I'm no expert, I'm just now researching this topic for my Master's thesis, and I might miss a bunch of important stuff, but in my opinion, a huge breakthrough might come from the ecological economics side of the debate.
A new (to my knowledge) way of integrating resources and energy into economic growth is to use the so called Exergy (usefull energy) variable in the model. When you use this Exergy (as opposed to simply cost of energy) variable in growth accounting, (again, to my knowledge) it explains away most of the Solow residual. So, ideas, human capital, physical capital and institutions start looking as a really minor factor when you compare them to the amount of calories/watts that we have been using to our advantage throughout the 20-th century...
The implications of the model (I'm speculating a bit here, people who know better could correct me, if I'm wrong) are that it's not that the physical capital per se drives growth, but rather the ability of the capital to extract natural (renewable and expendable) energy resources AND, most importantly, convert those resources into useful work.
You can look up works by Steve Keen
and this paper:
Aggregate production functions: How does the Solow Residual change when introducing quality-adjusted values for capital, labour, and energy?
Authors: João Santosa; Matthew Kuperus Heunb, Paul E. Brockwayc, Marco Sakaic; Tiago Domingosa
Adding: a quote from Steve Keen "Capital without energy is a statue, labour without energy is a corpse."