# Current and conductance from the Landauer formula

The Landauer formula for a one dimensional quantum system (potential step scattering) can be written as

$$I(V)=\frac{2e}{h}\int_{-\infty}^\infty dE T(E) (f_S(E) - f_D(E)),$$

where $$T(E)$$ is the transmission probability and $$f_i(E)$$ is the Fermi function of source $$S$$ or drain $$D$$. In Cuevas it is claimed that if the temperature is zero (Fermi functions are potential steps) and if low voltages is assumed, the expression reduces to

$$I = GV,$$

where the conductance is given by $$G=(2e^2/h)T$$.

What is the low voltages assumption? In other words, if I assume low voltages, along with zero temperature, what is left to compute in the integral?

In general, the relationship $$I(V)$$ for arbitrary voltages is nonlinear in the voltage difference $$V$$, and the assumption of low voltage allows one to write the linear approximation $$I \approx GV$$ by expanding at first order the difference of the Fermi-Dirac functions.
At zero temperature, the relationship $$I(V)$$ is still nonlinear unless $$T(E)$$ is a constant independent of $$E$$.