Diodes – Selecting Diode with Vf~300mV at 100uA Forward Current for Zero Crossing Circuit

It is actually possible to get very low impurity Si now. It's just that if you look at it crossed eyed it gets doped.

Your question really comes down to answering the question of what is a minority carrier when there is no doping? There will be carriers of both types present because of the intrinsic concentration but as the doping goes down these become more and more balanced. At the limit, the net polarity of the material is determined by impurity ionization from an applied field.

To form a junction you need to look at the fermi level and do the volume integration to balance the charge through out the semi-conductor. If that doesn't make sense to you, then think about what it means to balance the charge when there is very low charge density. That means that the volume must increase. In simplest terms this means the the depletion volume has to extend to include sufficient # of impurities for the fermi level to establish itself. If you have low impurities the depletion layer moves through the whole volume available.

A case in point is the deep depleted CCD's built on 1e10 /cm^3 substrates with depletion regions that are 500 um or more thick.

There's not just one "ideal" diode.

The simplest type (what I think you are talking about) has 0V across it if the polarity is such that current would flow in the direction of the arrow, and doesn't conduct at all if the polarity is reversed.

A slightly more complex model has a fixed voltage drop (usually taken to be 0.6V or 0.7V) if it is conducting current, and does not conduct current if the voltage is less than the fixed drop. If you consider the voltage drop to have a temperature coefficient (around -2mV/°C) then you can roughly incorporate temperature effects.

In either of the above two cases, to find out what the diode is doing, remove the diode as a thought experiment and see (it may be obvious or you may need to resort to math) whether the bias would be positive or negative.

An even more complex type of ideal diode obeys the nonlinear and temperature-sensitive Shockley diode equation, with ideality factor 1. Most diodes are not that ideal, and the ideality factor is more like 2. You can also have different Is for positive and negative bias.

Finally, you can include ohmic resistance in the equation (which has a positive temperature coefficient), and you'll be fairly close to modelling diode behavior in most real situations.