the electron gas. While the discovery of the quantum Hall effect was made using silicon field-effect transistors, where a potential on the gate forced the electrons in a narrow 2d channel, most metrological applications used Gallium-Arsenide/ Gallium-Aluminum-Arsenide heterostructures. Here, the electrical band structure is engineered such that the electrons are confined at a 10 nm thin interface layer between the two materials. The research activity in the metrological application of the quantum Hall effect received a large impetus in 2004 with the discovery of graphene [17]. Graphene is a naturally 2D material and shows a much sparser set of submultiples of RK, which helps the quantized resistance to be maintained at higher temperatures. In recent years, the research in graphene devices for resistance realization has made significant strides. Initially, the devices were made from exfoliated graphene. This tedious process involved carefully isolating a single layer from natural crystals of graphite and allowed only for small devices capable of handling tiny currents. These days, epitaxial graphene layers grown on Silicon-Carbide substrates are widely used to fabricate devices. The graphene-based quantum Hall devices operate at lower magnetic fields, higher currents, and larger temperatures than the GaAs devices. Hence, the graphene devices are more practical and less costly to operate than the early conventional devices. For example, a graphene device can operate at a temperature of 4 K, and a magnetic flux density of 5 T, while a conventional GaAs might require 1.2 K and 9 T. While the differences in the respective operating parameters are small, the technical benefits are enormous: the graphene device can be operated in a closed-cycle cryostat, and this overall decrease in complexity and cost enables primary laboratories to acquire and operate their own quantum Hall standard. In fact, at NIST, a table-top graphene quantum Hall device has been used as part of the calibration services in resistance metrology since 2017. There are two exciting developments in this research field that we can look forward to in 2020: pn-junctions and topological insulators [18]. While the order of magnitude of RK/2 is practical, the numerical value 12.906 kΩ, is far away from the decadal sequence of standard resistors. An interesting solution to this problem is an array device that combines parallel and/or serial connection of quantum Hall devices. With gated graphene devices, researchers can control the density and even the polarity of charge carriers. For example, by applying an electric potential to gates electrically insulated from graphene, charge carrier density in the graphene can be adjusted (Fig. 6). With large enough electric potential difference on gates in different regions of the graphene, pn-junctions are created. Such junctions allow either multiples or fractions of RK. Hence, by changing the voltages on the various top gates the total resistance of the device can be programmed. Clearly, a programmable quantum Hall effect will have a huge benefit for the realization of the ohm. Broadly speaking, the quantum Hall effect falls in the category of topological quantum effects. In another class of 16 Fig. 6. An example of a complex pn-junction graphene device the potential of the top gates can be changed to externally program various values of resistance (image courtesy of Albert Rigosi, NIST). materials, called topological insulators, electrical conduction occurs strictly along the surface and not in the bulk of the material. In the quantum Hall devices, the current travels along the edge of the device, in a so-called edge state, because the strong magnetic field restricts where current flow can occur. Interestingly, some new composite materials such as Cr-doped (Bi,Sb)2Te3 are topological insulators and can maintain similar edge effects without external magnetic fields. Several reports on observing the quantum Hall effect in such materials have been published in the literature. Currently, the required temperatures to achieve the quantum anomalous Hall effect in topological insulators are very low, tens of mK [9]. A focus of research is to find material systems that are topological insulators at higher temperature without an external magnetic field, which would substantially simplify the realization and dissemination chain of the unit of resistance. So far, we discussed the application of the quantum Hall effect in direct current applications. What about applications in alternating current circuits? The dc unit of resistance can be transferred to ac by using calculable resistors [20]. These devices can be calibrated at dc values, and the calibration value can be extended to low frequencies because the frequency dependence of the resistor can be calculated from first principle. Once an ac value of a resistance is known, it can be transferred to capacitors and inductors using quadrature bridges. A second way to realize the farad is via the calculable capacitor, also known as the Thompson-Lampard capacitor [21]. In this capacitor the capacitance per unit length is given by C 0 = ln 2, (17) L π where ϵ0 is the electric constant. The Thompson-Lampard capacitor is an electro-mechanical device that is similarly complex as the Kibble balance. In theory, both the Kibble balance and the Thompson-Lampard capacitor seem simple, but most of the complexity of each experiment lies in managing the small imperfections that the real apparatus has in contrast to the idealized geometry on paper. Hence, using the quantum IEEE Instrumentation & Measurement Magazine May 2020

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