Electric field and potential plots across a pn junction diode

Question

Below shows how the charge density, electric field and potential is varying along the horizontal axis of a pn junction diode under equilibrium:

I can understand the the charge density plot easily because the left part of the depletion region is negatively charged and right part is positively charged after diffusion of electrons and the holes end, and the rest part of the diode(besides the depletion region) is neutral hence zero charge density there.

How are rest of the plots i.e electrical field and potential plots are derived/obtained from this charge density plot? Some mathematical or physical interpretation would help.

Answer 1

Calculating the electric field from the charge density can be done using the first Maxwell equation:

$$\nabla\cdot \vec E = \frac{\rho}{\epsilon}$$

Where \$\rho\$ is the charge density, and \$\epsilon\$ is the permittivity of the material. In a single dimension, this equation is simply:

$$\begin{align} \frac{dE}{dx} &= \frac{\rho(x)}{\epsilon} \\ &\Downarrow \\ E(x) &= \frac{1}{\epsilon}\int_{-\infty}^{x} \rho(u)du \end{align}$$

The potential is defined as:

$$\begin{align} V &= -\int \vec E \cdot d\vec l \\ &\Downarrow \\ V(x) &= -\int_{-\infty}^{x} E(u)du \end{align}$$

So the integral of a constant charge density results in a linear electric field dependency.

Integrating the linear electric field, results in a quadratic potential.

Answer 2

The charge density is an equilibrium between diffusion (=thermal motion) and electricl attraction of charged particles.Its derivation is actually quite complex differental equation problem which needs statistical electron physics and field theory.

The curve for electric field is a brute force one dimensional solution of the basic electricity fact that the vector divergence of the field density in a given point is = charge density.in that point.

The potential curve is a calculation how much work is needed to move an unit charge through the given uneven electric field - it's got directly from the definition of the potential.

If you know those math laws, you should immediately notice that electrical field stength is the charge density integrated along X axis. The potential curve is the electric field integrated (=double integrated charge density curve).

This all needs about half year math and physics studies in lowest university level.

About measuring the potential:

If you have a diode, there's a potential difference (=voltage) between the terminals. In theory it could be detected if you had a sensitive enough contactless measuring method. Unfortunately as soon as you insert a normal voltmeter, the voltage vanishes - there's nothing that could produce continuous current that a normal voltmeter needs.