Traceably Calibrated Scanning Hall Probe Microscopy at Room Temperature

Fabrication, characterization and comparison of gold and graphene micro- and nano-size Hall sensors for room temperature scanning magnetic field microscopy applications is presented. The Hall sensors with active areas from 5 $\mu$m down to 50 nm were fabricated by electron-beam lithography. The calibration of the Hall sensors in an external magnetic field revealed a sensitivity of 3.2 mV/(AT) $\pm$ 0.3 % for gold and 1615 V/(AT) $\pm$ 0.5 % for graphene at room temperature. The gold sensors were fabricated on silicon nitride cantilever chips suitable for integration into commercial scanning probe microscopes, allowing scanning Hall measurements under ambient conditions and controlled sensor-sample distance. The height dependent stray field distribution of a magnetic scale was characterized using a 5 $\mu$m gold Hall sensor showing good agreement with numerical simulations within the uncertainty budget.


I. Introduction
High resolution quantitative magnetic field measurements at the micrometer scale are increasingly important for research and development in areas like magnetic sensors and magnetic positioning. However, the characterization of microscale magnetic structures entails new challenges for magnetic stray field measurement techniques, since the generated magnetic stray fields locally change their direction on the nanometer range, and the field amplitude decreases rapidly with an increasing distance to the sample surface. Thus, suitable magnetic sensors not only need to be small to avoid averaging over different stray field directions but also must be precisely positioned close to the sample.
Here, micro-and nano-scale Hall sensors for traceable scanning Hall probe microscopy (SHPM) are characterized, and one technique to integrate them into a commercial atomic force microscope (AFM) is presented. AFM based SHPM 1-3 (AFM-SHPM) has certain advantages compared to other magnetic imaging techniques. In comparison to magnetic force microscopy 4,5 , the AFM-SHPM is non-invasive due to the use of non-magnetic materials and the measurement directly generates quantitative results albeit with reduced spatial resolution. It is applicable to a broader field range than magneto-optical indicator film techniques 6,7 , which are limited by the saturation field of the sensor film. Unlike magneto-resistive sensors 8,9 , Hall probes show excellent linearity without hysteresis and measure a well-defined field component perpendicular to the sensor plane. Furthermore, SHPM enables a low sample-probe distance and therefore a spatial resolution limited by the sensor size only. A three-dimensional mapping of the stray field can be performed by repeating the two-dimensional plane scans at defined heights 10 and first commercial systems employing sub-micron Hall sensors for room temperature (RT) measurements are available 11,12 . Concerning the choice of the Hall sensor material different material classes have been considered: Standard Hall sensor devices are typically based on semiconductor technology. They show an outstanding performance at low temperatures 13 , and micron-sized sensors are also working at room temperature 1,14,15 . In contrast, nano-scale semiconductor Hall sensors show a weak signal to noise (S/N) ratio at RT. Besides the semi-metal bismuth 3 , graphene 2,16-19 has the potential to bridge this gap. The semi-metallic graphene can be prepared with a low carrier density at RT resulting in a high Hall coefficient. It could also enable the application of Hall sensors at high temperatures 16 , in contrast to semiconductor sensing with a limited range of operational temperatures. Moreover, a one atom thin active layer prevents field averaging perpendicular to the sensor area allowing high spatial resolution in the corresponding direction. Another option is the use of metals as Hall sensor materials. Here, in particular, gold is very stable, radiation hard 20 , simple to manage in the fabrication process and its carrier density is widely insensitive to surface contaminations. Additionally, it was reported that gold Hall sensors with sizes below 500 nm have a better noise figure at RT than sensors based on two-dimensional electron gases 21 .
In this work, the fabrication and investigation of gold-and graphene-based micro-and nano-Hall sensors with respect to SHPM applications is presented. The sensor sensitivity and stability is characterized and discussed. Gold Hall sensors are fabricated on silicon nitride (SiN) cantilever chips suitable for AFM-SHPM. After traceable calibration, they allow quantitative stray field measurements of magnetic scales with few micron resolution. The uncertainty budget of the measurements is discussed, and the setup is validated by comparing the measurement results to stray field and Hall voltage signal calculations.

II. Fabrication of Hall sensors
The Hall sensor cross-structures with active square areas from 5 x 5 µm² down to 50 x 50 nm² were fabricated using electron-beam lithography (EBL). The substrate of the gold sensors consists of a silicon wafer covered on both sides with a 1 µm thick SiN layer deposited by low-pressure chemical vapor deposition. The inset of Figure 1a shows a gold cross-shaped Hall sensor that is produced through two lithography steps. First, the Hall cross defining the sensor's active area is structured, after electron beam deposition of the metals, by EBL and lift-off through a PMMA resist. The active sensor material consists of a 5 nm titanium adhesion layer followed by a 30 nm gold layer. In the second step, an additional 50 nm gold layer is deposited to support the outer extended contact regions. After fabrication of the Hall sensors, cantilever chips with Hall sensors on the tips of the cantilevers, as depicted in Figure 1  The graphene samples were grown on silicon carbide (SiC) (0001) substrates with a size of 5 x 10 mm 2 using a so-called polymer-assisted sublimation growth technique [22][23][24] . The high morphological and electronic homogeneity of the graphene samples utilizes scalable realization of Hall sensors on true two-dimensional carbon sheets without bilayer inclusions. The graphene Hall sensors were patterned with EBL and AC plasmaetching through a resist mask. The fabrication of graphene Hall cross-structures requires four steps. Initially, the electrical contacts are defined by depositing Ti/Au (10/30 nm) layers in a lift-off process. In the next step, the monolayer graphene is structured by plasma etching through an EBL resist mask, leading to well-defined geometry and sizes of the small active area. Finally, the outer contact areas and leads are defined using a 50 nm gold layer. To avoid the environmental influences, especially the rapid change of the carrier density by surface absorption on the graphene, the graphene sample was encapsulated with 50 nm co-polymer. Note that within this work the graphene sensors were only fabricated on SiC wafer dies. The fabrication of graphene sensors on cantilevers will be subject of future studies.

III. Characterization of the Hall sensors
For the Hall sensor calibration, an electromagnet driven by a Kepco power supply with a pole shoe diameter of 92 mm was used to provide a spatially homogeneous magnetic flux density up to 450 mT at a pole shoe distance of 18.5 mm. The operation current for the Hall sensors was generated by a Keithley source meter 2400. The Hall voltage was measured with Keithley Nanovoltmeter 2182A. During the Hall sensor calibration, the magnetic flux density was simultaneously measured with a traceably calibrated commercial Hall probe FH55 from Magnet-Physik Dr. Steingroever GmbH. As a consequence, the Hall sensor calibration is traceable to the SI units.
The typical output from the characterization of graphene and gold sensors is presented in Figure 2. The Hall voltage was measured as a function of the magnetic flux density and corrected for the offset. Both sensors show a linear dependence of the Hall voltage Hall on as expected from Hall = × /( × × ), where is the supply current, is the electron density, is the electron charge and is the thickness of the active layer. For the 5 µm gold Hall sensor operated at 10 mA, the output is in the µV range for between -150 mT and 150 mT. This leads to a sensitivity of 3.2 mV/(AT) ± 0.3 %. For the same field range, the Hall voltage of the 500 nm graphene sensor is in the mV range using an operating current of 50 µA. Fitting the data reveals a sensitivity of 1615 V/(AT) ± 0.5 %. The sensitivity of the graphene sensor is six orders of magnitude higher due to the lower carrier density of graphene in comparison to gold. Similar results were observed in the measurements on several other 5 µm large Hall sensors. The mean sensitivity of all gold sensors is 3.1 mV/(AT) with a maximum deviation of 0.2 mV/(AT) within the sensors in this study. Moreover, the time stability as well as fabrication reproducibility of the sensors were investigated. To this end, the sensors were frequently characterized within one year and compared with nominally identical sensors from different batches. For the gold sensors, the long-term stability was very high with a deviation over time below 0.6 %. Graphene sensors showed sensitivity deviations of up to 9.3 % from one day to another. Also, the overall variation in sensitivity was larger for graphene sensors ranging from 500 V/(AT) to 1700 V/(AT) depending on the carrier density in the respective graphene material and actual sensor. Based on the Hall voltage deviation of measured data points from the expected value given by the linear characterization fit, a typical resolution of 2 mT for gold sensors and 0.45 mT for the graphene sensors is calculated. This resolution includes, besides the sensor properties, also influences and noise contributions from devices and cables in the circuitry. Noise measurements revealed a detectivity of 60 µT/√Hz at 1 Hz for a graphene Hall cross. The property data are summarized in Table I. Due to the small resistance of the gold Hall sensors, it was not possible to measure their noise characteristics. This also means that the noise properties of the electronics have a larger influence on the S/N than the sensor itself. With the described measurement equipment, it was possible to calibrate gold sensors with active areas down to 1 µm. For 50 nm gold sensors on cantilevers, the background noise of the setup is larger than the expected Hall voltage of 1.5 nV per 10 mT at the operating current of 50 µA. Because of higher sensitivity and thus larger Hall voltage, graphene sensors with a size of 100 nm still show an overall linear dependence on the applied magnetic flux density, as shown in the inset of Figure 2. The resolution is decreased to 8.5 mT because of the growing impact of carrier fluctuations for smaller sensor sizes. Furthermore, the lower current supply leads to smaller Hall voltages, thereby the overall S/N is reduced.

IV. AFM based SHPM -setup and measurement
AFM-SHPM is realized by integration of the manufactured cantilever chips with gold Hall sensors into a commercial AFM (Nanoscope IIIa, Dimension 3000 scanner). The cantilevers have a typical resonance frequency of about 50 kHz and can be used in standard tapping mode operation, and thus in close contact with the sample surface. As shown in the upper part of Figure 3, the Hall sensors are positioned at the bottom side of the cantilever and close to its tip to achieve a small distance between the sensor and sample. This is significant for improving spatial resolutions in measuring nanostructures due to the fast decay of stray fields with increasing distance to the sample surface. The cantilevers are mounted at an angle of 10° given by the cantilever holder. This leads to minimal measurement heights of 400 nm and 3366 nm for the 50 nm and 5 µm sized sensors, respectively, for an ideal alignment of the Hall sensor on the cantilever. The electrical connection to the Hall sensor is realized by bond wires from the cantilever chip to a printed circuit board (PCB) that is fixed to the cantilever holder. The current source, voltmeter and PCB are connected via soldered cables. To increase the scan area up to millimeter range, additional piezo tables were added to the setup that allows scanning the sample with a fixed cantilever position. The Hall sensor was calibrated in the electromagnet before and after the AFM-SHPM measurement. As a test sample, a commercially available magnetic scale SST250HFA-04 from Sensitec was chosen. The scale is made of a wet pressed strontium ferrite with a remanence magnetization of = 395 mT. The material was magnetized into alternating up and down magnetized stripes with a width of nominally 250 µm and several millimeters length.
The bottom part of Figure 3 displays the results of AFM-SHPM on the scale using a 5 µm gold sensor with a Hall sensitivity of 2.3 mV/(AT) ± 13 %, measured under an applied operating current of 1 mA. Line scans with ten repetitions each were performed at seven different measurement heights. The closest line scan to the sample was attained in the tapping mode and thus followed the sample topography at a distance of approximately 4 µm. The scans for higher distances were carried out at fixed heights ranging from 19 µm to 169 µm (with 30 µm intervals) by moving the sample with a 3-axis piezo scanning system and fixed probe position. The sample has a granular structure and thereby height variations of around 10 µm and locally tilted surface areas. Therefore, during the measurement in tapping mode, the z-piezo table was utilized to keep the sensor-sample distance in a range that is controllable by the AFM head. The measurement was performed at a scan speed of 50.5 µm/s. 400 points were measured per line with an averaging time of 20 ms. All plots in Figure 3 show the expected 500 µm periodicity of the scale. The decay of the stray field amplitude with increasing distance to the sample is clearly visible in both actual measurement values and simulation results, as presented in the next paragraph. Furthermore, cantilever and sensor appeared to be very robust, allowing the characterization of rather rough samples, as demonstrated in this study.

V. Comparison with simulations
Here, two modeling approaches are presented to validate the measurement procedure. The first one uses a Fourier transform method to calculate the z-component Bz of the stray field produced by the magnetic scale. Assuming that the sample is perfectly parallel to the Hall sensor surface and neglecting the 10° cantilever tilt, the Hall sensor response is mainly dominated by the perpendicular component. From now on, this will be called the perpendicular or z-component in contrast to the in-plane components lying parallel to the sample surface. The calculated field profile is then compared to the experimental one, which is derived from the measured Hall voltage as = ( × × × )/ after the application of offset corrections. In the following, the stray field simulation procedure is described. To simulate the perpendicular stray field component, the underlying sample magnetization has to be known. To this end, the magnetization is guessed from the measured Hall signal based on the following assumptions: (i) The transition between up and down magnetized poles can be found at zero transitions of the Hall signal after a performed offset and drift correction. (ii) The magnetization pattern of the scale, therefore, can be found by discrimination between areas with positive and negative Hall voltages. (iii) Areas with + Hall (− Hall ) have purely perpendicular and, over the thickness, homogenous magnetization of + r (− r ). To account for the pole writing process, the step-like transitions are then additionally smoothed as visualized in Figure 4a. For the stray field calculation, a transfer function (TF) approach 25,26 was pursued due to its numerical simplicity. It can be shown, that in a partial Fourier space of only the in-plane spatial components, as for the transformation from (x,y,z) to ( , ,z), the calculation of the stray field above a perpendicular magnetization distribution can be performed through a multiplication by a to TF.
The stray field of the scale at measurement height was calculated for a magnetic layer with an assumed thickness of = 75 µm. Similarly, the impact of the finite sensor dimensions might be considered by introducing a multiplication by an appropriate sensor sensitivity TF. However, for the relatively slowly varying field of the scale with 250 µm pole width, this is expected to have a minor impact and was therefore neglected. Inverse discrete Fourier transformation was used to obtain the value of the perpendicular component plotted in Figure 4a for each SHPM measured point in real space.
A good agreement between the measured stray field and simulated data was obtained, giving evidence of the validity of the measured quantitative magnetic field distribution, as shown exemplarily in Figure 4a for the sensor-sample distance of 49 µm. The simulation confirms very well the measurement results in terms of maximal and minimal magnetic flux density as well as spatial periodicity. However, the drift and slight decrease in amplitude cannot be explained by this stray field simulation. One reason would be an unstable temperature during the measurement and thus a drift of the offset voltage. Another explanation would be an angular misalignment of the sample if it is not flat or placed perfectly horizontal on the table. Moreover, the implementation of the cantilever chip in the AFM with a 10° canting angle, resulting in a canted Hall sensor with respect to the sample surface, would lead to an asymmetric signal. As a proof, a second modelling approach was implemented, in which the Hall voltage signal due to magnetic scale scanning is numerically calculated, considering an angular misalignment of about 1° between the magnetic sample and the Hall device. The spatial distribution of the electric potential within the sensor is derived from the finite element solution 27 of the following equation where ̈( )is the conductivity tensor In (3) µ is the electron mobility, assumed equal to 8.7 x 10 -4 m 2 /(Vs) from 4-point resistance measurement, n = 1.92 x 10 21 m -2 and ⊥ is the orthogonal component to the sensor of the stray field from the scale below, which also includes the component of parallel to the sample surface, due to the sensor-sample relative angular orientation. The formulation is completed by the boundary conditions in correspondence of the current and voltage contacts.
The stray field from the scale, which is discretized in N 10 µm size hexahedra with imposed uniform magnetization, is calculated as where e is the surface of the e-th hexahedron having normal unit vector ne and barycentre with vector position re 28 .
The drift effect in the measured Hall voltage signal is well reconstructed by the numerical results, which also support the validity of the linear dependence of Hall on B for all the scanning points, due to the large width of the pole scale with respect to the Hall cross size. The agreement with experimental results is highlighted in Figure 4b for an average sensor-sample distance of 49 µm and 139 µm. The peaks reduce in amplitude during scanning, as a consequence of the increase in the sensor-sample distance.
For further verification, the behavior of the stray field with an increasing distance to the sample surface, as shown in Figure 3, was quantitatively analyzed. Therefore, the maximal measured stray field amplitudes over the poles for each measurement height were compared in Figure 5 with the values expected from simulations with the first approach. For the two largest measurement heights and corresponding two lowest expected magnetic flux densities, systematic uncertainties from evaluating the extrema have a more significant influence on the result due to an enlarged contribution of noise. However, for all measurements, the simulation result overlaps with the uncertainty squares of the data points. More details about the uncertainty range are given in the next paragraph. From these results, the validity of the quantitative AFM-SHPM method using a gold sensor is concluded.

VI. Evaluation of uncertainty budget
Four major contributions entering into the net measurement uncertainty are: (i) The Hall sensor itself, where its stability, sensitivity, offset, temperature dependence and noise must be considered. (ii) The Hall sensor calibration via the electromagnets magnetic field homogeneity, stability and repeatability. (iii) The Hall sensor driving and read-out electronics, including the stability of the current source and the voltmeter noise as well as thermoelectric voltages. (iv) The positioning accuracy of the scanning system. The multiplicity of uncertainty sources and the fact that standard uncertainty analysis is not sufficient for linear regression tasks 29 , as used in the Hall sensor calibration, rule out a conventional uncertainty propagation calculation. Therefore, the uncertainties of the main contributions were analyzed separately to evaluate their impact on the measurement.
By repeated sensor calibrations and statistical analyses of the results, a calibration uncertainty for the sensor sensitivity of 13 % was found. The characterization of several gold sensors, as shown above, leads to the expectation of lower sensitivity uncertainties for the gold sensors in general. The uncertainty includes contributions from the electronic components and the applied magnetic flux density. The scanning system uncertainty is 7 mT after the correction of drifts by offset subtraction. This was evaluated from ten repeated line scans. The different measurement heights were realized by an additional z-piezo that has a positioning accuracy of 15 µm. This leads to an extended uncertainty of ± (7 mT + 13 %) for the SHPM using the amplification factor k = 2.

VII. Conclusion
In summary, SiN based AFM cantilevers equipped with micro and nano-scale gold Hall sensors were fabricated, which facilitate accurate traceably calibrated scanning magnetic field microscopy (AFM-SHPM) at room temperature. The measurement data were in good agreement with simulation results, which underline the reliability of the presented approach. The gold sensors exhibit a sensitivity of 3.2 mV/(AT) with high long-term stability. Also, Hall sensors out of epitaxial, zero-band gap, semi-metallic graphene (on SiC) were fabricated and studied. In contrast to the metallic gold sensors, the graphene samples show an outstanding high sensitivity of 1700 V/(AT), but low time-stability. This suggests that by proper isolation of the graphene sensors from environmental influences (e.g., using hexagonal boron nitride or aluminum dioxide) a higher performance could be achieved. This, in addition to the implementation of the graphene Hall sensor into the AFM cantilever, are the subjects of future studies. For the 5 µm gold AFM-SHPM, the uncertainty budget of room temperature measurement was determined to be ± (7 mT + 13 %). This method enables a direct quantitative characterization of magnetic microstructures in ambient conditions with the capability of generating three-dimensional maps of the sample's out of plane stray fields within a range from mT to few T. The fabrication process is scalable thus in principle allowing high volume sensor production. Finally, the AFM-SHPM is a non-destructive and robust method for both scientific research as well as industrial applications, e.g., quality control within industrial processes.