If the concentration gradient is high, then the diffusion current density is also high. Similarly, if the concentration gradient is low, then the diffusion current density is also low. The concentration gradient for n-type semiconductor is given by. The concentration gradient for p-type semiconductor is given by. Where D n is the diffusion coefficent of electrons. The diffusion current density due to holes is given by. The total current density due to electrons is the sum of drift and diffusion currents. The total current density due to holes is the sum of drift and diffusion currents.
Where D p is the diffusion coefficent of holes. The mean free path is the average length a carrier will travel between collisions. These three averages are related by:. Shown is a variable carrier density, n x. Where the flux due to carriers moving from right to left is subtracted from the flux due to carriers moving from left to right. We now replace the product of the thermal velocity, v th , and the mean free path, l , by a single parameter, namely the diffusion constant, D n , so that:.
We now further explore the relation between the diffusion constant and the mobility. At first, it seems that there should be no relation between the two since the driving force is distinctly different: diffusion is caused by thermal energy while an externally applied field causes drift.
However one essential parameter in the analysis, namely the collision time, t c , should be independent of what causes the carrier motion. Applied to a one-dimensional situation, this leads to:. We use equations 2. Using the definition of the diffusion constant we then obtain the following expressions which are often referred to as the Einstein relations:.
The hole diffusion current density equals:. The total electron current is obtained by adding the current due to diffusion to the drift current, resulting in:. The total current is the sum of the electron and hole current densities multiplied with the area, A , perpendicular to the direction of the carrier flow:.
Diffusion in Silicon - Semantic Scholar
Whenever drift or diffusion of carriers occurs, the semiconductor is no longer in thermal equilibrium. As a result we can no longer use a constant Fermi energy throughout the semiconductor. We therefore generalize the concept of the Fermi energy by allowing the Fermi energy to vary throughout the material and by assigning a different Fermi energy, namely the Quasi-Fermi energies, Fn and Fp, to electrons and holes.
This approach is based on the notion that the electron and hole distributions van still be approximated with the same distribution function, but that electrons are no longer in thermal equilibrium with holes. The equations for the carrier densities are then:. The physical interpretation of the quasi-Fermi energies can be clarified by inserting equations 2. From this equation, we conclude that the gradient of the quasi-Fermi energy represents the total force acting on the carriers including both the force due to the applied electric field and the force due to the carrier gradient.
The Hall effect describes the behavior of the free carriers in a semiconductor when applying an electric as well as a magnetic field. The experimental setup shown in Figure 2. A voltage V x is applied between the two contacts, resulting in a field along the x -direction. The magnetic field is applied in the z -direction. As shown in Figure 2. The magnetic field causes a force to act on the mobile particles in a direction dictated by the right hand rule. As a result there is a force, F y , along the positive y -direction, which moves the holes to the right.
In steady state this force is balanced by an electric field, y , so that there is no net force on the holes.
As a result there is a voltage across the sample, which can be measured with a high-impedance voltmeter. This voltage, V H , is called the Hall voltage. For the sign convention shown in 2. The behavior of electrons is shown in 2. The electrons travel in the negative x-direction. Therefore the force, F y , is in the positive y-direction due to the negative charge and the electrons move to the right, just like holes. The balancing electric field, y , now has the opposite sign, which results in a negative Hall voltage.
Effect of vacancy reduction on diffusion in semiconductors
We now assume that the carriers can only flow along the x -direction and label their velocity v x. The Lorentz force then becomes:. Since the carriers only flow along the x -direction, the net force must be zero along the y and z direction. As a result, the electric field is zero along the z direction and:.
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Equation 2. This electric field is called the Hall field. The Hall coefficient, R H , is defined as the Hall field divided by the current density, J x , and magnetic field, B z :. Alternatively, one can calculate the Hall coefficient from the measured current, I x , and measured voltage, V H :. A measurement of the Hall voltage is often used to determine the type of semiconductor n -type or p -type the free carrier density and the carrier mobility. Repeating the measurement at different temperatures allows one to measure the free carrier density as well as the mobility as a function of temperature.
Since the measurement can be done on a small piece of uniformly doped material it is by far the easiest measurement to determine the carrier mobility. It should be noted that the scattering mechanisms in the presence of a magnetic field are different and that the measured Hall mobility can differ somewhat from the drift mobility. A measurement of the carrier density versus temperature provides information regarding the ionization energies of the donors and acceptor that are present in the semiconductor as described in section 2.
While the interpretation of the Hall measurement is straightforward in the case of a single dopant, multiple types of impurities and the presence of electrons and holes can make the interpretation non-trivial.
Where n and p are the electron and hole density in the semiconductor. Random motion of carriers in a semiconductor with and without an applied electric field. Combining both relations yields an expression for the average particle velocity:. Calculate the average time between collisions.
Calculate the distance traveled between two collisions also called the mean free path. The collision time, t c , is obtained from: where the mobility was first converted in MKS units. The mean free path, l , equals:. Temperature dependence of the mobility in germanium, silicon and gallium arsenide due to phonon scattering. Electron and hole mobility versus doping density for silicon. The mobility at a particular doping density is obtained from the following empiric expression:. Resistivity of n -type and p -type silicon versus doping density. Sheet resistance of a 14 mil thick n -type and p -type silicon wafer versus doping density.
Closely enough to notice that this picture contains a very basic mistake that doesn't influence what has been discussed, however? If you didn't notice, you may want to activate the link. For direct semiconductors the diagram would be very similar, except that the extrema of the bands are now on top of each other. Recombination now is easily possible, the energy liberated will be in the form of a photon - recombination thus produces light.
Life times in the case of direct semiconductors thus tend to be short - typically nanoseconds - and are dominated by the properties of the semiconductor itself.
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What exactly determines the life time in indirect semiconductors like Si? In our simplified view of "perfect" crystals recombination would simply be impossible? There are several mechanisms that allow recombination in real crystal. Generally, a third partner is needed to allow momentum and energy transfer. Usually, this third partner is a defect of some kind. Most notorious are certain atomic defects, often interstitial atoms as, e. But the doping atoms and coarser defects like dislocations and grain boundaries also help recombination along.
In summary , the life time of indirect semiconductors is dominated by defects, by impurities, by anything that makes the crystal imperfect. It is thus a property that can vary over many orders of magnitude - some examples can be found in the link. In very perfect indirect semiconductors, however, it is a very large time for electrons , and can easily be found in the millisecond range. In an important generalization of what has been said so far, we realize that a minority carrier does not "know" how it was generated.
Generation by a thermal energy fluctuation in thermal equilibrium or generation by a photon in non-equilibrium - it's all the same! The minority carrier, once it was generated, will recombine on average after the life time t. This is valid for all minority carriers at least as long as their concentration is not too far off the the equilibrium value. This implies that the minority carriers will disappear within fractions of a second after they were generated!
However, since we have a constant concentration in thermal equilibrium, we are forced to introduce a generation rate G for minority carriers that is exactly identical to the recombination rate R in equilibrium. In other words, the carrier concentrations in the valence and conduction bands are not in static, but in dynamic equilibrium see thermodynamic script as well. Think of your bank account. Its average balance will be constant as long as the withdrawal rate is equal to the deposition rate.
If we know the life time t , however, we can immediately write down the recombination rate:. With the relations from above we obtain the following expression. This equation is a good approximation for the temperature range where the doping atoms are all ionized but little band-band transitions take place, i. And do not forget: it is only valid in thermodynamic equilibrium.