I was wondering if Zener Breakdown is a special case of Avalanche Breakdown which just works at a lower potential due to heavy doping.
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It is common to use the terms "Avalanche breakdown" and "Zener breakdown" interchangeably, however they are not exactly the same.
You ask if zener breakdown is just a special case of avalanche, the answer is Yes and No.
Yes because the starting mechanism is the same. We apply a reverse voltage on both, this causes an electrons to break away from its covalent bonds.
The answer is also No because everything that happens after the start is different. In an avalanche breakdown the electric field is so large that the dislodged electrons gain enough energy to ionize. This has a multiplicative effect. This can damage the device.
In a zener breakdown on the other hand, as you said the large doping causes the depletion layer to be small. The depletion layer is so small that the electrons cannot gain enough velocity to cause impact ionization. The electrons simply tunnel through the depletion region. Therefore it is possible to achieve zener breakdown without damaging your device.
