Carrier Transport

The topic of carrier transport will now be analyzed. First, carrier transport as a result of drift will be studied followed by an analysis of carrier transport caused by diffusion. Important equations of state will be derived. This topic will lead us directly into the analysis of the PN junction in the next section.

Drift

Pierret defines drift as the charged-particle motion in response to an applied electric field. From our earlier analysis of charge carriers in semiconductors, properties of the various electron states near the bottom of the conduction band and near the top of the valence band and the introduction to the concept of holes, we can make the following generalization:

When an electric field ( E ) is applied across a semiconductor, negatively charged electrons will accelerate in a direction opposite E and positively charged holes will accelerated in a direction parallel to E.

The carriers will not accelerate indefinitely because of scattering from various sources such as impurity atoms (both ionized and neutral), phonon scattering, carrier-carrier scattering and other scattering mechanisms. Averaged over time, the carriers will tend to have a certain time averaged drift velocity v_d that, for electric fields that are not excessively large, is linearly proportional to the applied field E. This relation can be written as:

where m is called the mobility and will have a different value for conduction band electrons and valence band holes denoted by m_n and m_p respectively. The mobilities are also very dependent on the semiconductor material. For instance, GaAs has a significantly higher electron mobility but comparable hole mobility relative to Si. For very high fields, v_d is no longer linearly proportional to E and tends to acquire a constant value denoted by v_saturation.

The electron and hole drift current density can easily be shown to have the following forms:

The mobility of a perfect crystal without any defects and at very low temperature should approach infinity. However, because of a variety of defects and scattering processes, the mobility in reality is finite. A few of the scattering processes are:

Phonon scattering
Ionized impurity scattering
Scattering by neutral impurity atoms and defects
Carrier-carrier scattering
Piezoelectric scattering

These scattering processes limit the mobility and is accounted for by assigning component mobilities to each process and using Matthiessens rule (similar to adding up resistances in an electrical circuit) to obtain the total mobility:

where m_n is the total electron mobility, m_Ln is the phonon scattering mobility component, m_In is the ionized impurity scattering mobility component. Likewise for second equation for hole mobility. Typically phonon scattering and ionized impurity scattering are the dominant factors limiting mobilities. These scattering processes have the following dependencies:

where

NI=NA+ND

Resistivity

The resistivity (r) is an important electrical property of materials. It is defined as the proportionality constant relating current density and electric field:

where s is called the conductivity. The resistivity and conductivity can be determined from previous equations we have studied:

Thus, it is easily seen that:

Hall Effect

The hall effect is used to determine information about the carriers in semiconductor materials. Using the Hall effect, it is possible to determine whether electrons or holes are primarily responsible charge transport.

The Hall coefficient is defined as:

To get a handle on what R_H is, let us derive from first principles the equation for R_H. The lorentz force on a moving positively charged hole in a magnetic field is:

Hence, we have that

Using the relation

for holes, we have:

Finally, we see that for and p-type semiconductor with the primary charge carriers as holes, R_H is a positive quantity with a value of:

Similarly for n-type materials with the primary charge carriers as electrons, R_H is a negative quantity with a value of:

Diffusion

Diffusion is a physical phenomenon generally thought of as residing in the thermodynamics field of physics and plays a crucial role in most physical systems you can envision. Hence, it should come as no surprise that it is of central importance in the electronic behavior of semiconductors. If you have not studied thermodynamics, dont fret, for we have been introducing concepts from different areas of physics throughout this course. We have brought together concepts from different areas of physics to produce a synthesis upon which we have developed a powerful foundation to predict the physical, electronic and optical properties of semiconductors. First it was the very geometrical field of crystallography joined with electromagnetism to predict x-ray diffraction, then it was crystallography and electromagnetism joined with quantum mechanics to describe energy bands and electron and hole states, finally it will be crystallography, electromagnetism and quantum mechanics joined with thermodynamics to predict carrier transport and electronic devices. Onwards we go...

Pierret defines diffusion to be a process whereby particles tend to spread out or redistribute as a result of their random thermal motion, migrating on a macroscopic scale from regions of high particle concentration into regions of low particle concentration. If there is no external applied force or chemical potential (i.e., the system is uniform throughout), then diffusion leads to a uniform distribution of particles. Figure 6.12 in Pierret gives a good visual description of diffusion and the resulting electrical current.

Pierrent gives a simplified proof of the equations for the diffusion current resulting in the common sense answer that the diffusion current is proportional to the gradient of the carrier concentration. The result is: