### MOS Capacitor

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#### Introduction

The Metal-Oxide-Semiconductor interface is one of the most basic element of today's microelectronic devices. It is fundamental to understand its basic properties to investigate integrated circuits. The example describes a simple MOS capacitor Capacitance-Voltage behavior, and focuses on its most known regions. Dedicated theoretical background can be found in \citep{sze2006physics}

#### Structure

It is built up from $Si(p+)|SiO_2|Si(n)$ layers as depicted in figure 1. The left contact, the highly p-doped Si layer is used as a metal contact.

The influence of the strain to the band-structure is not considered, and the bandgap of the $SiO_2$ layer was chosen to be arbitrary high ^{1)}. As it is shown in figure 1, that the screening length in the **p** type layer is much longer then in the **n** type layer due the lower doping.

#### Voltage characteristics

If we apply bias to the structure there is no current flow due the lack of charge carriers in the oxide. The Fermi-levels are flat in the doped regions. We can separate the voltage characteristics of the structure in three regions:
**accumulation, depletion, inversion**

#### Accumulation

The density of electrons in the **n** doped layer changes
linearly with the applied bias. The resulted charge density distribution is depicted in ref 2. It shows that the concentration of majority type charge carriers(holes) in the **p** doped layer is higher at the interface, than in bulk. $\rightarrow$ Accumulation.

#### Depletion

As the bias voltage gets lower the density of holes in the substrate gets lower, and the density of the electrons gets higher. After a certain point (flat-band condition) the two densities become equal $\rightarrow$ Depletion. Close to the flat band condition the densities are plotted in figure 3.

#### Inversion

For even lower negative bias voltage the concentration of electrons in the $p$ type layer gets more than the density of holes $\rightarrow$ Inversion, which is depicted in figure 4.

#### Capacitance relation

As the accumulated charge strongly depends on the applied bias, one can calculate the differential capacitance of the structure: \begin{equation} C = \frac{\partial Q}{\partial U} \end{equation}

Plotting the calculated $Q$ in the **n** type region, with derivation one gets the capacitance, which is plotted in figure 5.

^{1)}As long as there is no current flow the fermi-levels in the doped Si layers remain flat, and the voltage drop is at the oxide.