The 2D Electron Gas and the Integer Quantum Hall Effect

The quantum well freezes one direction of an electron's motion and leaves a two-dimensional electron gas (2DEG) in the plane, with the flat step density of states $g_{2D} = m^{*} / π ℏ^{2}$ . Cool that gas below $1 K$ and apply a magnetic field of several tesla perpendicular to the plane, and the flat density of states shatters into a comb of hugely degenerate levels. The Hall resistance, which classically rises linearly with field, instead develops flat plateaus pinned to $h / ν e^{2}$ for integer $ν$ , reproducible to nine significant figures regardless of the sample's shape, mobility, or material. Klaus von Klitzing's 1980 discovery of this quantization turned a mesoscopic transport measurement into a definition of the ohm.

The classical Hall effect

Drive a current $I$ along $\overset{x}{^}$ through a 2DEG of areal density $n_{s}$ in a field $B = B \overset{z}{^}$ . In steady state the transverse electric field balances the Lorentz force, $e E_{y} = e v_{x} B$ , and with $j_{x} = n_{s} e v_{x}$ the transverse (Hall) resistivity is

ρ_{x y} = \frac{E _{y}}{j _{x}} = \frac{B}{n _{s} e} .

In two dimensions resistivity and resistance carry the same units, and for a Hall bar the measured Hall resistance equals $ρ_{x y}$ . Classically this rises linearly with $B$ and measures the carrier density and sign. The quantum result replaces the straight line with a staircase.

Landau quantization

An electron of effective mass $m^{*}$ and charge $- e$ in a field $B \overset{z}{^}$ has Hamiltonian $H = (p + e A)^{2} /2 m^{*}$ . Choose the Landau gauge $A = (0, B x, 0)$ , so the Hamiltonian commutes with $p_{y}$ and eigenstates take the form $ψ = e^{i k_{y} y} ϕ (x)$ . Substituting reduces the problem to a one-dimensional oscillator in $x$ centered at $x_{0} = - ℏ k_{y} / e B$ :

[- \frac{ℏ ^{2}}{2 m ^{*}} \frac{d ^{2}}{d x ^{2}} + \frac{1}{2} m^{*} ω_{c}^{2} (x - x_{0})^{2}] ϕ = E ϕ, ω_{c} = \frac{e B}{m ^{*}} .

The frequency $ω_{c}$ is the classical cyclotron frequency. The spectrum is that of a harmonic oscillator, independent of $k_{y}$ :

E_{n} = ℏ ω_{c} (n + \frac{1}{2}), n = 0, 1, 2, \dots

These are the Landau levels. The continuous 2D band has condensed into a set of discrete, equally spaced levels separated by $ℏ ω_{c} = ℏ e B / m^{*}$ .

Landau levels fan out linearly with field as E_n grows like B; as the field rises the levels sweep past a fixed Fermi energy, emptying one at a time.

Degeneracy and the filling factor

The degeneracy of each Landau level follows from counting the guiding centers $x_{0} = - ℏ k_{y} / e B$ that fit in the sample. With periodic boundary conditions in $y$ over length $L_{y}$ , the allowed $k_{y}$ are spaced by $2 π / L_{y}$ , so the centers $x_{0}$ are spaced by $Δ x_{0} = 2 π ℏ/ (e B L_{y})$ . Requiring $0 \leq x_{0} \leq L_{x}$ gives the number of states per level in the sample,

N_{L} = \frac{L _{x}}{Δ x _{0}} = \frac{e B L _{x} L _{y}}{2 π ℏ} = \frac{e B}{h} A .

The degeneracy per unit area is therefore

n_{B} = \frac{e B}{h} = \frac{B}{Φ _{0}}, Φ_{0} = \frac{h}{e} = 4.136 \times 1 0^{- 15} Wb,

exactly one state per flux quantum $Φ_{0} = h / e$ threading the plane. The ratio of the electron density to the level degeneracy is the filling factor,

ν = \frac{n _{s}}{n _{B}} = \frac{n _{s} h}{e B},

the number of Landau levels occupied. As $B$ rises at fixed $n_{s}$ , each level holds more electrons and $ν$ falls; whenever $ν$ passes through an integer, an integer number of levels is exactly filled and the gas has a gap to the next level.

The quantized plateaus

Substitute the classical Hall result at the fields where $ν$ is an integer. With $n_{s} = ν e B / h$ ,

ρ_{x y} = \frac{B}{n _{s} e} = \frac{B}{( ν e B / h ) e} = \frac{h}{ν e ^{2}} .

At integer filling the Hall resistance takes the value $h / ν e^{2}$ with no material parameters remaining. The remarkable experimental fact is that $ρ_{x y}$ does not merely pass through this value — it sits on a flat plateau over a finite range of field, and across each plateau the longitudinal resistivity $ρ_{xx}$ collapses to zero. Dissipationless current flows while the Hall resistance is quantized.

Hall resistance versus field forms flat plateaus at h over integer times e squared; the longitudinal resistance vanishes on each plateau and spikes only in the transitions between them.

Disorder makes the plateaus

A clean Landau level is a single delta function in energy, so its filling would change discontinuously and $ρ_{x y}$ would step without a plateau. Disorder is what produces the flat plateaus. A random potential broadens each delta function into a band of states. The states in the tails of each broadened level are spatially localized on hills and valleys of the potential and carry no current; only the extended states near the center of each level conduct. As the field sweeps the Fermi level through the localized tail of a level, the number of occupied extended states — and hence $σ_{x y}$ — does not change, so the Hall resistance stays pinned. It moves to the next quantized value only when $E_{F}$ crosses the narrow band of extended states at the level center, which is also where $ρ_{xx}$ spikes. The plateaus are wide precisely because most states in a disordered Landau level are localized.

A disorder-broadened Landau level splits into localized states in the tails (shaded, carrying no current) and extended states at the center; the Hall plateau persists while the Fermi level lies in the localized tails.

Edge states

The dissipationless current has a real-space picture at the sample boundary. In the bulk, an electron in a magnetic field executes a closed cyclotron orbit and drifts nowhere. At the edge the orbit collides with the boundary and bounces along it in a series of skipping orbits, always advancing in the same direction set by $B$ . These edge trajectories are one-dimensional conducting channels, one per filled Landau level, and they are chiral: electrons on a given edge move only one way, so there are no counter-propagating states to scatter into. Backscattering would require an electron to cross the insulating bulk to the opposite edge, which the localized bulk forbids. With no backscattering the channels are ballistic, each contributing one conductance quantum $e^{2} / h$ , and $ν$ filled levels give exactly $σ_{x y} = ν e^{2} / h$ with $ρ_{xx} = 0$ .

In the bulk the cyclotron orbits are closed and carry no net current; at the edges the orbits skip along the boundary as chiral one-way channels, one per filled Landau level, and cannot backscatter.

The von Klitzing constant

The plateau value at $ν = 1$ defines the von Klitzing constant,

R_{K} = \frac{h}{e ^{2}} = 25 812.807 Ω.

Its reproducibility across silicon MOSFETs, GaAs heterostructures, and even graphene, independent of material and geometry, made it a resistance standard; since the 2019 redefinition of the SI in terms of fixed values of $h$ and $e$ , $R_{K}$ is an exact number and the quantum Hall effect realizes the ohm.¹ The quantization is exact because it counts filled edge channels, a property immune to the microscopic details that disorder and geometry would otherwise spoil.

The Hall resistance measured in units of R_K falls as a staircase in one over the filling factor, landing on 1, 1/2, 1/3 as successive Landau levels empty.

The integer effect rests on filling non-interacting Landau levels. When the lowest level is only partly filled, the non-interacting picture predicts a metal with no gap, yet experiments find plateaus at fractional $ν$ . The next lesson shows those plateaus are a many-body effect: Coulomb interactions among the electrons in a single Landau level open a gap and produce excitations carrying a fraction of the electron charge.

NIST, von Klitzing constant $R_{K} = h / e^{2}$ , physics.nist.gov/cgi-bin/cuu/Value?rk. ↩