A Benchmark of Integrated Magnetometers and Magnetic Gradiometers

This survey paper compiles and benchmarks the performance of integrated magnetic sensors, in terms of their noise and full-scale magnetic field. In total, we collected performance metrics of 31 sensors realized in various technologies. For each sensor, we derived a noise model to estimate the noise under the same conditions. We obtained a comprehensive benchmark with a focus on integrated devices. We also measured the noise spectrum of 11 of those sensors to confirm our noise model. Given that for many emerging applications, such as tactile, current, torque and biomagnetic sensing, the signal of interest is the field gradient to reject magnetic stray fields, we extended the benchmark to gradiometers. We developed a low-noise measurement setup and a low-field gradient (full scale: $10 ~\mu \text{T}$ /mm) generation setup to characterize magnetic gradiometers in this regime. The paper discusses the key sensor trade-off between precision and range, and summarizes the technology trends graphically on a trade-off curve, providing broad insight.


I. INTRODUCTION
About 10 billion magnetic sensors are shipped annually [1] to supply, for the most part, the established automotive applications.Given the global electrification trend and the proliferation of magnetic materials, an increasing number of applications require differential field sensing schemes to reject stray magnetic fields.Instead of measuring the magnetic field at a single spot, it is desirable that the sensor measures the field differences between at least two spots.The sensor essentially measures a component of the magnetic field gradient ∂B i /∂j, and rejects uniform background stray fields (like the Earth's magnetic field).These types of sensors are referred to as ''magnetic gradiometers''.
To illustrate the use of magnetic gradiometers, consider the following four applications.First, in contactless current sensing [2] the two sensing spots are placed near the current-carrying wire or bus bar engineered in a serpentine shape to enhance the field gradient [3], while rejecting the influence of unrelated nearby currents.Second, in torque sensing based on the magnetoelastic effect, multiple magnetometers are used to sense differentially the magnetic The associate editor coordinating the review of this manuscript and approving it for publication was Montserrat Rivas.field from a shaft under torsion [4].Given that the typical torque sensitivity is in the order of 10 µT/Nm, the Earth's magnetic field would induce a noticeable error without the differential configuration.Third, magnetic sensing is also emerging as one of the most promising sensing modalities for tactile sensors [5].The gradiometric configuration was shown to be highly effective at rejecting stray fields [6].Fourth, magnetocardiography is a medical technique with the potential to diagnose severe cardiac issues by monitoring the magnetic field pattern from the heart near the chest.
The gradiometric configuration enables sensor operation in an unshielded room, despite the minute fields (100 pT), as demonstrated in [7].The signal amplitudes of the four mentioned applications cover 9 orders of magnitude (100 pT to 100 mT).These applications cannot be addressed by a single universal technology.They share one commonality though: they all require a gradiometric measurement.
For compact integrated magnetometers, the distance d between magnetic sensing spots should be mm-scale or cmscale.Such characteristic distances can be realized on a single chip by combining two magnetometers, as in Figure 1(a).For the cm-scale, two magnetometer chips can be combined on a PCB, like in Figure 1(b).
The gradiometric requirement was ignored in previous reviews and benchmarks [8], [9], [10].There is then a need to revisit them to cover the gradiometric configuration, and recent developments such as high-mobility (GaAs, graphene) Hall and giant magneto-impedance (GMI) sensors.This paper addresses this need by benchmarking magnetic sensors covering different technologies.We focus on highly integrated magnetic sensors, ideally with all components on a single chip (called thereafter a ''fully integrated'' device).We also consider ''partially integrated'' devices, for which the transducer is chip-scale (the longest dimension < 1 cm).Discrete sensors are not in focus, but are still referenced to illustrate performance limits.
The paper is structured as follows.Section II explains the device selection rationale, and how we extracted the benchmark metrics.Next, Section III provides measurement for most of the commercially available integrated devices.Section IV consolidates the benchmarking results in summary plots.Finally, we discuss the key trade-off and trends in Section V.

A. DEVICE SELECTION
The list of the devices is provided in Table 1, grouped by technology.Given that silicon Hall magnetic sensors account for the majority (about 80%) of sensors shipped [1], commercial silicon Hall 3D magnetometers [11], [12], [13] were used as baselines.To illustrate the Hall-effect limits, two recent R&D prototypes were included [14], [15] with optimized signalconditioning circuits and GaAs Hall plates, respectively.Two recent high-purity graphene Hall transducers were also included: one academic prototype [16], and a commercial one [17].We then selected various magnetoresistance (xMR: AMR, TMR, GMR) devices from the same manufacturer (Sensitec) [18], [19], [20].These three devices are bare transducers, arranged as passive resistive bridges.We added a fully integrated 3D anisotropic magnetoresistance (AMR) magnetometer [21], and a fully integrated tunneling magnetoresistance (TMR) magnetometer [22].The next group consisted of fluxgates: a commercial fully integrated device [23], a recent academic prototype with the coil on a separate chip [24].We added a 3D magneto-impedance (MI) sensor [25], partially integrated with 3 coils off-chip, and treated it as part of the fluxgate category.We also incorporated giant magneto-impedance (GMI) devices [26], [27] for their superior high-speed capability than fluxgate.We also added a recent GMI development [28], partially integrated with the mm-scale magnetoimpedance element off-chip.One 3D MEMS magnetometer [29] was also added to acknowledge its single-chip (or single-package) integration potential.To broaden the perspective and illustrate the performance limits of non-integrated magnetometers, we also selected a few discrete sensors with salient characteristics.Reference [30] features a quantum-well GaAs Hall sensor.References [31] and [32] achieve one of the lowest reported noise densities for xMR and GMI respectively, while [33] also addresses the issue of 1/f noise and offset instability in GMI.
For the gradiometers, the choice was more restricted, as the differential field sensing is an emerging function.We selected one device from each major technology family.As a baseline, we took the dual-sensing spot silicon Hall sensor [34] illustrated earlier in Figure 1(a).We included a prototype planar Hall effect gradiometer device [35] built along a single strip of material.This device is only the passive transducer.Note that planar Hall effect, unlike what the name suggests, is related to the AMR effect, and hence we classified this device as such.We also included a commercial fully integrated TMR gradiometer [36] optimized for current sensing.We added the fluxgate gradiometer [37] from Figure 1(b), assembled on a PCB.For diversity, two more fully integrated gradiometers were included, based on GMI [38] and MEMS [39] technology, respectively.Like for magnetometers, we included discrete gradiometers [40], [41] to illustrate the performance achievable without the integration constraint.

B. COMPARISON METRICS
For each device, we collected two primary performance metrics: the equivalent RMS magnetic noise B n and the maximum full-scale field B max .The input numbers for our comparison analysis were derived from the references (either a datasheet or a paper).The maximum field is usually directly specified in the references by a statement of the form ''range = ±B max ''.Apart from the usual issue of dealing with various magnetic units ([T], [Gauss], [A/m]), there is no ambiguity.
On the other hand, there is considerable variability in the way noise is specified in datasheets and papers.Some references specify the noise at the output (in [V] or [LSB] for analog and digital sensors, respectively).In addition, the noise can be specified in terms of the RMS noise over a given bandwidth B n , or as a noise density B n (in [T/

√
Hz] in SI units) at a specific frequency (usually 1 Hz).A complete specification should also include the shape of the noise spectral density.The noise can be ''white'' with a uniform density B n , or exhibit a low-frequency signature B nd (f ) ∝ 1/ √ f (and 1/f in terms of the power spectral density).This noise is referred to as ''pink'' noise by analogy to the color spectrum.For gradiometers, there is an extra source of TABLE 1. Devices considered in benchmark.The column IC captures the degree of integration (full:+, partial:•, discrete:−).B max is the maximum full-scale field (rounded to 2 significant digits to save space).B n(,meas.) is the noise calculated (or measured) in a 500-Hz bandwidth.The right arrow indicates that the calculated noise was fitted to the measured noise.NFR is the noise-free resolution.For the gradiometers, the field values B max , B n(,meas) should be interpreted as per mm.
possible confusion.The maximum signal and the noise are either specified as field difference [T], or as field gradient [T/m].The scaling factor is the distance d between the two sensing spots (also known as the gradiometric baseline).
To handle systematically all these cases, we collected for each device a specification string, in a human-and machinereadable format.The encoded strings are shown in the column ''Raw spec.'' in Table 1.These strings remain as close as possible to the original formulation in the references, using the exact physical quantities (the values and the units) quoted in the paper.We coded a Noise_Model class in Python decoding the specification string, and performing the calculation of the RMS noise in the same 500-Hz bandwidth for all sensors.As an example, the specific case of [28] is treated in the Appendixes.All devices are treated in the Supplementary Notes.
We restricted the noise models to two cases: either pure white noise, or pure pink noise.The equations given the RMS noise in a bandwidth [f min , BW] are given below [42].
When the noise had both a white and pink components, we took the dominant one in the bandwidth of interest, thereby slightly underestimating the noise.We also assumed that the bandwidth was limited by an ideal brick-wall filter.These assumptions are justified because we are providing estimates of the noise performances for benchmarking purpose.In an actual sensor design project, where meeting the noise requirement would be critical, higher fidelity models should be used.We discuss more advanced noise model equations, involving the equivalent noise bandwidth and the 1/f corner frequency, in the Appendixes.
We used the noise models instantiated from the specification of each device, to calculate the RMS in the [1, 500 Hz] bandwidth, by convention.This is equivalent to a 1-kHz sampling frequency (Nyquist frequency).This bandwidth is sufficient for the majority of applications mentioned earlier (i.e.torque sensing, tactile sensing, magnetocardiography) in the generic fields of mechatronics, robotics and digital health.For applications requiring a different bandwidth, the calculations should be adjusted accordingly.We will come back to this point in the Section V.
More generally, when performing an assessment of fit for a specific application, other metrics (cost, size, power, drift, 3D capability . . . ) must be considered.Nevertheless, the two key metrics {B n , B max } are helpful to screen out devices in an early design phase.

III. MEASUREMENT SETUPS AND RESULTS
To verify the noise models and the underlying assumptions, especially about the shape of the noise spectrum, we also measured the noise spectral density of 13 of the commercially-available devices The setup, shown in Figure 2(a), consisted of a magnetically shielded chamber, a low-noise supply (filtering the 50-Hz main frequency and its harmonics), a signal acquisition chain, and an offline spectral analysis.We obtained the noise density at the sensor outputs, in unit of [V] or [LSB].For magnetometers with operating field in the 1-mT range, we measured the sensitivity (in [V/T] or [LSB/T]) using our standard set of Helmholtz coils for 3D Hall sensors characterization.For magnetometers with lower operating fields, we took the values from the reference.Just like the maximum field, the sensitivity is always specified without ambiguity.
For the gradiometers, we measured the sensitivity of the three commercially-available gradiometers [34], [37], [38] using a newly-developed setup, illustrated in Figure 2(b).This setup was designed to generate a controlled field gradient ∂B x /∂x in a cm-scale cube.It consists of a pair of Maxwell coils with opposite currents (also known as gradient coil [43]).The magnetic constant was designed to be 3 nT/(mm.mA),to achieve around 10 µT/mm at the 400-mA full-scale current.
Figure 3(a) shows the measured input-referred noise spectral density for the magnetometers.Note that the setup was able to measure the flat spectrum without spurious peaks of the ultrasensitive GMI device [32] with a noise density around 10pT/

√
Hz. Similarly, Figure 3(b) shows the noise spectrum density for the gradiometers (note the unit: T/mm/ √ Hz).For the 8 devices for which we had not prior noise specification, we used the measured noise to build the model.By construction, the model calculation and the measurement are identical.These cases are represented in Table 1 by a right arrow in column B n , to indicate that the measured value in the next column was used.Generally, the measurement agreed with the model projections.For the other 5 devices, the measurement was intended as a confirmation of the specification and the corresponding model.The agreement between the model calculation and the measurement can be assessed by comparing the two columns B n and B n,meas. .Most often the agreement is within 10-50%, except for device [13], because it was configured with a bandwidth lower than 500 Hz during the measurement (as seen by the low-pass characteristic of its spectrum in Figure 3).This discrepancy is resolved in the Supplementary Notes.

IV. BENCHMARKING RESULTS
The last three columns of Table 1 summarize all the results.We first focus on the with partial or full integration.To visualize and compare their dynamic range, Figure 4 shows the range as a segment from the minimum detectable signal at the noise level B n , all to the way to the maximum field B max .To quantify the dynamic range in bits, the segments are also annotated with the Noise-Free Resolution (NFR).It indicates the number of stable bits under static conditions.It is directly related to the ratio of the two metrics (their peak-to-peak ratio): The x-axis of Figure 4 is annotated with a few relevant magnetic sources for reference.This clarifies, visually, which devices can be excluded for a given application, because the relevant source is not in the operating range.Similarly, Figure 5 plots the dynamic range of the partially or fully integrated gradiometers (in T/mm).The vertical line shows the full-scale gradient that the setup in Figure 2(b) can generate.It is well suited for applying a controlled gradient in the range [100 nT, 10 µT], and characterizing the most sensitive gradiometers (not the Hall-based gradiometers).

V. DISCUSSION
To visualize the key trade-offs and generate broad insight, Figure 6 aggregates all devices (magnetometers and gradiometers) in a scatter plot, where each device is a point (B n , B max ) expressed in T or T/mm depending upon the measurand.This representation highlights the classic sensor trade-off between the precision quantified by B n , and the full scale defined by B max .A Pareto front restricting the feasible region for fully integrated magnetometers is also    The discrete devices can break the Pareto front.They are not constrained by the voltage range of an on-chip ADC chain.In addition, discrete magnetic transducers are relatively bulky with respect to chip-scale dimensions, like in the discrete fluxgate [40] and GMI sensor [33].The magnetization fluctuations, a fundamental noise source, are then smaller, given the larger magnetic volume [44].Hence, discrete sensors are favored and do achieve better magnetic performance (at the expense of cost, size, power).Note that the ultra-clean graphene Hall devices [16], [17] are also lying past the front (top-right corner).They provide an impressive 13-bit of noise-free resolution, higher than the silicon Hall devices.These are bare Hall plates supporting maximum full-scale field around 1 T. Few high-volume applications exist in that range.As of today, their noise is no better than silicon Hall devices, but they have room for improvement.In the Supplementary Notes, we calculate their projected noise improvement should they incorporate a classic current-spinning electronic readout like in most Hall devices, to suppress their dominant pink noise.The Supplementary Notes also show that the quantum-well GaAs Hall device from [30] can also operate up to 7 T with a relatively modest noise increase.Here, we retained the lowest noise configuration with B max = 35 mT.
Concerning the partially and fully integrated devices near the front, we can make the following general statements about the state of the art.These statements are only based on the two metrics considered here in the [1, 500 Hz] bandwidth and do not replace a proper system design and analysis considering all application requirements.If the maximum signal is around 1 T, graphene and quantum-well GaAs Hall devices are well suited.They remain linear in that range.As of today, the noise performance of Graphene devices is no better than silicon Hall devices.Hence, for signal in the 10 mT-100 mT (the bulk of today's automotive applications), silicon Hall dominates.It offers up to 11b of noise-free resolution for the 3D components of the magnetic field in fully integrated mainstream chips.The silicon Hall devices are on a line [11], [12], [13], meaning that they exhibit similar NFR and are trading noise for range.They are all constrained by the same technological limits set by the mobility of carriers in silicon and the thermal noise (eq.7 in [45]).Using the same notation as above, this equation becomes: where the 6%/T is the ultimate limit of the voltage-related sensitivity at room temperature for a silicon Hall plate.For a 4-k Hall plate biased at 3.3V, the above equation yields 41 nT/ √ Hz.As silicon Hall is a mature technology, multiple sensor suppliers or designers approach this limit.For example, in a noise-optimized design [14] using the parameters above, the noise in a 500-Hz bandwidth reaches 1.2 µT, and the noise density 50ÂnT/

√
Hz (numerically 20 times less due to the division by √ 500).The GaAs Hall device [15] provides up to 3 extra bits.
xMR and fluxgate devices can resolve even smaller fields with noise approaching 10 nT at the expense of the range due to the saturation effect in the soft magnetic layers.As mitigation, the fluxgate devices [23], [24] and the GMI device [28] employ feedback via another coil wrapped around the same magnetic core, thereby improving the NFR, and are optimum in the range [10 nT, 1 mT].
Below this range, discrete sensors must be used, highlighting the need for further research to realize integrated sensors capable of resolving 1 nT and lower.With such ultra-sensitive sensors, stray field rejection becomes a must outside of a lab environment.In this scenario, gradiometers are desirable and can be realized by taking the field difference between two sensing spots.An alternative approach consists of a singlepoint gradiometer, directly sensitive to ∇B B B like the expression of the force exerted on a magnet [39].
If the application bandwidth exceeds largely 500 Hz, the calculations should be adjusted.For example, if the required bandwidth is 100x larger, the noise of the devices limited by white noise will increase by one order of magnitude.On the other hand, the noise of the devices limited by 1/f pink noise (typically the xMR devices) would only see a modest increase due to the log function in Equation 1.Hence, for high-speed applications, a device like [36] would shine (and it is actually marketed as such).
Conversely, if the minimum bandwidth is below 1 Hz, and stability over time becomes critical, the devices with white noise (Hall, Fluxgate, GMI) will shine.The xMR devices would suffer as their low-frequency noise is dominating [46].Our choice of the [1,500 Hz] bandwidth lies between these two extremes, and provides a starting point, where all devices are competitive.

VI. CONCLUSION
In summary, we surveyed 31 magnetometers and gradiometers, with a combination of literature review and experimental characterization.We developed a generic noise model and implemented it in Python for consistency and scalability.We illustrated the state of the art in terms of the precision and range.We derived technology trends in integrated magnetometers and gradiometers.To visualize these trends and steer technology development, we plotted a trade-off curve: a concise summary of the state of the art to inspire and challenge sensor developers.
Apart from silicon Hall which is a mature technology, for which the magnetic improvement potential is rather limited, the other technologies are still progressing.For example, the equivalent magnetic noise of graphene Hall devices has been improving over the years [47], and could soon rival GaAs-Hall devices, especially if the established current-spinning technique is used to suppress 1/f noise like in other Hall sensors.In particular, device [17] with its substantial 1/f noise would benefit (see the Supplementary Notes).
Academic groups have been revisiting fluxgate [24] and GMI [28] in highly integrated devices, paving the road towards a full integration with the magnetic core and coil onchip.The MEMS device [39] has also substantial potential as its calculated thermal noise floor (110 fT/cm −1 √ Hz) is several decades lower than its current performance.Beyond the devices surveyed here, quantum sensors based on nitrogen-vacancies in diamond have emerged.This technology has seen rapid progress, with the most recent device [48] achieving < 1 nT/

√
Hz (competitive with Fluxgate and xMR) in a volume of about 3.6 cm 3 .New nanos-scale magnetic transducers (e.g.50 nm in [49]) are also emerging from the progress in the field of spintronics.Consequently, we expect this benchmark to evolve.To streamline further benchmarking by the magnetics community, we released a computational notebook online [50].Our computational notebook reproduces an interactive version of Figure 6, that can be expanded as new devices become available.

APPENDIX A NOISE MODEL EXTENSION
We assumed a brick-wall filter, whereas in practice some out-of-band noise beyond the bandwidth contribute.The net result is that the equivalent noise bandwidth ENBW is higher than BW.In addition, we have assumed that the noise was either pure white or pure pink.We excluded the case where both noise components are present and contribute significantly.This case happens when the 1/f corner frequency f C is a substantial fraction of the bandwidth.The extra pink noise contribution can also be formulated as an apparent increase of the white-noise bandwidth.This is expressed by the following equation (eq.7.6 from [42]) For a higher fidelity noise model, one could incorporate the above equation.The noise model would become more generally applicable over different bandwidths, applicable for both offset-critical and speed critical applications Another possible extension would be to include non-linear distortions and other parasitic environmental influences (temperature, aging, hysteresis. . . ) into a ''total noise'', not just the random variation.This concept is well established when characterizing analog-to-digital converters (ADC).''Total noise'' is defined [51] as ''any deviation between the output signal (converted to input units) and the input signal . . .''.Total noise can be generalized to a complete sensor measurement chain, which is a form of an ADC (measurand-to-digital converter).This logic is followed in the texbook [42] and yields to the definition of the effective number of bit (ENOB) of the complete sensor.This metric captures the number of accurate bits, in the sense that the quantization error of an otherwise ideal ADC would achieve the same RMS ''total noise''-the same uncertainty due to all random and systematic errors.The author refers to this metric as ENOB a (eq.2.137 in [42]).This is a more comprehensive representation of the sensor performance than the noise-free resolution used here.It would require to extract specification for many additional parameters (offset & gain error, non-linearity, hysteresis and their respective thermal drift and drift over lifetime).This more extensive procedure is recommended for actual design project, when a limited shortlist of transducers are considered.

FIGURE 3 .
FIGURE 3. Measured spectra for (a) magnetometers and (b) gradiometers with the RMS noise in a 500-Hz bandwidth annotated.

FIGURE 4 .
FIGURE 4. Range plot for the surveyed magnetometers.
sketched.This is a concise representation of the state of the art.It approximately aligns with the line representing 16-bit of noise-free resolution.

FIGURE 6 .
FIGURE 6. Trade-off curve.Maximum full-scale field vs noise (in the considered 500-Hz bandwidth).