Abstract
Graphene is a promising material for sensing magnetic fields via the Hall effect due to its atomicscale thickness, ultrahigh 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 twochannel transport complicates the sensor operation and causes performance tradeoffs that demand careful examination. Here, we examine the operation of graphene sensors operated near the charge neutrality point (CNP) where twochannel 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 µ_{c}, which scales with magnetic field.
Introduction
Graphene 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 lowcost 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 (hBN)^{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 currentrelated 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 twodimensional electron gas (2DEG) based Hall sensors^{15}. This record value was achieved in a structure comprised of exfoliated graphene and hBN 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 hBN^{2}, CVD hBN^{6}, and SiO_{2}^{3,5}. Advanced “encapsulated” designs based on an allCVD hBN/graphene/hBN 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 graphenebased 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 graphenebased MR and Hall sensors. The study is based on a twochannel 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}/pSi 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 welldescribed by the model, including nonlinearities and unusual gatebias 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 highquality graphene. Of particular interest in our results are tradeoffs 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
Twocarrier 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 twocarrier 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. 1–4). 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 driftdiffusion^{21}, and the transport involves a longer effective pathlength 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 twocarrier magnetoresistance expressions^{12} with electrostatic carrier density expressions^{23} for the gatebias 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, \(n\left[ {V_{\mathrm{g}}} \right]\) is the gatebias 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 tradeoffs due to linearity constraints and to predict the sensor performance for highquality 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 twochannel 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 electronhole 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 \((\rho _{xy}  \rho _{xy}^0)/\rho _{xy}^0\), where \(\rho _{xy}^0\) 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 twocarrier 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 counterclockwise 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 twocarrier 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 twocarrier transport, we fit the MR–B and ρ_{xy}–B experimental data in Fig. 1a, b simultaneously with the twocarrier 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 twochannel 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}/pSi 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 gatebias range, as shown in Fig. 2. (The theoretical curves in Fig. 2 will be discussed later.)
Here we experimentally examine the influence of twocarrier transport on the Hall sensitivity. The currentrelated sensitivity S_{I} is defined as V_{xy}/IB = ρ_{xy}/B^{14.} Twocarrier 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 twocarrier 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 twocarrier 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 electronside. (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 materialquality 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)^{2}^{14}, 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 tradeoff, 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 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 tradeoff. Because of this tradeoff, 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 \(S_I^{\mathrm{R}}\). For example, if α_{max} is constrained to 10% and µ = 2000 cm^{2} V^{−1} s^{−1}, then Fig. 5d shows that \(S_I^{\mathrm{R}}\) 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}, \(S_I^{\mathrm{R}}\) 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 \(S_I^{\mathrm{R}}\)) 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 \(S_I^{\mathrm{R}}\) 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 highquality graphene on hBN^{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 currentrelated sensitivity \(S_I^{\mathrm{R}}\), 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^{1/2}^{14}, where n_{s} is sheet carrier density and P is power, and thus S_{A} is the most important parameter for powerlimited applications. The realizable value of S_{A}, which we refer to as \(S_{\mathrm{A}}^{\mathrm{R}}\), is constrained by both the needed linearity and the power limitation for the particular application. The linearity constraint affects the \(S_I^{\mathrm{R}}\) at high mobility values, as illustrated in Fig. 7a, b, which show \(S_I^{\mathrm{R}}\) 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 \(S_I^{\mathrm{R}}\) 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 \(S_I^{\mathrm{R}}\). 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 \(S_I^{\mathrm{R}}\) 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 powerlimited at a lower value. Although \(S_I^{\mathrm{R}}\) is commonly used as a figure of merit of Hall sensors, \(S_I^{\mathrm{R}}\) does not take power into account. Thus, \(S_{\mathrm{A}}^{\mathrm{R}}\), which depends on both linearity constraints and power limitations, is a better figure merit for powerlimited applications. Figure 8a, b shows \(S_{\mathrm{A}}^{\mathrm{R}}\) 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 \(S_I^{\mathrm{R}}\), which decreases for higher mobilities, we can see that \(S_{\mathrm{A}}^{\mathrm{R}}\) increases monotonically with µ. However, the linearity constraints greatly reduce \(S_{\mathrm{A}}^{\mathrm{R}}\) 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 \(S_{\mathrm{A}}^{\mathrm{R}}\) by 58% and 27%, respectively, compared with the unconstrained value. The results in Fig. 8a show that an \(S_{\mathrm{A}}^{\mathrm{R}}\) of 4.5 VT^{−1} at 1 mW (equivalent 0.14 VT^{−1} at 1 µW) should be possible for highquality 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}).
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 \(S_I^{\mathrm{R}}\) and \(S_{\mathrm{A}}^{\mathrm{R}}\) to lower values. Since α_{max} is proportional to (µB)^{2}^{14}, 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 \(S_{\mathrm{A}}^{\mathrm{R}}\) is not limited by a 10% linearity constraint; while for larger (µB)^{2}, \(S_{\mathrm{A}}^{\mathrm{R}}\) 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, \(S_I^{\mathrm{R}}\) 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. \(S_{\mathrm{A}}^{\mathrm{R}}\) is the relevant figureofmerit for powerlimited applications and lower mobility limits \(S_{\mathrm{A}}^{\mathrm{R}}\) to lower values. The advantage of graphene for achieving high \(S_{\mathrm{A}}^{\mathrm{R}}\) is that it offers both high mobility and low sheet carrier density. For example, for a simple grapheneonSiO_{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 \(S_{\mathrm{A}}^{\mathrm{R}}\) 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 \(S_{\mathrm{A}}^{\mathrm{R}}\) 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 hBN 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 \(S_{\mathrm{A}}^{\mathrm{R}}\) 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 hBN encapsulated CVD graphene^{26}) and n_{0} = 1 × 10^{11} cm^{−2} (a typical value for highquality graphene). Our model shows that an equivalent \(S_{\mathrm{A}}^{\mathrm{R}}\) 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 \(S_{\mathrm{A}}^{\mathrm{R}}\) increases by about 20% to 4.9 VT^{−1} at 1 mW. Thus, our model indicates that the record experimental \(S_{\mathrm{A}}^{\mathrm{R}}\) 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 \(S_{\mathrm{A}}^{\mathrm{R}}\) 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 \(S_I^{\mathrm{R}}\) 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 \(S_{\mathrm{A}}^{\mathrm{R}}\) 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 electronhole 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 twocarrier nature of graphene is a disadvantage for linearity, this does not mean that graphene is inferior to singlecarrier semiconductor Hall sensors. Our calculations for graphene with n_{0} = 1 × 10^{11} cm^{−2} give an \(S_I^{\mathrm{R}}\) of 3470 VA^{−1} T^{−1} with 10% linearity, which is comparable with the best experimental \(S_I^{\mathrm{R}}\) 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 highquality 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 \(S_{\mathrm{A}}^{\mathrm{R}} = 4.5\,{\mathrm{VT}}^{  1}\) which is about two times higher than the best values reported for narrowgap 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
Backgated sixarm Hall bar structures were fabricated for these experiments using commercial graphene deposited by CVD and transferred to a SiO_{2}(285 nm)/pSi substrate. Photolithography followed by oxygen plasma etching was used to pattern the graphene channel region. A standard photoresist liftoff 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 gatebias 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 nanovoltmeter (Model 2182A, Keithley Inc.). Currentreversal 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 pSi 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.
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Acknowledgements
The authors would like to acknowledge Z. Huang for his help in preliminary analysis of experimental data and J.R. Lindemuth (Lake Shore Cryotronics) for helpful discussions on multicarrier Hall measurements. R.K. also thanks R. Macedo for her contribution to device fabrication.
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R.K. created and supervised the project. G.S. did most of the device fabrication and materials characterization. R.K. designed the experiments and modeling studies. G.S. and M.R. carried out the magnetotransport measurements. G.S. implemented and carried out the modeling study. G.S. and R.K. wrote the manuscript with comments from M.R.
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Song, G., Ranjbar, M. & Kiehl, R.A. Operation of graphene magnetic field sensors near the charge neutrality point. Commun Phys 2, 65 (2019). https://doi.org/10.1038/s4200501901615
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