What is the meaning of "current" in IV curves of periodic structures?

[qweasel]

What is the meaning of current in periodic structures that are infinite in A and B directions?
Why is it "current" and not "current density"?

[Daniele Stradi]

1. The current is actually the electric current evaluated using the Landauer-Buttiker formalism.
2. The current density is the current evaluated in 1. divided by the cross section of the device system. 

[qweasel]

What I don't understand is why "current" is correct terminology if the device is infinite in both directions orthogonal to transport. How can you possibly calculate an integral quantity across an infinite cross section?

[Petr Khomyakov]

The current through a device, which has infinite, periodic atomic structure in the directions perpendicular to the transport direction,  is calculated through the lateral unit cell of this periodic structure. Even so the total current is infinite, it is however finite per lateral unit cell. The current density is then defined as the current per lateral unit cell divided by the lateral unit  cell area, and that is equivalent to the definition given in #2 in the post by Daniele.

In other words, if we calculate the current density as the total current through N-lateral unit cells divided by the total cross section of these N lateral unit cells when N->infinity, the obtained current density is finite, while the device cross section is infinite indeed.

[qweasel]

So if in the lateral directions I only have one unit cell, the current should be identical to current density?

[Petr Khomyakov]

They are not and cannot be identical, even just because they are given in different units; otherwise, apples and oranges would be identical as well. The current density is the current (e.g., in units of Ampere) through lateral unit cell area divided by the lateral unit cell area (e.g., in units of m^2), meaning that the current density is in units of Ampere/m^2.