There is a type of Diode called a Schottky diode, which is basically a metal-semiconductor junction, so it begs the question, how do you form a metal contact with any semiconductor device, not just a diodes.

The answer lies in why a metal-semi junction exhibits diode behaviour in some circumstances. When you have a discrete junction between the metal and semiconductor regions, in simplistic terms it causes bending of the band structure. The energy bands in the semiconductor curve to match those of the metal at the junction. Depending on how the bands bend will determine whether and an inbuilt energy barrier forms (a diode).

If the metal has a higher work function than an n-type semiconductor, the bands of the semiconductor bend upwards to meet it. This causes the lower edge of the conduction band to rise up causing a potential barrier (diode) which must be overcome in order for electrons to flow from the conduction band of the semiconductor into the metal. 

Conversely if the metal has a lower work function than the n-type semiconductor, the bands of the semiconductor bend down to meet it. This results in no barrier because electrons don't need to gain energy to get into the metal.

For a p-type semiconductor, the opposite is true. The metal must have a higher work function that the semiconductor because in a p-type material the majority carriers are holes in the valence band, so electrons need to flow from the metal out into the semiconductor.

As an alternative, it is possible to form an Ohmic contact by making the barrier larger - sounds odd, but its true. When you make the barrier larger, it gets thinner. When the barrier is thin enough, quantum effects take over. The electrons can basically tunnel through the barrier and junction loses its diode behaviour.

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Once you have an Ohmic connection with the semiconductor, you can simply deposit a metal bond pad onto the connection point, and then wire bond those to the diodes metal pads (SMD) or legs (through-hole).