Making a simple transistor shouldn't be too difficult. Of course, the problem is that the lithography and other optical equipment needed to make ICs that can actually do anything are not only expensive, they're also huge and consume a lot of electricity.

Incidentally, garage transistor would be an excellent name for a rock band.

Making a homebrew transistor should be possible, it probably would be quite expensive and time-consuming and the final product would be quite crappy, but it would be cool.

I would start with a diode though, since it is a simpler device. Getting a n+ doped silicon wafer from a nearby university lab should be possible, if it isn't considered cheating. An EE friend of mine has a whole 10 cm wafer of Schottky diodes from a lab he did once, so it can't be too expensive. I really don't know how to make a Schottky diode, but I think it is just a silicon-metal junction, so I'll consider a normal pn-junction diode, although a Schottky diode maybe much simpler.

Having a n+ substrate, one would need to p dope the surface with Boron in a diffusion furnace. In the garage maybe a pyrex tube in a kiln would do the trick. First of all the parts of the wafer which isn't supposed to be doped have to be masked, maybe a cut piece of glass would do. The wafer is placed inside the tube, which is sealed and has a supply of inert gas and Boron connected to it. The kiln should be set to around 1000 °C and it should probably be left on for more than 4 hours. I don't think overdoing it would be too bad.

Then , there's the contacts. Vaporizing aluminum in a vacuum chamber and allowing it to condense over the (masked) wafer should make adequate contacts. A good vacuum would be needed, roughing with a mechanical pump and finishing with a sorption pump should do the trick.

Finished. Does it work? Probably not. Still, something that I just have to try out sometime.

Oops, this turned into a general semiconductor processing writeup. Anyway, this is a ASCII drawing of what a finished diode would look like:

        __metal__
_______/_________\________
!    \   p doped   /      !
!     \___________/       !
!                         !
!                         !
!       n+ substrate      !
!_________________________!
!_________metal___________!
Sorry, this was my first attempt at ASCII drawings in HTML.

Back in the 1950s, hobbyists made crude germanium transistors.  Below are some suggestions...

 

 

 

It is possible to make a transistor using Galena (lead sulfide, PbS). Galena is often available from rock shops and science museum stores. You can even make your own by melting sulfur and lead powder over a flame. Look up keywords such as "cat's whisker diode" and "crystal radio" to find out more.

The trick to making a transistor is to use a hyper-clean, freshly-cleaved crystal face, to sharpen your cat's-whisker contacts by dissolving the tips using electrolysis, and then to put the tips within 0.05mm of each other (or preferably within 0.01mm). Obviously the latter is the hardest part. Better use a microscope! The authors of the following article found that the base/emitter junction was critical: it HAD to act as a good rectifier. The base/collector junction wasn't as important. They got some power gain, but their beta was in the single digits. Others have mentioned that if you break open a 1N34 glass diode to expose the Germanium chip, you can make a crude transistor with a similar procedure. Old Germanium audio power transistors probably do the same, while giving much larger semiconductor area on which to play.

Crystal Triode Action in Lead Sulphide, P. C. Banbury, H.A. Gebbie, C. A. Hogarth, pp78-86. SEMI-CONDUCTING MATERIALS, Conference proceedings, H.K. Henisch (ed), 1951 Butterworth's scientific publications LTD 1951.

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