Ohmic contacts conduct the same for both polarities. (They obey Ohm's Law).
There are two ways to make a metal-semiconductor contact look ohmic enough to get signals into and out of a semiconductor (or doing the opposite makes a good Schottky contact).
Lower the barrier height
The barrier height is a property of the materials used, ie. materials whose barrier height is small.
Annealing can create an alloy between the semiconductor and the metal at the junction, which can also lower the barrier height.
Make the barrier very narrow
One very interesting property of very tiny particles like electrons and holes is that they can "tunnel" through barriers that they don't have enough energy to just pass over. The probability of tunneling becomes high for extremely thin barriers (in the tens of nanometers).
The barrier is made very narrow by doping it very heavily (1019 dopant atoms/cm3 or more).
Later, Gil.