I am writing a short story.
One of the pre-existing conditions of the story is a world 75-100 years from now where no viable source of energy comparable to fossil fuels has been found to replace the energy density of fossil fuels.
In this model, the loss of access to concentrated and relatively low-cost energy will have severe impacts on manufacturing processors. The high cost of energy will be reflected in the cost to the consumer for the processing power. This increase in cost will, more likely than not, decrease the demand for chips. The decrease in demand will provide a feedback loop which will require further price increase to offset volume of sales.
Chips would necessarily require the lowest amount of energy possible in their manufacture in order to retain the volumes required to economically maintain production
What sort of processors would fit into such a economic model, and what ARM processor, if any, would be the precursors of such a line of computing technology?
Please note that I am replying here in my capacity as a SF reader - don't mistake me for someone who is actually involved in the manufacturing process!
That disclaimer aside, I would imagine that all forms of manufacturing would evolve to use lower-energy methods. Already we see prototype circuits being printed using low-temperature organic semiconductors. These are currently large, and relatively slow, but the approach shows some promise for limited applications. With energy much more expensive, the economics of this kind of process could change, and the incentives for improving it would increase. But so also would the incentives for developing new energy technologies - you might find it hard to find a plausible reason why (for example) nuclear or solar don't take off once the cost of electricity rises sufficiently.
Anyway, that's just my 2 cents.