Drift Current Density

If we have a positive volume charge density ρ moving at an average drift velocity v_d, the drift current density is given by

(1)

where J is in units of C/cm²-s or amps/cm². If the volume charge density is due to positively charged holes, then

(2)

where J_p|drf is the drift current density due to holes and v_dp, is the average drift velocity of the holes.

The equation of motion of a positively charged hole in the presence of an electric field is

(3)

where e is the magnitude of the electronic charge, a is the acceleration, E is the electric field, and m^*_p is the effective mass of the hole. If the electric field is constant, then we expect the velocity to increase linearly with time. However, charged particles in a semiconductor are involved in collisions with ionized impurity atoms and with thermally vibrating lattice atoms. These collisions, or scattering events, alter the velocity characteristics of the particle.

As the hole accelerates in a crystal due to the electric field, the velocity increases. When the charged particle collides with an atom in the crystal, for example, the panicle loses most, or all, of its energy. The particle will again begin to accelerate and gain energy until it is again involved in a scattering process. This continues over and over again. Throughout this process, the particle will gain an average drift velocity which, for low electric fields, is directly proportional to the electric field. We may then write

(4)

where μ_p is the proportionality factor and is called the hole mobility. The mobility is an important parameter of the semiconductor since it describes how well a particle will move due to an electric field. The unit of mobility is usually expressed in terms of cm²/v-s.

By combining Equations (2) and (4). we may write the drift current density due to holes as

(5)

The drift current due to holes is in the same direction as the applied electric field.

The same discussion of drift applies to electrons. We may write

(6)

where J_{n|dr f} is the drift current density due to electrons and v_dn is the average drift velocity of electrons. The net charge density of electrons is negative.

Table 1.1 Typlcal mobility values at T = 300 K and luw dopmg concentraion.

The average drift velocity of an electron is also proportional to the electric field for small fields. However, since the electron is negatively charged, the net motion on the electron is opposite to the electric field direction. We can then write

(7)

where μ_n is the electron mobility and is a positive quantity. Equation (6) may now be written as

(8)

The conventional drift current due to electrons is also in the same direction as the applied electric field even though the electron movement is in the opposite direction.

Electron and hole mobilities are functions of temperature and doping concentrations, as we will see in the next section. Table 1.1 shows some typical mobility values at T = 300 K for low doping concentrations.

Since both electrons and holes contribute to the drift current, the total drift current density is the sum of the individual electron and hole drift current densities, so we may write

(9)