Landauer-Büttiker formula ($I_i=(e^2/h)[N_i{V}_i-{\sum }_j{T}_{ij}{V}_j]$)

A multi-reservoir quantum transport relation. Here $I_i$ is conventional current injected from reservoir $i$, $N_i$ counts outgoing spin-resolved modes and $T_{ij}$ counts transmission from reservoir $j$ into $i$. Ideal voltage probes draw zero net current and equilibrate incident carriers. On a chiral integer Hall edge with $N$ modes, $I_i=(N{e}^2/h)(V_i-V_{i-1})$ for the stated circulation order.

Let $T_{ij}$ count transmitted modes from injector $j$ to collector $i$, excluding reflection $R_{ii}$. Scattering conservation gives ${\sum }_{i\ne j}{T}_{ij}=N_j-R_{jj}$; unitarity also gives the corresponding row balance. The conductance matrix has zero row sum, making equal lead potentials carry no current, and zero column sum, conserving total current.

An ideal voltage probe is a reservoir that equilibrates incident carriers but draws zero net average current. Its chemical potential is fixed by that zero-current condition, and can differ from a nearby geometrical point's inferred diffusive potential in a ballistic device. Reading the probe therefore requires the transmission network, not just a local Ohm law.