Landauer formula

A transport expression relating current to occupied incident modes and their transmission probabilities. For spin-degenerate independent electrons, $I=(2e/h){\sum }_i\int T_i(E)[f_L(E)-f_R(E)]dE$, with the direction convention chosen consistently. In linear response a perfectly transmitted mode contributes $2e^2/h$.

An element $t_{nm}$ is the amplitude for an incident unit-flux channel $m$ to emerge in outgoing unit-flux channel $n$. Then $|t_{nm}{|}^2$ is a transmission probability and $\mathrm{tr}(t^{†}t)$ sums transmitted channels. Unnormalized wave amplitudes require velocity ratios.

A perfectly transmitted spin-resolved channel contributes $e^2/h$ to two-terminal linear conductance. Two degenerate spin channels give $2e^2/h$. Flux normalization cancels group velocity against the longitudinal state density.

For perfect transmission above threshold $E_n$, integrating $-df/dE$ gives $G_n=(e^2/h)f(E_n)$. Thermal occupation rounds a zero-temperature conductance step over an energy range of order $k_{B}T$.

For one directed spin-resolved channel, the density of longitudinal states per unit length and wave number is $1/(2\pi )$. Since $v={\hslash }^{-1}dE/dk$, velocity times density per energy is $1/h$. A reservoir occupation imbalance therefore contributes particle flux $\int \delta f(E)dE/h$; in zero-temperature linear response it is $\delta \mu /h$. Multiplying by charge and transmission gives the Landauer current per mode.

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.

Stepwise conductance as transverse propagating modes open in a clean constriction. Each fully transmitted spin-degenerate subband contributes $2e^2/h$; finite temperature, imperfect transmission and unresolved modes round the steps.