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In quasi-one-dimensional quantum transport, transverse confinement makes discrete subbands while motion along a quantum wire remains continuous. Only occupied propagating subbands conduct. For elastic scattering between equilibrated reservoirs, the Landauer formula isThe sum explicitly counts spin. A perfectly transmitting spin-degenerate subband gives $2e^2/h$ at low temperature; a quantum point contact shows steps as a gate opens additional subbands. The one-dimensional density of states cancels the carrier velocity in the current, giving a conductance quantum rather than a material-dependent velocity factor. Finite two-terminal contact resistance remains even without internal scattering.

Transport is ballistic transport for length $L\ll \ell$, where $\ell$ is the mean free path. In a multichannel wire with $\ell \ll L$ below its localization length, scattering reduces transmission and gives approximately Ohmic behaviour. If the phase coherence length exceeds the sample length, amplitudes interfere; weak localization and sample-dependent universal conductance fluctuations can appear. At greater lengths, generic coherent disordered wires undergo Anderson localization, with localization length of order the channel number times $\ell$, up to symmetry-dependent factors. Thermal averaging and dephasing smear the quantum features.

Strong interactions require more than independent-electron transmission. A clean single-channel interacting wire is described at low energy by a Luttinger liquid, with collective modes and possible spin-charge separation; impurities can generate temperature-dependent power laws. Ideal contacts to noninteracting leads can still fix the two-terminal conductance of a clean wire, so it must not automatically be multiplied by a bulk Luttinger parameter.

A GaAs–AlGaAs semiconductor heterostructure confines electrons normal to the interface in a semiconductor quantum well. Remote modulation doping supplies a high-mobility two-dimensional electron gas. Negative split gates deplete it laterally, leaving a narrow quantum point contact with discrete transverse subbands and continuous longitudinal momentum. A single occupied subband requires transverse excitation gaps larger than thermal and applied-bias energies.

Take $e>0$ and count one spin-resolved channel. In a length $L$, right-moving states have density $Ldk/(2\pi )$; each contributes charge current $-e{v}_k/L$. ThusThe second expression is the right reservoir's left-moving contribution expressed in the same left-to-right conventional-current sign. Since $v_{k}dk=dE/\hslash$,The limit needs $k_{B}T,|eV|\ll \mu -E_0$ and ballistic transport. Velocity cancels the one-dimensional state density. Two degenerate spin channels give $2e^2/h$, not an extra factor hidden in the quoted single-channel result.

For elastic phase-coherent scattering, expand each incident flux-normalized mode in outgoing lead modes. Its transmitted fraction is $|(t_E{)}_{nm}{|}^2$; sum over all open outgoing $n$ and occupied incident $m$. Linearizing the reservoir occupations gives the Landauer formulaThe source's $T_E^{+}$ denotes this amplitude matrix, not temperature or a probability matrix. Flux normalization is essential: ordinary wave-amplitude ratios need a velocity factor. Reservoirs equilibrate, contacts supply incoming Fermi distributions, and inelastic processes within the channel are neglected. Spin is included among channel labels. At $T\to 0$, only transmissions at $\mu$ contribute; perfect channels each add one quantum. This is a two-terminal conductance including contact resistance, even with no internal backscattering.

Trace I has steps near integer $2e^2/h$ as less-negative gate voltage opens successive spin-degenerate subbands. At the printed parallel field $11\mathrm{T}$, Zeeman splitting separates the two spin thresholds, allowing approximately half-sized steps in the plotted units; the onset also shifts. Thermal broadening, imperfect transmission and unresolved thresholds round the plateaux. A parallel field primarily resolves spin in the ideal thin-interface model; finite layer thickness can also shift subband energies. The plot alone does not determine a unique effective $g$ factor or microscopic disorder model.

For a clean, adiabatically connected GaAs channel at low temperature and small bias, opening successive spin-degenerate subbands givesConductance therefore shows a staircase of plateaux, rather than the continuously geometry-dependent classical Ohmic result. Each fully transmitted mode contributes approximately $77.5\mu \mathrm{S}$. More positive, or less negative, depletion-gate voltage opens a conventional GaAs constriction; more negative voltage closes it.

Require subband spacings larger than thermal and bias broadening and a mean free path exceeding the constriction length. The general Landauer formula is $G=(e^2/h){\sum }_{\alpha }{T}_{\alpha }$ with each spin/valley channel counted separately. A material with additional valley degeneracy need not have the same step sequence as GaAs, and imperfect transmission or contact resistance blurs the ideal plateaux.

Use the one-way density per unit length $g_{+}/L=2/(h|v|)$. The electron-flux magnitude in an interval is charge times velocity times line density:If $g$ is instead defined for a finite length $L$, the traversal frequency is $|v|/L$; the same cancellation follows. The printed $I=Nev$ therefore requires $N$ to be a line density, not an unscaled total particle number.

Subtract the opposite reservoir's incoming flux. With the stated electron electrochemical potentials, conventional positive current is opposite to electron-number flux, soAt low temperature and small bias, each fully transmitted occupied spin-degenerate subband contributes $2e^2/h$. Hence $\boxed{G=N_{b}2e^2/h}$. More generally, the Landauer formula weights each mode by its transmission probability; imperfect transmission removes exact integer plateaux.

A quantum point contact is made by using negative surface split gates to leave a short narrow channel in a clean two-dimensional electron gas. Its length should be smaller than the transport mean free path; transverse confinement should be strong enough that subband spacing exceeds thermal and lifetime broadening. Low-resistance Ohmic contacts connect large reservoirs. Apply a small source–drain bias, measure current and differential conductance, and sweep the gate to change the channel width or barrier height. Contact and wiring series resistance must be accounted for when comparing plateau values.

The Landauer formula explains the conductance without treating the short channel as a diffusive wire. For each propagating one-dimensional mode, the density of states and group velocity cancel in the incident flux, leaving $e/h$ per spin per energy interval. The reservoir occupation difference supplies the net current:The factor two assumes unresolved spin degeneracy. In zero-temperature linear response,A well-transmitted channel has $T_n\simeq 1$, giving steps $G=N(2e^2/h)$ as modes open. A finite two-terminal resistance persists even for a perfectly ballistic channel because reservoirs inject only a finite number of modes; it is not evidence of resistive scattering inside the channel. Gate-controlled point-contact conductance quantization was demonstrated in a two-dimensional electron gas.^[ref]

The saddle-point model of a quantum point contact resolves how modes open. Near the bottleneck, choose the channel direction as $x$ and writeTransverse harmonic confinement gives barrier tops $E_n=U_0+\hslash {\omega }_y(n+1/2)$. Scattering through the inverted longitudinal harmonic barrier givesSmall longitudinal curvature makes a sharp transmission threshold compared with the transverse subband spacing; finite curvature rounds each conductance step. This is the explicit smooth-barrier model developed for a saddle constriction.^[ref]

At larger source–drain bias, the current samples a finite energy window. For a rigid transmission spectrum at zero temperature,With symmetric bias, ${\mu }_{S,D}=E_F\pm eV/2$, the nonlinear conductance of a quantum point contact isIf both window edges sample $N$ open modes, the usual plateau remains. If one edge samples $N+1$ and the other $N$, differential conductance approaches $(N+1/2)(2e^2/h)$: a half-integer plateau relative to the zero-bias step. It is a bias-window effect, not half an electron spin degeneracy. Ridges in gate–bias transconductance occur when a reservoir chemical potential crosses a subband threshold; their separations can measure subband energies.

The half-plateau expression assumes symmetric bias and an approximately bias-independent barrier. An asymmetric voltage drop or self-consistent shift of $U_0$ changes the numerical relation. At still larger biases, heating and inelastic scattering can invalidate a rigid elastic saddle model. Thus the zero-bias quantization counts transmitted modes, while nonlinear spectroscopy compares mode counts at the two ends of the bias window.

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.