Why Does Germanium Spontaneously Extrude Long Screw Dislocations? 
Pure germanium is known to spontaneously extrude very long screw
  dislocations. They are one of the primary reasons for the failure of
  older diodes and transistors made from germanium; depending on what
  they eventually touch, they may lead to an electrical short.
  -Wikipedia


I'm not sure what is going on in the below figure: the caption is confusing.


FIG. 7. Point contact between tungsten-molybdenum whisker and
  germanium surface: (a) Before forming; (b) After forming; (field of
  view measures 50 μm from left to right; Smith 1956).

Silicon Carbide also form screw dislocations which are more pipe-like. Pictures of SiC screw dislocation. 

A team led by University of Wisconsin-Madison chemist Song Jin,
  writing this week (April 23, 2010) in the journal Science, shows that
  a simple crystal defect known as a "screw dislocation" drives the
  growth of hollow zinc oxide nanotubes just a few millionths of a
  centimeter thick.
Dislocations are fundamental to the growth and characteristics of all
  crystalline materials. As their name implies, these defects prompt the
  creation of spiral steps on an otherwise flawless crystal face. As
  atoms alight on the crystal surface, they form a structure strikingly
  similar in appearance to the spiral ramps of multistory parking
  structures. In earlier work, Jin and his research group showed that
  screw dislocations drive the growth of one-dimensional nanowire
  structures that looked like tiny pine trees. That, says Jin, was a
  critical clue to understanding the kinetics of spontaneous nanotube
  growth.
It turns out that "making the structure hollow and making it twist are
  two good ways of relieving such strain and stress," Jin explains. "In
  some cases, the large screw dislocation strain energy contained within
  the nanomaterial dictates that the material hollow out its center
  around the dislocation, thus resulting in the spontaneous formation of
  nanotubes."

Is it stress that causes screw dislocations in Germanium? What causes the stress?
 A: The screw dislocations are a result of the stress you mention and the fact that the crystal is vulnerable due to being cleaved.  It is also important to remember that all crystals, when first grown, contain some dislocations.
If the germanium crystal extended over all space it would be much more robust.  However, in order to use semiconductors we normally need "chips" which are long flat slabs.  This means cleaving a larger crystal along one of it planes.  The surface of the cleaved plane obviously interrupts the crystal lattice so the unfortunate atoms on the edge have to make their own bonding arrangements, so to speak.  Each atom would like four bonds, and this is not available at the surface.  This makes the surface special and makes the crystal vulnerable to reacting in a "layered way" along the cleavage plane.
If stress is applied to the surface, say a contact is pushing down while some other area is being supported, then a screw dislocation can nucleate at the surface and relieve a bit of the stress by allowing the crystal to respond as a layered structure where part of the layer can "give way".  (There may also be a dislocation present from before the crystal was cut obviating the need for a nucleation.)  Further stress can be relieved as atoms then migrate through the very unstable dislocation (where atoms are also deprived of their four-bond diamond lattice and become more mobile) and arrive at the surface.
I think the images you show have a contact "pushing" on the surface and creating the problem.  Also, that contact is likely to heat the immediate vicinity and make all atoms more mobile.
A: Dislocations are difficult to avoid in bulk materials. They are probably already created when a large crystal is cooled down to room temperature. The difference in temperature between center and perifery causes thermal stresses, even when cooling slowly.
After slicing in wafers, some dislocation will intersect the surface. This is an atomic step. Isolated adsorbed atoms diffuse to the steps. For screw dislocation, its position will not change, and the process will continue indefinitely.
A comment mentioned tin whiskers. There are also iron whiskers, essentially single crystals of Fe with one screw dislocation along its center. These are grown from the vapor. Adsorbed atoms diffuse to the ledge. This creates the pattern of the schematic figure.
In the case of germanium, the adatoms must be coming from the bulk somehow. Then they diffuse to an edge at the dislocation core, and it will grow outward.
