Quantum Hall resistance standards from graphene grown by chemical vapour deposition on silicon carbide

Abstract

Replacing GaAs by graphene to realize more practical quantum Hall resistance standards (QHRS), accurate to within 10(-9) in relative value, but operating at lower magnetic fields than 10 T, is an ongoing goal in metrology. To date, the required accuracy has been reported, only few times, in graphene grown on SiC by Si sublimation, under higher magnetic fields. Here, we report on a graphene device grown by chemical vapour deposition on SiC, which demonstrates such accuracies of the Hall resistance from 10 T up to 19 T at 1.4 K. This is explained by a quantum Hall effect with low dissipation, resulting from strongly localized bulk states at the magnetic length scale, over a wide magnetic field range. Our results show that graphene-based QHRS can replace their GaAs counterparts by operating in as-convenient cryomagnetic conditions, but over an extended magnetic field range. They rely on a promising hybrid and scalable growth method and a fabrication process achieving low-electron-density devices.

Figure 1. Magnetic field dependence of the Hall quantization in graphene grown by CVD on SiC.

(a) Hall resistance deviation measured on the ν=2 plateau at 1.4 K (black), 2.2 K (magenta) and 4.2 K (violet). (b) Longitudinal (R_xx) and Hall (R_H) resistances (a 100-nA current circulates between I₁ and I₂ terminals, the longitudinal and Hall voltages are measured using (V₁, V₂) and (V₂, V₃) terminal-pairs, respectively.) for B varying from −1 to 19 T for the graphene sample (red and blue curves, respectively) and varying from 8 to 13 T for the GaAs sample (wine and grey curves, respectively). is the LL filling factor calculated from the carrier density n₀ determined at low magnetic fields. Inset: optical image of the sample with terminal labels, scale bar=100 μm. A very wide Hall resistance plateau is observed in the large Hall bar device. (c) Accurate measurements of the longitudinal resistance versus B at 1.4 K (black), 2.2 K (magenta) and 4.2 K (violet). Error bars represent combined standard uncertainties, given with a coverage factor k=1 corresponding to 1 s.d. A perfect quantization of the Hall resistance, without significant deviations with regards to the relative standard measurement uncertainty of 10⁻⁹, is observed over a magnetic field range of 9 T from 10 T. It coincides with values lower than (30±20) μΩ.

Figure 2. Dissipation in (B, I) space in the ν=2 plateau.

Colour rendering of the longitudinal resistance per square R_xx (measured using terminal-pair (V₃,V₅)) as a function of I (circulating between I₁ and I₂ terminals) and B at T=1.4 K. I_C(B) (black solid line) corresponds to the evolution of the critical current (breakdown current) leading to R_xx=0.25 mΩ as a function of B. The horizontal black dashed line indicates the current used for the accurate measurements. I_C(B) can be well adjusted by ξ(B)⁻² data (red square), where ξ(B) is the localization length. There is no significant dissipation in the QHE regime (R_xx<0.25 mΩ) for currents lower than 40 μA in the magnetic field range from 10.5 to 19 T.

Figure 3. Relationship between Hall and longitudinal resistances.

(a) ΔR_H/R_H (filled circles, left axis) and R_xx/R_H (empty circles, right axis) as a function of the temperature T for two magnetic fields B=10 T (blue) and B=19 T (red). (b) |ΔR_H/R_H| as a function of R_xx/R_H for several magnetic fields B in log-log scale. (c) ΔR_H as a function of R_xx in linear scales. A linear relationship, ΔR_H=−0.67 × R_xx independent of the magnetic field in the range from 10 to 19 T, is found. It is concluded that relative deviations of the Hall resistance from R_K/2 are smaller than 10⁻⁹ for longitudinal resistance values lower than 15 μΩ.

Figure 4. Analysis of the dissipation based on the VRH mechanism.

(a) Tσ_xx as a function of T^−1/2 in a semi-log scale for magnetic fields from 7 to 19 T and in a temperature range from 4 to 40 K. (b) Temperature parameter T₀ (black circles, right axis) and localization length ξ (blue squares, left axis) obtained from the adjustment of curves of a by the VRH model as a function of the magnetic field. The dissipation in the QHE regime is well described by a VRH mechanism with a soft Coulomb gap. The continuous decrease of the localization length as the magnetic field increases, without showing a minimal value, explains the robustness of the Hall resistance plateau towards high magnetic fields.

Figure 5. Current effect on dissipation.

(a) R_xx as a function of the current at 1.4 K for different magnetic fields. The horizontal black dashed line indicates R_xx=0.25 mΩ. (b) Effective temperature T_eff as a function of I (giving σ_xx(T_eff)=σ_xx(I)) for several magnetic fields B. The linear relationship between T_eff and I shows that the VRH mechanism with soft Coulomb gap explains the dissipation increase caused by the current.

Figure 6. Correlation between quantization and localization.

(a) R_xx versus B at T=25 K and I=1 μA where we observe the presence of a tiny minimum at ∼15 T. The ratio of the localization length to the magnetic length ξ/l_B (b) and the prefactor of the conductivity σ₀ (c) extracted from the VRH analysis as a function of B. The minimum of R_xx occurs at the highest magnetic field where both ξ/l_B and σ₀ have the lowest values. ξ(B) is locked to the magnetic length l_B within 10% over the magnetic field range, from 10 to 19 T, where the Hall resistance is accurately quantized with a 10⁻⁹-relative standard uncertainty.

Figure 7. Complementary structural and electronic characterization.

(a) Colour-scale map of the ARPES intensity of the sample after outgassing at 500 °C. The intensity is plotted as a function of binding energy E_b and momentum k_⊥ taken along the direction perpendicular to the direction in reciprocal space. The momentum reference is at the point. The photon energy was 36 eV. The light was p polarized. The black and red solid lines are fits for monolayer and bilayer graphene, respectively. (b) ARPES intensity taken at E_b=−1.2 eV, along k_⊥, evidences the small bilayer contribution at k_⊥=±0.19 Å⁻¹. These ARPES measurements show that a graphene monolayer covers the whole SiC surface, but ∼10% is covered by a second graphene layer. (c) Longitudinal (R_xx, in dashed lines) and transversal (R_H, in solid lines) resistances as a function of B for another Hall bar sample (400 μm by 1,200 μm) fabricated from another piece (5 × 5 mm² size) of the same graphene wafer as the first sample (same graphene growth run). The carriers density of the sample is n_s=3.3 × 10¹¹ cm⁻² and the carrier mobility is μ=3,300 cm² V⁻¹ s⁻¹. The observation of similar QHE and the measurement of similar electronic properties in the first and third samples demonstrated the large-scale homogeneity of the graphene growth and the repeatability of the fabrication process of samples having a few 10¹¹-cm⁻² n-doping.

Figure 8. Schema of the R_xx measurements using a CCC.

The sample is biased with a d.c. current I and a 2,065-turn winding of a CCC is connected to the two voltage terminals. The voltage drop V_xx gives rise to the circulation of a current i in the winding. The longitudinal resistance R_xx is given by R_xx=(i/I)R_H.