**Branch :**Computer Science and Engineering

**Subject :**Fundamental of Electronic Devices

## Gradients in the Quasi-Fermi Levels

We can use the results of the equations of the power of the concept of quasi-Fermi levels in semiconductors. If we take the general case of nonequilibrium electron concentration with drift and diffusion, we must write the total electron current as

where the gradient in electron concentration is

Using the Einstein relation, the total electron current becomes

giving a direct cancellation of and leaving

Thus, the processes of electron drift and diffusion are summed up by the spatial variation of the quasi-Fermi level. The same derivation can be made

for holes, and we can write the current due to drift and diffusion in the form of a modified Ohm's law

Therefore, any drift, diffusion, or combination of the two in a semiconductor results in currents proportional to the gradients of the two quasi- Fermi levels.

Conversely, a lack of current implies constant quasi-Fermi levels. One can use a hydrostatic analogy for quasi-Fermi levels and identify it as water pressure in a system.

Just as water flows from a high-pressure region to a low-pressure region, until in equilibrium the water pressure is the same everywhere, similarly electrons flow from a high- to low-electron quasi-Fermi level region, until we get a flat Fermi level in equilibrium.

Quasi-Fermi levels are sometimes also known as electrochemical potentials because, the driving force for carriers is governed partly by gradients of electrical potential (or electric field), which determines drift, and partly by gradients of carrier concentration (which is related to a thermodynamic concept called chemical potential), giving rise to diffusion.