---
title: "The 2D Electron Gas and the Integer Quantum Hall Effect"
module: Nanostructures
moduleNumber: 11
lessonNumber: 2
order: 1102
summary: >
  A two-dimensional electron gas in a strong perpendicular magnetic field has its
  continuous density of states collapse into macroscopically degenerate Landau
  levels. As the field is swept, the Hall resistance locks onto exact plateaus at
  h over an integer times e squared, while the longitudinal resistance drops to
  zero. This lesson derives the Landau levels and their degeneracy, explains the
  plateaus through disorder-localized states and current-carrying edge channels,
  and states why the von Klitzing constant is now a resistance standard.
topics: [Nanostructures]
draft: false
sources:
  - book: Kittel
    ref: "Ch. 18 — Nanostructures; quantum Hall effect"
  - book: Ashcroft & Mermin
    ref: "Ch. 1, Ch. 12 — the Hall effect and semiclassical dynamics"
---

The [quantum well](/condensed-matter/nanostructures/quantum-wells-wires-and-dots)
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
$g_{\text{2D}} = m^\ast/\pi\hbar^2$. Cool that gas below $1\ \text{K}$ 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 $h/\nu e^2$ for integer $\nu$, 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 $\hat x$ through a 2DEG of areal density $n_s$ in a
field $\vec B = B\hat z$. In steady state the transverse electric field balances
the Lorentz force, $eE_y = e v_x B$, and with $j_x = n_s e v_x$ the transverse
(Hall) resistivity is

$$
\rho_{xy} = \frac{E_y}{j_x} = \frac{B}{n_s e}.
$$

In two dimensions resistivity and resistance carry the same units, and for a Hall
bar the measured Hall resistance equals $\rho_{xy}$. 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 $m^\ast$ and charge $-e$ in a field $B\hat z$ has
Hamiltonian $H = (\vec p + e\vec A)^2/2m^\ast$. Choose the Landau gauge
$\vec A = (0, Bx, 0)$, so the Hamiltonian commutes with $p_y$ and eigenstates
take the form $\psi = e^{ik_y y}\phi(x)$. Substituting reduces the problem to a
one-dimensional oscillator in $x$ centered at $x_0 = -\hbar k_y/eB$:

$$
\left[-\frac{\hbar^2}{2m^\ast}\frac{\d^2}{\d x^2}
+ \tfrac{1}{2}m^\ast \omega_c^2 (x - x_0)^2\right]\phi = E\,\phi,
\qquad \omega_c = \frac{eB}{m^\ast}.
$$

The frequency $\omega_c$ is the classical cyclotron frequency. The spectrum is
that of a harmonic oscillator, independent of $k_y$:

$$
E_n = \hbar\omega_c\left(n + \tfrac{1}{2}\right), \qquad n = 0, 1, 2, \dots
$$

These are the **Landau levels**. The continuous 2D band has condensed into a set
of discrete, equally spaced levels separated by $\hbar\omega_c = \hbar eB/m^\ast$.

> **Definition (Landau level).** A Landau level is one of the discrete,
> uniformly spaced energy eigenvalues $E_n = \hbar\omega_c(n+\tfrac12)$ of a
> charged particle confined to a plane in a perpendicular magnetic field. Every
> state in a level shares the same energy but differs in the guiding-center
> coordinate $x_0$, making the level macroscopically degenerate.

$$
% caption: Landau levels fan out linearly with field as E_n grows like B; as the
% field rises the levels sweep past a fixed Fermi energy, emptying one at a time.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, ->] (0,0) -- (6.6,0) node[below] {magnetic f\/ield $B$};
  \draw[black, ->] (0,0) -- (0,4.0) node[left] {energy};
  % fan of Landau levels E_n = (n+1/2) hbar omega_c, slope increases with n
  \foreach \m in {0,1,2,3,4}{
    \pgfmathsetmacro{\s}{0.133*(\m+0.5)}
    \draw[acc, thick] (0,0) -- (6.2,{\s*6.2});
  }
  \node[acc, anchor=west] at (6.2,{0.133*0.5*6.2}) {$n=0$};
  \node[acc, anchor=west] at (6.2,{0.133*1.5*6.2}) {$n=1$};
  \node[acc, anchor=west] at (6.2,{0.133*2.5*6.2}) {$n=2$};
  % fixed Fermi energy
  \draw[black, very thick, dashed] (0,2.2) -- (6.2,2.2);
  \node[black, anchor=south east] at (6.2,2.2) {$E_F$};
\end{tikzpicture}
$$

## Degeneracy and the filling factor

The degeneracy of each Landau level follows from counting the guiding centers
$x_0 = -\hbar k_y/eB$ that fit in the sample. With periodic boundary conditions in
$y$ over length $L_y$, the allowed $k_y$ are spaced by $2\pi/L_y$, so the centers
$x_0$ are spaced by $\Delta x_0 = 2\pi\hbar/(eB L_y)$. Requiring $0 \le x_0 \le
L_x$ gives the number of states per level in the sample,

$$
N_L = \frac{L_x}{\Delta x_0} = \frac{eB\,L_x L_y}{2\pi\hbar} = \frac{eB}{h}\,A.
$$

The **degeneracy per unit area** is therefore

$$
n_B = \frac{eB}{h} = \frac{B}{\Phi_0},
\qquad \Phi_0 = \frac{h}{e} = 4.136\times10^{-15}\ \text{Wb},
$$

exactly one state per flux quantum $\Phi_0 = h/e$ threading the plane. The ratio
of the electron density to the level degeneracy is the **filling factor**,

$$
\nu = \frac{n_s}{n_B} = \frac{n_s h}{eB},
$$

the number of Landau levels occupied. As $B$ rises at fixed $n_s$, each level
holds more electrons and $\nu$ falls; whenever $\nu$ 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 $\nu$ is an integer.
With $n_s = \nu e B/h$,

$$
\rho_{xy} = \frac{B}{n_s e} = \frac{B}{(\nu eB/h)\,e} = \frac{h}{\nu e^2}.
$$

At integer filling the Hall resistance takes the value $h/\nu e^2$ with no
material parameters remaining. The remarkable experimental fact is that
$\rho_{xy}$ 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 $\rho_{xx}$ collapses to zero. Dissipationless current
flows while the Hall resistance is quantized.

$$
% caption: Hall resistance versus field forms flat plateaus at h over integer
% times e squared; the longitudinal resistance vanishes on each plateau and
% spikes only in the transitions between them.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, ->] (0,0) -- (7.4,0) node[below] {magnetic f\/ield $B$};
  \draw[black, ->] (0,0) -- (0,4.0) node[left] {resistance};
  % classical linear reference
  \draw[black, densely dotted] (0,0) -- (7.0,3.7);
  \node[black, anchor=west] at (5.6,2.9) {classical};
  % Hall plateaus (rising staircase)
  \draw[acc, very thick]
    (0.4,0.7) -- (1.6,0.7)
    (1.6,0.7) .. controls (2.1,1.0) and (2.1,1.2) .. (2.6,1.5)
    (2.6,1.5) -- (3.9,1.5)
    (3.9,1.5) .. controls (4.4,1.9) and (4.4,2.1) .. (4.9,2.5)
    (4.9,2.5) -- (6.2,2.5)
    (6.2,2.5) .. controls (6.6,2.9) and (6.6,3.1) .. (7.0,3.4);
  \node[acc, anchor=south] at (3.2,1.5) {$R_{xy}$ plateaus};
  % longitudinal Rxx peaks between plateaus
  \draw[black, thick]
    (0.4,0.1) -- (1.9,0.1)
    (1.9,0.1) .. controls (2.2,0.9) and (2.5,0.9) .. (2.6,0.1)
    (2.6,0.1) -- (4.2,0.1)
    (4.2,0.1) .. controls (4.5,0.9) and (4.8,0.9) .. (4.9,0.1)
    (4.9,0.1) -- (6.8,0.1);
  \node[black, anchor=south west] at (3.0,0.15) {$R_{xx}$};
\end{tikzpicture}
$$

## Disorder makes the plateaus

A clean Landau level is a single delta function in energy, so its filling would
change discontinuously and $\rho_{xy}$ 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 $\sigma_{xy}$ — 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 $\rho_{xx}$ spikes. The plateaus are wide precisely because most states in a
disordered Landau level are localized.

$$
% caption: A disorder-broadened Landau level splits into localized states in the
% tails (shaded, carrying no current) and extended states at the center; the Hall
% plateau persists while the Fermi level lies in the localized tails.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, ->] (0,0) -- (7.4,0) node[below] {energy};
  \draw[black, ->] (0,0) -- (0,3.0) node[left] {density of states};
  % two broadened Landau peaks
  \draw[very thick, domain=1.0:3.0, samples=60, variable=\x]
    plot ({\x},{2.4*exp(-(\x-2.0)*(\x-2.0)/0.16)});
  \draw[very thick, domain=4.0:6.0, samples=60, variable=\x]
    plot ({\x},{2.4*exp(-(\x-5.0)*(\x-5.0)/0.16)});
  % extended-state central bands
  \draw[black, very thick] (1.95,0) -- (1.95,2.45);
  \draw[black, very thick] (4.95,0) -- (4.95,2.45);
  \node[black, anchor=south] at (2.0,2.5) {extended};
  \node[black, anchor=south] at (5.0,2.5) {extended};
  % shaded localized tails
  \fill[acc!14] (1.0,0) .. controls (1.6,1.0) and (1.7,1.2) .. (1.8,1.3)
    -- (1.8,0) -- cycle;
  \fill[acc!14] (2.2,1.3) .. controls (2.3,1.2) and (2.4,1.0) .. (3.0,0)
    -- (2.2,0) -- cycle;
  \node[black, anchor=north] at (1.3,-0.05) {localized};
\end{tikzpicture}
$$

## 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 $\vec B$. 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 $\nu$ filled levels give exactly $\sigma_{xy} = \nu e^2/h$
with $\rho_{xx} = 0$.

$$
% caption: In the bulk the cyclotron orbits are closed and carry no net current;
% at the edges the orbits skip along the boundary as chiral one-way channels,
% one per filled Landau level, and cannot backscatter.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % sample boundary
  \draw[black, very thick] (0,0) rectangle (7.0,3.0);
  % bulk closed cyclotron orbits
  \foreach \x in {2.0,3.5,5.0}{
    \foreach \y in {1.1,1.9}{
      \draw[black] (\x,\y) circle (0.28);
      \draw[black, ->] (\x+0.28,\y) arc (0:60:0.28);
    }
  }
  \node[black, anchor=south] at (3.5,2.15) {closed bulk orbits};
  % top edge skipping orbits (moving right)
  \foreach \x in {0.5,1.3,2.1,2.9,3.7,4.5,5.3,6.1}{
    \draw[acc, very thick] (\x,3.0) arc (180:360:0.4 and 0.32);
  }
  \draw[acc, very thick, ->] (6.4,2.7) -- (6.9,2.7);
  % bottom edge skipping orbits (moving left)
  \foreach \x in {0.5,1.3,2.1,2.9,3.7,4.5,5.3,6.1}{
    \draw[acc, very thick] (\x,0.0) arc (180:0:0.4 and 0.32);
  }
  \draw[acc, very thick, ->] (0.6,0.3) -- (0.1,0.3);
  \node[acc, anchor=north] at (3.5,-0.15) {chiral edge channels};
\end{tikzpicture}
$$

## The von Klitzing constant

The plateau value at $\nu = 1$ defines the **von Klitzing constant**,

$$
R_K = \frac{h}{e^2} = 25\,812.807\ \Omega.
$$

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.[^rk] The
quantization is exact because it counts filled edge channels, a property immune
to the microscopic details that disorder and geometry would otherwise spoil.

$$
% caption: The Hall resistance measured in units of R_K falls as a staircase in
% one over the filling factor, landing on 1, 1/2, 1/3 as successive Landau levels
% empty.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, ->] (0,0) -- (7.0,0) node[below] {magnetic f\/ield $B$};
  \draw[black, ->] (0,0) -- (0,3.6) node[left] {$R_{xy}$ in units of $R_K$};
  % plateaus at 1/3, 1/2, 1
  \draw[acc, very thick] (0.4,0.9) -- (2.4,0.9);
  \node[acc, anchor=south] at (1.4,0.92) {$\frac{1}{3}$};
  \draw[acc, very thick] (2.4,0.9) -- (2.4,1.35) -- (4.2,1.35);
  \node[acc, anchor=south] at (3.3,1.37) {$\frac{1}{2}$};
  \draw[acc, very thick] (4.2,1.35) -- (4.2,2.7) -- (6.4,2.7);
  \node[acc, anchor=south] at (5.3,2.72) {$1$};
  \draw[black, dashed] (0,2.7) -- (4.2,2.7);
  \node[black, anchor=east] at (-0.05,2.7) {$R_K$};
\end{tikzpicture}
$$

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 $\nu$. The
[next lesson](/condensed-matter/nanostructures/fractional-quantum-hall-and-topology)
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.

[^rk]: NIST, von Klitzing constant $R_K = h/e^2$,
[physics.nist.gov/cgi-bin/cuu/Value?rk](https://physics.nist.gov/cgi-bin/cuu/Value?rk).