Drift

As noted in the movement of carriers page, in the absence of an electric field, carriers move a certain distance at a constant velocity in a random direction. However, in the presence of an electric field, superimposed on this random direction, and in the presence of thermal velocity, carriers move in a net direction.  There is an acceleration in the direction of the electric field if the carrier is a hole or opposite to the electric field if the carrier is an electron. The acceleration in a given direction causes a net motion of carriers over a certain net distance, as shown in the animation below. The direction of the carrier is obtained as a vector addition between its direction and the electric field. The net carrier movement in the presence of an electric field is characterized by mobility, which varies between different semiconductor materials. Values for silicon, the most commonly used semiconductor material for PV applications, are given in the appendix.

The animation shows how the presence of an electric field will introduce a net distance into the movement of a carrier. In this animation, the carrier is a hole so that it moves in the same direction as the electric field.

Transport due to the movement of carriers due to the presence of an electric field is called "drift transport". Drift transport is the type of transport that occurs not only in semiconductor material, but also in metals. The following animation shows the motion of carriers in random direction with, and without an electric field. The carrier in this case is an electron. Since the electron has a negative charge, it will tend to move in the direction opposite to the electric field. Notice that in the majority of cases, the electron moves in the direction opposite to the electric field. In some cases, for example if the electron follows a sequence of moves in the direction of the electric field, the net motion may actually be in the direction of the electric field for a short distance.

The animation above shows randomly carriers with and without an electric field. The application of an electric field causes a net movement of an electron towards the right and the hole to the left.

In the next animation, an intrinsic semiconductor with an equal number of electrons and holes is pictured. Without the electric field the electrons and holes move around randomly in the semiconductor. When the field is turned on the electrons and holes drift in opposite directions.

For clarity, the effect of the electric field is greatly exaggerated. In a typical semiconductor the electric field has only a very minor effect on the random movement of the carriers.

Drift Equation. Conductivity and Mobility.

One-dimensional drift equation is given by the following formula.

where J_x is the current density in the x-direction, E_x - electric field applied in the x-direction, q - electron charge, n and p - electron and hole concentrations, µ_n and µ_p - electron and hole mobilities.

To derive the drift equation let's consider the bulk of semiconductor.

If the electric field E_x is applied in the x-direction each electron experiences a net force which leads to additional acceleration in the direction opposite to the direction of the field.

The net acceleration in the case of steady state current flow is balanced by the decelerations of the collision processes. If N(t) is the number of electrons that have not undergone a collision by time t, then the rate of decrease N(t) is proportional to the number  left unscattered at t.

Where τ represents the mean time between scattering events.

The probability that an electron has a collision in dt is , then the differential change in p_x due to collisions in dt is

where n is the electron concentration.

And average momentum per electron is

The net drift speed is equal to

The current density is the number of electrons crossing the unit area per unit time

,

Where is the conductivity of a semiconductor and is the mobility of carriers.

Rearranging gives

Finally considering both hole and electron conduction