The constant random motion of carriers can lead to a net movement of carriers if one particular region has a higher concentration of carriers than another region. When this happens, a carrier gradient exists between the high-concentration region and the low-concentration region. Carriers flow from the high-concentration region to the low-concentration region. This carrier flow, called "diffusion", is caused by the random motion of carriers. In all regions of the device, the same probability exists for carriers to move in any particular direction. Near a high carrier concentration region, a large number of carriers move in all directions, including towards the low-concentration region. However, the lower number of carriers around the low carrier concentration region means that fewer carriers move towards the high-concentration region. This imbalance in carrier movement causes a net carrier flow from a high carrier concentration towards a low carrier concentration region and is shown in the animation below.

The rate at which diffusion occurs depends on the speed at which carriers move and on the distance between scattering events. It is termed diffusivity and measured in cm²s^-1. Values for silicon are given in the appendix. Since raising the temperature will increase the thermal velocity of the carriers, diffusion occurs faster at higher temperatures.

One major effect of diffusion is that it evens out carrier concentrations in a device, such as those induced by generation and recombination, without an external force being applied to the device. This is shown in the animation below in which one region of the device has a high concentration of electrons and the other has a high concentration of holes. Due purely to the random movement of carriers, the two concentrations will become uniform.

The animation shows how a high concentration of carriers in one part of the semiconductor tends towards an even distribution. The carriers fill the available space solely through random motion. Electrostatic repulsion has negligible effect since the carriers are so far apart. The holes (coloured blue) have a lower diffusivity than the electrons (coloured red), and so take longer to fill the full space.

Diffusion Equation

As it was said before, diffusion in semiconductors is the motion of charge carriers due to their concentration gradient. It is known from the molecular physics that the flux of diffusing particles is proportional to the concentration gradient. The one-dimensional diffusion equations for electrons and holes can be written as follows

,

where J_n and J_p are the diffusion current densities, q - electron charge, D_n and D_p - diffusion coefficients for electrons and holes, n and p - electron and hole concentrations.

Let’s derive the equation of diffusion for carriers in the bulk of semiconductor. The spreading of a pulse of electrons by diffusion is shown below.

Spreading of a pulse of electrons by diffusion. Arbitrary part of n(x) is divided into the segments of length equal to a mean free path for the electrons. Concentration is supposed to be constant for every segment.

Consider arbitrary electron concentration n(x) at time t. We can divide x into small segments with the width l, where l is the mean free path. As the l is small we can assume the concentration to be constant for each segment. Then the number of electrons passing x₀ per unit time per unit area (electron flux density) from left to right is composed of electrons came from left region and electrons came from right region and is equal to

where is mean free time.

The difference in electron concentration can be written as

And in the limit of small l

The factor in front of the carrier concentration gradient is defined as

D_n is called electron diffusion coefficient with units cm²/s. The minus sign arises from the fact that the vector of the concentration gradient is directed toward the increase of the concentration, while the particles diffuse to the area with lower concentration.

Direction of the concentration gradient is opposite to the direction of the carrier motion with the result that the formula for the carrier flux density should have a minus sign.

Thus .

The same derivation is hold for holes which leads to .

The diffusion current is the carrier flux density multiplied by the carrier charge. That is why the electron and hole currents are in opposite directions.  

A more detailed derivation and properties of the carrier diffusion equation can be found in almost every textbook on solid state physics as (B. G. Streetman, S. Banerjee).

Continuity Equation

Continuity equations give the rate of carriers buildup in the bulk of semiconductor.

,

where U is carrier recombination rate, G - generation rate.

Currents entering and leaving a volume AΔx.

Consider the length dx of semiconductor and movement of holes through it. The net increase in hole concentration per unit time is the difference between the flux of holes entering and leaving the volume AΔx plus generation rate and minus recombination rate.

In the derivative form

,

Finally, plugging in the diffusion equations one can get

,