Operation of graphene magnetic field sensors near the charge neutrality point

Graphene is a promising material for sensing magnetic fields via the Hall effect due to its atomic-scale thickness, ultra-high carrier mobilities and low cost compared to conventional semiconductor sensors. Because of its Dirac band structure, graphene sensors differ from semiconductor sensors in that both electrons and holes participate in the carrier transport. This two-channel transport complicates the sensor operation and causes performance trade-offs that demand careful examination. Here, we examine the operation of graphene sensors operated near the charge neutrality point (CNP) where two-channel transport prevails. We find that, while the largest magnetoresistance occurs exactly at the CNP, the maximum realizable Hall sensitivities occur away from the CNP and depend on linearity constraints and power limitations. In particular, a more stringent linearity constraint reduces the realizable sensitivities for mobilities above a critical value µc, which scales with magnetic field. Graphene is expected to be of particular use for Hall sensors but aspects of its electronic band structure and transport properties still require careful examination. Here, the authors analyse the performance of graphene sensors near the charge neutrality point and demonstrate the required trade-offs required to maximise the performance of a graphene-based Hall sensor.

G raphene is a promising material for magnetic field sensors because of its fundamental advantages of high carrier mobility, low sheet carrier density and weak temperature dependence [1][2][3][4][5] , and its practical advantages of simple and low-cost fabrication processes [5][6][7][8][9] . Studies of graphene for both magnetoresistance (MR) [10][11][12] and Hall sensors 2,3,13 have demonstrated performance outpacing traditional magnetic sensors based on semiconductors. For example, a large MR of~2000% at 9 T has been demonstrated in multilayer graphene on hexagonal boron nitride (h-BN) 12 . An even larger MR of 55,000% at 9 T was obtained in extraordinary magnetoresistance devices, where the geometrical MR is enhanced by an embedded metal structure 10,11 . A record value of current-related sensitivity S I of 5700 VA −1 T −1 has been reported 2 for a graphene Hall sensor, which is nearly two orders of magnitude higher than that of commercial Silicon Hall sensors (~100 VA −1 T −1 ) 14 and twice as high as that for the best two-dimensional electron gas (2DEG) based Hall sensors 15 . This record value was achieved in a structure comprised of exfoliated graphene and h-BN stacks designed for high mobility. S I values in the range of about 1000-3000 VA −1 T −1 have been reported for more practical structures based on chemical vapor deposition (CVD) deposited graphene transferred onto various insulators, including exfoliated h-BN 2 , CVD h-BN 6 , and SiO 2 3,5 . Advanced "encapsulated" designs based on an all-CVD h-BN/graphene/h-BN sandwich structure have also been reported 8 , but have been limited thus far to about 100 VA −1 T −1 due to high carrier densities, emphasizing the importance of obtaining low carrier density, in addition to high mobility, for high sensitivity.
The MR, sensitivity and linearity of graphene sensors are closely related to the presence of both electrons and holes near the charge neutrality point (CNP). The existence of a CNP is a unique feature of graphene's Dirac band structure and distinguishes graphene sensors from conventional semiconductor based sensors. Despite the general interest and demonstrated promise of graphene-based sensors, there has been little detailed investigation of the magnetoresistance characteristics and sensor performance of graphene near the CNP, which is essential for optimizing graphene sensor operation.
Here we present a study of the potential performance and optimization of graphene-based MR and Hall sensors. The study is based on a two-channel model that combines the longitudinal and Hall resistivities of parallel electron and hole channels with electrostatic carrier density expressions for graphene. Our primary focus is the optimization at biases near the CNP (i.e., the Dirac point), where the transport is complicated by the presence of both electrons and holes. We begin by validating our model by examining the experimental characteristics of sensors based on commercial CVD graphene transferred to a SiO 2 /p-Si substrate over wide ranges of gatebias V g , magnetic field B, and temperature T. We show that the experimental MR-B, Hall resistivity ρ xy -B and ρ xy -V g characteristics are well-described by the model, including nonlinearities and unusual gate-bias dependence in the ρ xy -V g characteristic near the CNP. We use the model to extract carrier mobilities and densities and show that model agrees with experimental data over wide ranges in gate bias and magnetic field. The characteristics are also examined over a wide temperature range (10-300 K). We then make use of the validated model to study the optimization of the sensitivity, linearity, and MR and to estimate the realizable performance in high-quality graphene. Of particular interest in our results are trade-offs due to the linearity constraints of an application, which are especially important at high mobilities and high magnetic fields-a regime mostly neglected in prior work.

Results
Model. Two-carrier magnetoresistance expressions were used to analyze the experimental data and predict optimized performance. For a single carrier, longitudinal resistivity ρ xx and Hall resistivity ρ xy are given by: for electrons, for holes, where e is the elementary charge and n, p, µ e , µ h represent carrier densities and mobilities for electrons and holes, respectively. For parallel electron and hole channels with equal mobilities, ρ xx and ρ xy are given by: Expressions for unequal mobilities are given by Gopinadhan et al. 12 . While for a single carrier ρ xx is independent of B, Eq. (5) shows that ρ xx is dependent on B 2 for two carriers of comparable density. Therefore, a large MR is possible near the CNP. While for a single carrier ρ xy depends linearly on B and tends to infinity as the carrier density approaches zero, Eq. (6) shows that ρ xy depends nonlinearly on B for two carriers of comparable density and tends to zero as the net carrier density approaches zero. Therefore, the linearity and sensitivity of a graphene Hall sensor are strongly influenced by two-carrier transport.
For simplicity in discussing the model, we have given the expressions in the case of µ e = µ h in Eqs (5) and (6), whereas the expressions for unequal mobilities 12 were used in the fitting analysis presented later (Figs. [1][2][3][4]. Examination of the expressions for unequal mobilities reveals that the deviations from the single carrier case occur when the two channels have comparable conductivities 12 , not comparable densities. Nevertheless, the deviations occur near the CNP even for unequal mobilities due to the strong dependence of carrier densities on gate bias. Near the CNP, inhomogeneities may occur due to charge impurities, intrinsic structural wrinkles, and substrate roughness 1,16 . It is believed that the resulting random potential fluctuations in the channel modulate the Dirac band structure so strongly that interspersed electron and hole puddles form, as has been observed experimentally 17,18 . Various theoretical treatments have been proposed to provide a detailed understanding of transport in this situation 1,[19][20][21] . A simple physical picture 21,22 is that the carriers move along percolation paths while scattering from puddle interfaces. Since the puddles are larger than the mean free path, the conductivities of individual puddles can be described by drift-diffusion 21 , and the transport involves a longer effective path-length and additional scattering (higher resistance). Thus, although puddles complicate the physical picture at the CNP and modify the effective transport parameters, the twochannel model still provides a useful approximation and has been used successfully for the study of graphene sensors by several groups 3,12,19 .
To develop a simple theory (i) for checking the accuracy of the electron and hole densities extracted from the data in our analysis and (ii) for predicting the performance of MR and Hall sensors, we combined the two-carrier magnetoresistance expressions 12 with electrostatic carrier density expressions 23 for the gate-bias dependent electron and hole carrier densities in graphene, similar to that done by Chen et al. 3 . The key relationships in the electrostatic carrier density expressions are: where n tot is the total carrier density, n 0 is the minimum carrier density at the CNP, n V g h i is the gate-bias dependent net charge density, C ox is the gate capacitance and V CNP is the gate bias at the charge neutrality point. In contrast to Chen et al. 3 , who used a simplified model assuming that µB ≪ 1, we use the full twocarrier resistivity equations above. This allows us to study performance trade-offs due to linearity constraints and to predict the sensor performance for high-quality graphene (high µ and low n 0 ) and high magnetic fields.
Comparison of experimental and modeled results. Figure 1a shows the variation of the MR, which is defined as {ρ xx (B) − ρ xx (0)}/ρ xx (0), as a function of magnetic field B for different gate biases V g . The current and temperature for these and the other experiments were 1 µA and 100 K, respectively, unless stated otherwise (see Supplementary Note 1 and Supplementary Fig. 1 for 300 K results). For the biases farthest from the CNP (V g = −20 V), where one carrier dominates, the MR is very small as expected from the Drude model of single carrier transport. It can be seen that the highest MR is achieved at V g close to the CNP (V g = −2 V), where both carriers are present. The MR at 2 T and 100 K is~22%, which is comparable with that previously reported for graphene at the same field and temperature 12 .
The solid lines in Fig. 1a present the calculated MR using the two-channel model. It is important to note here that the calculated MR agrees well with the experimental data, except that the experimental data is more linear at high fields. Such linearity has been attributed to electron-hole recombination leading to an edge conductivity contribution 24,25 as well as to the existence of electron and hole puddles 20 .
A highly linear ρ xy -B characteristic is critical for many Hall sensor applications. Figure 1b shows the Hall resistivity ρ xy -B at various V g . It can be seen that the linearity of ρ xy degrades when the gate bias is close to the CNP (V g = −2.5, −2, −1.5, and −1 V). Following Xu et al. 5 , linearity error α is defined as ðρ xy À ρ 0 xy Þ=ρ 0 xy , where ρ 0 xy is the best linear fit value of the ρ xy -B curve. We define α max as the maximum of α over the B range. In the electron or hole dominated regimes (V g = −20 and 20 V), the value of α max for our data is very low, <2%. However, α max increases rapidly as V g approaches the CNP, reaching a value of 77% at V g = −2 V, which is much too high for Hall sensor applications. This large α max means that ρ xy is not linearly dependent on B in the two-carrier regime.
Close examination of the dependence of the ρ xy -B characteristic on V g shown in Fig. 1b reveals another type of unusual behavior near the CNP. The rotation of the curves around the origin with increasing V g changes direction near the CNP-i.e., the rotation is clockwise near the CNP (the slope ρ xy /B decreases with V g from −10 to +5 V) while it is counter-clockwise away from the CNP (the slope ρ xy /B increases with V g for |V g | >~10 V). In the analysis below, it will be shown that this unusual rotation reversal behavior is also a result of two-carrier transport.
If there are two distinct carrier species-either (i) two carriers of the same type (both electrons or both holes) with different mobilities or (ii) two carriers of different type (one electrons, the other holes) with the same (or different) mobilities-then each species has a different contribution to the overall MR-B and ρ xy -B characteristics. To examine whether the nonlinearity and rotation effects in Fig. 1b can be explained by two-carrier transport, we fit the MR-B and ρ xy -B experimental data in Fig. 1a, b simultaneously with the two-carrier magnetoresistance expressions in Eqs (5) and (6). The fitting parameters were the electron and hole mobilities, which were taken to be independent of V g , and the carrier densities, which were taken to be dependent on V g . This fitting procedure allows us to extract density and mobility values for the electrons and holes.
Calculated results for the two-channel model using the extracted parameters are in close agreement with the experimental results, as shown in Fig. 1. The accuracy of the theory is shown by the good agreement between the experimental and modeled resistivity curves in Fig. 1. However, the validity of our model also requires that the extracted mobilities and densities are accurate. The extracted electron and hole mobilities are 2598 and 2168 cm 2 V −1 s −1 , respectively, which are reasonable values for commercial quality graphene on SiO 2 /p-Si substrates. It is also important to examine the extracted carrier densities in order to validate our model. While the extraction of carrier density from Hall data is simple for a single charge carrier, the presence of two carriers makes the extraction difficult. Nevertheless, we find that the extracted values of electron and hole density are also reasonable throughout the entire gate-bias range, as shown in  Here we experimentally examine the influence of two-carrier transport on the Hall sensitivity. The current-related sensitivity S I is defined as V xy /IB = ρ xy /B 14. Two-carrier transport near the CNP will affect S I through its impact on the ρ xy -V g characteristic. The variation of ρ xx , ρ xy , and S I with V g is shown in Fig. 3. The increase in ρ xx with increasing B near the CNP shown in Fig. 3a is due to two-carrier transport. Figure 3b shows that |ρ xy | peaks on either side of the CNP and that ρ xy crosses zero at the CNP. This is different from the case of a single carrier, where ρ xy is proportional to the inverse of the carrier density and therefore tends to infinity as the carrier density approaches zero. In graphene, however, the total carrier density (p + n) does not approach zero. Instead, (p + n) reaches a minimum value n 0 and the net density (p − n) changes sign at the CNP. This peaking of | ρ xy | places an important limitation on the realizable sensitivity and affects the optimum bias condition for a graphene Hall sensor, as will be considered in detail later. While the temperature for the data presented in this paper was 100 K, the ρ xx -V g and ρ xy -V g characteristics were examined over a range in temperature from 10 to 300 K and confirm the excellent thermal stability of graphene Hall sensors 5 (see Supplementary Note 2 and Supplementary Fig. 1).
It is also interesting to note that the S I behavior in Fig. 3c indicates that ρ xy /B decreases with increasing V g near the CNP (V g from −7 to +5 V) while it increases with increasing V g away from the CNP, which is consistent with our conclusion that the rotation reversal in Fig. 1b is the result of two-carrier transport.
To test the capability of our model for predicting MR and Hall sensor performance, we attempted to fit our experimental data over wide ranges in gate voltage and magnetic field. Following Peng et al. 3 , n 0 was extracted by fitting the S I -V g characteristic in Fig. 3c. The extracted n 0 = 4.65 × 10 11 cm 2 was then used to calculate the electron and hole densities as a function of V g using the electrostatic carrier density expressions in Eqs (7) and (8). The calculated densities are plotted along with the experimental values in Fig. 2 and are seen to be in good agreement with the experiment. From the calculated densities and extracted mobilities, we worked backward to calculate theoretical ρ xy -V g characteristics for various B, which are plotted along with the experimental data in Fig. 3b. Working backward in a similar way, the theoretical MR, S I , and α max characteristics were determined and found to be in good agreement with experimental results, as shown in Fig. 4a-c, respectively. The agreement between the theory and experiment in Figs. 3b and 4b is excellent, except near the peak on the electron-side. (The better agreement on the hole side of the characteristics simply reflects the better fit obtained on the hole side in Fig. 2.) The agreement shows that our model should be useful for estimating the optimized performance of graphene Hall sensors over wide ranges in mobility and magnetic field.
Realizable sensitivities. We now use our model to explore the influence of material-quality parameters (µ and n 0 ) on sensitivity, linearity, and MR for graphene magnetic sensors. Figure 5a, b shows modeled results for n 0 over the range 6.1 × 10 10 -1 × 10 12 cm −2 with µ and B set equal to 2000 cm 2 V −1 s −1 and 2 T, respectively. (The lower limit of n 0 was chosen to be equal to the theoretical thermal limit at 300 K 3 .) It is seen that n 0 has a strong influence on the peak value of S I , which occurs at about 1 V on either side of the CNP and reaches a value of 5360 VA −1 T −1 as n 0 decreases to 6.1 × 10 10 cm −2 . As depicted in Fig. 5c, the peak S I is proportional to 1/n 0 , which agrees with Chen et al.'s conclusion in previous work 3 . The peak value of α max occurs exactly at the CNP and reaches a value 8.8% independent of n 0 , while the width of the α max decreases with decreasing n 0 .
Since α max is proportional to (µB) 214 , the linearity is improved at lower B. Thus, both higher sensitivity and higher linearity occur for lower n 0 . At this moderate mobility, α max is always low enough that operation at the peak S I point is possible. However, for high mobilities the strong variation in both S I and α max with V g introduces an important performance trade-off, which we examine next. Figure 5d, e shows results for µ over the range of 1000-20,000 cm 2 V −1 s −1 for n 0 and B set equal to shows the fit to the data that was used to extract the residual carrier density n 0 used in our model. The peaking of |ρ xy | places a limitation on sensitivity and affects the optimum bias condition 1 × 10 11 cm −2 and 2 T, respectively. It is seen that the peak values of both S I and α max increase with increasing µ. Since increased S I is beneficial while increased α max is not, this represents a performance trade-off. Because of this trade-off, the peak S I cannot be realized and is limited to a value that depends on the α max constraint; we call this realizable value S R I . For example, if α max is constrained to 10% and µ = 2000 cm 2 V −1 s −1 , then Fig. 5d shows that S R I occurs at 1.3 V (square symbol), which is the peak of S I and is equal to 3270 VA −1 T −1 . On the other hand, for the same α max but µ equal to 20,000 cm 2 V −1 s −1 , S R I occurs at 3.7 V (triangle symbol), which is away from the peak S I , and is therefore limited to only 2180 VA −1 T −1 . At this operating point, |S I | is only about 20% of its peak value of 9730 VA −1 T −1 . The reason behind this counterintuitive effect (viz., that increased mobility can degrade S R I ) and its implications will be considered later.
In the above examples, the electron and hole mobilities were assumed to be equal. Figure 5f, g shows results for carrier mobility ratios μ h /μ e over the range of 1 to 1/10. As can be seen from Fig. 5f, g, decreasing μ h while keeping μ e constant causes the peak S I to increase and shift to the left while α max decreases near the peak (both desirable). In the case of μ h /μ e = 1000/ 5000, for example, the peak S I is about two times that for equal mobilities while the α max at the peak is 1% compared with 22% for equal mobilities. These changes in the S I and α max characteristics result in an improvement in S R I of nearly three times that for α max = 10% (see symbols in Fig. 5f). The reason for this improvement is that the transport for a higher mobility ratio is more similar to that for a single carrier, where S I tends to be high while α max is low. It is interesting that a difference in carrier mobilities can have a significant influence on both S I and α max as well as the optimum V g bias point. The impact of this on performance will also be considered in a later section.  We have also used our model to explore how the MR is affected by µ and n 0 . Figure 6a shows the modeled results for MR-V g when n 0 is varied from 6.1 × 10 10 to 1 × 10 12 cm −2 with µ and B set equal to 2000 cm 2 V −1 s −1 and 2 T, respectively. Contrary to what was seen for S I in Fig. 5a, the peaks of MR occur exactly at the CNP. While the peak MR value is independent of n 0 , a larger n 0 gives a wider peak and hence a larger V g operating range. Figure 6b shows the calculated MR for µ varied from 1000 to 20,000 cm 2 V −1 s −1 with n 0 and B set equal to 1 × 10 11 cm −2 and 2 T, respectively. The MR peak value increases dramatically with µ, reaching nearly 1600% as µ approaches 20,000 cm 2 V −1 s −1 , a mobility representative of that for high-quality graphene on h-BN 26,27 . Figure 6a, b shows that, in contrast to Hall sensitivity, the key to achieving high MR in graphene is having a high µ rather than a low n 0.
Here we examine the details of how a linearity constraint influences the realizable current-related sensitivity S R I , as well as the absolute sensitivity S A = V xy /B = S I I. S A is proportional to (µ/n s ) 1/2 P 1/214 , where n s is sheet carrier density and P is power, and thus S A is the most important parameter for power-limited applications. The realizable value of S A , which we refer to as S R A , is constrained by both the needed linearity and the power limitation for the particular application. The linearity constraint affects the S R I at high mobility values, as illustrated in Fig. 7a, b, which show S R I and α max vs µ at B = 2 T for various constraints on α max (1, 2, 5, and 10%). As shown in Fig. 7b, the α max constraint is active for mobilities above a critical value, which we define as µ c . For example, the value of µ c for α max = 10% is 3270 cm 2 V −1 s −1 . It can be seen in Fig. 7a that S R I is near its maximum value of 3470 VA −1 T −1 when µ is less than about 1000 cm 2 V −1 s −1 for all shown values of α max . However, increasing the mobility above 1000 cm 2 V −1 s −1 results in an α max -dependent decrease in S R I . This is because the operating bias must be moved further away from the CNP to meet the α max constraint when µ > µ c , as is shown in Fig. 7c. While S R I barely changes when µ is lower than 1000 cm 2 V −1 s −1 , the channel resistivity ρ xx at the operating bias increases rapidly with decreasing µ to values too high for lowpower operation, as illustrated in Fig. 7d. At a higher resistivity the maximum current I max is power-limited at a lower value. Although S R I is commonly used as a figure of merit of Hall sensors, S R I does not take power into account. Thus, S R A , which depends on both linearity constraints and power limitations, is a better figure merit for power-limited applications. Figure 8a, b shows S R A and α max vs µ for various α max constraints values (1, 2, 5, 10%; and unconstrained) with an assumed power limitation of 1 mW. In contrast to S R I , which decreases for higher mobilities, we can see that S R A increases monotonically with µ. However, the linearity constraints greatly reduce S R A compared with the unconstrained case in the high mobility regime when µ > µ c . For example, the value of µ c for α max = 10% is 4040 cm 2 V −1 s −1 . At µ = 20,000 cm 2 V −1 s −1 , for instance, α max constraints of 1 and 10% lead to reductions of S R A by 58% and 27%, respectively, compared with the unconstrained value. The results in Fig. 8a show that an S R A of 4.5 VT −1 at 1 mW (equivalent 0.14 VT −1 at 1 µW) should be possible for high-quality graphene with good linearity over a large magnetic field range (α max = 10%, B = 2 T, n 0 = 1 × 10 11 cm −2 , µ = 100,000 cm 2 V −1 s −1 ). Mobility (cm 2 V -1 s -1 ) 100 1000 10,000 100,000 d Fig. 7 Dependence of realizable current-related sensitivity on mobility and linearity. Modeled a realizable current-related sensitivity S R I , b maximum linearity error α max , c |V g -V CNP |, and d longitudinal resistivity ρ xx vs mobility for various α max constraints for n 0 = 1 × 10 11 cm −2 and B = 2 T. Gate-bias V g is optimized for the highest S R I within the α max constraints. V CNP is the charge neutrality point. These results show the interplay between the linearity constraint and mobility in determining the realizable S I values; in particular a stringent linearity constraint together with a high mobility results in a substantial reduction in S R

Discussion
Several points about the results presented above deserve further discussion. As was seen in the calculations, higher mobility and higher magnetic field result in poorer linearity, which limits S R I and S R A to lower values. Since α max is proportional to (µB) 214 , the calculated results for B = 2 T in Figs. 5-8 can easily be extended to other B values. In the insert of Fig. 8b we have plotted α max vs (µ c B) 2 , as determined from the data in Fig. 8. As shown in Fig. 8, α max is <10% for (µ c B) 2 < 0.65, which means that S R A is not limited by a 10% linearity constraint; while for larger (µB) 2 , S R A is reduced for a 10% linearity constraint. Thus, determining µ c (the mobility above which the realizable sensitivity is constrained by linearity) for the α max and B required by a particular application can be useful in designing graphene Hall sensors. This is obviously important when both the mobility and magnetic field are high. It is important to note that the nonlinearities are also significant in the mT range when the mobility is very high, as is possible in optimized graphene devices. For example, for a mobility of 100,000 cm 2 V −1 s −1 , (µB) 2 is equal to 1 at 100 mT and α max is over 10% (see inset of Fig. 8b), and the nonlinearity cannot be neglected. Thus, the linearity constraints discussed in this paper can also be important for low fields used in many Hall sensor applications.
As was seen in Fig. 7a, S R I is near its maximum value for µ below about 1000 cm 2 V −1 s −1 . This does not mean that a low mobility is sufficient for good sensor performance since power limits must also be considered in many applications. S R A is the relevant figure-ofmerit for power-limited applications and lower mobility limits S R A to lower values. The advantage of graphene for achieving high S R A is that it offers both high mobility and low sheet carrier density. For example, for a simple graphene-on-SiO 2 structure with µ and n 0 values of 7800 cm 2 V −1 s −1 and 1 × 10 11 cm −2 3 , respectively, the equivalent S R A value based on the reported S I and current-voltage data is 0.9 VT −1 at a power of 1 mW with a linearity error of 4% for B = 0.4 T. This S R A value, which is slightly lower than our calculated value of 1.4 VT −1 for the same parameters, is the best reported result for this simple structure. For an advanced h-BN encapsulated, exfoliated graphene 2 structure, record sensitivity values of voltagerelated sensitivity S V = 2.8 T −1 and S I = 5700 VA −1 T −1 have been reported, which correspond to an equivalent S R A value of 4.0 VT −1 at 1 mW. Although neither the linearity nor the mobility were reported with this record data, we can use our model to estimate the sensitivity and linearity by assuming µ = 80,000 cm 2 V −1 s −1 (the mobility reported for similar h-BN encapsulated CVD graphene 26 ) and n 0 = 1 × 10 11 cm −2 (a typical value for high-quality graphene). Our model shows that an equivalent S R A of 4.0 VT −1 at 1 mW should be possible with α max = 10% for B up to 2 T. If we increase µ to 120,000 cm 2 V −1 s −1 , the best value reported near room temperature in exfoliated and suspended graphene 28 , then S R A increases by about 20% to 4.9 VT −1 at 1 mW. Thus, our model indicates that the record experimental S R A reported for advanced graphene Hall structures is 80% of what can be achieved with good linearity (α max = 10%). If excellent linearity (α max = 1%) is required, however, our model indicates that an S R A value of 2.8 VT −1 at 1 mW is the best that can be expected.
An important part of this study has been to take into account how the linearity constraint of an application influences the achievable performance of a graphene Hall sensor. The basic issue is that, even though graphene offers high mobility with low residual carrier density at biases near the CNP (both beneficial for Hall sensing), linearity is reduced because of comparable conductivities for the electron and hole channels in this bias regime. Thus, schemes for providing that one channel conductivity dominates over the other could be useful for improving linearity. The obvious approach of biasing the device away from the CNP so that the density of one carrier dominates can improve linearity, but seriously degrades sensitivity due to the increased carrier density. However, the alternative scheme of reducing the mobility of one carrier compared to the other does not suffer from this drawback. While electron and hole mobilities in graphene are usually similar, carrier mobility ratios of~0.3 have been reported for graphene FETs 29,30 and attributed to asymmetric scattering for electrons and holes 30 . Higher ratios might be possible in engineered structures. Calculated results on the effect of the mobility ratio μ h /μ e on Hall sensor sensitivity and linearity were presented in Fig. 5d, where it can be seen that S R I improves by a factor of nearly 3 for μ h /μ e = 0.2 and α max = 10%. In the powerlimited case, our calculations show that an improvement in S R A of about 50% is possible under the same assumptions. Another scheme for providing that one conduction channel dominates over the other is to use an electrical contact technology having different contact resistances for electrons and holes. Previous studies have reported electron-hole conduction asymmetry for various metal/graphene contacts [31][32][33] , and this effect might also be engineered to improve Hall sensor linearity.
It is important to realize that although the two-carrier nature of graphene is a disadvantage for linearity, this does not mean that graphene is inferior to single-carrier semiconductor Hall sensors. Our calculations for graphene with n 0 = 1 × 10 11 cm −2 give an S R I of 3470 VA −1 T −1 with 10% linearity, which is comparable with the best experimental S R I value of 2745 VA −1 T −1 reported for graphene on SiO 2 3 . These values are much higher than those for Si and GaAs sensors 14 : 100 and 700 VA −1 T −1 , respectively. Our calculations for high-quality graphene (n 0 = 1 × 10 11 cm −2 , µ = 100,000 cm 2 V −1 s −1 , α max = 10%, B = 2 T, P = 1 mW) give S R A ¼ 4:5 VT À1 which is about two times higher than the best values reported for narrow-gap III-V heterostructure sensors (S I = 2750 VA −1 T −1 ; S A = 2.17 VT −1 at 1 mW) 15 . Thus, graphene provides performance much better than simple semiconductor structures and comparable to the best complex III-V heterostructure designs.

Methods
Device fabrication and characterization. Back-gated six-arm Hall bar structures were fabricated for these experiments using commercial graphene deposited by CVD and transferred to a SiO 2 (285 nm)/p-Si substrate. Photolithography followed by oxygen plasma etching was used to pattern the graphene channel region. A  Fig. 8 The dependence of realizable absolute sensitivity on mobility and linearity. a Modeled realizable absolute sensitivity S R A and b maximum linearity error α max vs mobility for various α max constraints for minimum carrier density n 0 = 1 × 10 11 cm −2 , magnetic field B = 2 T and power P = 1 mW. Gate-bias V g is optimized for the highest S R A within the α max constraints. The inset shows the dependence of α max on (µ c B) 2 , where the critical mobility µ c is determined from the break points in (b) and the dashed line is a linear fit. The results show that, when mobility is higher than µ c , the linearity constraints substantially reduce S R A compared with the unconstrained case standard photoresist lift-off process was performed to form metal contacts comprised of Ti/Au (10/120 nm) layers deposited by electron beam evaporation. Examination of the current-voltage characteristics confirmed that the fabricated Hall bars exhibited excellent electrical properties (see Supplementary Fig. 3b). Raman spectroscopy was used to examine the quality of the graphene and its monolayer thickness (see Supplementary Note 3 and Supplementary Fig. 3c, d).
Transport measurements. The longitudinal resistivity ρ xx and Hall resistivity ρ xy were measured as a function of magnetic field B and gate-bias V g by the van der Pauw method 34 under vacuum using a cryogenic probe Hall measurement system (Model 8425, Lake Shore Cryotronics Inc.) with a current source (Model 6220, Keithley Inc.) and a nano-voltmeter (Model 2182A, Keithley Inc.). Current-reversal averaging and geometry averaging techniques were included to remove unwanted contributions due to offset currents and offset voltages 35 . V g was applied to the p-Si substrate and the gate leakage current was monitored. The magnetic field was applied perpendicular to the sample plane over the range of −2 to 2T and the temperature of sample stage was varied from 10 to 300 K. The temperature sensitivity observed throughout these experiments was very small, and we only present data primarily for an intermediate temperature of 100 K.

Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.