Quantum Hall Effect

Quantization of Hall resistance in two-dimensional electron gases. Theory, formulas, worked examples and interactive graphs.

Conceptual context

The classical Hall effect already shows that a longitudinal $I$ in a conductor under a perpendicular $B$ produces a transverse $V_H$. The quantum Hall effect is stranger: in a clean two-dimensional system at low temperature, the response stops changing smoothly. As $B$ is swept, $R_H$ locks onto plateaus fixed by $h$ and $e$. The central issue is why a real material, full of microscopic details, returns universal numbers. The answer combines Landau levels, edge channels, localization, and the $\nu$.

🟢 Essential level

In a two-dimensional electron gas, a perpendicular $B$ reorganizes electron motion into quantum orbits called Landau levels.

As $B$ changes, the number of available states in each level changes, so the $\nu$ changes too. When the effective occupied level remains stable, $R_H$ no longer follows every microscopic variation of the sample; it locks to a universal scale. That stable region is a plateau. The $I$ flows through edge channels, while many bulk states become localized and do not carry charge across the sample. Therefore the transverse measurement, $V_H$, can ignore many imperfections. Quantization does not mean every response is always discrete; it means that under the right conditions the Hall response selects robust values that survive small changes in field, density, and disorder.

Klaus von Klitzing discovered this effect in 1980 while studying a silicon transistor at very low temperature, a finding that earned him the 1985 Nobel Prize in Physics. The von Klitzing constant, h/e², is about 25,812.8 ohms and is now used as the international standard for electrical resistance, precisely because the plateaus are extraordinarily reproducible across different laboratories.

🔵 Formal level

Basic formulas

The formal model starts from electrons confined to two dimensions, with $n_s$ and a perpendicular $B$. The effective occupation of Landau levels is summarized by $\nu$, which compares the electron population with the field-generated degeneracy of available quantum states. That comparison tells whether the sample lies near a plateau or near a transition, and it is the bridge between material tuning and the measured Hall response.

Filling factor of the 2D electron gas

When $\nu$ takes a stable integer value, the transverse $\sigma_{xy}$ is quantized in units set by $e$ and $h$.

Quantized Hall conductance

The dual quantity measured as a transverse ratio is $R_H$. On a plateau, it is controlled not by the fine geometry of the sample but by $R_K$ divided by $\nu$.

Quantized Hall resistance

Real student questions

Why can impurities help instead of only spoiling the experiment? Moderate disorder localizes bulk states and widens the interval of $B$ over which $R_H$ remains constant.

Why does the plateau not depend on the exact sample shape? The quantized value depends on $h$, $e$, and $\nu$, not on channel width, as long as the regime remains valid.

Is it the same as the classical Hall effect? No. The classical Hall effect varies smoothly with $B$ and $n_s$, whereas the quantum Hall effect produces robust plateaus.

Final synthesis

The quantum Hall effect shows that a two-dimensional material can turn an electrical measurement into a window onto fundamental constants. Plateaus appear because $\nu$, localization, and edge channels stabilize $R_H$ against smooth changes in $B$.