For an n-type region this yields the following equations: The product of the carrier densities can be expressed as a function of the intrinsic density in the following way: where it was assumed that the semiconductor is non-degenerate and that the difference between the electron and hole quasi Fermi energies in electron volt equals the applied voltage in volt. The first difference between the two diagrams is that the conduction band edge in the n-type graded region as well as the valence band edge in the p-type graded region are almost constant. Using equations (4.4.1) and (4.4.2), one finds that high injection occurs in a p+-n diode for the following applied voltage: or at Va = 0.55 V for the diode of Figure 4.4.5 as can be verified on the figure as the voltage where the ideality factor changes from one to two. The former approach was taken to obtain the electric field in Figure 4.4.12.
Current-Voltage characteristics of a silicon diode under forward bias. Energy band diagram of the graded p-n diode shown above under forward bias. For EI = Et and t0 = 1/Ntsvth this expression simplifies to: Throughout the depletion region, the product of electron and hole density is given by the "modified" mass action law: This enables to find the maximum recombination rate which occurs for. One can then use the linearized Poisson equation or solve Poisson's equation exactly. The I-V characteristics are plotted on a semi-logarithmic scale and four different regions can be distinguished as indicated on Figure 4.4.5. Here we have to distinguish between the different recombination mechanisms - band-to-band recombination and Shockley-Hall-Read recombination - as they lead to different current-voltage characteristics. As an example we now consider an abrupt one-sided p-n diode. Focusing on a diode with a graded bandgap we first assume that the gradient is indeed constant in the quasi-neutral region and that the doping density is constant. A numeric analysis reveals that the error is less than 10 % when using the short diode expression with Ln > 2 wp' and when using the long diode expression with Ln < wp'/2. Similarly, for a "short" diode, as illustrated by Figure 4.4.6 (b), one obtains: Where tr,p is the hole transit time given by: Again, the excess charge can be related to the current. Energy band diagram of a p-n diode. where the "short" diode expression was used for the capacitance associated with the excess charge due to electrons in the p-type region. Finding total flux of probability current through a sphere, Non-Relativistic Limit of Klein-Gordon Probability Density, Finding the expression for probability density (the Klein Gordon equation). Why Is there no effect in the mass of the bob on the period of the simple pendulum? During a time free charges move a distance The volume of charge that moves past a point is The number of free charges in the volume is The amount of free charge in the volume is In terms of this , the current is ̅ Define current density to be ⃗ ⃗ Just as for the homojunction we find that current in a p-n junction can only exist if there is recombination or generation of electron and holes somewhere throughout the structure. First, there is the ideal diode region where the current increases by one order of magnitude as the voltage is increased by 60 mV. For diodes with a quasi-neutral region shorter than the diffusion length, and assuming an infinite recombination velocity at the contacts, the diffusion length can simply be replaced by the width of the quasi-neutral region.
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