• cryogenics
• metrology
• Hall
• quantum
• magneto-transport

The Quantum Hall Effect (QHE) is the quantum limit of the Hall effect and its main manifestation is the quantization of the Hall coefficient in high-mobility two-dimensional electron systems, when subjected to low temperature conditions and to a strong perpendicular magnetic field. This effect emerges from the quantization of the cyclotron electron orbits into "Landau levels". The effect was discovered by Klaus von Klitzing while working on a high purity silicon-based MOSFET, and he observed this phenomenon in the MOSFET's inversion layer at liquid helium temperature and in a 15 T magnetic field. Quantized Hall conductance has found extremely useful metrological applications, since it occurs at integer multiples of $e^2/h$, where $e$ is the electron charge and $h$ is Plank's constant, i.e. it is a combination of two fundamental constants. Experimentally, resistance measurements yield this value with an astonishing precision (up to 12 decimal figures). Starting from the so-called metrological triangle, which involves the use of other quantum mechanical effects such as the Josephson effect and single electron transport, it was considered whether it is possible to obtain the values of $h$ and $e$ with metrological precision.
The importance of the resistance quantum value $R_K = h/e^2$ is not just connected to the fundamental constants but interesting as a resistance standard for electrical calibration. The quantum Hall resistance is equal to $R_H = R_K/\nu$, where the integer $\nu$ is called filling factor, which corresponds to the number of filled Landau levels. Ideally, one would like to have different filling factors on the same sample to achieve, in a single chip, different standards to calibrate resistances of different orders of magnitude. Many approaches have been proposed to achieve this goal. In particular: $(i)$ cryogenic current comparators (CCC), which allow current (and thus resistance) rescaling with metrological precision; $(ii)$ so called QHARSs (Quantum Hall Array of Resistance Standards), which consist of a network of Hall bars yielding a rational fraction q/p of the resistance $R_K$. Both these methods are affected by limitations. The networks used in QHARS rely on Ohmic contacts between individual Hall bars that introduce measurement errors due to stray voltages, parasitic resistances or other non-local effects. Differently, CCCs typically require an additional cryogenic system, and they can be affected by noise when trying to compare resistance values $<< R_K$. Finally, both these methods yield a fixed resistance standard and to obtain a different value one has to use a different QHARS or a CCC with a different number of windings. This thesis is based on a recently-proposed QH circuit architecture that, thanks to a precise and tunable voltage bisection scheme, can yield any four-wire resistance of the form $R=k2^n/(R_K/2)$. The integer $n$ indicates the number of bisection stages whose configurations set the value $1 <= k <= 2^n$. While the basic working principle of this QH circuit has been experimentally demonstrated, its ultimate precision is still unknown.

This thesis has two targets: $(i)$ to experimentally quantify the maximum bias voltage compatible with the QH effect, since this directly affects the top achievable precision; $(ii)$ to study the influence of a finite output resistance of the QH circuit as this again affects the precision of the resistance standards produced by the circuit.

We now summarize the main results of this thesis.
We study a novel quantum Hall circuit that uses an edge bisection scheme to obtain a custom fractional value of $R_K$. The edge bisection works with field-effect barriers (mixers or gates) on a GaAs/AlGaAs two dimensional electron gas (2DEG). During the thesis, Hall bar devices integrating these barriers have been fabricated and characterized at filling factors $\nu=2$ and $\nu=4$. The fractional values of $R_K$ obtainable via the bisection scheme have been assessed.
A key element to determine the precision of the studied QH bisection circuit is the maximum current or voltage bias that the Hall bar can handle before a QH breakdown. To study this effect it was important to check if we obtained the correct behaviour from the gates. We obtained the correct number of fractional values for $R_K$ and for this reason, the QH breakdown of the bar has been explored by biasing a single mixer with different voltage values. It was found that the device works safely at all filling factor configurations for voltage bias values < 10 mV and breakdown currents of the order of 1 muA. It is interesting to stress that the architecture studied in this thesis was found to be more susceptible to break down with respect to ungated structures. Even if this QH breakdown current value appears to be low compared to bias currents used in top precision metrology that do not have gating features, it can still be useful for calibration applications.
Unfortunately, the ultimate precision of the bisection mechanism could not be determined, presumably due to a parallel conduction of the sample, as revealed by SdH oscillations that do not display a dissipation-less regime ($\rho_{xx} = 0$) despite the cryogenic temperatures (T=300 mK) achieved with a $^{3}$He cryostat.
In addition, the influence of a finite output resistance on the QH circuit has been studied, both with a numerical and an analytical approach. In the last case, a resistive model of the QH circuit has been developed and discussed. From these studies we found a general formula for $R_{out}$ and shown that despite a growth as a function of $n$, $R_{out}$ remains limited even in the $n \rightarrow \infty$ limit. The resistive analytical model turns out (with some caveats) to be a very useful tool that allows to simplify the calculation of voltage drops for this kind of bisection circuit, by mapping them on a simple resistive network.