation, metal deposition techniques and/or post-metallization annealing treatments [15,16]. In particular, the electronic properties of the metal/semiconductor contact are affected by the presence of surface states, which can be related to roughness, surface contaminants, residual thin interfacial oxide layers and so on, and are responsible for a deviation from the Schottky–Mott prediction [17,18].

The experimentally measured value of the Schottky barrier height *φ<sup>B</sup>* also depends on the methods used for its determination [19]. Generally, the barrier height in a metal/ semiconductor system can be determined by means of different techniques, such as electrical characterizations (current–voltage (I–V) and capacitance–voltage (C–V) measurements) or internal photoemission (IPE) measurements that are based on the photo-generated current detection [19,20]. Obviously, each method shows advantages and drawbacks. The electrical characterization methods require the fabrication of appropriate test patterns, namely Schottky diodes, and provide interesting insights of the Schottky barrier nature. For example, the I–V characterization is very sensitive to the presence of inhomogeneity of the barrier, with some well-established models, such as the Tung's model [17]) or the Werner's and Güttler's model [21], developed to take this aspect into account. On the other hand, the C–V characterization supplies information about the space-charge region width [22]. As a consequence, for a given Schottky contact, the barrier height value extrapolated by I–V analysis is typically lower than that derived from the C–V characteristics. This aspect can be explained by the fact that lower barrier height regions are preferential paths for the current, while C–V characteristics account for an overall electrical behavior with the largest regions (usually with highest barrier) dominating in the barrier extraction [22]. On the other hand, IPE measurements are independent of the local barrier inhomogeneity and supply a reliable value for the Schottky barrier. However, the photocurrent detection requires special equipment and semi-transparent front or back contacts, thus making these kind of measurements less common with respect to the electrical characterizations.

The Schottky barrier height determines the electrical behavior of a metal/semiconductor contact by governing the current flow through the metal/semiconductor interface. Generally, for doping density *<sup>N</sup><sup>D</sup>* in the range 1 <sup>×</sup> <sup>10</sup><sup>15</sup> <sup>&</sup>lt; *<sup>N</sup><sup>D</sup>* < 1 <sup>×</sup> <sup>10</sup><sup>17</sup> cm−<sup>3</sup> , under forward voltage the current transport mechanism through the metal/4H-SiC interface is ruled by the thermionic emission (TE) theory. In this model the current *ITE* can be expressed by [12]

$$I\_{TE} = AA^\* T^2 \exp\left[\left(-\frac{q\phi\_B}{k\_B T}\right)\right] \exp\left[\left(\frac{qV\_F}{nk\_B T}\right) - 1\right] \tag{2}$$

where *<sup>A</sup>* is the device area, *A\** is the effective Richardson's constant (146 A·cm−<sup>2</sup> ·K −2 for 4H-SiC [23]), *k<sup>B</sup>* is the Boltzmann's constant, *q* is the elementary charge, *V<sup>F</sup>* is the applied forward voltage and *T* is the absolute temperature.

As can be seen in Equation (2), besides the Schottky barrier height *φB*, another important electrical parameter that characterizes the Schottky contact is the ideality factor *n*.

These parameters can be determined by the intercept and the slope of a linear fit in semilog scale of the forward current–voltage characteristic, using Equation (2) for *V<sup>F</sup>* > 3 *kBT*/*q*, where the term −1 can be neglected.

Basically, the TE theory assumes a temperature-independent ideality factor and barrier height. However, in order to justify the experimentally observed temperature dependence of these parameters in real 4H-SiC Schottky contacts, the TE theory was modified, introducing models taking into account local fluctuations (inhomogeneity) of the Schottky barrier over the contact interface, as discussed in the Tung's [17] model or in the Werner's and Güttler's [21] model. Specifically, the Tung's model [17] assumes a local lateral inhomogeneity at nanometric scale by considering the presence of low barrier regions (patches) embedded in a high-barrier background, while the Werner's and Güttler's [21] model considers a Gaussian distribution of barrier heights around an apparent temperature-dependent barrier height.

On the other hand, when the doping concentration of the semiconductor exceeds 10<sup>17</sup> cm−<sup>3</sup> , the high electric field at the interface and the thin barrier width make dominant for the current transport a thermionic filed emission (*TFE*) mechanism, which involves a tunneling component for thermally excited electrons [19,24]. Specifically, for doping in the range 1 <sup>×</sup> <sup>10</sup><sup>17</sup> <sup>&</sup>lt; *<sup>N</sup><sup>D</sup>* < 10<sup>19</sup> cm−<sup>3</sup> , the *TFE* describes the electrical behavior of the system [24], according to the following current–voltage relationship [25]:

$$I\_{TFE} = I\_{0,TFE}(V\_F) \times \exp\left(q \frac{V\_F}{E\_0}\right) \tag{3}$$

The term *I*0,*TFE (VF)* corresponds to the saturation current and is given by

$$I\_{0,TFE}(V\_F) = \frac{AA^\*T}{k\_B \cosh(qE\_{00}/k\_B T)} \times \sqrt{\pi E\_{00}(\phi\_B - \Delta E\_F - V\_F)} \times \exp\left(-\frac{q\Delta E\_F}{k\_B T} - \frac{\phi\_B - \Delta E\_F}{E\_0}\right) \tag{4}$$

with *<sup>E</sup>*<sup>0</sup> <sup>=</sup> *<sup>E</sup>*<sup>00</sup> <sup>×</sup> *coth qE*00 *kBT* dependent on the doping concentration *N<sup>D</sup>* through the parameter *E*<sup>00</sup> = *<sup>h</sup>* 4*π* × q *N<sup>D</sup> m*∗*εSiC* . The other symbols are the Planck's constant *h*, the effective mass *m\** = 0.38 *m*<sup>0</sup> (with m<sup>0</sup> the electron mass) [26] and the dielectric constant of the semiconductor *εSiC* = 9.76 *ε*<sup>0</sup> (with *ε*<sup>0</sup> the vacuum permittivity) [4]. ∆*E<sup>F</sup>* is the difference between the bottom of the conduction band and the semiconductor Fermi level. In this case, the barrier *φ<sup>B</sup>* and the doping concentration *N<sup>D</sup>* can be considered the parameters to be determined from a best-fit procedure of the experimental detected current–voltage characteristics.

Figure 2 depicts a schematic energy band diagram for a metal/semiconductor junction under forward bias *VF*, when the predominant current transport mechanism is ruled by the TE (Figure 2a) or by the *TFE* (Figure 2b) model, according to the doping of the semiconductor layer at the interface. Noteworthy, as shown in Equations (2)–(4), in both models, the current has an exponential dependence on the Schottky barrier height. Thus, a good control of the *φ<sup>B</sup>* must be guaranteed, as it significantly affects the current level through the contact.
