The Einstein relationship is a relationship between the diffusion coefficients (D_P and D_N) and the mobilities of the carriers (m_N and m_P). Pierret then mentions three facts to which I will add an additional one:

Under equilibrium conditions, the Fermi level inside a material (or inside a group of materials in intimate contact) is invariant as a function of position; that is . This fact is obvious once the concept of the Fermi level is understood as the level where electron states with energies below this level are predominantly filled and electron states with energies above this level are mostly empty (analogous to a lake of water where the top of the water is the Fermi level, most of the water is below the surface Fermi level of the lake in its low energy available states).
Fact number 2 states what we found in chapter 4 that as one dopes a semiconductor more and more n-type, the Fermi level departs from the intrinsic Fermi level and approaches the conduction band. Similarly, as one dopes a semiconductor more and more p-type, the Fermi level moves towards the valence band.
One third fact I will insert is fact that an applied electric field causes a bending in the valence and conduction band and hence the intrinsic Fermi level according to the following equation:

See chapter 4.3.2 for a detailed explanation of this but it is fairly obvious. From electromagnetism, the electric field is equal to the negative gradient of the potential and the potential is proportional to the potential energy. Hence:

Figure 6.14 of Pierret is a good example of band bending as a result of a nonuniformly doped semiconductor:

These facts being stated, we can continue. Under equilibrium conditions, the current density is zero. Hence:

Using information from fact #3 above, and for nondegenerate semiconductors we have:

Using this result in the equation for J_N, we have:

Similarly for J_P, we find:

These two relations between the diffusion coefficients and the mobility for electrons and holes are called Einstein relationships. These will prove useful.

Equations of State

Collecting the results from the first two sections, we have that the total current is a sum of the conduction band electrons and valence band holes: J = J_N + J_P where:

These last two equations can be cast into a simpler form by the introduction of the concept of quasi-Fermi levels F_N and F_P for electrons and holes respectively. Pierret states that these energy levels are by definition related to the nonequilibrium carrier concentration in the same way E_F is related to the equilibrium carrier concentration. Stated mathematically, this is:

In general F_N and F_P will be two distinct values that will tend back to E_F as the semiconductor goes back to equilibrium. It can be shown by differentiating p and n above to get and, substituting this in for J_P and J_N and using the Einstein relationships that:

These equations are extremely useful for analysis of energy bands and current transport in electronic devices.

Continuity Equations

The charge continuity equations will now be studied, first in general and then applied to minority carriers under low level injection and low electric fields. Continuity equations are used throughout physics and engineering and relate the time dependence of some concentration to other functional relationships of the concentration. For instance, concentration of mass can be written as the time dependence of mass density which can be shown to be:

where is the mass current where v is the velocity of the mass particles. A good description of this concept, the derivation and proof is given in Vector Calculus by Jerrold E. Marsden pg. 544-545:

The same type of continuity equation can be applied to charge carriers with the realization that charge concentration can increase or decrease because of other mechanisms besides carrier transport, namely recombination and generation. Hence we can write:

where the divergence of the electron and hole current account for the drift and diffusion current contribution to rate of change in time of the charge concentration, the terms r_N and r_P are the electron and hole recombination rates respectively and the terms g_N and g_P are other electron and hole generation processes respectively (such as photoexcitation, ....). These are the general rate equations that will be used in quite often in device analysis. Let us apply these equation to a simplified situation of minority carrier diffusion.

Minority Carrier Diffusion Equations

Let us make the following assumptions:

The system is one-dimensional.
The analysis is restricted to only minority carriers.
The electric field is negligible.
The equilibrium carrier concentrations are not a function of position.
Low level injection conditions are held throughout the system.
No other recombination or generation processes except for photoexcitation.

Under these conditions, we have:

g_N= G_L

where G_L is the number of electron-hole pairs generated per sec-cm³ by the absorption of externally introduced photons. If the semiconductor is not subjected to illumination, then G_L = 0. Using these simplifications, we obtain the minority carrier diffusion equations: