3.1.6.1 Ohmic Contact

An ohmic contact is defined as a metal-semiconductor contact that has a negligible contact resistance relative to the bulk or series resistance of the semiconductor. A satisfactory ohmic contact should not significantly degrade device performance and can pass the required current with a voltage drop that is small compared with the drop across the active region of the device. The metal quasi-FERMI level (which is specified by the contact voltage $\phi_{\mathrm{m}}$) is equal to the semiconductor quasi-FERMI level. The contact potential $\phi_{\mathrm{s}}$ at the semiconductor boundary reads

$$\phi_{\mathrm{s}}= \phi_{\mathrm{m}}+ \psi_{\mathrm{bi}},$$ (3.30)

here, the built-in potential $\psi_{\mathrm{bi}}$ is obtained from [25]

$$\psi_{\mathrm{bi}}= \frac{{\mathrm{k_B}}\cdot T_\mathrm{L}}{{\mat... ...rac{1}{2\cdot N_2}\cdot \left(-N+ \sqrt{N^2+4\cdot N_1\cdot N_2}\right)\right),$$ (3.31)

where $N$ is the net concentration of dopants and other charged defects at the contact boundary. The auxiliary variables $N_1$ and $N_2$ are defined by

$$N_{1}=N_\mathrm{c}\cdot \exp\left(\frac{-\ensuremath{E_\mathrm{c}}}{{\mathrm{k_B}}\cdot T_\mathrm{L}}\right)$$ (3.32)

$$N_{2}=N_\mathrm{v}\cdot \exp\left(\frac{E_\mathrm{v}}{{\mathrm{k_B}}\cdot T_\mathrm{L}}\right).$$ (3.33)

The carrier concentrations in the semiconductor are pinned to the carrier concentrations at the contact. They are expressed as

$$n_\mathrm{s}=N_\mathrm{c}\cdot \exp \left(\frac{-\ensuremath{E_\mathrm{c}}+{\mathrm{q}}\cdot \psi_{\mathrm{bi}}}{{\mathrm{k_B}}\cdot T_n}\right),$$ (3.34)

$$p_\mathrm{s}=N_\mathrm{v}\cdot \exp \left(\frac{E_\mathrm{v}-{\mathrm{q}}\cdot \psi_{\mathrm{bi}}}{{\mathrm{k_B}}\cdot T_p}\right),$$ (3.35)

where $T_n$ and $T_p$ are carrier temperatures for electrons and holes, respectively, and set equal to the lattice temperature $T_\mathrm{L}$.

$$T_\nu = T_\mathrm{L}. \hspace{1cm}\nu = n,p$$ (3.36)

In the case of a thermal contact the lattice temperature $T_\mathrm{L}$ is calculated using a specified contact temperature ${\mathrm{T}}_C$ and thermal resistance $R_{\mathrm{T}}$. The thermal heat flow density ${\mathbf{S}}_{\mathrm{L}}$ at the contact boundary reads:

$${\mathbf{n}}\cdot{\mathbf{S}}_{\mathrm{L}}= \frac{T_\mathrm{L}-T_C}{R_{\mathrm{T}}}$$ (3.37)

If no thermal resistance is specified, an isothermal boundary condition will be assumed, and the lattice temperature $T_\mathrm{L}$ will be set equal to the contact temperature $T_C$.

$$T_\mathrm{L}= T_C$$ (3.38)

In the case of drift diffusion simulation with self-heating an additional thermal energy is accounted for. This thermal energy is produced when the carriers have to surmount the potential difference between the conduction or valence band and the metal quasi-FERMI level. The energy equation reads:

$${\mathbf{J}}_n \cdot \left(\frac{\ensuremath{E_\mathrm{c}}}{{\mat... ...mathrm{q}}}+ \phi_{\mathrm{m}}\right) = \mathrm{div}\,{\mathbf{S}}_{\mathrm{L}}$$ (3.39)

The expression $\mathrm{div}\,{\mathbf{S}}_{{\mathrm{L}}}$ denotes the surface divergence of the thermal heat flux at the considered boundary. T. Ayalew: SiC Semiconductor Devices Technology, Modeling, and Simulation