Question 5.219: Explain the P-N diode characteristics equation?...

Let us consider a P-N junction with a donor concentration ND and acceptor concentration NA. Let us also assume that all the donor atoms have donated free electrons and become positive donor ions and all the acceptor atoms have accepted electrons and created corresponding holes and become negative acceptor ions. So, we can say the concentration of free electrons (n) and donor ions ND are the same and similarly, the concentration of holes (p) and acceptor ions (NA) are the same. Here, we have ignored the holes and free electrons created in the semiconductors due to unintentional impurities and defects.

n=N_{D}\quad a n d\quad p=N_{A}

Across the P-N junction, the free electrons donated by donor atoms in n-type side diffuse to the p-type side and recombine with holes. Similarly, the holes created by acceptor atoms in the p-type side diffuse to the n-type side and recombine with free electrons. After this recombination process, there is a lack of or depletion of charge carriers (free electrons and holes) across the junction. The region across the junction where the free charge carriers get depleted is called the depletion region. Due to the absence of free charge carriers (free electrons and holes), the donor ions of n-type side and acceptor ions of the p-type side across the junction become uncovered. These positive uncovered donor ions towards the n-type side adjacent to the junction and negative uncovered acceptors ions towards the p-type side adjacent to the junction cause a space charge across the P-N junction. The potential developed across the junction due to this space charge is called the diffusion voltage. The diffusion voltage across a P-N junction diode can be expressed as

V_{D}={\frac{k T}{e}}\ln{\frac{N_{A}N_{D}}{n_{i}^{2}}}

The diffusion potential creates a potential barrier for further migration of free electrons from the n-type side to the p-type side and holes from the ptype side to the n-type side. That means diffusion potential prevents charge carriers to cross the junction. This region is highly resistive because of the depletion of free charge carriers in this region. The width of the depletion region depends on the applied bias voltage. The relation between the width of the depletion region and bias voltage can be represented by an equation called the Poisson Equation.

W_{D}=\sqrt{\frac{2\epsilon}{e}(V_{D}-V)\left(\frac{1}{N_{A}+\frac{1}{N_{D}}}\right)}

Here, ε is the permittivity of the semiconductor and V is the biasing voltage. So, on an application of a forward bias voltage, the width of the depletion region i.e. P-N junction barrier decreases and ultimately disappears. Hence, in absence of potential barriers across the junction in the forward bias condition free electrons enter into the p-type region and holes enter into the n-type region, where they recombine and release a photon at each recombination. As a result, there will be a forward current flowing through the diode. The current through the PN junction is expressed as:

I=I_{s}\Bigl(e^{\frac{e V}{k T}}-1\Bigr)

Here, voltage V is applied across the P-N junction and total current I, flows through the pn junction. Is = reverse saturation current, e = charge of the electron, k is Boltzmann constant and T is the temperature in Kelvin scale.