Graphene, universality of the quantum Hall effect and redefinition of the SI system

The Syst\`eme Internationale d'unit\'es (SI system) is about to undergo its biggest change in half a century by redefining the units for mass and current in terms of the fundamental constants h and e, respectively. This change crucially relies on the exactness of the relationships which link these constants to measurable quantities. Here we directly compare the integer quantum Hall effect in epitaxial graphene with that in GaAs/AlGaAs heterostructures. We find no difference of the quantized resistance value within the relative standard uncertainty of our measurement of 8.6\times10-11, being the most stringent test of the universality of the quantum Hall effect in terms of material independence.

The new quantum SI units for mass and current will be based on the fundamental constants of nature h, Planck's constant, and e, the electron charge. The confidence in the new definition crucially relies on the ability to experimentally confirm the exactness of the relationships which link these constants to measurable quantities. The quantum Hall effect (QHE) defines one such a relationship through the theoretical argument that the Hall resistance is quantized in units of h/Ne 2 where N is an integer. The QHE is a fascinating macroscopic quantum effect occurring in two-dimensional conductors that has become one of the cornerstones of the worldwide reference system for scientific and industrial measurements. 1 Yet, the hypothesis of resistance quantization units of h/Ne 2 and its independence of material implementation has to be tested experimentally. The appearance of an unusual half-integer variation of the QHE in graphene 2,3 confirmed the unique electrical properties of this 2-dimensional carbon material, where the charge carriers behave as massless Dirac fermions. As well as providing an experimental system for the study of new transport physics, graphene holds out the prospect of a more robust implementation of the QHE resistance standard. 4 We report here the result of a highest precision direct comparison of the quantized resistance, R=h/2e 2 , realised in an epitaxial graphene QHE sample with the matching N=2 plateau of the QHE in a traditional GaAs/AlGaAs heterostructure device. Demonstrating the equivalence of this resistance in different devices is a vital step in proving the suitability of graphene for metrological use, but is also a useful test of the theory that predicts no corrections to the simple relation R=h/Ne 2 . The quantum Hall resistance is considered to be a topological invariant, not altered by the electronelectron interaction, spin-orbit coupling, or hyperfine interaction with nuclei, and insensitive to much more subtle influences of gravity. 5 Recently, a quantum electrodymical correction to the von Klitzing constant of the order of 10 -20 has been predicted for practical magnetic field values. 6 The fundamental nature of the Hall resistance quantization makes experimental tests of its universality of the utmost importance, in particular, for improving our knowledge of two fundamental quantities of nature: the electron charge and Planck's constant. The precision obtained through a universality test as presented here is much greater than is possible by a comparison to the values of the constants h and e. 7 Analysis of the complete set of published results carried out by CODATA 7 showed no deviation from h/e 2 to within 2x10 -8 , which calls for more accurate measurements.
Soon after the first observations of the QHE in graphene 2,3 , Giesbers et al. 8 reported an evaluation of the accuracy of the resistance quantization in exfoliated graphene flakes. Unfortunately, the small size of the flakes and electrical contacts along with the low breakdown current in their devices made these measurements very difficult. An accuracy of only a few parts in a million could be obtained (4 orders of magnitude below the state-of-the-art in GaAs and Si) and so no meaningful conclusions on the universality of the QHE could be drawn. Our own work in Ref. 9 reported the first accurate observation of the QHE in large epitaxial graphene devices. We achieved an accuracy of 3 parts in 10 9 via an indirect method whereby both quantum Hall devices were separately measured against a room temperature standard resistor. Recently, we reported 10 an unusually strong pinning of the ν=2 quantum Hall state in epitaxial graphene due to charge exchange with the localised states in the substrate resulting in a very robust resistance quantization and demonstrated invariance of the resistance quantization to 0.3 parts in 10 9 over a field range of 3.5 T. Importantly for precision metrology, the extraordinarily robust quantum Hall state in these devices sustains very high non-dissipative currents ensuring a large signal-to-noise ratio.
Our graphene sample was produced by epitaxial growth on a SiC substrate 9 and shows the properties (such as low contact resistance and negligible longitudinal resistivity) required for accurate metrological use. Its resistance was compared to that of the GaAs device in a null measurement using the standard methods of resistance metrology. (The 4-terminal nature of QHE resistors means that some form of bridge circuit is needed, even to compare identical resistors; here a cryogenic current comparator 11 was used to establish an exact 1:1 current ratio.) A summary of the results is shown in Figure 1 (for details see Methods section). The weighted average of all our data is (R GaAs/AlGaAs -R Graphene ) / (h/2e 2 ) = (-4.7 ± 8.6) x 10 -11 . The relative standard uncertainty of 8.6×10 -11 represents a factor of 35 improvement on our prior result obtained via an indirect measurement. 9,10 In an indirect measurement the accuracy is limited by the properties of the resistor used as a transfer standard. 1 Here we directly compare both devices against each other thereby eliminating many systematic effects. Previously our knowledge of the universality of the QHE has been limited to the level of 2 or 3×10 -10 for comparisons between GaAs and Si or between identical GaAs devices. 1 However both GaAs and Si are traditional semiconductors with a parabolic bandstructure and governed by the same physics. Graphene is a semimetal with a linear bandstructure and is described by Dirac-type massless charge carriers and so universality in terms of material independence goes well beyond the comparison between two semiconductors. In our universality experiment the maximum source-drain current that the GaAs device can sustain without dissipation limits the measurement uncertainty, whereas a potentially lower uncertainty can be obtained in a consistency check of two graphene devices.
Our results on material independence is the strongest evidence yet of the hypothesis that the resistance is quantized in units of h/Ne 2 is correct and thereby supports the pending redefinition of the SI-units for kilogram and ampere in terms of h and e. 12 Judging from the robustness of the quantization and wide operational parameter space, epitaxial graphene should be the material of choice for quantum resistance metrology.

Methods
The epitaxial graphene sample used in the reported experiment was produced on the Si-face of SiC. 9 The graphene Hall bar was encapsulated in a polymer bilayer, a spacer polymer followed by an active polymer able to generate acceptor levels under UV light. More fabrication details can be found elsewhere. 13 The sample had an electron density, n S, of 4.6x10 11 cm -2 and mobility, µ, of 7500 cm 2 V -1 s -1 . Note that this mobility is rather low compared to that achieved in exfoliated or suspended graphene and much lower than that obtained in the best GaAs. Fortuitously, in the quantum Hall effect disorder is in fact necessary to provide localisation of the electron states and for precision metrology the mobility should not be too high in order to provide a wide quantum Hall plateau. A standard 8-contact Hall bar geometry was patterned on the device with dimensions 160 µm x 35 µm. The graphene sample was placed in system 1 at 300 mK and 14 Tesla. The two GaAs samples used were traditional GaAs-AlGaAs heterostructures obtained from the PTB (device 1) and LEP (device 2).
Device 1 had n S = 4.6x10 11 cm -2 and µ = 4x10 5 cm 2 V -1 s -1 , the size of the chip was 6000 µm x 2500 µm and contacts were made from small tin balls at the edge of the chip. Device 2 had n S =5.1x10 11 cm -2 , µ = 5x10 5 cm 2 V -1 s -1 ; the chip had an etched Hall-bar geometry of 2200 µm x 400 µm and AuNiGe alloyed contacts. Both GaAs devices were placed in system 2 at 1.5 K and either 9.5 T (device 1) or 10.5 T (device 2). Before commencing the high-accuracy measurements all devices were fully characterised according to the guidelines for quantum Hall resistance metrology 1 (i.e. we confirmed that the three-terminal contact resistance measured on the N=2 plateau was of the order of a few ohms for all contacts used and that the longitudinal resistivity at the measurement current was below 10 µΩ). For the graphene device the maximum source-drain current, I C , at which the device remains in the non-dissipative state was approximately 500 µA. For the GaAs devices I C was ≈150 µA for device 1 and ≈100 µA for device 2.
The measurements were made with a cryogenic current comparator (CCC) bridge 11 illustrated in simplified form in Fig. 2. Isolated current sources 1 and 2 separately drive current through samples S1 and S2 and associated windings A and B on the CCC. The current ratio can be set via electronics to a few parts in 10 6 and this ratio is improved to a level of 1 part in 10 11 by forming a negative feedback loop from the SQUID sensing the net flux in the CCC to one of the current sources. The potential contacts on S1 and S2 are closed in a loop via winding C on a second CCC. This device is configured with just a single winding to measure a current null rather than two windings to establish a current ratio. Data are collected alternately in forward and reverse current direction so as to eliminate electrical offsets. Measurement uncertainty arises from leakage currents in the connecting cables, residual error in the ratio A:B, accuracy of the negative feedback loop and random noise. The random noise of 8.7 parts in 10 11 dominates over the other components, estimated to have a combined standard uncertainty of 1.6 parts in 10 11 .