In a PN junction, under no bias, when the depletion region forms, an electric current is also formed, in the direction, from N side to P side.
It would be more correct to say that under no bias conditions, there are TWO currents formed, a drift current and a diffusion current, and they exactly cancel out. Or it would be correct to say that under no bias conditions there is NO net current.
This electric field, also has a potential difference, obviously, which is known as the barrier potential.
By "this electric field" I think you obviously mean the electric field that forms spontaneously when N and P type semiconductors form a junction, i.e. the "built-in" electric field. You refer to this as "the barrier potential", and I think it is important to stop here. I think the term "barrier" is perhaps misleading. While a PN junction can be, in some cases, a "barrier" to net current flow, it is, in no sense a "barrier" to the movement of individual carriers. It is not as though an electron, traveling somewhere near 100 km / sec comes up to the depletion region and sees it as an insurmountable barrier and suddenly turns around. Rather, individual carriers can and do cross the depletion region regularly, but if they cross with equal frequency in both directions, there is no net current.
This barrier potential is required to be overcome by the majority carriers on both sides in order to cross the depletion region in order to diffuse,
No. The diffusion current density depends only on the gradient of the carrier density. Carriers do not need to "overcome" some "barrier" in order to diffuse. The diffusion current is affected by the electric field only in the following way. The electric field controls the width of the depletion region. The wider the depletion region, the lower the slope of the density function, i.e. the lower the density gradient. Hence the lower the diffusion current. Conversely, the more narrow the depletion region, the steeper the carrier density function, the higher the gradient, and the higher the diffusion current.
provides a force opposite to the diffusion of the majority charge carriers.
The electric field does provide a "force", and that "force" is responsible for the drift current that flows in the opposite direction of the diffusion current. But the "force" doesn't stop the diffusion. It creates a counter-current, making the net current zero in the no-bias case. The electric potential difference across the depletion region affects the width of the depletion region, which affects the carrier density gradient, which affects the diffusion current.
When the electric field is induced, due to the uncovered ions on the both sides, it resists it's diffusion to the other side.
"it resists it's diffusion"? I think you are trying to say that the electric field "resists" the diffusion of carriers. But, as I am trying to explain, the individual carriers still diffuse. They diffuse at a lower rate the wider the depletion region.
Now, about the minority carriers that are formed in the P and N region due to thermal energy, if they manage to reach to the edge of the Depletion region[,] will they not be repelling the holes back into the N region? Why do the holes still drift to the N region?
Carriers will still diffuse across the junction, AND they will still drift back. It isn't an either or.
Also, is barrier potential just another name for junction built-in voltage?
I think they are generally used interchangeably, but I'm not an expert in the precise use of physics language. A physicist might draw a distinction that I am unaware of.
So from what you're saying is: The magnitude of the diffusion current works to manage the width of the depletion region which in turn affects the magnitude of the electric field induced in that region. And that induced electric field regulates the magnitude of the drift current which cancels out the diffusion current in equilibrium. (from comment in chat)
I think the actual physics goes like this.
The applied voltage modifies the width of the depletion region. The wider the depletion region, the smaller the slope in the carrier concentration curves, and therefore the smaller the diffusion current.
Also, the wider the depletion region, the greater the distance for the voltage drop. The electric field in the depletion region depends on both the voltage drop across the depletion region, and the width of the depletion region. In negative bias, and also forward bias with low applied voltage, the change in width and change in voltage across the drop work in tandem to keep the electric field, and hence the drift current, fairly constant.