How do diodes work in an AM band application?
That's not a very good crystal radio.  Try as long a wire as you can find and the strongest AM station in your area.  You might be able to get more than the 300 mV it takes to turn on the diode.  By the way, nowadays you use a Schottky diode instead of a germanium one.  They have about the same forward drop, but are a lot easier to find.

Better crystal radios have two coils that form a transformer.  The primary is the smaller coil, which is driven from antenna to ground.  The larger coil is the secondary, which is used as you show, except that the antenna is not directly connected to it.

This does two useful things.  First, it increases the output voltage.  This get past the diode drop more easily, and causes less distortion because the diode drop is a smaller fraction of the output signal.  Second, it raises the impedance of the output signal.  This is a better match to the type of headphones that a crystal radio is meant to be used with.

I go into more detail on crystal radios here: https://electronics.stackexchange.com/a/150307/4512.  Search for "crystal radio" here and you will find a number of posts with more information.