A Highly Sensitive and Wide-Range Resonant Magnetic Micro-Sensor Based on A Buckled Micro- Beam


 We experimentally demonstrate a miniature highly sensitive wide-range resonant magnetic Lorentz-force micro-sensor. The concept is demonstrated based on the detection of the resonance frequency of an in-plane electrothermally heated straight resonator operated near the buckling point. The frequency shift is measured with optical sensing and the device is operating at atmospheric pressure. The magnetometer demonstrates a sensitivity (S) of 33.9/T, which is very high compared to the state of the art. In addition, the micro-sensor shows a good linearity in wide range and low power consumption around 0.2 mW. The above performances make the proposed micro-sensor promising for various low-cost magnetic applications.

The intensive demand of MEMS structures has led to magnetic sensors based on Lorentz force MEMS resonators thanks to the high frequency shifting and resonance frequency tunability [20][21][22][23][24] . The frequency shift resonant magnetometer yields high accuracy, high outstanding stability, high sensitivity, low power consumption, and immunity to noise. Exposing heated structures to magnetic eld creates uniform distributed forces perpendicular to the test structures affecting their stiffness. Hence, the eld strength can be measured by detecting the resonance frequency shift of resonating structures.
Various techniques of magnetic eld micro-resonators have been used, such as capacitive, piezoresistive, piezoelectric, and optical sensing techniques 14,23,[25][26] . Despite the intrinsic losses from the device imperfections, optical sensing requires simple read-out electronic circuits, is more robust technique, and presents immunity to electromagnetic interference. In addition, this technique does not demand a vacuum package 14 .
The static and dynamic behaviours of electrothermally actuated in-plane micro-beams, have been extensively investigated [27][28][29][30][31][32] . In a previous work 2 , we exploited the buckling point of heated straight micro-beam to demonstrate highly sensitive and scalable pressure sensor. The concept is based on tracking the frequency shift of the buckled micro-beam upon changing the surrounding air pressure.
Based on the same concept, we extended our study to demonstrate highly sensitive gas sensors 8 . The method is based on the simultaneous recording of the frequencies of the rst and second modes while changing the gas concentration and type.
A small and low-cost magnetic sensor with high sensitivity in wide range, which is also simple in fabrication, operation, and sensing scheme, would be highly desirable. In this paper, we aim to realize a highly sensitive and wide range magnetic sensors based on electrothermally heated straight micro-beam operated near the buckling point. The method is based on measuring the frequency shift of the fundamental resonance frequency of micro-beam upon changing the magnetic eld. Operating near the buckling point offers signi cant shifting in frequency, and thus high sensitivity in wide range of magnetic eld.

Background And Measurement Method
The concept is illustrated in Fig. 1. First, the variation in the resonant frequency of the rst mode is monitored under DC current (I Th ). As shown in Fig. 1a, by increasing I Th , the resonant frequency of the micro-beam decreases due to increase in its stiffness until the buckling point where the micro-beam's stiffness is almost zero. After buckling, an increase in the resonant frequency is observed due to the stretching mechanism, which increases the stiffness of the buckled micro-beam. Next, as shown in Fig.  1b, a low DC electrostatic voltage (V DC ) induces a static de ection that leads to decrease in the dip of the rst resonance frequency. Note here that the straight micro-resonator takes advantage to high frequency shifting by operating around the buckling bifurcation at which the micro-beam is very sensitive to a small stiffness change. By applying a magnetic eld while operating around buckling point, it is possible to add a de ection which leads to an increase in the frequency shifting, Fig. 1c. Hence, the concept is based on tracking the fundamental natural frequency of the electrothermally buckled straight micro-beam upon exposing it to a wide range of magnetic eld.
A schematic of the micro-sensor is shown in Fig 2a. To demonstrate the concept, we utilize a microresonator fabricated from a highly doped silicon device layer of silicon-on-insulator (SOI) wafer from MEMSCAP wafer. The micro-beam is of length (L) 800 μm, width (h) 2 μm, 25μm depth (b), and is separated with two adjacent electrodes of 8 μm gap (g). A schematic of the proposed Lorentz-force magnetic sensor is shown in Fig. 2b. The electrothermal voltage V Th is applied between the two anchors inducing a current I Th owing through the micro-beam that generates heat. In addition, the straight beam is electrostatically actuated by a DC voltage V DC . As seen in Fig. 2b, with the presence of a magnetic eld (B Z ) and with a DC current I Th , which ows through the micro-beam, Lorentz-force (F Y ) is generated normal to the straight micro-beam in Y-axis.
Schematic of the experimental setup used to test the micro-resonator is shown in Fig. 2c. The resonant frequency shifting is measured using a laser Doppler vibrometer from Polytec 33 (MSA 500) as actuating the straight micro-beam with a white noise signal. The driving electrode of the resonator is electrically connected to a DC and AC harmonic voltage sources provided by the MSA. Also, the micro-beam is electrothermally actuated by a separate DC voltage source V Th . To generate a DC magnetic eld in z-axis (B Z ), a Neodymium permanent magnet is positioned above the sensor. All experiments are conducted at pressure atmospheric.
Results And Discussions Figure 3a shows the variation in the measured fundamental resonance frequency of the micro-beam, while changing the input current I Th and for 0 magnetic eld (B Z = 0 mT). It can be observed that the resonance frequency decreases with the increase in I Th and reaches a minimum value around 0.27 mA, which corresponds to the buckling point. By increasing I Th , the compressive axial load induced through the micro-beam increases via Joules heating, which also causes decrease in its stiffness. After buckling, a sharp increase in the resonant frequency is observed, which increases the stiffness of the buckled micro-beam. Note here that the buckling bifurcation is utilized since the micro-beam becomes very sensitive to any small stiffness change. A small electrostatic voltage (V DC = 16 V) is applied between the lower driving electrode and micro-beam, which creates a small de ection in Y-axis, and thus causes decrease in the dip (above zero) of the rst resonance frequency 27 . Note that the thermal time constant of the proposed magnetic sensor is around 176 μs 8 . For faster operation, the micro-sensor needs to be placed in vacuum.
The variation in the resonance frequency, while changing I Th and for varying magnetic eld strength (B Z ), is shown in Fig. 3b. By increasing B Z , the Lorentz-force (F Y ) increases, which also causes increase in initial de ection, and thus increases the resonant frequency of the resonator. The results show that the resonance frequency dip is almost eliminated for high value of B Z (400 mT), which explains that the micro-beam experiences a high perturbed bifurcation due to the existence of magnetic eld. In addition, we observe that the frequency responses can be measured with wide range of B Z strengths from 4 mT to 400 mT at atmospheric pressure. However, our proposed micro-sensor can detect a magnetic eld lower than 4 mT.
Next, we discuss the sensitivity of the micro-sensor S (1/T), which is de ned as the relative change in the resonant frequency (Δf/f 0 ) over the variation of an input magnetic (∂B Z ) 20 . The frequency shift (Δf) is de ned as (f -f 0 ), where f 0 and f, are respectively, the frequency of the micro-beam at 0 mT and during the measurement with B Z . We rst measured Δf with B Z as xing I Th at 0.14 mA (before buckling) and 0.27 mA (at the buckling point), respectively, Fig. 4a. For both values of I Th , the results show two linear trends, which separate the magnetic strengths in two ranges; low (B Z ≤ 8 mT) and high range (B Z ≥8 mT). As shown, the frequency shift (resulting from the same eld strength) is found higher at I Th =0.27 mA where the micro-beam stiffness reaches almost zero. Hence, operating the resonator near the buckling point maximizes the frequency shift.
As we mention above, the measured linear coe cient of the relative change in the resonant frequency as a function of increasing magnetic eld can represent the sensitivity (S) of the micro-sensor. Figs. 4b and  4c present Δf/f 0 measurements against B Z at a bias current of 0.14 mA, away from the buckling point, for both low and high magnetic eld ranges. As shown in Figs. 4(b,c), the sensitivity at low range is 8.46/T, and for high range is 0.35/T. It is observed that, through linear tting, the micro-sensor shows high linearity for both ranges. We next plot the results for both ranges around the bucking point (0.27 mA), Figs. 4(d,e). The slopes of the plots yield a sensitivity of 33.9 /T for the low range and 2.56/T for the high range. Hence, the device can sense with high sensitivity a wide range of magnetic eld. These values are four and seven times larger than the values at 0.14 mA. This con rms well with the concept that by operating at the buckling point, the proposed micro-sensor is very sensitive to any external force including from magnetic elds. The results indicate that the dependence of S on B Z becomes more signi cant in low range compared to high ranges. The improved high sensitivity at low ranges encourages the efforts toward low cost magnetic sensors applications.
Next, we show the results of S with I Th to have an understanding of the relationship between sensitivity and power consumption. Figs. 5(a,b) plot the sensitivity versus I Th for the two ranges. The results show that S increases as a cubic polynomial with I Th . Increasing the current from 0.1 mA to 0.27 mA (before buckling point), the sensitivity can be further improved by 870 % for the low range and 800 % for the high range. This again con rms that the sensor sensitivity becomes high as operating around the buckling point. It also shows that the operating current point of the magnetic sensor can be tuned to achieve higher sensitivity.
One should mention that the proposed sensor might be promising for some applications, which require high sensitivity. In addition to having high sensitivity, low power consumption is also an important factor. At bucking point (0.27 mA), the sensor consumes power around 0.2 mW due to the electrothermal actuation. This value can be reduced to half by operating at 0.135 mA while there is signi cant reduction in S (81 %). Hence, with a straight micro-beam, high sensitivity is achieved even for a low input current.
However, the power of the device can be improved by reducing the cooling effect between the micro-beam and the surrounding air (e.g., operating at low pressure using vacuumed package).
Note here that the current detection method using optical readout system (laser) does not suffer considerably from noise, electronic circuitry, and weight. In other sensing methods, where resonances are detected electrically, such as capacitive sensing, they suffer from parasitic capacitances and other sources of noise, which can have high impact on the resolution of measurements. Furthermore, the proposed sensing technique does not compensate the environmental temperature variation, which represents one limitation of the frequency based magnetic sensors compared to capacitive technique.
Moreover, calibration experiments may be conducted to compensate the variation of ambient temperature. Table 1 below shows comparison for the performance of the proposed micro-sensor to some recent works on Lorentz force resonant magnetic eld micro-sensors. As listed in Table 1, the sensitivity of the device is signi cantly higher and with smaller dimensions compared with other previously reported magnetic sensors. Also, it is noted that the results show high sensitivity in wide range of magnetic eld strengths. In addition, the proposed micro-sensor is useful to reduce the device power and its size, and thus the cost of the MEMS magnetometer.

Conclusions
In this paper, we demonstrated experimentally a highly sensitive in-plane Lorentz-force magnetic microsensor by exploiting the buckling point bifurcation of a straight beam. The concept is based on the measured resonance frequency shift of the rst mode under different magnetic strengths. The microsensor demonstrates high sensitivity compared to reported magnetic sensors. The main advantages of the proposed sensor are the simplicity of fabrication, low-cost, excellent linearity, low power consumption, scalability, and wide range measurements. The high performances above motivate in-depth studies for the implementation of the proposed magnetic micro-sensor in low magnetic elds strengths around µT. In conclusion, we demonstrated that a single electrothermal straight micro-resonator could be potentially used as a magnetic sensor by measuring the frequency shift near the buckling point. Future studies are planned for measurements in vacuum to reduce the power consumption and lower the minimal detectable magnetic elds.