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The other difference between drift current and diffusion current, is that the direction of the diffusion current depends on the change in the carrier concentrations, not the concentrations themselves. In the equation, the signs are reversed as we are used to seeing them. We usually assign a +q to holes and q to electrons. In the case of diffusion current, they are reversed to be opposite of the derivative of the concentrations. This occurs because the carriers are diffusing from areas of high concentrations to areas of low concentrations.
For example, if the derivative of p with respect to x is positive, then the concentration of holes is growing as you move towards the +x direction. Diffusion current will be the opposite of that, the holes will be diffusing in the x direction to where there's a lower concentration of holes. If the derivative is negative, the opposite will occur. The concentration of holes is decreasing as you go from the x to +x direction. Therefore, holes will diffuse to the +x direction where there's a lower concentration of holes. This is why the negative sign is needed in the equation for the hole diffusion current.
The same goes for electrons, but in this case, the signs cancel for a positive derivative because the electrons, carrying q, diffuse to the x direction where there's less electrons. The sign remains if the derivative is negative, because electrons will be diffusing to the +x direction carrying a q charge. For these reasons it's not included in the equation for the electron diffusion current.
Both drift current and diffusion current make up the total current in a semiconductor. They may not be occurring at the same time, but the equation is still valid. Under equilibrium conditions, the current density should be zero because there shouldn't be any drastic changes occurring, like applying an electric field or changing the carrier concentrations by a large margin. Even so, if the doping is not completely uniform, there will be a change in concentration is some places in the semiconductor, resulting in a gradient. This gradient can in turn give rise to an electric field, which in turn can give rise to nonzero current densities.
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