A wedge will not work. A wedge by its shape presupposes extended area behind the point, which will meet the resistance of the ground as the shape widens. A wedge is useful in latticed material, where it can break bonds and it can trigger a break by supplying the necessary energy. The ground is anything but an organized lattice. It is the reason one uses drills in disorganized solids, as in making holes in concrete, or to reach oil and water underground.
This article on new drilling technologies might give a lead .
From the list I would favor lasers, and plasma, though it would be a very dangerous environment and when going towards the core, the path would not be stable.