### Schottky Contact

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#### Introduction

The interface of two joint semiconductor and metal layer is an interesting subject. If the work function of the metal and the electron affinity of the semiconductor is different is going to form a barrier ^{1)}, called Schottky-barrier\citep{monch1994metal}. This barrier under bias acts as a rectifier due its asymmetric I-V characteristics.

#### Structure

The band diagram of the structure at zero bias is plotted in figure 1. the barrier height has been set to $U_B = 0.5V$, for an n-doped **GaAs** sample. The doping of the was set to be $N_d = 1E23\frac{1}{m^3}$ for a two-level dopant.

The screening-length(warping of the bands) is related to the doping density: $w_ {screen} = \sqrt{\frac{2 \Phi_B \epsilon}{q N_d }}$, where $\Phi_B$ is the Schottky barrier thickness, and $\epsilon$ is the permittivity of the material.

The induced electron density at the interface due the band warping is plotted in figure 2. It shows that the amount of electrons is lower at the interface which results lower conductivity in the layer. This conductivity can be change with the applied voltage of the contacts.

#### Voltage characteristics

If we apply bias on the structure, it lowers the barrier, which induces higher electron concentration at the metal-semiconductor interface. Thus higher current is going to flow over the structure.

As in figure 3. depicted the current grows exponentially with the applied bias, according to the diode equation.

\begin{equation} I = I_0 \left( exp(\frac{q U_{bias}}{k_B T}) -1 \right) \end{equation}

##### Mott-Schottky plot

If we calculate the charge density in the sample, and differentiate in the function of voltage we get the capacitance of contact, which can be measured also with the Mott-Schottky plot technique. According to this the capacitance is related to the doping with the following formula:

\begin{equation} \frac{d C^{-2}}{dV} = \frac{2}{q A^2 N_d(w) \epsilon}. \end{equation}

Which should be linear for constant doping.

^{1)}Schottky-Mott rule: the barrier height is going to be the difference. In reality the barrier height is just weakly dependent on the work-functions, but the chemical interaction of the metal and the semiconductor interface - induces additional states in the middle of the band-gap - called Fermi-level pinning