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A sufficiently clean two-dimensional electron gas at low temperature in a perpendicular magnetic field has quantized orbital Landau levels. In the integer quantum Hall regime, the longitudinal resistivity is nearly zero while the Hall resistance has plateauxThe Hall sign depends on carrier charge and lead convention. Fractional plateaux can occur in strongly interacting states, but the following single-particle counting and calibration use the integer effect.

One resolved spin branch of a Landau level has one state per flux quantum $h/e$: in area $A$, its guiding-centre degeneracy is $Ae{B}_z/h$ for positive $B_z$. Defining the nominal filling factor as the number of electrons per available orbital state givesThe same count follows in Landau gauge: a $2\pi /L_y$ increment in $k_y$ moves the orbit centre by $2\pi {\ell }_B^2/L_y$, giving $A/(2\pi {\ell }_B^2)=Ae{B}_z/h$ states. Include each spin branch separately; an unresolved spin-degenerate orbital level contains twice this count, and common plateau indices are then even.

Plateau qualification: $\nu$ as defined by density is continuous when $B_z$ changes at fixed $N_S$. The integer plateau index $p$ can remain unchanged over a field interval because localized states accommodate the changing occupation. Therefore $N_S=pe{B}_z/h$ is not an exact identity across an entire plateau. It is the ideal filled-level relation, often used approximately at a suitable plateau centre; confusing the nominal filling with the plateau index would contradict fixed density.

The plateau resistance realizes the von Klitzing constant $R_K=h/e^2\simeq 25812.8\Omega$, divided by a known integer. A four-terminal Hall measurement with negligible longitudinal voltage and contacts in the quantized regime supplies a reproducible resistance reference, which can be compared to other resistors using precision bridges. Its robustness comes from the quantized transport state, not from measuring the Hall bar's width accurately.

As magnetic field grows from zero, a clean two-dimensional electron gas first has an ordinary finite longitudinal resistivity, then develops Shubnikov–de Haas oscillations as broadened Landau levels pass the chemical potential. Thermal and disorder broadening obscure the low-field oscillations; their visibility grows when the level spacing exceeds those broadening scales. In a well-developed integer quantum Hall regime, ${\rho }_{xx}$ has minima close to zero on Hall plateaux and maxima during transitions between them, when extended states cross the chemical potential. Detailed peak heights and envelopes depend on disorder and scattering and are not fixed by quantization alone.

At fixed density the nominal filling is $\nu (B)=N_{S}h/(eB)$, so level-filling features are approximately periodic in $1/B$, with period $e/(N_{S}h)$ for resolved spin levels or $2e/(N_{S}h)$ for an unresolved spin pair. Increasing $B$ lowers the filling; spin splitting can create additional odd-index features. The original schematic below shows these relations without pretending to reconstruct the missing data.

TODO — missing source graph: the actual $0$–$12\mathrm{T}$ positions and integer plateau labels require $N_S$ or surviving plateau fields from part (b). Neither is provided independently. Thus a uniquely labelled field-axis sketch for this sample cannot be completed. The following DOS diagram distinguishes extended centres and localized tails generically; its chemical-potential markers are illustrative and do not stand for the requested 8 T or 10 T assignments.

A common implementation uses a gallium arsenide/aluminium gallium arsenide heterostructure with a two-dimensional electron gas near the interface. electrons supplied by remote donors occupy quantized normal-direction states, while an undoped spacer reduces ionized-impurity scattering. Surface metallic gates, insulated from the conducting layer, deplete selected regions when negatively biased.

Leave a small conducting island bounded by depleted regions and connect it weakly to source and drain reservoirs through two tunable constrictions. A plunger gate controls the island potential/population; barrier gates control tunnel coupling. Ohmic contacts connect the reservoirs to the measurement circuit. Vertical heterostructure confinement plus lateral depletion force defines the dot. A 2DEG without lateral gates would remain extended in its plane.

The original device diagram labels the gates, island, tunnel barriers and reservoirs. Charge stability and spectroscopy versus gate/bias should verify that an isolated weakly coupled island has actually formed; a lithographic outline alone is insufficient.

A larger-gap AlGaAs barrier has a higher conduction-band edge than GaAs. Donors in AlGaAs supply electrons that transfer into lower-energy GaAs states, leaving a positively charged donor region. The resulting electrostatic field bends the GaAs conduction band into a triangular interface well. Motion normal to the interface is quantized, while motion parallel to it remains approximately free, producing a two-dimensional electron gas.

The confined envelope peaks just inside GaAs and decays into the AlGaAs barrier; a triangular-well treatment gives an Airy-function-like envelope with boundary matching. An undoped spacer separates the electrons from the donors, strongly reducing ionized-impurity scattering compared with uniformly doped 3D material and enabling high mobility. Band offsets, donor supply, interface roughness and electrostatics jointly determine whether a single occupied subband is obtained.

A gallium arsenide/aluminium gallium arsenide semiconductor heterostructure combines lower-gap GaAs with a higher-gap AlGaAs barrier. In a modulation-doped design, AlGaAs donors supply electrons to lower conduction-band states on the GaAs side. An undoped spacer separates the channel from ionized donors and reduces impurity scattering. The interface barrier and band bending quantize motion normal to the interface while allowing in-plane motion: a two-dimensional electron gas.

The normal confining potential is commonly triangular and finite rather than an infinite square well. At suitable density and low temperature, only the lowest normal subband is occupied. modulation doping reduces an important scattering source; it does not eliminate all disorder.

Use negatively biased surface gates to deplete the surrounding electron gas and leave a rectangular central region. Gate voltage controls lateral dimensions and barrier heights. Two constrictions can weakly couple the region to reservoirs, with a plunger gate controlling population. Etched boundaries are possible but may introduce edge disorder. A flat interior and steep edges approximate a box, givingFringing fields, finite barriers and penetration change the ideal result. A 2DEG confines only the normal direction; gates supply the extra lateral confinement. Gate-dependent spectroscopy can test level spacings, while charging measurements separate orbital confinement from Coulomb blockade.

For a parabolic two-dimensional electron gas with one valley and two spin states, the density of states per area is $g_{2D}=m^{\ast }/(\pi {\hslash }^2)$. Integrating from the subband edge to the zero-temperature Fermi energy gives $n_s=g_{2D}{E}_F$, henceHere $m^{\ast }=0.067(9.109\times 10^{-31})=6.103\times 10^{-32}\mathrm{kg}$, and the energy is measured from the occupied subband bottom. Equivalently $k_F=\sqrt{2\pi n_s}=1.3722\times 10^8{\mathrm{m}}^{-1}$. ThereforeThe spin factor is already included in $g_{2D}$; counting it again would halve the energy incorrectly.

A two-dimensional electron gas is confined into discrete subbands perpendicular to an interface while retaining nearly free motion in its plane. It is not a literal zero-thickness electron sheet. In a GaAs/AlGaAs semiconductor heterostructure, the band offset and self-consistent band bending confine electrons on the GaAs side. Donors in AlGaAs can supply electrons while an undoped spacer separates them from the ionized dopants, reducing impurity scattering. This modulation-doping arrangement yields a high-mobility channel, with the lowest confinement subband occupied when the confinement spacing exceeds the Fermi and thermal scales. Gates tune the density or locally deplete the channel into constrictions. Low-temperature Hall and longitudinal measurements test the resulting sheet density and transport quality.

A gallium arsenide/aluminium gallium arsenide heterojunction has a conduction-band offset that can confine electrons on the GaAs side. Donors in the wider-gap AlGaAs supply electrons; band bending produces a narrow interfacial potential well. Motion normal to the interface is quantized into subbands, while motion within the plane remains approximately free with GaAs effective mass. Occupying the lowest confinement subband creates a two-dimensional electron gas.

Modulation doping spatially separates the ionized donors from the mobile electrons. An undoped spacer between the donor layer and the channel reduces Coulomb scattering from donor ions while still allowing charge transfer. This can produce higher mobility than doping the conducting channel itself. The spacer thickness balances lower scattering against weaker carrier supply and electrostatic control; interface roughness, alloy disorder, phonons and background impurities can still limit mobility. The channel is a finite-width quantum wavefunction, not a mathematically zero-thickness sheet.

The integer quantum Hall effect requires a conducting two-dimensional electron gas with well-resolved Landau levels. In the simple parabolic-band model their orbital spacing isLow temperature and sufficiently low disorder broadening are needed so that $k_{B}T$ and the relevant level width $\Gamma$ are small compared with this spacing. A high-mobility sample with ${\omega }_c\tau \gg 1$ is therefore useful.

Some disorder has a constructive role: localized states between extended Landau-level states allow the Fermi energy to move through a range of carrier populations while the transport response stays on a plateau. The longitudinal response is then very small and the transverse response is quantized. Whether odd or only even integer plateaux are resolved also depends on whether spin splitting is resolved.

Take the two-dimensional electron gas in the $xy$ plane, the Hall bar along $x$, and $\mathbf{B}=B\hat{\mathbf{z}}$ with $B>0$ perpendicular to it. An electron has charge $-e$. The confining electron potential energy $U(y)$ is approximately flat inside a wide strip and rises at either boundary; smooth quenched disorder adds hills and valleys. For a slowly varying edge, each spin-resolved Landau level approximately followsHere $s=\pm 1$, and $Y$ is the guiding centre. For $\mathbf{A}=(-By,0,0)$, $Y={\ell }_B^2{k}_x$ and the edge group velocity isThus electrons move along $+x$ at the upper edge and $-x$ at the lower edge. The Lorentz force gives counterclockwise bulk cyclotron motion in this convention. At a hard boundary its interrupted circles become directed skipping orbits; a smooth boundary gives the corresponding guiding-centre drift. Smooth bulk quenched disorder generally gives closed drift contours around localized hills or valleys, with extended trajectories appearing near a broadened level's critical energy. Realistic electrostatic screening can produce alternating compressible quantum Hall edge strips and incompressible quantum Hall edge strips, so the unscreened bending curves are an effective sketch rather than an exact self-consistent edge profile.

In the original diagram, contacts are ordered around the electron propagation direction: upper-left $1\to 2\to 3$, then lower-right $4\to 5\to 6$. Let $I_1=I$, $I_4=-I$ and the other leads be ideal equilibrating voltage probes. Define $R_H=(V_2-V_6)/I$ and $R_{xx}=(V_2-V_3)/I$. These choices fix the voltage sign; reversing the field reverses chirality and the Hall polarity. For spin-resolved plateau index $N$,At fixed sheet carrier density and increasing field, $N$ decreases, giving successively higher Hall plateaus; longitudinal peaks occur between them. Unresolved spin branches can suppress odd plateaus. The sketch uses resistance magnitudes, and every integer step is not promised in an arbitrary sample.

For Fermi-energy oscillations in a two-dimensional electron gas, first distinguish fixed density from fixed reservoir chemical potential. At negligible temperature the former is determined by state counting. Each resolved Landau level contains $eB/h$ states per unit area, soIgnoring spin and broadening for the counting illustration, if $N<\nu <N+1$ then $N$ levels are full and level $N$ is partly full: $\mu =E_N$. At fixed $n_s$, increasing $B$ increases both level energy and degeneracy. The partly filled level rises in energy until the lower levels can contain all electrons; $\mu$ then drops to the next lower level. The resulting sawtooth is periodic in $1/B$, with $\Delta (1/B)=e/(n_{s}h)$ per resolved branch. At an exactly integer filling in an ideal gap, any chemical potential within the gap gives the same zero-temperature particle number; the plotted discontinuity denotes its limiting values.

At fixed $B$ while $n_s$ increases, the ideal chemical potential stays at a partly occupied level and steps upward after each additional density $eB/h$. It therefore has a staircase superposed on the rising zero-field trend, not a repeated downward sawtooth. Landau-level broadening and finite temperature smooth both ideal graphs. With an externally pinned chemical potential, density changes instead; the fixed-density sawtooth cannot simultaneously be imposed. In a realistic disordered sample, the Fermi energy sweeps localized states between transport-carrying level centres. The following schematic omits spin for clarity and does not attempt an interaction-corrected density of states.

For $N$ transmitted spin-resolved quantum Hall edge states, the Landauer-Büttiker formula reduces, in that contact order, toZero probe currents imply $V_2=V_3=V_1$ and $V_5=V_6=V_4$. Hence $I=(N{e}^2/h)(V_1-V_4)$ and the boxed resistance results follow. The spin-resolved conductance quantum arises because the one-dimensional density of states times velocity is $1/h$ per unit energy; a reservoir imbalance supplies $e^2\Delta V/h$ per channel. Its value does not depend on edge slope or sample width.

On a plateau, opposite-moving edge channels are separated by a wide bulk with a mobility gap. Ordinary elastic scattering on one edge has no counterpropagating channel into which to reflect; closed bulk orbits do not connect the contacts. Changing $B$ or $n_s$ can alter the occupancy of localized states without changing the number of transmitting edge channels. This accounts for the finite plateau width, which an entirely clean set of sharp levels would not explain. Accuracy requires negligible backscattering between opposite edges, low enough current to avoid breakdown, equilibrating contacts and negligible probe loading. The finite-width plateau and the precision of its quantized value are related but distinct observations.

An antidot is a depleted potential hill, not a potential well. Its closed edge circulates oppositely to the outer boundary. In a constriction, tunnelling through an antidot orbit can transfer an electron from a forward edge into the returning edge: resonant antidot backscattering produces conductance dips. For $N$ otherwise open spin-resolved channels and one active resonance,At symmetric coupling a single resonance can remove $e^2/h$; asymmetry or thermal averaging reduces its depth. If the background constriction is closed and the resonant route instead supplies forward transmission, peaks occur. The lead connectivity must therefore accompany the conductance sketch. In the fixed-area single-particle limit, successive alignments change enclosed flux by ${\Phi }_0=h/e$, giving the Aharonov-Bohm antidot periodThe effective orbit area, not merely the etched island area, enters. Charging, screening and spin can modify this simple sequence; two coupled spin branches can yield a near-$h/(2e)$ pattern without changing the elementary flux quantum.^[ref]

A trajectory-resolving ballistic experiment. Use a high-mobility modulation-doped GaAs-AlGaAs heterostructure with a two-dimensional electron gas. Grow the layers by molecular-beam epitaxy, separate the donor layer from the conducting interface with an undoped spacer, and pattern an isolated mesa with ohmic reservoir contacts. Electron-beam lithography defines metal split gates; negative gate voltage depletes the underlying electron gas, leaving narrow adjustable injector and collector quantum point contacts along an edge. Keep the relevant path shorter than the momentum-relaxation mean free path, cool to suppress phonon scattering and thermal smearing, and use a small excitation to avoid heating.

Inject a low-frequency current through one contact to a distant drain and measure the collector's open-circuit voltage with a high-input-impedance amplifier. Sweep perpendicular magnetic field and gate settings. The cyclotron radius is $r_c=\hslash k_F/(e|B|)$, with $k_F=\sqrt{2\pi n_s}$ for a spin-degenerate single-valley gas. For contact separation $L$, semicircular or skipping trajectories focus into the collector whenThe appropriate field sign bends carriers toward the collector; the opposite sign suppresses the peaks. Density-dependent focusing positions, suppression with increased scattering and reproducible gate control distinguish directional ballistic propagation from an ordinary diffusive voltage. Quantum interference may add fine structure, but the basic focusing signal does not require phase coherence over every path. Split-gate injection and collection in a two-dimensional gas were directly demonstrated in the original focusing experiment.^[ref]

Multi-probe current bookkeeping. For one directed spin-resolved mode, velocity times one-dimensional density of states gives a particle flux $dE/h$. A small reservoir electrochemical-energy shift ${\mu }_j$ therefore supplies excess flux ${\mu }_j/h$. Let $N_i$ be the number of propagating modes at lead $i$, $R_{ii}$ their summed return probability, and $T_{ij}$ the total transmission from reservoir $j$ into lead $i$. Multiplying outgoing-minus-incoming electron flux by charge $-e$ gives the source's convention,The reservoirs equilibrate arriving carriers; an ideal voltage probe adjusts its chemical potential until its net current vanishes. Here ${\mu }_i$ is an energy, not a voltage. Choose the lead-voltage convention ${\mu }_i=-e{V}_i$ and positive conventional current injected into the junction. The working equation is thenConservation of scattering probability and gauge invariance require both column and row balance: every outgoing channel is accounted for and equal voltages give zero current. The counts $M,N$ below include every occupied internal channel; an extra factor of two must not be inserted if spin is already counted. If one instead counts orbital channels with unresolved spin degeneracy, double all transmission counts consistently.

Zero-field routing. The original diagram labels lead 1 at the left, 2 above, 3 at the right and 4 below. With columns denoting injectors and rows collectors, straight-through ballistic routing and the stipulated excess-wide-channel rule giveThere is no reflection in this ideal routing model. Set $V_4=0$, $I_2=I_3=0$, and $I_1=I=-I_4$. The two probe equations giveThusThe derivation assumes $N>M>0$. At $N=M$, the ideal network splits into two disconnected opposite-lead pairs and cannot carry the imposed 1-to-4 current at finite voltage; the divergence diagnoses failure of that limiting transport configuration, not an infinite dissipative resistance of each channel. Negative bend resistance is a nonlocal voltage sign, compatible with positive total dissipated power.

Positive field. At $+B_0$, left-deflected incoming carriers go $1\to 2$, $2\to 3$, $3\to 4$. Lead 4 supplies $M$ modes to narrow lead 1; the remaining $N-M$ enter wide lead 3. The balanced transmission matrix isTherefore $V_2=V_1$, $V_3=M{V}_1/N$ and $I=g_{0}M{V}_1$, givingNegative field. Reversing the chirality gives ${\mathsf{T}}_{-}={\mathsf{T}}_{+}^{\mathsf{T}}$, as expected from reciprocity for this spin-independent routing model. With $V_4=0$, the zero-current conditions become $V_3=V_4=0$ and $V_2=V_3=0$. ThusThis does not contradict reciprocity: the general relation interchanges the current and voltage lead pairs when reversing field; it does not require the same fixed-contact four-terminal resistance to be even in $B$.

Comparison with the measured trace. The PDF shows a negative region around zero field, approximately zero signal at sufficiently negative field, and a positive rising resistance near $+0.4\mathrm{T}$ and above. These signs match the three routing limits. Near zero field the depth is roughly a few tenths of a kilohm; the graph does not supply $M,N$, so there is no justified exact amplitude fit. Rounded corners, collimation and finite cyclotron radius continuously change which outgoing lead is reached between the ideal limits. Skipping trajectories and geometrical focusing can produce shoulders or resonances. At larger positive fields magnetic depopulation and spin resolution can change the mode counts, increasing the positive resistance or producing structure; disorder, incomplete equilibration and finite temperature round ideal features. These are candidate mechanisms, not uniquely established assignments of every small feature from one trace.

Integer Hall transport and filling. A clean two-dimensional electron gas at low temperature in a strong perpendicular magnetic field forms Landau levels with orbital spacing $\hslash {\omega }_c$, where ${\omega }_c=e|B|/m^{\ast }$. Examples include semiconductor heterojunctions and inversion layers. On integer plateaus,with integer plateau index $\nu$. The density of states in one resolved Landau branch is $e|B|/h$ per unit area, so the actual filling factor isIt varies continuously with density and field. Within a disorder-broadened plateau, localized states can change occupation without changing the integer number of transporting edge branches; the filling need not equal the plateau integer at every point in that interval.

At a boundary, the confining potential bends the Landau energies through the Fermi energy, giving chiral edge states, interpreted semiclassically as skipping orbits. Opposite edges carry opposite propagation directions. An ideal spin-resolved transmitted edge channel contributes $e^2/h$; $\nu$ such channels give $I=\nu (e^2/h)(V_S-V_D)$. Chirality and suppressed backscattering between well-separated edges explain vanishing longitudinal voltage. Ideal reservoirs and appropriate probe equilibration are part of this edge-state transport model.

A gated antidot. A local repulsive gate potential depletes an island inside the Hall bar. Edge states circulate around its boundary as closed antidot orbits. When a localized orbit is weakly coupled to extended channels on opposite sides, resonance at the reservoir electrochemical potential permits backscattering and produces conductance dips on an otherwise quantized transmitting plateau. If the path instead adds a transmission route between otherwise disconnected channels, resonant peaks can occur: the sign depends on the device's tunnel connectivity and what conductance is measured. An isolated antidot too far from every conducting edge has negligible transport signature.

For the simplest opposite-edge backscattering picture with one active branch,where $f$ branches would otherwise transmit. A gate moves the resonance through the Fermi energy by changing both confinement and orbit area. Stronger depletion of the constrictions also changes $f$, producing broad conductance plateaus or pinch-off, with narrow resonances superposed where coupling is finite. Low temperature resolves individual resonances; thermal and tunnel broadening can merge them.

For a fixed-area single-electron orbit, a phase change of $2\pi$ corresponds toSweeping gate voltage at fixed field shifts enclosed area and level energy, giving oscillatory resonance tracks. This period is not automatically set by the lithographic gate area. Charging energy, self-consistent screening and spin branches modify spacings; in particular coupled spin branches in a charging model can generate two resonances per $h/e$ cycle without changing the electron flux quantum.^[ref]

No antidot dimensions, capacitances, lever arms or measured spectrum are supplied, so absolute periods and resonance voltages cannot be extracted. The sketch illustrates the mechanisms and regimes requested, rather than inventing a numerical device data set.

Improving the bare edge potential. A picture of noninteracting Landau levels simply intersecting a prescribed smooth confinement ignores the redistribution and screening of electron charge. A self-consistent electrostatic calculation combines gate and donor potentials, electron density and Coulomb interactions. Where a partially occupied Landau level has a large density response, electrons screen strongly and create a compressible strip with nearly pinned level energy. Regions with the chemical potential in a gap are incompressible and support much of the potential change. Consequently the screened edge alternates between compressible and incompressible strips, rather than preserving an arbitrary externally drawn slope. Finite temperature, disorder, exchange and edge reconstruction can require refinements beyond the simplest electrostatic model.^[ref]

Fractional states and composite fermions. Very clean two-dimensional systems show plateaus at selected fractional fillings, with small longitudinal resistance and a many-electron excitation gap. Electron-electron correlations are essential; partial filling of a noninteracting Landau level is not enough. The simplest composite fermion binds two statistical flux quanta to each electron. This is a representation of correlated electron motion, not a physical solenoid attached to a particle. With $B>0$, ${\Phi }_0=h/e$, and density $n_s$, the mean effective field isDefine a signed effective filling ${\nu }^{\ast }=n_s{\Phi }_0/B^{\ast }$. ThenFor ${\nu }^{\ast }=p>0$, filled composite-fermion levels produce $\nu =p/(2p+1)=1/3,2/5,3/7,\dots$. For reversed effective field write ${\nu }^{\ast }=-p$, giving $\nu =p/(2p-1)=1,2/3,3/5,\dots$; the nontrivial fractional upper sequence starts with $p=2$. If effective filling is defined as nonnegative using $|B^{\ast }|$, retain these as separate field-direction branches instead of using the signed expression blindly. At $\nu =1/2$ the mean $B^{\ast }$ vanishes and the effective filling diverges, so it is not an integer composite-fermion Hall plateau in this simplest construction. Attaching two flux quanta preserves fermionic statistics; the emergent integer-like gaps organize a prominent sequence of fractional electronic states.^[ref]

A high-mobility modulation-doped GaAs-AlGaAs electron gas is grown by molecular beam epitaxy as a clean layered semiconductor heterostructure. An undoped gallium arsenide region is followed by an undoped aluminium gallium arsenide spacer and a donor-doped barrier region, commonly with a protective cap. The closely matched crystal structures allow a high-quality heterojunction. The spacer supplies a crucial separation: the mobile electrons reside away from the ionized donors, reducing impurity scattering compared with putting the donors directly in the conducting layer.

The larger conduction band energy on the aluminium gallium arsenide side permits donor electrons to transfer into gallium arsenide. The resulting charge separation creates band bending; together with the band offset, it forms a roughly triangular quantum well at the interface. The normal-direction wavefunction is confined, while motion parallel to the interface remains approximately free:When only the lowest confinement subband is occupied, this is a two-dimensional electron gas. It is the quantized normal motion, not a physically zero layer thickness, that makes the low-energy system two-dimensional.

At zero magnetic field, one occupied spin-degenerate parabolic subband has constant density of states per area per energy, $D_0=m^{\ast }/(\pi {\hslash }^2)$ above its edge. A perpendicular field quantizes the in-plane orbital motion into Landau levels:Here $g^{\ast }$ is the material's effective spin factor and $e$ is the positive elementary charge; no free-electron $g=2$ assumption is made for this semiconductor. Each resolved spin branch contains $e|B|/h$ states per unit area. The ideal density of states becomeswith Landau-level broadening in a real sample. The Landau filling factorcounts the carrier density in units of one spin-resolved level's degeneracy. Two unresolved spin branches have twice that degeneracy; resolved branches must not each be counted twice.

Disorder localizes states in the broadened level tails, while extended states occur near level centres. As $B$ varies, occupation of localized states can change without changing the number of current-carrying edge channels. This produces a finite-width quantum Hall plateau, rather than quantization only at isolated fields. In a mobility gap, longitudinal resistivity approaches zero and the Hall magnitude is $R_K/N=h/(N{e}^2)$ for $N$ occupied transmitting branches. When the chemical potential passes through extended states, longitudinal dissipation peaks and the Hall value changes to another plateau. A spectral density of states can remain nonzero where transport is localized. Very strong disorder can destroy the resolved quantum Hall regime; the statement is not that any amount of disorder improves it.

The original plot is schematic, not fitted data. At fixed sheet density, increasing $|B|$ decreases Landau filling factor, so the Hall magnitude climbs through $R_K/N$ plateaus while longitudinal peaks separate the plateaus. The signed Hall voltage depends on carrier and probe conventions.

For the gate calculation, the target sheet density is fixed by the desired Landau filling factor:Use top-gate density tuning in the parallel-plate approximation. The geometric capacitance per unit area isThereforeThe gate must become more negative relative to the electron gas, reducing its density. The voltage is the change from the gate setting associated with the stated initial density; if that setting is defined as zero, it is the required gate voltage. Work-function offsets, trapped charge and quantum capacitance are neglected. An absolute voltage referenced to some other uncalibrated potential cannot be inferred from geometry alone. Numerical values use exact SI $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$ and $e=1.602176634\times 10^{-19}\mathrm{C}$, and ${ϵ}_0\simeq 8.8541878188\times 10^{-12}\mathrm{F}{\mathrm{m}}^{-1}$.^[ref]

For the subsequently specified five-channel plateau, the chemical potential between the fifth and sixth resolved branches means $N=5$, not ten. The physical contact order in the PDF is $1,2,3,4$ along one side of the mesa. Use ideal equilibrating Ohmic contacts and the Landauer-Büttiker formula, rather than inferring a longitudinal measurement merely from the contacts being drawn on one edge.

Choose one field polarity so the chiral order is $1\to 2\to 3\to 4\to 1$. With $G_H=5e^2/h$,Contacts 2 and 4 are voltage probes, so $I_2=I_4=0$. With $V_3=0$, this gives $V_2=V_1$, $V_4=V_3=0$, and $I_1=G_H{V}_1$. Hencefor this circulation. Reversing the field reverses the contact order and gives $R_{13,24}=-h/(5e^2)$ with the same voltage ordering. The resistance magnitude is $5.16\mathrm{k\Omega }$; it is not zero. Contact 2 samples the outgoing source potential and contact 4 the outgoing drain potential for the chosen chirality.

The antidot is a depleted island supporting closed quantum Hall edge states. Tunnelling between those states and opposite propagating edges can produce resonant backscattering and resistance peaks as $B$ varies. One electronic Aharonov-Bohm phase winds by $2\pi$ when flux through a fixed effective orbit area $S$ changes by $h/e$. Using the supplied area as that effective area,With the two spin-split branches of the lowest orbital level occupied, the common Coulomb-dominated antidot oscillation model has two charge rearrangements per added $h/e$ flux. Both branches contribute to charge buildup; screening and their capacitive coupling can make tunnelling through the outer edge occur twice during the cycle. It then givesThis is two electron-tunnelling resonances per flux cycle, not evidence that the mobile particles have charge $2e$. The primary charging model specifically addresses this two-spin regime.^[ref]

The sketch shows resistance peaks in a weak-backscattering configuration, with arbitrary height and linewidth because tunnel couplings and temperature are unspecified. A two-spin count alone does not force two independent noninteracting resonance series to be exactly half a flux period apart: their relative positions also depend on spin splitting and confinement. Thus $51.7\mathrm{mT}$ describes the stated charging-model interpretation; $103.4\mathrm{mT}$ is the single-electron fixed-area flux period. The actual enclosed edge-orbit area can differ from the geometrically depleted area.

For bulk filling two and total forward transmission $\mathcal{T}$ in spin-resolved units, ideal chiral probes give the longitudinal four-terminal resistanceResonant loss of transmission therefore gives peaks. Their size, background and detailed shape cannot be calculated from the supplied area and bulk filling alone. If the gates instead make a tunnelling path that enhances forward transmission, the resonant contrast can be reversed; the plotted peak convention is explicitly the backscattering one.

Use two collinear quantum point contacts in a clean two-dimensional electron gas. Surface split gates deplete the electron gas beneath them, leaving two short constrictions facing one another across an open region. The first injects a directional beam from a biased reservoir; the second collects carriers into a downstream reservoir, with electric current flowing in the same spatial direction through both constrictions. Additional surrounding contacts can absorb carriers that miss the collector. A direct collector signal at zero magnetic field, sensitive to alignment and to obstruction of the intervening path, demonstrates ballistic electron transport across the separation.

An illustrative GaAs device uses gate gaps of a few hundred nanometres and electrically tunable constriction widths of tens to a few hundred nanometres, with injector–collector separation of a few micrometres. These are typical design scales, not dimensions supplied for a particular chip in the question. The electrical width can be much smaller than the lithographic gap because depletion extends laterally. A one-mode opening has width comparable to roughly half a Fermi wavelength; it is not simply any opening below one micrometre. Low-temperature operation suppresses inelastic scattering and resolves transverse modes. The direct trajectory length should be comfortably below the relevant transport mean free path.

Collimation is provided by the smooth exit from a constriction. In adiabatic electron collimation, transverse mode action is approximately conserved, giving $p_{\perp }W\simeq \mathrm{constant}$. As the channel opens from width $W_0$ to $W_{\mathrm{out}}$, $p_{\perp ,\mathrm{out}}\simeq p_{\perp ,0}{W}_0/W_{\mathrm{out}}$. At the same time the bottom of the electrostatic potential falls, increasing longitudinal kinetic energy. Thusand the angular distribution narrows. The reciprocal collector selects this forward-directed set of trajectories. A narrow abrupt aperture by itself may diffract into a broad fan; the smooth funnel and its potential profile are important. Here “focusing” refers primarily to collimation and efficient injector–collector coupling, not a claim that every emitted trajectory converges to one exact geometric point.

In scanning gate microscopy, a conducting tip biased sufficiently negatively creates a local repulsive potential and, when strong enough, a small depleted region in the electron gas beneath it. Raster the tip and record the collector current or differential conductance. Placing the perturbation on a strongly contributing path deflects or backscatters electrons and changes collection, producing spatial contrast. A primary scanning-probe experiment records conductance changes as a negatively charged tip depletes and backscatters the two-dimensional gas.^[ref]

The resulting image is a map of transport sensitivity to the tip, not a literal nonperturbing photograph of an electron trajectory. Broad electrostatic gating of either constriction should be distinguished from local beam disruption. Weak smooth disorder can create branches even in a high-mobility sample. Where electron phase coherence is preserved, interference between alternative paths can add fringes, often with a scale comparable to half a Fermi wavelength for a round-trip path-length change. Ballistic motion and phase-coherent interference are related but distinct requirements.

A small perpendicular magnetic field curves the imaged paths through the Lorentz force, $\mathbf{F}=-e\mathbf{v}\times \mathbf{B}$. Reversing $B$ reverses the bending, and a fixed aligned collector generally loses the initially straight beam unless a curved path connects it. For a circular orbit at the Fermi velocity,For one valley with two nearly equally occupied spin species at small field, state counting in the Fermi disc gives $n_s=2\pi k_F^2/(2\pi {)}^2=k_F^2/(2\pi )$. HenceSpin polarization or extra valley degeneracy changes the density-to-wavevector relation; the stated low-field GaAs estimate uses the ordinary spin-degenerate case.

The electron mobility gives the momentum relaxation time in a parabolic-band Drude model, $\mu =e{\tau }_{\mathrm{tr}}/m^{\ast }$. ThereforeThe effective mass cancels from both ${\ell }_{\mathrm{tr}}$ and $r_c$ when expressed in terms of density and mobility. For the supplied density,With $m^{\ast }=0.067m_0$, $m_0\simeq 9.1093837139\times 10^{-31}\mathrm{kg}$, one also obtains $v_F\simeq 1.937\times 10^5\mathrm{m}{\mathrm{s}}^{-1}$ and ${\tau }_{\mathrm{tr}}\simeq 38.1\mathrm{ps}$. The requested lengths areA useful independent ratio is ${\ell }_{\mathrm{tr}}/r_c=\mu |B|=3$. The standard constants used here are explicitly tabulated, rather than inferred from the sample.^[ref]

Mobility measures momentum relaxation. Predominantly small-angle elastic scattering can occur more frequently than that transport time, and the phase-coherence time is different again. Thus the calculation supports a several-micrometre transport scale, without proving that every electron experiences literally no perturbation over that distance.

As temperature rises, the directed ballistic collector signal decreases. Increased electron-phonon scattering and other inelastic processes reduce survival of the directed beam; thermal averaging broadens the mode and angular selection, and interference contrast also diminishes. A diffuse background current can remain, so disappearance of beam contrast need not mean all current through the device vanishes.

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.

An electrostatically defined quantum dot uses depletion gates to confine a small island laterally in a two-dimensional electron gas. Two narrow barriers allow weak tunnelling to source and drain; a plunger gate tunes the island's electrochemical levels, and additional gates can form a double quantum dot. Adjusting the gates resolves successive charge states and eventually reaches the single-electron regime. An electrical dot is defined by confinement and tunnel coupling, not simply by the drawn metal area.

In a constant interaction model,Here $Q_g$ is gate-induced charge and $C_{\Sigma }$ total island capacitance. Successive additions cost $E_{\mathrm{add}}=e^2/C_{\Sigma }+{ϵ}_{N+1}-{ϵ}_N$, with spin, exchange and correlations modifying the orbital term. Coulomb blockade appears when no allowed charge transition lies in the source–drain energy window. Gate sweeps give charge steps and transport peaks; finite-bias Coulomb diamonds constrain charging and orbital energies under a stated bias convention. Resolved single-electron operation requires thermal and tunnel broadening smaller than the relevant addition energies.

Quantum-dot charge sensing avoids relying on a measurable current through a nearly isolated dot. A nearby quantum point contact biased on a sensitive conductance slope, or a single-electron transistor, responds capacitively when one electron enters or leaves the dot. A time trace of sensor current can therefore distinguish $N$ from $N-1$ and resolve individual tunnelling events. The sensor measures charge; a spin-dependent step is needed to measure spin.

For one electron, Zeeman splitting gives a lower and a higher spin level separated by $E_Z=|g^{\ast }|{\mu }_B|B|$. In energy-selective spin readout, initialize an empty dot and pulse both levels below the reservoir chemical potential to load one electron. Then move to a readout configurationAn electron in the higher state can tunnel out, temporarily leaving the dot empty, after which a lower-state electron can refill it. An electron initially in the lower state remains trapped. The charge sensor records a pulse for the first outcome and no escape pulse for the second. This is spin-to-charge conversion; it measures the spin projection along the field through a state-dependent tunnelling probability. A primary single-shot experiment uses this conversion with a quantum-point-contact charge detector.^[ref]

The readout window must be long enough to detect escape but short compared with excited-spin relaxation:Detector bandwidth must also resolve the empty-state dwell time before refilling. Thermal activation, relaxation before escape, missed fast events and sensor backaction cause errors. Since the excited electron can leave and be replaced, this simple sequence does not preserve an arbitrary initial spin state.

For coupled dots, label charge occupation by $(n_L,n_R)$. Gate-controlled detuning can make $(1,1)$ and $(0,2)$ accessible. Two electrons in the lowest orbital of one dot must form a singlet state; a triplet state requires a different orbital or another higher-energy configuration. Choose a measurement window where the $(0,2)$ singlet is available but the $(0,2)$ triplet is not. Thenunder spin-conserving tunnelling. The second process is forbidden by spin symmetry and the Pauli exclusion principle, producing Pauli spin blockade. Charge sensing distinguishes transfer from a blocked $(1,1)$ state. This measures singlet versus triplet content; without a reference spin or additional control it is not a measurement of each electron's absolute spin direction separately.

The exchange interaction in a double quantum dot provides control of these states. In an effective spin description,For a symmetric weak-tunnelling two-site model with onsite charging cost $U$, $J\simeq 4t_c^2/U$. Gates change tunnel coupling $t_c$ and detuning, allowing preparation and evolution followed by a charge-conversion measurement. Spin mixing from local magnetic-field differences or spin-orbit effects can lift blockade, so the sensing interval must be chosen before unwanted relaxation obscures the initial state. Single-dot readout uses spin-dependent escape; double-dot readout can use spin-dependent charge transfer. In both cases the measurable signal is a resolved charge change.

Physical Review Letters 60, 848–850. Full title: Quantized conductance of point contacts in a two-dimensional electron gas. Gate-controlled GaAs-AlGaAs ballistic point contacts show conductance steps from transverse quantization.

A depleted potential hill inside a two-dimensional electron gas. In a strong perpendicular magnetic field, closed quantum Hall edge states circulate around it. Coupling these states to propagating edge channels can produce resonant backscattering; Coulomb blockade can substantially alter the single-particle resonance pattern.

The quantum Hall effect connects an electrical measurement to a quantum counting rule: a two-dimensional electron gas in a perpendicular magnetic field can exhibit plateaus of transverse electrical resistance. The integer case supplies a resistance scale built from the Planck constant and elementary charge, rather than the sample's dimensions. The fractional quantum Hall effect involves additional many-electron physics.

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.