When an external voltage holds the p-type material at a higher potential than the n-type material, we say that the p-n junction diode is under forward bias. A representation of this condition is shown below.
A p-n junction is forward biased by the voltage source.
The motions of the holes and electrons in the p-n junction diode are different under forward bias than they are under reverse bias. As before, the holes in the p-type material experience a diffusive force to the right; the electrons experience a diffusive force to the left. However, now the holes experience two opposing drift forces: a drift force to the right, due to the position of the positively-charged cathode, and a drift force to the left, due to the electrical field from the space charge region. Due to the relative weakness of this electrical field, the net result of these opposing drift forces is a force to the right. This means that both drift and diffusion will push the holes to the right; drift and diffusion will also work together to push the electrons in the n-type section to the left.
Because of this, the width of the space charge region will shrink and the barrier potential will fall. As (the absolute value of) V increases towards the (absolute value of the) barrier potential, the space charge region will shrink until it becomes so thin that it its associated electric field is too weak to hold the holes and electrons back. The diffusion and larger drift forces will dominate, and majority carriers will cross the junction. The crossing of majority carriers onto the other side is known as carrier injection.
The net effect of forward bias, then, is a current to the right that increases as voltage increases. Unlike reverse bias, the current that flows under forward bias is due to the movement of majority carriers, rather than minority carriers.
"Chapter 6: Diodes." Fundamentals of Electrical Engineering. 2nd ed. New York, New York: Oxford UP, 1996. 362-63. Print.