Why the current across a diode is zero when the forward bias voltage is less than barrier potential?
Here's what the 1N4148 signal diode characteristic looks like in the foothills: -



As you can see, below ~0.8 volts it's a pretty regular shaped characteristic. The current is approximately falling by a factor of ten for each 100 mV reduction in base-emitter voltage.

On a linear graph it looks like there is a real definable knee voltage: -



But, that linear graph hides the reality/subtlety of things but who cares - most diode circuits are nicely approximated by saying current is zero below 0.6V (or 0.7V) and the diode behaves like a short at this voltage.

There is an equation called the Shockley diode equation that adequately describes current versus voltage based on temperature and reverse leakage - often a sim will use this type of equation but rough and ready calculations on a scrap of paper don't need to.