Quantum transport

Transport of particles or charge when quantum mode structure, transmission and occupation of reservoir states determine the measured current.

A coherent disordered mesoscopic conductor has reproducible, sample-dependent conductance variations when magnetic field, carrier energy or disorder configuration changes. Their typical scale is of order $e^2/h$ in a fully coherent diffusive sample, with numerical factors determined by symmetry, dimensionality and geometry. Thermal averaging and finite phase coherence length reduce the measured amplitude.

The characteristic distance over which phase-breaking processes suppress electron-path interference. For diffusion, $L_{\varphi }\sim \sqrt{D{\tau }_{\varphi }}$; ballistic phase propagation instead gives a scale $v{\tau }_{\varphi }$. This differs from the elastic mean free path, since elastic scattering can change momentum while retaining coherence.

A narrow conductor has discrete transverse modes and nearly continuous longitudinal motion. Its conductance depends on the transmissions of occupied quantum wire subbands, spin counting and contacts. The Landauer formula connects these transmissions to two-terminal conductance. Disorder, interference, dephasing and interactions distinguish ballistic, diffusive, localized and interacting regimes.

A transverse confinement eigenstate that defines a threshold for propagation along a narrow conductor. Above that threshold, its longitudinal dispersion supplies an incident channel from each reservoir. spin can label separate channels; an unresolved spin-degenerate mode contributes twice the single-spin spin-resolved conductance quantum. This physical use of mode is unrelated to the most probable value of a probability distribution.

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.