Operation of graphene magnetic field sensors near the charge neutrality point

Abstract

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.

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 dependence1,2,3,4,5, and its practical advantages of simple and low-cost fabrication processes5,6,7,8,9. Studies of graphene for both magnetoresistance (MR)10,11,12 and Hall sensors2,3,13 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 structure10,11. A record value of current-related sensitivity SI of 5700 VA−1 T−1 has been reported2 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 sensors15. This record value was achieved in a structure comprised of exfoliated graphene and h-BN stacks designed for high mobility. SI 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-BN2, CVD h-BN6, and SiO23,5. Advanced “encapsulated” designs based on an all-CVD h-BN/graphene/h-BN sandwich structure have also been reported8, 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 SiO2/p-Si substrate over wide ranges of gate-bias Vg, magnetic field B, and temperature T. We show that the experimental MR–B, Hall resistivity ρxy–B and ρxyVg characteristics are well-described by the model, including nonlinearities and unusual gate-bias dependence in the ρxy–Vg 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 ρxx and Hall resistivity ρxy are given by:

for electrons,

$$\rho _{xx} = \frac{1}{{en\mu _{\mathrm{e}}}},$$
(1)
$$\rho _{xy} = \frac{B}{{en}};$$
(2)

for holes,

$$\rho _{xx} = \frac{1}{{ep\mu _{\mathrm{h}}}},$$
(3)
$$\rho _{xy} = \frac{B}{{ep}};$$
(4)

where e is the elementary charge and n, p, µe, µh represent carrier densities and mobilities for electrons and holes, respectively. For parallel electron and hole channels with equal mobilities, ρxx and ρxy are given by:

$$\rho _{xx} = \frac{1}{e}\frac{{\left( {p + n} \right)\left( {1 + \left( {\mu B} \right)^2} \right)}}{{\mu \left( {\left( {p + n} \right)^2 + \left( {\mu B} \right)^2\left( {p - n} \right)^2} \right)}},$$
(5)
$$\rho _{xy} = \frac{1}{e}\frac{{B( - p + n)(1 + (\mu B)^2)}}{{((p + n)^2 + (\mu B)^2(p - n)^2)}}.$$
(6)

Expressions for unequal mobilities are given by Gopinadhan et al.12. While for a single carrier ρxx is independent of B, Eq. (5) shows that ρxx is dependent on B2 for two carriers of comparable density. Therefore, a large MR is possible near the CNP. While for a single carrier ρxy depends linearly on B and tends to infinity as the carrier density approaches zero, Eq. (6) shows that ρxy 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 µe = µh in Eqs (5) and (6), whereas the expressions for unequal mobilities12 were used in the fitting analysis presented later (Figs. 14). Examination of the expressions for unequal mobilities reveals that the deviations from the single carrier case occur when the two channels have comparable conductivities12, 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 roughness1,16. 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 experimentally17,18. Various theoretical treatments have been proposed to provide a detailed understanding of transport in this situation1,19,20,21. A simple physical picture21,22 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-diffusion21, 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 two-channel model still provides a useful approximation and has been used successfully for the study of graphene sensors by several groups3,12,19.

Fig. 1
figure1

Comparison of experimental and modeled results. Dependence of magnetoresistance MR and Hall resistivity ρxy on magnetic field B for various values of gate-bias Vg: a MR–B and b ρxyB. Good agreement is obtained between the experiment (dots) and model (solid lines) over wide ranges in Vg and B. (The current and temperature for these and the other experiments were 1 µA and 100 K, respectively, unless stated otherwise.)

Fig. 2
figure2

Comparison of experimental and calculated carrier densities. The extracted densities (solid) and calculated densities (dashed) as a function of gate-bias Vg are in good agreement with each other, which is important for validation of our model. The color bands in this figure indicate hole dominated (light blue), electron dominated (pink), and two-carrier (yellow) regions

Fig. 3
figure3

Gate-bias Vg dependence of resistivity and sensitivity. a Longitudinal resistivity ρxxVg and b Hall resistivity ρxyVg and c the current-related sensitivity SIVg for various magnetic field B. The dashed lines in (b) are theoretical characteristics calculated from the model, which show good agreement with the experiment, especially on the left side of the CNP. The dashed line in (c) shows the fit to the data that was used to extract the residual carrier density n0 used in our model. The peaking of |ρxy| places a limitation on sensitivity and affects the optimum bias condition

Fig. 4
figure4

Comparison of experiment and theory. a Magnetoresistance MR, b current-related sensitivity |SI|, and c maximum linearity error αmax as a function of gate-bias Vg − VCNP. VCNP is the charge neutrality point

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 expressions12 with electrostatic carrier density expressions23 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:

$$n_{{\mathrm{tot}}} = p + n \cong \sqrt {n_0^2 + n[V_{\mathrm{g}}]^2} ,$$
(7)
$$n\left[ {V_{\mathrm{g}}} \right] = p - n = - \frac{{C_{{\mathrm{ox}}}}}{e}(V_{\mathrm{g}} - V_{{\mathrm{CNP}}})$$
(8)

where ntot is the total carrier density, n0 is the minimum carrier density at the CNP, \(n\left[ {V_{\mathrm{g}}} \right]\) is the gate-bias dependent net charge density, Cox is the gate capacitance and VCNP 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 two-carrier 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 n0) and high magnetic fields.

Comparison of experimental and modeled results

Figure 1a shows the variation of the MR, which is defined as {ρxx(B) − ρxx(0)}/ρxx(0), as a function of magnetic field B for different gate biases Vg. 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 (Vg = −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 Vg close to the CNP (Vg = −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 temperature12.

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 contribution24,25 as well as to the existence of electron and hole puddles20.

A highly linear ρxyB characteristic is critical for many Hall sensor applications. Figure 1b shows the Hall resistivity ρxyB at various Vg. It can be seen that the linearity of ρxy degrades when the gate bias is close to the CNP (Vg = −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 ρxyB curve. We define αmax as the maximum of α over the B range. In the electron or hole dominated regimes (Vg = −20 and 20 V), the value of αmax for our data is very low, <2%. However, αmax increases rapidly as Vg approaches the CNP, reaching a value of 77% at Vg = −2 V, which is much too high for Hall sensor applications. This large αmax means that ρxy is not linearly dependent on B in the two-carrier regime.

Close examination of the dependence of the ρxyB characteristic on Vg shown in Fig. 1b reveals another type of unusual behavior near the CNP. The rotation of the curves around the origin with increasing Vg changes direction near the CNP—i.e., the rotation is clockwise near the CNP (the slope ρxy/B decreases with Vg from −10 to +5 V) while it is counter-clockwise away from the CNP (the slope ρxy/B increases with Vg for |Vg| > ~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 ρxyB 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 ρxyB 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 Vg, and the carrier densities, which were taken to be dependent on Vg. 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 cm2 V−1 s−1, respectively, which are reasonable values for commercial quality graphene on SiO2/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 Fig. 2. (The theoretical curves in Fig. 2 will be discussed later.)

Here we experimentally examine the influence of two-carrier transport on the Hall sensitivity. The current-related sensitivity SI is defined as Vxy/IB = ρxy/B14. Two-carrier transport near the CNP will affect SI through its impact on the ρxyVg characteristic. The variation of ρxx, ρxy, and SI with Vg is shown in Fig. 3. The increase in ρxx with increasing B near the CNP shown in Fig. 3a is due to two-carrier transport. Figure 3b shows that |ρxy| peaks on either side of the CNP and that ρxy crosses zero at the CNP. This is different from the case of a single carrier, where ρxy 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 n0 and the net density (pn) changes sign at the CNP. This peaking of |ρxy| 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 ρxxVg and ρxyVg characteristics were examined over a range in temperature from 10 to 300 K and confirm the excellent thermal stability of graphene Hall sensors5 (see Supplementary Note 2 and Supplementary Fig. 1).

It is also interesting to note that the SI behavior in Fig. 3c indicates that ρxy/B decreases with increasing Vg near the CNP (Vg from −7 to +5 V) while it increases with increasing Vg 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, n0 was extracted by fitting the SIVg characteristic in Fig. 3c. The extracted n0 = 4.65 × 1011 cm2 was then used to calculate the electron and hole densities as a function of Vg 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 ρxyVg characteristics for various B, which are plotted along with the experimental data in Fig. 3b. Working backward in a similar way, the theoretical MR, SI, and αmax 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 n0) on sensitivity, linearity, and MR for graphene magnetic sensors.

Figure 5a, b shows modeled results for n0 over the range 6.1 × 1010–1 × 1012 cm−2 with µ and B set equal to 2000 cm2 V−1 s−1 and 2 T, respectively. (The lower limit of n0 was chosen to be equal to the theoretical thermal limit at 300 K3.) It is seen that n0 has a strong influence on the peak value of SI, which occurs at about 1 V on either side of the CNP and reaches a value of 5360 VA−1 T−1 as n0 decreases to 6.1 × 1010 cm−2. As depicted in Fig. 5c, the peak SI is proportional to 1/n0, which agrees with Chen et al.’s conclusion in previous work3. The peak value of αmax occurs exactly at the CNP and reaches a value 8.8% independent of n0, while the width of the αmax decreases with decreasing n0. Since αmax is proportional to (µB)214, the linearity is improved at lower B. Thus, both higher sensitivity and higher linearity occur for lower n0. At this moderate mobility, αmax is always low enough that operation at the peak SI point is possible. However, for high mobilities the strong variation in both SI and αmax with Vg introduces an important performance trade-off, which we examine next.

Fig. 5
figure5

The influence of material-quality parameters on sensitivity and linearity. a Modeled current-related sensitivity |SI| and b maximum linearity error αmax vs Vg – VCNP at various minimum carrier density n0 for mobility µ = 2000 cm2 V−1 s−1, magnetic field B = 2 T. c |SI| vs 1/n0, as extracted from (a). d modeled |SI| and e αmax vs Vg – VCNP for equal electron and hole mobilities (µe = µh = µ). f modeled |SI| and g αmax vs Vg – VCNP for unequal electron and hole mobilities (µe ≠ µh) for n0 = 1 × 1011 cm−2, B = 2 T. Vg is the gate bias and VCNP is the charge neutrality point. Symbols in (c) and (d) show the realizable SI for αmax = 10% for different mobilities. The modeling shows that the realizable SI values, which are inversely proportional to n0, are reduced by linearity constraints

Figure 5d, e shows results for µ over the range of 1000–20,000 cm2 V−1 s−1 for n0 and B set equal to 1 × 1011 cm−2 and 2 T, respectively. It is seen that the peak values of both SI and αmax increase with increasing µ. Since increased SI is beneficial while increased αmax is not, this represents a performance trade-off. Because of this trade-off, the peak SI cannot be realized and is limited to a value that depends on the αmax constraint; we call this realizable value \(S_I^{\mathrm{R}}\). For example, if αmax is constrained to 10% and µ = 2000 cm2 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 SI and is equal to 3270 VA−1 T−1. On the other hand, for the same αmax but µ equal to 20,000 cm2 V−1 s−1, \(S_I^{\mathrm{R}}\) occurs at 3.7 V (triangle symbol), which is away from the peak SI, and is therefore limited to only 2180 VA−1 T−1. At this operating point, |SI| 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 μh/μe over the range of 1 to 1/10. As can be seen from Fig. 5f, g, decreasing μh while keeping μe constant causes the peak SI to increase and shift to the left while αmax decreases near the peak (both desirable). In the case of μh/μe = 1000/5000, for example, the peak SI is about two times that for equal mobilities while the αmax at the peak is 1% compared with 22% for equal mobilities. These changes in the SI and αmax characteristics result in an improvement in \(S_I^{\mathrm{R}}\) of nearly three times that for αmax = 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 SI tends to be high while αmax is low. It is interesting that a difference in carrier mobilities can have a significant influence on both SI and αmax as well as the optimum Vg 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 n0. Figure 6a shows the modeled results for MR–Vg when n0 is varied from 6.1 × 1010 to 1 × 1012 cm−2 with µ and B set equal to 2000 cm2 V−1 s−1 and 2 T, respectively. Contrary to what was seen for SI in Fig. 5a, the peaks of MR occur exactly at the CNP. While the peak MR value is independent of n0, a larger n0 gives a wider peak and hence a larger Vg operating range. Figure 6b shows the calculated MR for µ varied from 1000 to 20,000 cm2 V−1 s−1 with n0 and B set equal to 1 × 1011 cm−2 and 2 T, respectively. The MR peak value increases dramatically with µ, reaching nearly 1600% as µ approaches 20,000 cm2 V−1 s−1, a mobility representative of that for high-quality graphene on h-BN26,27. 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 n0.

Fig. 6
figure6

The influence of material-quality parameters on magnetoresistance. a Modeled magnetoresistance MR vs gate-bias Vg – VCNP at various minimum carrier density n0 for mobility µ = 2000 cm2 V−1 s−1, magnetic field B = 2 T and b modeled MR vs Vg – VCNP for various mobilities (µe = µh = µ) for n0 = 1 × 1011 cm−2, B = 2 T. VCNP is the charge neutrality point. The modeling shows that the key to achieving a high MR in graphene is having a high µ, rather than a low n0

Here we examine the details of how a linearity constraint influences the realizable current-related sensitivity \(S_I^{\mathrm{R}}\), as well as the absolute sensitivity SA = Vxy/B = SI I. SA is proportional to (µ/ns)1/2P1/214, where ns is sheet carrier density and P is power, and thus SA is the most important parameter for power-limited applications. The realizable value of SA, 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 αmax vs µ at B = 2 T for various constraints on αmax (1, 2, 5, and 10%). As shown in Fig. 7b, the αmax constraint is active for mobilities above a critical value, which we define as µc. For example, the value of µc for αmax = 10% is 3270 cm2 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 cm2 V−1 s−1 for all shown values of αmax. However, increasing the mobility above 1000 cm2 V−1 s−1 results in an αmax-dependent decrease in \(S_I^{\mathrm{R}}\). This is because the operating bias must be moved further away from the CNP to meet the αmax constraint when µ > µc, as is shown in Fig. 7c. While \(S_I^{\mathrm{R}}\) barely changes when µ is lower than 1000 cm2 V−1 s−1, the channel resistivity ρxx at the operating bias increases rapidly with decreasing µ to values too high for low-power operation, as illustrated in Fig. 7d. At a higher resistivity the maximum current Imax is power-limited 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 power-limited applications. Figure 8a, b shows \(S_{\mathrm{A}}^{\mathrm{R}}\) and αmax vs µ for various αmax 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 > µc. For example, the value of µc for αmax = 10% is 4040 cm2 V−1 s−1. At µ = 20,000 cm2 V−1 s−1, for instance, αmax 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 high-quality graphene with good linearity over a large magnetic field range (αmax = 10%, B = 2 T, n0 = 1 × 1011 cm−2, µ = 100,000 cm2 V−1 s−1).

Fig. 7
figure7

Dependence of realizable current-related sensitivity on mobility and linearity. Modeled a realizable current-related sensitivity \(S_I^{\mathrm{R}}\), b maximum linearity error αmax, c |Vg – VCNP|, and d longitudinal resistivity ρxx vs mobility for various αmax constraints for n0 = 1 × 1011 cm−2 and B = 2 T. Gate-bias Vg is optimized for the highest \(S_I^{\mathrm{R}}\) within the αmax constraints. VCNP is the charge neutrality point. These results show the interplay between the linearity constraint and mobility in determining the realizable SI values; in particular a stringent linearity constraint together with a high mobility results in a substantial reduction in \(S_I^{\mathrm{R}}\)

Fig. 8
figure8

The dependence of realizable absolute sensitivity on mobility and linearity. a Modeled realizable absolute sensitivity \(S_{\mathrm{A}}^{\mathrm{R}}\) and b maximum linearity error αmax vs mobility for various αmax constraints for minimum carrier density n0 = 1 × 1011 cm−2, magnetic field B = 2 T and power P = 1 mW. Gate-bias Vg is optimized for the highest \(S_{\mathrm{A}}^{\mathrm{R}}\) within the αmax constraints. The inset shows the dependence of αmax on (µcB)2, where the critical mobility µc 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 µc, the linearity constraints substantially reduce \(S_{\mathrm{A}}^{\mathrm{R}}\) compared with the unconstrained case

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 αmax is proportional to (µB)214, the calculated results for B = 2 T in Figs. 58 can easily be extended to other B values. In the insert of Fig. 8b we have plotted αmax vs (µcB)2, as determined from the data in Fig. 8. As shown in Fig. 8, αmax is <10% for (µcB)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 µc (the mobility above which the realizable sensitivity is constrained by linearity) for the αmax 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 cm2 V−1 s−1, (µB)2 is equal to 1 at 100 mT and αmax 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 cm2 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 figure-of-merit for power-limited 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 graphene-on-SiO2 structure with µ and n0 values of 7800 cm2 V−1 s−1 and 1 × 1011 cm−2 3, respectively, the equivalent \(S_{\mathrm{A}}^{\mathrm{R}}\) value based on the reported SI 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 h-BN encapsulated, exfoliated graphene2 structure, record sensitivity values of voltage-related sensitivity SV = 2.8 T−1 and SI = 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 cm2 V−1 s−1 (the mobility reported for similar h-BN encapsulated CVD graphene26) and n0 = 1 × 1011 cm−2 (a typical value for high-quality graphene). Our model shows that an equivalent \(S_{\mathrm{A}}^{\mathrm{R}}\) of 4.0 VT−1 at 1 mW should be possible with αmax = 10% for B up to 2 T. If we increase µ to 120,000 cm2 V−1 s−1, the best value reported near room temperature in exfoliated and suspended graphene28, 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 (αmax = 10%). If excellent linearity (αmax = 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 FETs29,30 and attributed to asymmetric scattering for electrons and holes30. Higher ratios might be possible in engineered structures. Calculated results on the effect of the mobility ratio μh/μe 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 μh/μe = 0.2 and αmax = 10%. In the power-limited 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 electron-hole conduction asymmetry for various metal/graphene contacts31,32,33, 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 n0 = 1 × 1011 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 SiO23. These values are much higher than those for Si and GaAs sensors14: 100 and 700 VA−1 T−1, respectively. Our calculations for high-quality graphene (n0 = 1 × 1011 cm−2, µ = 100,000 cm2 V−1 s−1, αmax = 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 narrow-gap III–V heterostructure sensors (SI = 2750 VA−1 T−1; SA = 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 SiO2(285 nm)/p-Si substrate. Photolithography followed by oxygen plasma etching was used to pattern the graphene channel region. A 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 ρxx and Hall resistivity ρxy were measured as a function of magnetic field B and gate-bias Vg by the van der Pauw method34 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 voltages35. Vg 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.

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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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Correspondence to Richard A. Kiehl.

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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/s42005-019-0161-5

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