*3.3. Numerical Simulation*

To investigate the influence of the surface-near increased Al concentration as well as the temperature dependent behavior of sample A a numerical simulation model was developed by using Sentaurus TCAD (Version O\_2018.06). Figure 4 shows the scheme of the used simulation model. This model includes Ti3SiC<sup>2</sup> based ohmic contact pads with a height of 100 nm and a pad distance *d* between the ohmic contacts. The model includes further a homogeneous n−-doped 4H-SiC epitaxial layer, a p<sup>+</sup> -doped region and a p ++-doped region beneath the ohmic contacts. The p<sup>+</sup> -doped region was created by using a Monte Carlo simulation of the implanted Al profiles. The surface near p++-doped region was created by adding a Gaussian distributed Al profile with diffused Al dose *NAl*,*dose* and the associated diffusion length *LAl*,*di f f* . Furthermore the model assumes complete activation of the Al atoms and takes account of incomplete ionization as well as doping and temperature dependent carrier mobility.

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**Figure 4.** Scheme of the simulation model.

An additional virtual N profile was added in order to model the concentration of ation centers. The distribution of this additional virtual N profile is iden- An additional virtual N profile was added in order to model the concentration of carrier compensation centers. The distribution of this additional virtual N profile is identical with the distribution of the implanted Al profile and can be scaled by using the compensation ratio *fcomp*. Due to these additional donor atoms the concentration of free holes can be reduced similarly to compensation by donor-like traps which increases the associated sheet resistance and allows to fit *Rsh* .

ℎ Using this numerical simulation model, I-V characteristics depending on the diffused Al dose *NAl*,*dose* , the associated diffusion length *LAl*,*di f f* and the compensation ratio *fcomp* for each temperature and each pad distance *d* can be obtained. This allows to simulate I-V characteristics for TLM structures at different temperatures.

 ℎ ℎ Based on these I–V characteristics the sheet resistance *Rsh* and the contact resistance *R<sup>C</sup>* of the modelled TLM structures were determined. Figure 5a compares the sheet resistance *Rsh* , Figure 5b compares the contact resistance *R<sup>C</sup>* determined from the electrical measurement data with simulated ones of sample A. It can be seen that both fits are in decent agreement with the measurements. It should be mentioned here that no adjustments on the parameters *NAl*,*dose* and *LAl*,*di f f* determined by the SIMS analysis were necessary.

**Figure 5.** (**a**) Measured and simulated sheet resistance (**b**) Measured and simulated contact resistance.

 onization energy from nitrogen differs from the ioniza- The determined compensation ratio *fcomp* is dependent on the temperature as shown in Figure 6. It can be seen that *fcomp* increases slightly with increasing temperature from 8.3% at 300 K to 10.5% at 450 K which fits to temperature independent compensation ratios known from literature (10% to 27%) [30–32]. This temperature dependence might be explained by the fact that the ionization energy from nitrogen differs from the ionization energy of the actual compensation centers.

**Figure 6.** Temperature dependence of determined compensation ratio.

#### **4. Discussion**

Due to the well-fitted simulation results, it can be concluded that the numerical simulation model is suitable to describe Ti3SiC<sup>2</sup> based ohmic contacts on p-doped 4H-SiC temperature dependent. Considering the fact that the simulation model does not show ohmic behavior when not using the surface near Al profile it can be further concluded that the surface near Al profile is essential for the ohmic contact formation. Based on these results it is possible to propose a theory regarding the formation mechanism of Ti/Al based ohmic contact on p-doped 4H-SiC and the role of Ti3SiC<sup>2</sup> during contact formation.

onization energy from nitrogen differs from the ioniza-

During the Ti3SiC<sup>2</sup> formation a certain amount of Al diffuses in the SiC surface via lattice places and increases the surface near Al concentration significantly. This increase of the surface near Al concentration can significantly decrease the specific resistance *ρ<sup>C</sup>* (see Equation (10)) and is therefore the key in the ohmic contact formation.

Further investigations are necessary to verify this model and to obtain a better understanding of the conditions leading to the ohmic contact formation under various process conditions. Nevertheless, the fundamental effects are becoming accessible for process integration and process modelling.

**Author Contributions:** Conceptualization, M.K.; methodology, M.K.; formal analysis, P.M.; software, M.K.; writing—original draft preparation, M.K.; writing—review and editing, M.R. and T.E. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The datasets generated and analysed during the current study are available from the corresponding author on reasonable request.

**Conflicts of Interest:** The authors declare no conflict of interest.
