The performance limits of epigraphene Hall sensors

Epitaxial graphene on silicon carbide, or epigraphene, provides an excellent platform for Hall sensing devices in terms of both high electrical quality and scalability. However, the challenge in controlling its carrier density has thus far prevented systematic studies of epigraphene Hall sensor performance. In this work we investigate epigraphene Hall sensors where epigraphene is doped across the Dirac point using molecular doping. Depending on the carrier density, molecular-doped epigraphene Hall sensors reach room temperature sensitivities $S_V=0.23 V/VT$,$S_I=1440 V/AT$ and magnetic field detection limits down to $B_{MIN}=27$ $nT/\sqrt{Hz}$ at 20 kHz. Thermally stabilized devices demonstrate operation up to $T=150$ $^oC$ with $S_V=0.12 V/VT$, $S_I=300 V/AT$ and $B_{MIN}\approx 100$ $nT/\sqrt{Hz}$ at 20 kHz.

Based on the classical Hall Effect, solid-state Hall sensors represent a large portion of magnetometers which are extensively used in automotive, marine and consumer electronics applications. Hall sensors based on silicon see widespread use owing to well-established and lowcost production methods, 1-3 but increasing requirements placed on improved magnetic performance or resilience to harsh conditions like high temperatures, demand the exploration of other, even more suitable materials. 4 Hall sensors detect magnetic fields by measuring the Hall voltage VH induced by an external field B. High device sensitivity implies a large magnitude of VH response to an external field, for a given bias current IB or voltage VB. This leads to two important material-related metrics: the current-related sensitivity SI=|VH/(BIB)| (V/(AT)), which is essentially determined by the Hall coefficient RH (/T), and the voltage-related sensitivity SV=|VH/(BVB)| (V/(VT)) which is ultimately limited by the carrier mobility (m 2 /(Vs)), where XX is sheet resistance.
Graphene appears to be a natural candidate for highly sensitive Hall elements due to its high mobility, and the possibility to tune carrier density n down to zero towards charge neutrality (Dirac point). Low carrier density is desirable because it increases the Hall coefficient, = 1/( ). 5,6 Moreover, since the mobility = / of graphene is inversely proportional to carrier density as ∝ 1/√ , 7 decreasing n towards neutrality would increase both SI and SV. In principle, low n leads to an increase in XXwhich follows the relation ∝ 1/ , in the limit where charged impurity scattering dominates (Supplementary S1). 8,9 Yet, decreasing n can actually lead to a lower magnetic field detection limit, B MIN = (I B R H ) ⁄ (T/√Hz), where VN is the voltage noise spectral density (V/√Hz). If Johnson-Nyquist noise dominates, then = ∝ √4 , with Boltzmann constant kB, temperature T, and the detection limit scales as B MIN ∝ R H ⁄ ∝ √ for a fixed IB. Disorder in real graphene samples prevents it from reaching true charge neutrality, but high-quality graphene can approach low carrier densities. 10 The highest quality graphene is obtained by mechanical exfoliation of graphite and encapsulation in hexagonal boron nitride (hBN-G). As Hall sensor, hBN-G has shown ultra-high device sensitivities, and detection limits comparable to that of silicon. 11 However, this approach serves only as a proof of principle of the capabilities of graphene Hall sensors since device fabrication cannot be scaled-up. Graphene grown using chemical vapor deposition (CVD) is a more scalable technology which also can reach high sensitivities, but reported performance varies greatly, [12][13][14] perhaps due to variability in material growth and the need for subsequent transfer to suitable substrates. 15 Epitaxial graphene on SiC substrate (epigraphene) is another attractive scalable technology. The insulating substrate allows for direct mass-fabrication of devices over wafer-scales, 16,17 forgoing the need for graphene-transfer thus increasing reproducibility and yield. Epigraphene is also compatible with operation at temperatures exceeding common industrial requirements. 18,19 Despite these advantages, epigraphene remains relatively unexplored for Hall sensing in literature, 18 possibly owing to the difficulties in tuning carrier density due to high intrinsic n-doping, pinned by the substrate. [20][21][22] We report the exploration of the performance limits of epigraphene Hall sensors for varying doping levels across the Dirac point. Carrier density control is enabled by a molecular doping method using electron acceptors F4TCNQ assembled on the surface of epigraphene. 23 Devices doped using this method have already shown excellent electrical properties and low charge-disorder, albeit at low temperatures. 24,25 We investigate Hall sensor figures of merit BMIN, SV, and SI, and finally thermal stability in ambient conditions from room temperature and just above 200 ˚C. Furthermore, we establish the limits for optimal operation of epigraphene Hall devices under realistic operational conditions.
Epigraphene was grown on 4H-SiC chips encased in a graphite crucible and heated using RF heating to around 1850 ˚C in an inert atmosphere of 1 bar argon. 16 Transmission mode microscopy was used to select only samples with over 90 % monolayer coverage. 26 Device fabrication used standard electron beam lithography. Epigraphene is removed using oxygen plasma etching and the metal contacts are deposited using physical vapor deposition of 5 nm Ti and 80 nm Au. The finished device is spin-coated with molecular dopants and the final carrier density is tuned by annealing at T=160 ˚C, with varying annealing time depending on the desired final doping level. 23 Electrical characterization was performed primarily using the Van der Pauw (VdP) method, with samples measured at room temperature and under ambient conditions unless otherwise stated. A magnetic field perpendicular to the chip surface was applied using a coil electromagnet up to 100 mT). Noise measurements where performed by taking the power spectral density (PSD) using a voltage amplifier DLPVA-100-F-D from Femto, with bandwidth limited to 100 kHz and measured input noise level of 9 nV/√Hz. High-field measurements were performed in PPMS (Quantum design) cryostat (2-300 K) with a superconducting magnet providing fields up to 14 T. For heating experiments, the sample was mounted using epoxy on a ceramic heater, and temperature was monitored using a Pt100-resistor.
Seven epigraphene Hall sensors ( Fig. 1(a)), spread across four chips, were investigated in total. They were designed using symmetric square or cross shaped geometries optimized with respect to SV. 27,28 Cryogenic measurements on a molecular-doped sensor demonstrates a full transition to half-integer Quantum Hall regime, with vanishing sheet resistance XX and quantized Hall resistance = ℎ/2 2 ( Fig. 1(b)). These measurements verify that the devices are made of highquality monolayer graphene with uniform doping. Hall measurements of the transversal resistance = / serve as basis for the evaluation of epigraphene Hall magnetometers. Hall coefficient, carrier densities, and mobilities are calculated from measurements in low magnetic fields (B<0.5 T) as = / , = 1/( ), and = / , respectively. For the low-field range, the linearity error of RXY is below 1 %, which is determined by the percentage deviation of the raw data from the low-field linear fit ( Fig. 2(a)). The samples were tested up to B=13 T at room temperature. For low doping (RH=1284 /T, n=4.9x10 11 cm -2 ) the transversal resistance remains within 5% error in a range of B=±1.2 T, but for higher doping (RH=949 /T, n=6.6x10 11 cm -2 ) the 5% error range increases to B=±6 T. Figure  2(b) shows a summary of the carrier densities achieved in our experiments. The gap in data near charge neutrality (n=0) indicates the disordered charge-puddle regime, characterized by a nonlinear low-field RXY. 23 At room temperature the maximum measured values of RH and  are RH=1440 /T and 2300 cm 2 /(Vs), respectively. In terms of charge disorder, at room temperature, epigraphene is in puddle regime for doping levels |n|<5x10 11 cm -2 thus setting the maximum RH attainable in our epigraphene samples. Fig. 2(c) shows the linearity of VH at 100 mT up to 6 mA bias current, measured for highly (n=1.6x10 12 cm -2 ) and lowly (n=4.5x10 12 cm -2 ) doped devices. We find that for all carrier densities the current-voltage (I-V) characteristic is linear within 5% error for IB< 2.5 mA. The non-linearity is expected to be due to self-heating. For all subsequent measurements we limit the bias current to below 1.5mA to ensure a linear I-V behavior within 2% error. The measurements in magnetic fields are complemented with noise measurements to unveil the minimum detection limit BMIN. Fig. 3(a) shows the low-bias (IB=10 A) voltage noise spectral density VN measured at the Hall voltage terminals for different doping levels. In the low bias regime, the corner frequency of 1/f noise is around ~30 Hz. As epigraphene approaches the Dirac point, the sheet resistance of the devices increases as ∝ 1/  and consequently the larger input and output resistance of the devices increases thermal noise. Dotted lines in Fig. 3 are the thermal voltage noise VTH calculated using measured input resistance. The agreement with experimental noise data points to the fact that, at low bias, thermal noise dominates in our sensors. Fig. 3(b) shows the increase of the 1/f noise contribution at larger bias currents, which nearly follows the Hooge's empirical relation with Hooge parameter ≈ 0.015 (Fig 3( 30 implying that the excess noise is mostly due to resistance fluctuations. In practical devices, the excess noise can be alleviated by using spinning Hall current measurement techniques. 29

FIG. 3. (a) Noise performance for one Hall sensor measured at different doping levels. The dotted lines are calculated noise levels assuming pure thermal noise of a resistor. (b) Measured voltage noise spectral density vs bias current in another lowly doped device. Inset: The noise amplitude vs bias current at two different frequencies.
The measured sensitivities for epigraphene Hall sensors and their dependence on doping, collected across all measured devices, are summarized in Fig. 4(a). The highest SI is reached for low doping levels, close to the puddle regime (n~5x10 11 cm -2 ). The highest SV occurs slightly outside the puddle regime, at doping levels n~6x10 11 cm -2 . We have performed full noise spectrum characterization (e.g. Fig. 3b) for four doping levels to obtain B MIN = (I B R H ) ⁄ , which includes not only intrinsic noise of epigraphene (thermal and 1/f noise) but also amplifier noise. Fig. 4(b) shows  as a function of IB, measured at a frequency of 3 kHz for fair comparison to other graphene devices reported in literature. The best BMIN=47 nT/√Hz is attained at lowest doping n~5x10 11 cm -2 , for IB=400 A. At higher frequencies, where the 1/f noise contribution is lower, BMIN can be naturally lower with BMIN=27 nT/√Hz, for n~5x10 11 at 20 kHz (inset Fig. 4(b)).

FIG. 4. (a) SI (orange region) and SV (purple region) versus RH compiled from 7 Hall sensors across 4 chips (Sq=square shaped, Cr=cross shaped). The sequence of data points span high to low doping (starting from the leftmost point).(b) BMIN versus bias current calculated directly from measured noise data for 3 kHz. Inset also shows data for 20 kHz. (c) Investigation of thermal stability of RH by measuring RH at elevated sample temperatures, for different initial room temperature doping. The error bars represent 2 standard deviations for measured RH averaged over 10-15 min of measurements. The solid lines are linear fits taken at the lower temperatures before permanent doping change is induced.
Finally, we describe the thermal stability of the molecular-doped Hall sensor through the temperature coefficient ΔT, defined as the percentage change of RH from its room temperature value per degree Celsius. Fig. 4(c) shows that samples doped close to neutrality (RH=1400 T) are stable up to T=80 ˚C (Supplementary S3), with a temperature coefficient ΔT=-0.6%/˚C. We achieve highest thermal stability with samples annealed for ~4 hours at T=160 ˚C, after which the RH reached a stable value of RH~300 T due to partial desorption of dopants. 23 After this curing step at 160 ˚C, samples showed a fairly low ΔT=-0.03%/˚C up to T=150 ˚C, while still displaying respectable performance at this temperature, with SV~0.12 V/(VT), SI~300 V/(AT), and BMIN~100 nT/√Hz.   Table 1 shows a comparison of our devices with other Hall sensors reported in literature. The maximum current-related sensitivity in doped epigraphene is found to be on the order of SI~1.500 V/(AT) at room temperature. This value is limited by minimum n attained in our sample (|n|<5x10 11 cm -2 ), and is set by the disorder present in the as-grown material, combined with additional contributions from external doping and thermally excited carriers in the dopant layer and the SiC substrate. Decoupling epigraphene and substrate by hydrogen intercalation has led to high  at cryogenic temperatures. However, at room temperature, the lowest n reported for Hintercalated epigraphene are all above 1x10 12 cm -2 , with 1300-1700 cm 2 /(Vs). 35 These mobilities are lower than the highest reported for epigraphene at room temperature (=5500 cm 2 /(Vs)) 22 S2. Left: Uncompensated offset in Hall voltages at zero magnetic field measured for 7 different devices. In general, the offset voltages are on the order 1 mV, and tends to increase as samples are doped towards neutrality (puddle regime). Note that we estimate that the residual magnetization of the coil magnet is on the order of ~mT, further skewing the data to high offset values. In this limited dataset there is no observed correlation between device geometry and offset. The lowest offset voltage is achieved for cross geometry and high doping levels. Offset compensation can be achieved using orthogonal coupling of two or more Hall elements, in combination with Van der Pauw averaging, and can reduce the final offset to below 1 V. Note that this requires very homogenously doped devices, which we do achieve when using molecular dopant F4TCNQ mixed with PMMA. We study the thermal stability of the moleculardoped Hall sensor by measuring the room temperature performance of devices after repeated annealing. Initially, the devices is doped close to the Dirac point (n<5x10 11 cm -2 ) and is kept at elevated temperatures for 15 min, and left to cool down back to room temperature. Performance is assessed in real-time like in (a). This heating process is repeated, moving to successively higher temperatures. Only when the sample is annealed to above 80 ˚C, close to glass transition temperature of the polymer using for doping, does significant permanent change of room temperature n occur. There is a permanent increase in n-doping leading to a decrease in RH. Subsequent heating above 80 ˚C induces further permanent change in doping toward even higher n-doping.