go read this :
https://www.eevblog.com/forum/beginners/how-transistors-work/msg109166/#msg109166This is where i give the 'long ' explanation. read all subsequent posts in that topic. As it explains even more details.
Transistors are actually pretty easy to understand , if it is explained correctly. just slapping the equation down doesn't do you any good because it doesn't tell you what is happening in the material. A lot of the operation of the transistor is depeendent on its physical construction. the chemistry gets you to point a, the equiations get you to point b , but to get the thing working you need to carefully craft the shape of the electrodes in respect to each other to make sure the electrons go where you want them to go.
and then there is all the dissipations in and around the depletion area. that's why we have all these weird physical contructions like pinch-gates, mesa's , planar, gunning , tunneling and 400 other ways to stack this tower of NPN. none of whihc is covered by emers moll equations but by ton's of other material physics and field related theory.
in practice : you draw a bunch of shapes your gut tells you should work , spin the wafer and run them through the curve tracer and then pick the best ones. if you got a lot of time and money you will learn a lot and be able to draw a shape that works better than another. that is what semiconductor companies do. they spend a shitload of money designing better transistors. can;t change the laws of physics , you can just build better channels to conudct more electrons and create a better doorway mechanism... one that lifts smoother , has no backlash. ( look up 'snapback' for a transsitor.. you'll go fuzzy eyed trying to understand that one ... especially if you approach it from the mathematics side. the mathematics say it doesn't exist yet every transistor has it... the electron will damn well flow where it wants to go, even if a formula say's it can;t go there... electrons are deaf and blind. but they are damn good at sniffing out a place that has a different charge than they have)
oh and before anyone starts hammering on the 'one electron short' i explained here and the 4-short i explain in the above article ... Here i explained what happens between an intrinsic silicon system and a dopant. the dopant delivers a free electron.
in the intrinsic system the sharing happens between neighbours. you need to think 3 dimensionally ( i sound like Doc Brown... you gotta think four dimensionally Marty, in reality that billboard with the indians isn't there) this stuff sits in a 3d-lattice so electrons come from left, right, top, bottom, front and back. the exchange is not between 1 silicon atom and 1 donor ( there is only one donor per x silicon atoms. how many donors we shoot in controls conductivity capability . ) so this exchange happens between a silicon atom in the center and all its neighbours in the lattice. it is sharing electrons between all its neighbours. semiconductors are very wonky materials. they are essentially natures misfits... ( if you think about what i just said you will find something wrong. silicon wants to go from 4 to 8 electrons. but it has 6 neighbours... left, right , top,bottom , front and back... so why 4 and not 6 ? well because other crap going on with the weak nuclear forces in an atom )
here is a nice representation of how this sharing works ( in 2 dimensions):
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/sili.htmlin reality the sharing is 3 dimensional so if you really want to whack your brain out ( this shows why 4 electrons and not 6) :
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/sili2.html#c1And that is just intrinsic silicon ( non-conducting , no excess electron , no short an electron) shoot a dopant in there that steals one electron or delivers a free one and your brain will try to eat it's own ass and desintegrate in a big 'pfffwwrrrrt i'm outta here , this is too complex... '
kind like Data feeding the Borg that funny hyperdimensional figure that cannot exist..