Two-dimensional electron gas ($n_s=N/A$)

An electron gas confined in one spatial direction, with motion available in the other two. Its electron number per unit area is $n_s=N/A$. The ideal Fermi gas approximation neglects interactions between electrons; an actual two-dimensional electron gas can have important interactions. Confinement alone does not require a magnetic field or Landau levels.

State counting with total degeneracy $g$ gives a constant two-dimensional density of states $gm/(2\pi {\hslash }^2)$ above a parabolic conduction band edge. Integrating the Boltzmann occupation yields the displayed areal carrier density. For one valley and two spin states the coefficient is $m{k}_{B}T/(\pi {\hslash }^2)$. Applying the same count to holes gives $np=N_c{N}_v{e}^{-E_g/(k_{B}T)}$ and the intrinsic density $n_i=\sqrt{N_c{N}_v}{e}^{-E_g/(2k_{B}T)}\propto T{e}^{-E_g/(2k_{B}T)}$, distinct from the three-dimensional $T^{3/2}$ prefactor.

A gate separated by dielectric thickness $d$ has geometric capacitance per area $C/A=ϵ/d$ in the parallel-plate approximation. An electron-density change is $\Delta n_s=(ϵ/ed)\Delta V_g$ with the stated polarity/reference convention. Fringing, trapped charge, depletion and finite quantum capacitance can modify the relation. An absolute gate voltage additionally requires the zero-density offset or another calibration.

A magnetic field curves ballistic carrier trajectories between an injector and a collector. For openings separated along a specular edge by $L$, peaks occur near $L=2j{r}_c$ with $r_c=\hslash k_F/(e|B|)$, giving $|B_j|=2j\hslash k_F/(eL)$. Field polarity and carrier sign select the connection direction. Electron phase coherence can produce fine structure but is not required for the basic semiclassical focusing geometry.

In a parabolic band, transport mobility and mean free path obey $\ell =\mu \hslash k_F/e$. The rough condition $\ell \gtrsim L$ gives $\mu \gtrsim eL/(\hslash k_F)$, whereas surviving a first semicircle uses path length $\pi L/2$. Clear peaks usually need a larger margin and a specular boundary, so this is an observability scale rather than a universal sharp threshold.

A short electrostatically defined constriction connecting two electron reservoirs. Transverse confinement produces transport modes, and sufficiently narrow contacts can collimate carriers for electron-focusing experiments. Its quantum conductance depends on mode transmission; an opening described only as narrower than a micrometre is not automatically in a single-mode limit.

A parallel field lifts spin degeneracy and separates threshold energies. The step size may change from $2e^2/h$ to $e^2/h$ when the two spin branches are resolved. Finite layer thickness can additionally alter orbital confinement.

At zero temperature, a rigid spin-degenerate transmission spectrum with symmetric source/drain bias has ${\mu }_{S,D}=E_F\pm eV/2$ and the displayed differential conductance. Half-integer plateaus in units $2e^2/h$ occur when the two bias-window edges sample different channel counts. The bias convention and electrostatic response matter.

In a rigid spectrum, source and drain threshold lines in a bias-versus-gate map satisfy $E_n(V_g)={\mu }_S$ or ${\mu }_D$. Where adjacent thresholds align with opposite edges of the bias window, their spacing equals the window width $e|V|$. For symmetric bias and spin degeneracy, neighboring channel counts at those edges give a differential electrical conductance halfway between integer $2e^2/h$ plateaus. The dc ratio $I/V$ is a different observable; barrier motion and heating can modify the interpretation.

A channel supporting ballistic electron transport carries incident populations set by different source and drain chemical potentials, rather than an everywhere-equilibrated local distribution with an Ohmic voltage gradient. Its electrostatic barrier responds self-consistently to contacts, gates and electrostatic screening. A symmetric rigid-barrier model places reservoir energies at $E_F\pm eV/2$ relative to the barrier. Asymmetric partition or bias-dependent transmission changes the weights in the differential electrical conductance, and must be included when extracting an energy scale.

For rigid spin-degenerate channel transmissions and ${\mu }_{S,D}=E_F\pm eV/2$, differentiating $I=(2e/h){\sum }_n{\int }_{{\mu }_D}^{{\mu }_S}{T}_n(E)dE$ gives $G_d=(e^2/h){\sum }_n[T_n({\mu }_S)+T_n({\mu }_D)]$. If the two edges sample neighboring integer channel counts, $G_d$ is a half-integer multiple of $2e^2/h$. Explicit bias dependence of the potential adds terms; the apparent fraction does not imply fractional carriers.

A locally separable potential with an unstable longitudinal maximum and stable transverse confinement. Mode $n$ sees barrier top $E_n=U_0+\hslash {\omega }_y(n+1/2)$ and transmission $T_n(E)=[1+\exp (-2\pi (E-E_n)/(\hslash {\omega }_x)){]}^{-1}$. It is a single-particle smooth-barrier model; interactions and bias-induced reshaping require additions.

Slow widening of a constriction approximately conserves the transverse mode action, reducing transverse momentum as the width $W$ increases. A lowering band bottom also increases longitudinal kinetic energy. Together they can narrow the angular distribution of an emitted electron beam. An abrupt narrow aperture alone can instead diffract strongly.

An injector and collector aligned across an open two-dimensional electron gas. A direction-sensitive signal between them probes ballistic electron transport. Contact resistance and diffuse current backgrounds must be separated from the direct beam contribution.

Two surface electrodes whose depletion regions define a narrow conducting opening in an underlying two-dimensional electron gas. Gate voltage changes the transverse confinement and channel transmission without physically etching the conducting channel to its electrical width.