Transport of Charge Carriers

Diffusion and drift are the 2 mechanisms for charge carrier transport in solar cells.

Diffusion refers to the net movement of charge carriers due to a concentration gradient.  Charge carriers perpetually move in random directions, including almost colliding with each other, which changes the direction and velocity of movement.  These near collisions are due to Coulomb-Coulomb interactions, where each charge carrier will bend another charge carrier's movement.  A concentration gradient refers to a non-uniform charge carrier distribution, where charge carriers will move from a region of higher concentration to a region of lower concentration.  The flux or movement of charge carriers in the opposite direction is less.  This will happen until the density of charge carriers is uniformly distributed.

For electrons, the Fick's law of diffusion equates the J_e (electron current density) to the product of q (elementary charge), D_e (diffusion coefficient of electrons), and dn/dx (density gradient in direction x):

J_e = qD_e dn/dx

A similar equation applies for holes:

J_h = qD_h dp/dx

Drift refers to the movement of charged particles under the influence of electric fields.  Holes will experience a force and move in the direction of the electric field, while electrons move in the opposite direction.  The current density induced by the electric field is as follows:

J_e = nqμ_eE
J_h = pqμ_hE

where n and p are the density of electrons and holes respectively (at the start point of the drift), μ is the mobility constant, and E is the electric field.

In an electronic band diagram (see below), the electric field will induce a slope positive (sloping upwards) in the direction of the electric field.  This is applied over both the valence and conduction bands.  Hence on average, excited electrons will move down the slope of the conduction band, while excited holes will climb up the slope of the valence band.

Electrons and holes also suffer from 3 recombination mechanisms, namely radiative recombination, Auger recombination, and the Shockley-Read-Hall (SRH) recombination.  Loss mechanisms such as recombination determine the lifetime of charge carriers.  A high recombination rate will lead to shorter lifetimes τ and correspondingly shorter diffusion lengths L.  Diffusion length is the average distance covered by excited charge carriers, and is shown below:

L_e = √(D_eτ_e)
L_h = √(D_hτ_h)

In a doped semiconductor material, the majority charge carriers will have longer diffusion lengths as compared with minority charge carriers.



Reference:
2.3.3 Transport of Charge Carriers, Delft University of Technology, https://www.youtube.com/watch?v=2EhJh3BQvB0