Avalanche ionization rate

At still higher fields carriers can acquire enough energy from motion in an electric field to create electron—hole pairs by impact ionization. For silicon the electron ionization rate, which is the number of pairs generated per cm of electron travel, depends exponentially on electric field. It is about 2 x 103 cm-1 for a 50 kV/cm field at 300 K. The electric field causes electrons and holes so created to travel in opposite directions. They may create other electron—hole pairs causing positive feedback, which leads to avalanche breakdown at sufficiently high fields. 

Recent measurements have shown that laser ablation (or negatively spoken damage) becomes more deterministic when femtosecond laser pulses are applied [22, 23, 40, 41]. This observation is due to the generation of conduction band (seed) electrons by means of multiphoton ionization (MPI). Based on this knowledge, a model for optical breakdown that takes into account avalanche ionization and MPI was developed [40], The temporal behavior of the free electron density in the conduction band n(t) can be described by a rate equation... 

In solids, electronic breakdown is observed in semiconductors (Seeger, 1973). At a sufficiently high electric field strength, electrons from the valence band can tunnel directly into the conduction band. This effect is also called internal field emission and it forms the basic principle of the tunnel diode. A necessary condition for this type of electronic breakdown is a narrow band gap (1 to 2 eV) and a high electron mobility. Avalanche breakdown takes place in the depletion layer of a reversed p-n junction where electric field strengths up to 10 V/cm are obtained. Collisional ionization by electrons (and holes) across the band gap takes place. Here we only consider the effect of the electrons. The relative increase of the number of charge carrier pairs per unit of length is called the ionization rate, a (the first Townsend coefficient of the gas phase), defined as... 

Here, I is the light intensity, a parameterizes the avalanche free-electron generation, and b(I) represents the Multi Photon Ionization (MPI) rate that is a highly nonlinear function of the intensity. The last term describes plasma recombination. 

With reference to Equation 2.23, explain why the rate of ionizing collisions (those which generate additional electron hole pairs) in avalanche breakdown increases rapidly as the accelerated free carrier kinetic energy increases through energy gap. Note the specific portions of the equation that are most strongly affected by this increase. 

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