Closing the theory–experiment gap in the fractional quantum Hall effect

In the fractional quantum Hall effect (FQHE), a two-dimensional electron gas at low temperature and high magnetic field forms an incompressible liquid of quasiparticles that have fractional electric charge. As a result, its Hall resistance is quantized at fractional rather than integer intervals (see Physics Today, December 1998, page 17 ). A basic property of the system is the energy gap between its incompressible ground state and its excited states. But despite nearly 40 years of study, the theoretical predictions for the gap’s size are consistently larger than what’s measured experimentally.

The primary reasons for that discrepancy are known. Among them are the nonzero thickness of the electron layer and the heterogeneity, defects, and other forms of disorder in real samples. Although theorists have tried to account for those and other effects, their models still predict too large an energy gap.

Mansour Shayegan and his colleagues at Princeton University have now conducted the first experimental analysis of the relationship between electron-layer thickness and the FQHE energy gap. Their measurements provide a benchmark for theorists trying to close the gap in energy gap predictions.

Shayegan and his team prepared gallium arsenide quantum wells whose widths w, ranging from 20 nm to 80 nm, roughly set the electron-layer thickness w̃. When the researchers measured the devices’ longitudinal resistances as a function of an applied magnetic field, the resistances showed about a dozen minima, which corresponded to FQHE states with different fractional values of ν, the Landau filling factor. As the temperature approached zero, the longitudinal resistance for a specific filling factor dropped exponentially to zero at a rate that depended on the energy gap.

As shown in the graph, the researchers measured the ν = 1/3 energy gap (^1/3Δ) at an array of quantum well widths (red). As expected theoretically, the gap decreased with increasing thickness. But the measured energy gaps were still consistently lower than those calculated theoretically (green, blue, and orange).

The models, however, consider the role of nonzero electron-layer thickness but not disorder. For each quantum-well thickness, the researchers looked at the experimental energy gaps for a series of filling factors and found an energy offset—a disorder energy—that made the extrapolated trend match what’s expected at the midpoint of the range of ν. The samples were relatively high quality; the disorder energy for the 70 nm well was 1.2 ± 0.2 K, half to a 10th of the values measured in previous studies on similar and other FQHE systems. Overall, the disorder energies had a scattered range of comparable values and no clear relationship with thickness.

When offset by the disorder energies, the energy gaps for thicker samples approximately matched the theoretical values, although thinner samples’ energies still fell short of theory. Getting better agreement between theory and experiment will require a more rigorous analysis of disorder and direct comparison with the data collected by Shayegan and his group. (K. A. Villegas Rosales et al., Phys. Rev. Lett. 127, 056801, 2021 .)

Thumbnail image credit: Reinhard Dietrich , CC0 1.0