Quantum resistance standard accuracy close to the zero-dissipation state

We report on a comparison of four GaAs/AlGaAs-based quantum resistance standards using an original technique adapted from the well-known Wheatstone bridge. This work shows that the quantized Hall resistance at Landau level filling factor $\nu =2$ can be reproducible with a relative uncertainty of $32\times {10}^{-12}$ in the dissipationless limit of the quantum Hall effect regime. In the presence of a very small dissipation characterized by a mean macroscopic longitudinal resistivity $\overline{R_{xx}(B)}$ of a few $\mu \mathrm{\Omega }$, the discrepancy $\mathrm{\Delta }{R}_{\mathrm{H}}(B)$ measured on the Hall plateau between quantum Hall resistors turns out to follow the so-called resistivity rule $\overline{R_{xx}(B)}=\alpha B\times d(\mathrm{\Delta }{R}_{\mathrm{H}}(B))\mathrm{/}dB$. While the dissipation increases with the measurement current value, the coefficient $\alpha$ stays constant in the range investigated ($40-120\mu \mathrm{A}$). This result enlightens the impact of the dissipation emergence in the two-dimensional electron gas on the Hall resistance quantization, which is of major interest for the resistance metrology. The quantum Hall effect is used to realize a universal resistance standard only linked to the electron charge \emph{e} and the Planck’s constant \emph{h} and it is known to play a central role in the upcoming revised \emph{Syst`eme International} of units. There are therefore fundamental and practical benefits in testing the reproducibility property of the quantum Hall effect with better and better accuracy.