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. It is the reason one uses drills in making holes in concrete, or to reach oil and water underground.

[This article][1] 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][2], the path would not be stable.


  [1]: https://en.wikipedia.org/wiki/New_drilling_technologies
  [2]: https://en.wikipedia.org/wiki/Structure_of_the_Earth#Core