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:

1. Phonon scattering
2. Ionized impurity scattering
3. Scattering by neutral impurity atoms and defects
4. Carrier-carrier scattering
5. Piezoelectric scattering

These scattering processes limit the mobility and is accounted for by assigning component mobilities to each process and using Matthiessen’s 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 .

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, don’t 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:

where D_P and D_N are the hole and electron diffusion coefficients respectively.

Einstein Relationships

The Einstein relationship is a relationship between the diffusion coefficients (D_P and D_N) and the mobilities of the carriers (m_N and m_P). Pierret then mentions three facts to which I will add an additional one:

1. Under equilibrium conditions, the Fermi level inside a material (or inside a group of materials in intimate contact) is invariant as a function of position; that is . This fact is obvious once the concept of the Fermi level is understood as the level where electron states with energies below this level are predominantly filled and electron states with energies above this level are mostly empty (analogous to a lake of water where the top of the water is the “Fermi level”, most of the water is below the surface “Fermi level” of the lake in its low energy available “states”).
2. Fact number 2 states what we found in chapter 4 that as one dopes a semiconductor more and more n-type, the Fermi level departs from the intrinsic Fermi level and approaches the conduction band. Similarly, as one dopes a semiconductor more and more p-type, the Fermi level moves towards the valence band.
3. One third fact I will insert is fact that an applied electric field causes a bending in the valence and conduction band and hence the intrinsic Fermi level according to the following equation:
   
   
   
   See chapter 4.3.2 for a detailed explanation of this but it is fairly obvious. From electromagnetism, the electric field is equal to the negative gradient of the potential and the potential is proportional to the potential energy. Hence:
   
   
   
   Figure 6.14 of Pierret is a good example of band bending as a result of a nonuniformly doped semiconductor:
   
   

These facts being stated, we can continue. Under equilibrium conditions, the current density is zero. Hence:

Using information from fact #3 above, and for nondegenerate semiconductors we have:

Using this result in the equation for J_N, we have:

Similarly for J_P, we find:

These two relations between the diffusion coefficients and the mobility for electrons and holes are called Einstein relationships. These will prove useful.

Equations of State

Collecting the results from the first two sections, we have that the total current is a sum of the conduction band electrons and valence band holes: J = J_N + J_P where:

These last two equations can be cast into a simpler form by the introduction of the concept of quasi-Fermi levels F_N and F_P for electrons and holes respectively. Pierret states that “these energy levels are by definition related to the nonequilibrium carrier concentration in the same way E_F is related to the equilibrium carrier concentration”. Stated mathematically, this is:

In general F_N and F_P will be two distinct values that will tend back to E_F as the semiconductor goes back to equilibrium. It can be shown by differentiating p and n above to get and, substituting this in for J_P and J_N and using the Einstein relationships that:

These equations are extremely useful for analysis of energy bands and current transport in electronic devices.

Continuity Equations

The charge continuity equations will now be studied, first in general and then applied to minority carriers under low level injection and low electric fields. Continuity equations are used throughout physics and engineering and relate the time dependence of some concentration to other functional relationships of the concentration. For instance, concentration of mass can be written as the time dependence of mass density which can be shown to be:

where is the “mass current” where v is the velocity of the mass particles. A good description of this concept, the derivation and proof is given in Vector Calculus by Jerrold E. Marsden pg. 544-545:

The same type of continuity equation can be applied to charge carriers with the realization that charge concentration can increase or decrease because of other mechanisms besides carrier transport, namely recombination and generation. Hence we can write:

where the divergence of the electron and hole current account for the drift and diffusion current contribution to rate of change in time of the charge concentration, the terms r_N and r_P are the electron and hole recombination rates respectively and the terms g_N and g_P are other electron and hole generation processes respectively (such as photoexcitation, ....). These are the general rate equations that will be used in quite often in device analysis. Let us apply these equation to a simplified situation of minority carrier diffusion.

Minority Carrier Diffusion Equations

Let us make the following assumptions:

1. The system is one-dimensional.
2. The analysis is restricted to only minority carriers.
3. The electric field is negligible.
4. The equilibrium carrier concentrations are not a function of position.
5. Low level injection conditions are held throughout the system.
6. No other recombination or generation processes except for “photoexcitation”.

Under these conditions, we have:

g_N = G_L

where G_L is the number of electron-hole pairs generated per sec-cm³ by the absorption of externally introduced photons. If the semiconductor is not subjected to illumination, then G_L = 0. Using these simplifications, we obtain the minority carrier diffusion equations:

where the subscripts have been added as Pierret does to denote the fact that these are minority carrier concentrations.