For example, Can a silicon diode be designed to have a forward bias voltage drop of 0.4V or may be 0.9V?
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Without going into solid state physics more than I am competent to : no. You might be able to shift the bias by a few millivolts this way by changing doping, but you can do that more easily by changing the temperature or the bias current. Read up about the Shockley equation - you can use a diode to generate logarithms that way...
To get a bigger change you need to move to a different semiconductor material (germanium for about 0.2-0.3V, silicon carbide for higher voltages (dim memory says about a volt...) anyone know the numbers for 3-5 compounds like Ga-As, Ga-InP etc?
Finally, Schottky diodes involve a metal-semiconductor boundary and have quite a low bias voltage (about 0.3V)
If a gold dope injected in the PN junction , the minority carriers will gain speed and at the expense of the higher potential (the majority of investors) . ( pun intended) eg. 1N914 is gold doped.
E.g. undoped silicon is lower as in PIN diodes but doping more does not increase beyond standard silicon voltages. More examples
Composition appears to be a trade secret and for those who need to know.
"It consists of an active part of each device and is made of single crystal silicon. It contains deminimus amounts, usually in parts per billion (ppb) levels, of doped elements such as arsenic, phosphorus and boron. "