2.10 Carrier diffusion

Table of Contents - Glossary - Study Aids - Ź Ž

In this section:

Introduction: Carrier diffusion
Derivation of the diffusion current
Total current

2.10.1 Introduction: Carrier diffusion

Carrier diffusion is due to the thermal energy which causes the carriers to move at random even when no field is applied. This random motion does not yield a net motion of carriers nor yield a net current in material with a uniform carrier density as any carrier which leaves a specific location is on average replace by another one. However if a carrier gradient is present, the diffusion process will attempt to make the carrier density uniform: carriers diffuse from regions where the density is high to regions where the density is low. The diffusion process is not unlike the motion of sand on a vibrating table; hills as well as valleys are smoothed out over time.

In this section we will first derive the expression for the current due to diffusion and then combine it with the drift current to obtain the total drift-diffusion current.

2.10.2 Diffusion current

The derivation is based on the basic notion that carriers at non-zero temperature (Kelvin) posess an additional thermal energy which equals kT/2 per degree of freedom. It is this thermal energy which drives the diffusion process. At T = 0 Kelvin there is no diffusion.

While one should recognize that the random nature of the thermal energy would normally require a statistical treatment of the carriers, we instead will use average values to describe the process. Such approach is justified on the basis that a more elaborate statistical approach yields the same results.

We now introduce the average values of the variables of interest, namely the thermal velocity, v_th, the collision time, t, and the mean free path, l. The thermal velocity is the average velocity of the carriers going in the positive or negative direction. The collision time is the time during which carriers will move with the same velocity before a collision occurs with an atom or with another carrier. The mean free path is the average length a carrier will travel between collisions. These three averages are related by:

Consider now the situation illustrated with the figure below:

diffusio.gif

Fig. 2.10.1

Carrier density profile used to derive the diffusion current expression

Shown is a variable carrier density, n(x). Of interest are the carrier densities wich are one mean free path away from x = 0, since the carriers which will arrive at x = 0 originate either at x = -l or x = l. The flux at x=0 due to carriers which originate at x = -l and move from left to right equals:

where the factor 1/2 is due to the fact that only half of the carriers move to the left while the other half moves to the right. The flux at x = 0 due to carriers which originate at x = +l and move from right to left equals:

The total flux of carriers moving from left to right at x = 0 therefore equals:

Where the flux due to carriers moving from right to left is substracted from the flux due to carriers moving from left to right. Given that the mean free path is small we can write the difference in densities divided by the distance between x = -l and x = l as the derivative of the carrier density:

The current for electrons equals the flux times the charge of an electron, or:

Typically we will replace the product of the thermal velocity and the mean free path by a single parameter, namely the diffusion constant for electrons, D_n.

Repeating the same derivation for holes yields:

We now further explore the relation between the diffusion constant and the mobility. At first it seems that there should be no relation between the two since the driving force is distinctly different: diffusion is caused by thermal energy while drift is caused by an externally applied field. However one key parameter in the analysis, namely the collision time, t, should be independent of what causes the carrier motion.

We now combine the relation between the velocity, mean free path and collision time,

with the result from thermodynamics which states that electrons carry a thermal energy which equals kT/2 for each degree of freedom, or applied to a one-dimension situation:

and use these relations to rewrite the product of the thermal velocity with the mean free path as a function of the carrier mobility:

Using the definition of the diffusion constant we then obtain the following expressions which are often refered to as the Einstein relations:

2.10.3 Total current

The total electron current is obtained by adding the current due to diffusion to the drift current, yielding:

and similarly for holes:

The total current density is the sum of the electron and hole current densities:

while the total current equals the current density multiplied with the area perpendicular to the carrier flow or:

which can be combined with the previous equations, yielding:

2.9 Ź Ž 2.11

Š Bart J. Van Zeghbroeck, 1996, 1997