3.3.4.3 Schottky Contact

For the gate contact in the HEMT two models are commonly used to model the Schottky contact. The first model for the contact interface condition for the potential is:

$$\varphi_s = \varphi_m -\varphi_{W}$$ (3.111)

where:

$$\varphi_{W} = -\frac{E_{W}}{q}$$ (3.112)

Thus, the potential at the boundary in the semiconductor $\varphi_s$ is set to the difference of the quasi-Fermi level potential in the metal $\varphi_m$, which is equal to the potential specified at the contact, and the potential $\varphi_{W}$. The latter represents the potential corresponding to the work function difference energy $E_{W}$ of metal and semiconductor. The applied current equations are:

$$\mathbf{n}\cdot \mathbf{J_n}=-q \cdot v_n\cdot (n_0-n_s)$$ (3.113)

$$\mathbf{p}\cdot \mathbf{J_p} = q \cdot v_p\cdot (p_0-p_s)$$ (3.114)

The currents are proportional to the so-called recombination velocities $v_\nu$ and the difference of the carrier concentration $\nu_s$ and $\nu_0$. The carrier concentrations at the boundary read:

$$n_s = N_C \cdot \exp \bigg(\frac{-E_C-E_{W}}{{\it k}_{\mathrm{B}}\cdot {\it T}_\mathrm{L}}\bigg)$$ (3.115)

$$p_s = N_V \cdot \exp \bigg(\displaystyle \frac{E_V-E_{W}}{{\it k}_{\mathrm{B}}\cdot {\it T}_\mathrm{L}}\bigg)$$ (3.116)

while the equilibrium concentrations $n_0$ and $p_0$ can be expressed as:

$$n_0 = N_C \cdot \exp \bigg(\frac{- \phi_B}{{\it k}_{\mathrm{B}}\cdot {\it T}_\mathrm{L}}\bigg)$$ (3.117)

$$p_0 = N_V \cdot \exp \bigg(\displaystyle \frac{- \phi_B}{{\it k}_{\mathrm{B}}\cdot {\it T}_\mathrm{L}}\bigg)$$ (3.118)

For the Schottky barrier height $\phi_B$ typical values range between 0.61 eV and 0.8 eV. For the hydrodynamic case the carrier temperatures are fixed, thus thermal equilibrium is assumed similarly to the Ohmic contact. The thermal boundary conditions applied similar to the Ohmic contact, i.e. either by an thermal resistance or a isothermal boundary condition. The model presented so far can also be called a thermionic emission model: The recombination velocity $v_\nu$ in (3.113) is then to be called a thermionic emission velocity. Inserting (3.115) and (3.117) into (3.113) we obtain:

$$\mathbf{n}\cdot\mathbf{J_n} = - q \cdot v_s \cdot N_C \bigg[ \exp... ...igg(\frac{-E_C-E_{W}}{{\it k}_{\mathrm{B}}\cdot {\it T}_\mathrm{L}}\bigg)\bigg]$$ (3.119)

As stated by Schroeder in [249], (3.119) can thus be written as:

$$\mathbf{n}\cdot\mathbf{J_n} = \frac{q \cdot m_n {\it k}_{\mathrm{... ...-\frac{ E_{Fm}-E_{Fsemi}}{{\it k}_{\mathrm{B}}{\it T}_\mathrm{L}}\bigg)-1\bigg]$$ (3.120)

using $\phi_B$ = $E_C$ - $E_{Fm}$ which is already named the Schottky barrier height. Furthermore we use:

$$E_W =E_{Fm}- E_{Fsemi}$$ (3.121)

which says that the work function difference is equal to the difference of the Fermi levels of the semiconductor and the metal. The velocity $v_n$ is rewritten as (see (3.92)):

$$v_n = \sqrt{\frac{{\it k}_{\mathrm{B}}T}{2\pi m_n}}= \frac{q m_n ... ...ot \frac{{\it T}_\mathrm{L}^2}{q N_c}= A\cdot\frac{{\it T}_\mathrm{L}^2}{q N_c}$$ (3.122)

where A is the Richardson constant typical for thermionic emission processes. Thus, (3.122) represents a typical thermionic emission equation, which is determined by the Schottky barrier height $\phi_B$ and the difference of the Fermi levels. Neglecting the tunneling in (3.91) the similarities are obvious. Typically the Fermi level of the semiconductor is chosen as the reference energy. A second model for the metal semiconductor interface including a tunneling current component has been tested [130,174]. It is based of the equivalence of a tunneling current and a generation process. It was found that such a non-local model is physically sound and useful on the first glance, but its usefulness depends critically on the actual two-dimensional implementation of the integration path for the tunneling probability in general purpose simulators, as stated in Section 3.6.