There are many ways to get the timeline you want but the one that almost perfectly fits your description would be "Silicon Jumpstart"
Which goes like this:
1930: Experimenting with a new process to purify silicon for his
rectifier research, Bell Labs scientist Russell Ohl accidentally
cracks a cylinder of the stuff, and is surprised to find that, as
a result, it produces a voltage when exposed to light. Ohl shows
this to his boss, Mervin Kelly. New Bell Labs hire Walter Brittain
is introduced to the silicon mystery by his supervisor Joseph
Becker.
1931: Karl Lark-Horowitz adds the electronic theory of solids to
his research program for the Physics Department at Purdue
University, since it seems a fruitful area of research with little
competition that can be done reasonably cheaply.
1932: Becker and Brittain return to the silicon mystery after Ohl
manages to duplicate the process. They theorize that, somehow,
electron-rich impurities in the silicon can send their electrons
to regions made electron deficient by a different impurity. Mass
spectrometry seems to indicate the presence of phosphorus in the
negative (or 'n') end, which would fit the theory. Ohl tries
deliberately salting part of a run with boron, to produce a positive
(or 'p') end.
1933: Following the work of Peierls, Bloch, and Wilson, Nevill Mott
at Bristol University, James Franck at Johns Hopkins, and Karl
Lark-Horovitz with the visiting Wolfgang Pauli make deep advances
into the theory of conductors and semiconductors; Lark-Horovitz
follows up with experiments on germanium crystals.
1935: Becker, Brattain, and Ohl at Bell Labs, Lark-Horovitz at Purdue,
and Julius Edgar Lilienfeld at Ergon Laboratories in Medford, Mass.
independently invent the point contact transistor. Newspaper stories
publicize the precedence battle, but lab notebooks show BB&O have
clear priority.
Now you just need to kill off Arthur Jeffrey Dempster and you have your timeline.