### p-n Diode characteristics

#### Introduction

The p-n junctions are the elementary components of semiconductor devices, while they have been already successfully described in the 1940s. It is crucial to understand its working principles, characteristics. The example covers a simple Si p-n diode and investigates its rectifying behavior.

#### Structure

We Simulate the following p-n junction described in figure 1., where the donor density is $N_d = 1E21 \frac{1}{m^3}$, and the acceptor density in the p region $N_a = 1E21 \frac{1}{m^3}$. If the two contacts at the left and on the right are ohmic, than it results the following bandstructure in figure 2. at zero bias. Figure 1. Schematics of the diode Figure 2. Bandstructure at 0V bias

\begin{equation} V_{bi} = \frac{k_bT}{q} ln \left(\frac{N_a N_d}{n_i^2}\right), \end{equation}

where $n_i$ is the intrinsic charge density in the sample. Also the charge density in the sample, with the depletion region can be calculated and plotted in figure 3. Figure 3. Charge density distribution in the sample at zero bias

It shows the depletion region where the amount of both electron and hole charge carriers is several orders of magnitude lower than in the bulk materials.

\section{Voltage Characteristics}

If we apply voltage to the structure currents starts to flow, and it represented in the split of the quasi Fermi levels, as it is depicted in figures 4, 5. Figure 4. Energy levels at 0.2V bias (forward) Figure 5. Energy levels at -0.2V bias

In figure 4 on forward bias the electron quasi-fermi levels moves closer to the conduction-band on the p-doped material, introduces more minority carriers in the sample. Which lowers the built in potential at the junction.

The current can be calculated from the slope of the fermi levels and the carrier concentration in the sample.

\begin{equation} j = j_p+j_n = q \sigma_p(x) p(x) \nabla E_{fp} - q \sigma_n(x) n(x) \nabla E_{fn} \label{eq:driftdiffusion} \end{equation}

The Shockley diode equation covers the voltage characteristics of the diode:

\begin{equation} I = I_0 \left(e^{\frac{U}{k_bT}} -1 \right). \end{equation}

The voltage characteristics simulated according to the quasi-fermi level approach \cite{sze2006physics}, and the current is calculated from equation \ref{eq:driftdiffusion} is plotted in figure 6. Figure 6. Voltage characteristics