In the Star Trek Illustrated Handbook (based on The Official Star Trek Fact Files 1997 - 2002) I read:
Following the model established by Zefram Cochrane, Federation vessels' warp engines accomplish the transition into subspace by using a matter-antimatter reaction to generate a series of warp fields that exert force against one another.
It goes on to state just how this works:
In the simplest terms, the warp core works by burning deuterium to create gases, which are then forced together with antimatter in the form of antihydrogen. The reaction is controlled by dilithium crystals to create a plasma stream that is split in two and routed to the warp nacelles. In the nacelles, the plasma is used to energize the warp field coils, made of verterium cortenide.
From numerous episodes, we learn that naturally occurring dilithium is extremely rare and is mined on only a few planets, such as Coridan, Troyius and Rura Penthe. So I wonder, how did Cochrane obtain dilithium to make the first warp flight? Or, what can we speculate is the most probable alternative mechanism that could still make the first warp drive function, lacking dilithium? By the same token, based on the description given above, I can ask the same question about the crystals verterium and cortenide. I guess the latter materials are somewhat less essential, but they appear also not too readily available, cf. e.g. ST VOY s 2, ep 20.
Of course, I want to exclude the very paradoxical explanation that he obtained the material(s) from the Enterprise in First Contact.