Carrier drift in semiconductors

The motion of a carrier drifting in a semiconductor due to an applied electric field, , is illustrated in Figure2.7.1. The field causes the carrier to move on average with a velocity, v.

Figure 2.7.1 :

Drift of a carrier due to an applied electric field.

Assuming that all the carriers in the semiconductor move with the same average velocity, the current can be expressed as the total charge in the semiconductor divided by the time needed to travel from one electrode to the other, or:

(2.7.1)

where tr is the transit time of a particle, traveling with velocity, v, over the distance L. The current density, J, can then be rewritten as a function of the charge density, r :

(2.7.2)

If the carriers are negatively charged electrons, the current density equals:

(2.7.2a)

while for positively charged holes it is:

(2.7.2b)

Where n and p are the electron and hole density in the semiconductor.

It should be noted that carriers do not follow a straight path along the electric field lines. Instead they bounce around in the semiconductor and constantly change direction and velocity due to scattering. This behavior occurs even when no electric field is applied and is due to the thermal energy of the carriers. Thermodynamics teaches us that electrons in a non-degenerate and non-relativistic electron gas have a thermal energy of kT/2 per particle per degree of freedom. A typical thermal velocity at room temperature is around 107 cm/s, which exceeds the typical drift velocity in semiconductors. The carrier motion in the semiconductor in the absence and in the presence of an electric field can therefore be visualized as in Figure 2.7.2.

Figure 2.7.2 :

Random motion of carriers in a semiconductor with and without an applied electric field.

In the absence of an applied electric field, the carrier exhibits random motion and the carriers move quickly through the semiconductor and frequently change direction. When an electric field is applied, the random motion still occurs but in addition, there is on average a net motion along the direction of the field. Due to their different electronic charge, holes move on average in the direction of the applied field, while electrons move in the opposite direction.

We now analyze the carrier motion considering only the average velocity,  of the carriers. Applying Newton's law, we state that the acceleration of the carriers is proportional to the applied force:

(2.7.3)

The force consists of the difference between the electrostatic force and the scattering force due to the loss of momentum at the time of scattering. This scattering force equals the momentum divided by the average time between scattering events or collisions, tc, so that:

(2.7.4)

where a particle with charge q was assumed.

Combining both relations yields an expression for the average particle velocity:

(2.7.5)

We now consider only the steady state situation in which the particle has already accelerated and has reached a constant average velocity. Under such conditions, the velocity is proportional to the applied electric field and we define the mobility as the velocity to field ratio:

(2.7.6)

The mobility of a particle in a semiconductor is therefore expected to be large if its mass is small and the time between scattering events is large.

The drift current, described by (2.7.2), can then be rewritten as a function of the mobility, yielding:

(2.7.7)

For electrons and

(2.7.8)

For holes

Throughout this derivation, we simply considered the mass, m, of the particle. However in order to incorporate the effect of the periodic potential of the atoms in the semiconductor we must use the effective mass, m*, rather than the free particle mass, m0, so that:

(2.7.9)

Example 2.8	Electrons in undoped gallium arsenide have a mobility of 8,800 cm2/V-s. Calculate the average time between collisions. Calculate the distance traveled between two collisions (also called the mean free path). Use an average velocity of 107 cm/s.
Solution	The collision time, tc, is obtained from: where the mobility was first converted in MKS units. The mean free path, l, equals:

. Impurity scattering

Impurities are foreign atoms in the semiconductor. They are efficient scattering centers especially when charged. Ionized donors and acceptors in a semiconductor are a common example of such impurities. The amount of scattering due to electrostatic forces between the carrier and the ionized impurity depends on the interaction time and the number of impurities. Larger impurity concentrations result in a lower mobility. The dependence on the interaction time helps to explain the temperature dependence. The interaction time is directly linked to the relative velocity of the carrier and the impurity, which is related to the thermal velocity of the carriers. The thermal velocity increases with the ambient temperature so that the interaction time decreases. Thereby, the amount of scattering decreases, resulting in a mobility increase with temperature. To first order, the mobility due to impurity scattering is proportional to T 3/2/NI, where NI is the density of charged impurities.

. Lattice scattering

Scattering by lattice waves includes the absorption or emission of either acoustical or optical phonons. These phonons represent quanta of mechanical waves that travel through the semiconductor crystal. Since the density of phonons in a solid increases with temperature, the scattering time due to this mechanism will decrease with temperature as will the mobility. Theoretical calculations reveal that the mobility in non-polar semiconductors, such as silicon and germanium, is dominated by acoustic phonon interaction. The resulting mobility is expected to be proportional to T -3/2, while the mobility due to optical phonon scattering only is expected to be proportional to T -1/2. Experimental values of the temperature dependence of the mobility in germanium, silicon and gallium arsenide are provided in Table 2.7.1.

Table 2.7.1 :

Temperature dependence of the mobility in germanium, silicon and gallium arsenide due to phonon scattering

. Surface scattering

The surface and interface mobility of carriers is affected by the nature of the adjacent layer or surface. Even if the carrier does not transfer into the adjacent region, its wavefunction does extend over 1 to 10 nanometer, so that there is a non-zero probability that the particle is in the adjacent region. The net mobility is then a combination of the mobility in both layers. For carriers in the inversion layer of a MOSFET, one finds that the mobility can be up to three times lower than the bulk value as further discussed in section 7.6.5. This is due to the distinctly lower mobility of electrons in the amorphous silicon oxide. The presence of charged surface states further reduces the mobility just as ionized impurities would. Ider Guerrero EES Secc: 1