Super cool post Valentin!!!!!

I think something is wrong for the second python code, where you have the applied voltage.

I don't think that the way you applied the voltage will not change the lower limit of the band, it just moves it in the k-space.

The bottom of your band is -12eV, so even for E(k) = 11.5 eV, you get that some states are filled in the case with no applied voltage.

I think that for the case of an applied voltage of 2eV over the x-direction, the bottom of the band is  just moved a bit to the right, so you should have the same amount of filled states, as long as your E(k) < E_f.

But an applied voltage should actually lower (or rise) the bottom of the band so you should use a different b-value for the kx-section

\( E(k) = E_0 - scale \cdot b \cdot cos(a k_x) -  b \cdot cos(a k_y) \)
 where scale is the effect of the applied voltage.