Carrier diffusion is due to the thermal energy, kT, which causes the carriers to move at random even when no field is applied. This random motion does not yield a net flow of carriers nor does it yield a net current in material with a uniform carrier density since 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 even out the carrier density variations: 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.
The derivation is based on the basic notion that carriers at non-zero temperature (Kelvin) have an additional thermal energy, which equals kT/2 per degree of freedom. It is the thermal energy, which drives the diffusion process. At T = 0 K there is no diffusion.
The reader should recognize that the random nature of the thermal energy would normally require a statistical treatment of the carriers. Instead we 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. To further simplify the derivation, we will derive the diffusion current for a one-dimensional semiconductor in which carriers can only move along one direction.
We now introduce the average values of the variables of interest, namely the thermal velocity, vth, the collision time, tc, 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 Figure 2.7.7.
|Figure 2.7.8:||Carrier density profile used to derive the diffusion current expression|
Shown is a variable carrier density, n(x). Of interest are the carrier densities which 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 that 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 that 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 subtracted 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 electron diffusion current equals this flux times the charge of an electron, or:
We now replace the product of the thermal velocity, vth, and the mean free path, l, by a single parameter, namely the diffusion constant, Dn, so that:
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 an externally applied field causes drift. However one essential parameter in the analysis, namely the collision time, tc, 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, stating that electrons carry a thermal energy which equals kT/2 for each degree of freedom. Applied to a one-dimensional situation, this leads to:
Using the definition of the diffusion constant we then obtain the following expressions which are often referred to as the Einstein relations:
|Example 2.10||The hole density in an n-type silicon wafer (Nd = 1017 cm-3) decreases linearly from 1014 cm-3 to 1013 cm-3 between x = 0 and x = 1 mm. Calculate the hole diffusion current density.|
The hole diffusion current density equals:
where the diffusion constant was calculated using the Einstein relation:
and the hole mobility in the n-type wafer was obtained from Table 2.7.3 as the hole mobility in a p-type material with the same doping density.
The total electron current is obtained by adding the current due to diffusion to the drift current, resulting in:
and similarly for holes:
The total current is the sum of the electron and hole current densities multiplied with the area, A, perpendicular to the direction of the carrier flow:
Whenever drift or diffusion of carriers occurs, the semiconductor is no longer in thermal equilibrium. As a result we can no longer use a constant Fermi energy throughout the semiconductor. We therefore generalize the concept of the Fermi energy by allowing the Fermi energy to vary throughout the material and by assigning a different Fermi energy, namely the Quasi-Fermi energies, Fn and Fp, to electrons and holes. This approach is based on the notion that the electron and hole distributions van still be approximated with the same distribution function, but that electrons are no longer in thermal equilibrium with holes. The equations for the carrier densities are then:
The physical interpretation of the quasi-Fermi energies can be clarified by inserting equations (2.7.36) and (2.7.37) into the expressions for the electron and hole current density, (2.7.33) and (2.7.34):
which can be simplified by rewriting the gradient of the intrinsic energy as function of the electric field using (2.3.8), resulting in:
From this equation, we conclude that the gradient of the quasi-Fermi energy represents the total force acting on the carriers including both the force due to the applied electric field and the force due to the carrier gradient.