Magnetic field detection limits for ultraclean graphene Hall sensors

Solid-state magnetic field sensors are important for applications in commercial electronics and fundamental materials research. Most magnetic field sensors function in a limited range of temperature and magnetic field, but Hall sensors in principle operate over a broad range of these conditions. Here, we evaluate ultraclean graphene as a material platform for high-performance Hall sensors. We fabricate micrometer-scale devices from graphene encapsulated with hexagonal boron nitride and few-layer graphite. We optimize the magnetic field detection limit under different conditions. At 1 kHz for a 1 μm device, we estimate a detection limit of 700 nT Hz−1/2 at room temperature, 80 nT Hz−1/2 at 4.2 K, and 3 μT Hz−1/2 in 3 T background field at 4.2 K. Our devices perform similarly to the best Hall sensors reported in the literature at room temperature, outperform other Hall sensors at 4.2 K, and demonstrate high performance in a few-Tesla magnetic field at which the sensors exhibit the quantum Hall effect.

0.35 (1300, 2000) (440, 700) 4 K InGaAs [8] 1.5 800 1200 300 K InSb [5] 0.6 12000 7200 300 K InSb [7] 1.5 600 800 300 K InAsSb c [11] -58 -300 K Si [4] 5 1000 5000 300 K 10 1000 10000 300 K 20 470 9400 300 K Bi [1] 0.05 80000 4000 300 K Supplementary Table 1. Lateral size w and magnetic field detection limit S 1/2 B at 1 kHz extracted from the literature, used in Figure 1 in the main text. Entries expressed as a pair of numbers are estimates of lower and upper bounds (see footnotes). a S 1/2 B extrapolated to 1 kHz assuming 1/f noise scales as f −α (0.4 < α < 0.6). b Width inferred from optical image. c Ref. [11] does not clearly state the size of the device for which the detection limit is reported, and we do not include this work in Figure 1 in the main text. Although the authors show an image of a device with w = 1 µm, we estimate from the reported carrier density and carrier mobility that this device would have series resistance > 14 kΩ, a factor of 10 larger than the resistance stated for the device exhibiting the reported detection limit.

Supplementary Note 1. Performance of additional devices
Supplementary Table 2  In most devices, the charge inhomogeneity δn is comparable to that reported previously in hBN-encapsulated graphene devices [12,13]. We speculate that poorly screened charge disorder from the etched device edges increases the effective δn for G2 and G3 as compared to G1 [14,15]. Assuming that electrons and holes each contribute δn/2 to the total charge inhomogeneity, the Fermi wavelength is λ F = 2π/ π(δn/2). If λ F is bounded by the device size w = 1 µm, we can estimate δn ∼ 10 10 cm −2 for G2 and G3 in agreement with our measurements.
The effective δn (indicated by the maximum Hall coefficient R max H ) depends on both gating and bias conditions. As we discuss in the main text, increasing the bias current Supplementary Table 2. Summary of the performance of additional devices at 4.2 K. δn is estimated from the width of the peak in the two-point resistance.
Supplementary Figure 1. Characterization of additional devices. a Optical images of the devices summarized in Table 2. b-c Hall coefficient (R H ) measurements of devices under 100 nA DC bias at 4.2 K. The insets illustrate the layer structure of the devices represented in each panel.
d-e Magnetic field detection limit S 1/2 B at 1 kHz and 4.2 K, shown for the current that yields the minimum S 1/2 B for each device. We report measurements for both graphite-gated (b, d) and metal-gated (c, e) devices. f Current bias dependence of peak R H . g Reduction in peak R H upon applying voltage to the silicon gate of M1. h Reduction of dc two-point resistance and peak voltage noise at 1 kHz upon applying silicon gate voltage to G1.

Supplementary Note 2. Quantum hall resistance plateaus at low bias current and magnetic field
The appearance of well-defined quantum Hall resistance plateaus at low magnetic fields is a clear signature of small charge inhomogeneity in high-quality graphene devices [16].
In most of our measurements, we consider the regime of large bias current to increase the Hall voltage and do not observe resistance plateaus developing until ∼500 mT (Fig. 5a).
At low bias current, the exceptionally small charge inhomogeneity is evident through the appearance of quantum Hall resistance plateaus developiung at magnetic field as low as

Supplementary Note 3. Carrier density gradient under large current bias
Applying a large bias current to our devices strongly modifies the relationship between Hall coefficient and gate voltage. Here, we show that our measurements are consistent with carrier density gradients resulting from the large bias current.
We consider an L × L square device with contacts spanning the entire length of each The electron (n g ) and hole (p g ) densities away from the CNP depend on the potential difference between the gate and the graphene layer: where C g is the gate capacitance. Accounting for charge inhomogeneity δn near the Dirac point, the electron and hole densities become [17] n(x) = n g + n 2 g + δn 2 2 p(x) = p g + p 2 g + δn 2 2 .
Noting n 2 g = p 2 g and n g + p g = 0, the total carrier density is: Finally, using the resistivity ρ −1 = eµ(n + p), the Ohmic potential drop is given by: Solving this differential equation numerically with initial condition ψ(L) = 0 reveals that the potential ψ(x) drops nonlinearly along the device channel (Supplementary Figure 3b). We extract the electron and hole densities n(x) and p(x) (Supplementary Figure 3e), two-point resistance R 2p = ψ(0)/I, and average Hall coefficient R H using a two-carrier magnetoresist-ance model and average electron and hole densities in the channel [18]: Our calculation (Supplementary Figure 3d)  In our devices, the white Johnson noise is always smaller than the charge noise contriubtions from 1/f and random telegraph noise. The Johnson noise spectral density T R is at most ∼10 nV Hz −1/2 for a maximum R 2p of ∼250 kΩ at liquid-helium temperature (main text) or ∼18 nV Hz −1/2 for ∼20 kΩ at room temperature (Supplementary Figure 5c). Ref. [24] notes that because Johnson noise is independent of sensor size, the corresponding magnetic field detection limit S 1/2 B ∝ w −1 , similarly to 1/f noise.

Random telegraph noise
Although the general behavior of our devices remains the same between cooldowns, the specific amplitude of RTN and gate voltage region over which it is significant tend to change.
To illustrate this, we present noise measurements taken during two successive cooldowns, one in which RTN is only present for a small range of gate voltages and another in which RTN is almost completely absent. These measurements are performed in the same way as in the main text, but the wiring used for these measurements involves twisted pairs which add a parasitic capacitance to ground that may suppress the noise slightly at frequencies approaching 1 kHz.
In The total voltage noise spectral density can be modeled using [25] S V = 4δV 2 τ 1 + τ 2 where f is the frequency, τ −1 = τ −1 1 + τ −1 2 , A is the flicker noise amplitude, and α ∼ 1. We fit the uppermost spectrum in Supplementary Figure 4e Figure 5b), we observe that R max H shows weak temperature dependence at low temperature and decreases as T −2 at high temperature. Modeling the potential fluctuations due to charge disorder as a Gaussian distribution with amplitude ∆, the charge inhomogeneity at the Dirac point is approximately [27] whereh is the reduced Planck constant, v F = 10 6 m/s is the Fermi velocity, and k B T is the thermal energy. In Supplementary Figure 5b, we plot (δn(T )e) −1 for ∆ = 9 meV (closely matching the 10 nA data) and ∆ = 32 meV (closely matching the 20 µA data). For small bias, the crossover into the T −2 regime occurs at a lower temperature than predicted by the model, likely due to reduction of R H via thermal activation of holes [28].
At room temperature (∼300 K), we perform full characterization of device G1 using the same cryostat insert used for low-temperature measurements, instead positioned between the poles of a C-frame electromagnet (GMW Associates, model 5403). Notably, the bias current has little effect on R H below ∼20 µA because the thermal charge inhomogeneity exceeds the additional effective inhomogeneity from the bias current (Supplementary Figure 5c).
Supplementary Figure 5d illustrates that S  Figure 5. Influence of temperature on Hall coefficient and detection limit. a R H measured as a function of temperature. b Temperature dependence of peak R H (markers) and comparison to the theoretical temperature dependence of charge inhomogeneity (solid curves). c R H and R 2p at room temperature for 20 µA bias current. d S 1/2 V and S 1/2 B at room temperature. All measurements are performed on device G1.