Graphene quantum Hall effect parallel resistance arrays

As first recognized in 2010, epitaxial graphene on SiC(0001) provides a platform for quantized Hall resistance (QHR) metrology unmatched by other two-dimensional structures and materials. Here we report graphene parallel QHR arrays, with metrologically precise quantization near $1000\mathrm{\Omega }$. These arrays have tunable carrier densities, due to uniform epitaxial growth and chemical functionalization, allowing quantization at the robust $\nu =2$ filling factor in array devices at relative precision better than ${10}^{-8}$. Broad tunability of the carrier density also enables investigation of the $\nu =6$ plateau. Optimized networks of QHR devices described in this work suppress Ohmic contact resistance error using branched contacts and avoid crossover leakage with interconnections that are superconducting for quantizing magnetic fields up to 13.5 T. Our work enables more direct scaling of resistance for quantized values in arrays of arbitrary network geometry.

Figure 1

Device layout and features in a quantizing magnetic field. (a) Each of 13 array elements is 0.4 mm $\times$ 0.2 mm monolayer EG and has multiple contact branches. (b) Single array element contact layout, with a Pd/Au underlayer that provides EG surface adhesion and ensures a normal metal interface with NbTiN superconductor. (c) With strong quantization, boundaries of high and low electrochemical potential surround the EG, beginning at the two hotspots created by the power dissipation at the source and drain. These are labeled equipotential, although small changes may occur where current enters or leaves the device at intermediate contacts. The positions of hotspots depend on the orientation of $B$. In one field direction $B+$ (magnetic vector pointing into the page; equipotential boundary shown by the green line), the Hall voltage is nearly perpendicular to the source-drain current vector. This voltage is labeled $V_{H+xx}$ to account for $R_{xx}$ contributions due to the small diagonal offset of the voltage probes, as described in the text. The opposite field direction $(B-)$ (equipotential boundary shown by the dashed orange line) causes the voltage and current contact points to exchange positions and increases the sensitivity to longitudinal resistance due to the width of the regions labeled source and drain.

Figure 2

Resistance of array device 2 as a function of magnetic field $B$ measured using a current source and a digital voltmeter. The symmetric resistance profile for $B-$ and $B+$ typifies the combined voltage and current interconnections of a parallel array.

Figure 3

Symmetric cryogenic current comparator bridge data for two devices at 1.6 K, showing precise reciprocity with magnetic field reversal. Device 1 does not reach full quantization at $\left|B\right|=9\mathrm{T}$. The results obtained on device 2 are quantized at 7.5 and 9 T, with deviation from the quantized value $(-0.45\pm 4.47)\mathrm{n}\mathrm{\Omega }/\mathrm{\Omega }$ when averaged for both field directions. Standard deviations are smaller than the size of the markers.

Figure 4

(a) As-grown graphene. (b) Protection layer deposition. (c) Sacrificed layer deposition. (d) Definition of the device pattern with ion milling. (e) Superconductor electrode deposition in a sputter chamber. (f) Removal of the sacrificial layer with dilute aqua regia.

Figure 5

Measured QHARS resistance using a current source and digital voltmeter, for several carrier density levels ($n_0$). Starting with the highest resistance at $B=0$ (green), heating in vacuum was used to increase $n_0$ to around $7\times {10}^{11}{\mathrm{cm}}^{-2}$ (black). Subsequent heating cycles in vacuum were used to produce the blue curve with $n_0\approx 1.6\times {10}^{12}{\mathrm{cm}}^{-2}$ (not discussed in the text), and the yellow curve with $n_0\approx 1.7\times {10}^{12}{\mathrm{cm}}^{-2}$, where a broad plateau is observed near $330.93\mathrm{\Omega }$.

Figure 6

The region near $330.93\mathrm{\Omega }$ for the black curve of Fig. 5, with $n_0\approx 7\times {10}^{11}{\mathrm{cm}}^{-2}$. The $\nu =6$ plateau is not visible, but these points show the minimum slope region measured using the DCC bridge at resistance levels near $R_{\text{K}}/78$. The slope of the fitted line is $\approx 54.5(\mathrm{m}\mathrm{\Omega }/\mathrm{\Omega })/\mathrm{T}$ and the plateau center was estimated as $B\approx 4.8\mathrm{T}$.

Figure 7

The $\nu =2$ plateau measured at selected values of $B$ for device 1 at two temperatures (0.35 and 1.6 K) in a ${}^3\mathrm{He}$ cryostat using a room-temperature direct current comparator (DCC). The carrier density was determined as $n_0\approx 7\times {10}^{11}{\mathrm{cm}}^{-2}$ as shown by the black curve in Fig. 5. The inset shows data with a much finer scale, where the quantized plateau is seen to begin around 12.8 T for $B+$ and 13 T for $B-$. In general, the resistance values for $B+$ are closer to the quantized value for field values shown in the inset, but at somewhat lower fields the $B-$ results are displaced closer to the plateau by increased longitudinal resistance.

Figure 8

Resistance quantization measurements for device 1 at $T\approx 0.35\mathrm{K}$. Standard deviations are smaller than the size of the markers, as derived from DCC bridge ratios. (a) The $B+$ and $B-$ resistance for the $\nu =6$ plateau near $R_{\text{K}}/78\approx 330.933429\mathrm{\Omega }$, where fitted lines are offset from the quantized value over a range of 1 T, with the scale in $\mathrm{m}\mathrm{\Omega }/\mathrm{\Omega }$. (b) The $\nu =6$ resistance differs from the quantized value by $(0.247\pm 0.054)\mu \mathrm{\Omega }/\mathrm{\Omega }$ for $I=20\mu \mathrm{A}$ and by $(0.402\pm 0.027)\mu \mathrm{\Omega }/\mathrm{\Omega }$ for $I=50\mu \mathrm{A}$ over a range of 0.6 T in $B+$, from 11 to 11.6 T; the scale is in $\mu \mathrm{\Omega }/\mathrm{\Omega }$. For the plateau center of 11.2–11.4 T, the measured deviation is $(0.171\pm 0.076)\mu \mathrm{\Omega }/\mathrm{\Omega }$ at $20\mu \mathrm{A}$ and $(0.264\pm 0.078)\mu \mathrm{\Omega }/\mathrm{\Omega }$ at $50\mu \mathrm{A}$. The expanded uncertainties are for a $2\sigma$ confidence interval. (c) Comparing the magnetic field reciprocity at the $\nu =2$ and 6 filling factors.

Figure 9

Current dependence. (a) The $\nu =6$ region, for device 1 and device 2, measured at two carrier density values. Quantization at the $\nu =6$ plateau was not attained for device 2, even at the higher carrier density. (b) The $\nu =2$ plateau resistance in device 1 measured at $B=-13\mathrm{T}$ with source-drain current levels from 100 to $1000\mu \mathrm{A}$ for the carrier density $n_0\approx 7\times {10}^{11}{\mathrm{cm}}^{-2}$. Inset: The values obtained at and below $300\mu \mathrm{A}$ are consistent with full quantization at $R_{\text{K}}/26$ with error bars showing the standard uncertainty of the DCC bridge data. The increase in the deviation at higher currents is proportional to the increase in power dissipation, or $I^{2}R$.