Past exam of the condensed matter physics major option course of the University of Oxford / 2018 / 6 / a / Solution

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.