Abstract
We present a highly sensitive Lorentzforce magnetic microsensor capable of measuring low field values. The magnetometer consists of a silicon microbeam sandwiched between two electrodes to electrostatically induce inplane vibration and to detect the output current. The method is based on measuring the resonance frequency of the microbeam around the buckling zone to sense outofplane magnetic fields. When biased with a current of 0.91 mA (around buckling), the device has a measured sensitivity of 11.6 T^{−1}, which is five orders of magnitude larger than the stateoftheart. The measured minimum detectable magnetic field and the estimated resolution of the proposed magnetic sensor are 100 µT and 13.6 µT.Hz^{−1/2}, respectively. An analytical model is developed based on the Euler–Bernoulli beam theory and the Galerkin discretization to understand and verify the microsensor performance. Good agreement is shown between analytical results and experimental data. Furthermore, the presented magnetometer is promising for measuring very weak biomagnetic fields.
Introduction
In recent years, magnetometers have been extensively used in a wide range of applications, including biomedical, automotive industry, robotics, and nondestructive material testing^{1,2,3,4}. In addition, the significant improvement in the performance of magnetic sensors, such as excellent sensitivity, high resolution, minimum detectable magnetic field, low power consumption, high stability, wide bandwidth, small size, low cost, and excellent linearity, has allowed them to be used in navigation (1 nT–600 μT), biomagnetic (100 fT–0.1 μT), and archeology (1 pT–700 µT) applications^{5}. Some applications, such as inertial navigation system require high sensitivity and low noise^{6}, whereas small size sensors are wellsuited for probing microscopy^{7}. In scanning Hall probe microscopy, miniature magnetic sensors can be integrated with temperature sensors for temperature compensation, thereby reaching a better tradeoff between magnetic performance and spatial resolution^{1,7}.
Various types of magnetic sensors have been developed, such as magnetoresistive and Halleffect magnetometers, which have attracted attention due to their high sensitivity and reliability^{8,9}. However, these magnetometers suffer many challenges, such as hysteresis effect and incompatibility of the material with standard manufacturing processes^{10}. Microelectromechanical Systems (MEMS) magnetometers with low power consumption and low cost have emerged as a promising alternative^{11,12,13}. In addition, these magnetometers can be implemented on silicon without the need for any special magnetic materials^{13}. Recently, considerable efforts have been focused on resonant Lorentzforce MEMS magnetometers^{14,15,16,17,18,19}, which are based on measuring the resonance frequency of a microstructure using optical^{20,21}, piezoresistive^{17,22}, and capacitive techniques^{23,24}.
The optical sensing technique needs a simple readout circuitry and has good immunity to electromagnetic interference^{23}. However, this method is bulky and cannot be easily integrated with the sensor for mass production. The piezoresistive method is a simple sensing technique that requires an easy fabrication process^{23}. Nevertheless, the technique can have voltage offset and temperature dependence issues^{5}. The capacitive sensing technique is the most used owing to its low noise and low power consumption^{25}. However, it suffers from parasitic capacitances. Still it has low temperature dependence^{19}.
Efforts have been made to improve the sensor performance, such as its sensitivity and resolution, by using different approaches, which can be classified into extrinsic and intrinsic techniques. The extrinsic technique involves the new modulation and control systems used to drive the MEMS Lorentz force sensors, either using open or closedloop^{12,26,27,28,29,30}. The closedloop aims to operate the sensor exactly at mechanical resonance^{26}. Using the closedloop control improves significantly the dynamical behaviour of the MEMS sensors, which enhances the measurement accuracy^{31}. However, this approach needs a lockin amplifier that increases the circuit complexity and complicates the integration of the control system with the MEMS device^{32}. Compared to the closedloop, the openloop approach is compact with less complexity for the driving and sensing electronics in spite of the sensitivity to parameter uncertainties of the input signal^{32}. On the other hand, the intrinsic technique can be attributed to the device itself using advanced MEMS capabilities^{23}. Many research studies have been reported to amplify the Lorentz force and thus boost sensitivity using internal amplification^{17} or microleverage mechanism^{14}. Moreover, several MEMS dynamical phenomena have been utilized to improve the MEMS sensor sensitivity^{19,33,34}, such as veering and buckling.
In a previous work, we proposed a highly sensitive Lorentzforce magnetic microsensor based on an inplane electrothermally heated initially curved microbeam exhibiting the veering phenomenon between its first two symmetric vibration modes^{35}. In a recent work, we presented a miniature highly sensitive widerange resonant magnetic Lorentzforce microsensor based on measuring the resonance frequency of straight heated microbeam operating around the buckling zone^{36}. The resonance frequency is measured with optical sensing (laser) and the sensor is operated at ambient pressure. In this paper, we explore the performance of a similar sensor to detect experimentally low magnetic field using open loop control system (capacitive sensing). We show that the proposed sensor is capable of detecting a magnetic field of 100 μT. In addition, a Galerkinbased reducedorder model (ROM) is developed to understand and verify the microsensor performance, which can further help in the designing of ultrahigh sensitivity magnetometer.
Microsensor working description and principle of operation
Figure 1a shows an optical image of the microsensor. It consists of a clamped–clamped straight microbeam fabricated from SOIMUMPs process by MEMSCAP^{37}. The fabrication process is based on highly doping 30 µm Si device layer of Silicon OnInsulator SOI substrate. As seen in Fig. 1a, the microbeam is sandwiched between two electrodes to electrostatically induce inplane vibration and to detect the output AC current. Figure 1b shows a crosssectional view of a SOIMUMPs processed device similar to the studied one. The microbeam has a 600 μm in length (L), 2 μm in width (h), and 30 μm in thickness (b). The gap (g) between one of the electrodes and the resonating microbeam is 7 μm. Applying an electrothermal voltage (V_{Th}) between the anchors of the microbeam, generates an induced current (I_{Th}), through Joule’s heating, which causes a compressive axial stress^{38}. Increasing I_{Th} leads to increase in the compressive axial load, and thus the resonance frequency (stiffness) of the microbeam decreases. Increasing further I_{Th}, the resonance frequency decreases until reaching a minimal value (very low stiffness). This is known as the buckling phenomenon^{39}, see Fig. 1c. In addition, the microbeam is electrostatically actuated with a static DC voltage to avoid the dip of the resonance frequency near buckling (bias the buckling), Fig. 1c.
In this work, the proposed MEMS magnetometer is based on the Lorentz force principle. Hence, to generate a constant magnetic flux B, the device is placed inside two identical circular magnetic coils, which is measured using a Gaussmeter with a commercial Hall probe. A schematic of the experimental setup and a photo of the experiment platform are shown in Fig. 2a,b, respectively. As seen in the Fig. 2a, to provide an AC harmonic voltage (V_{AC}), the drive electrode is connected to the network analyzer output (Keysight N9913A). To sense the output AC motional current (I_{O}) due to the microbeam inplane motion, the sense electrode is connected to the input of the network analyzer. In addition, we use a lownoise amplifier (LNA SR560) to amplify I_{O}. Note that the existence of V_{DC} creates an initial deflection, which enhances the output motional current. As we can see from Figs. 1a and 2a, with the presence of a magnetic field (B) in the zaxis and with a DC current (I_{Th}) in the + xaxis, Lorentzforce (F_{L}) is generated normal to the microbeam in the + yaxis. As shown in^{21,36}, to achieve maximum sensitivity, the Lorentz force is chosen to be in the same direction of the initial deflection and buckling direction (+ yaxis).
Dynamic analytical model
Assuming an Euler–Bernoulli nonlinear beam model, the structural inplane behaviour of the microbeam is governed by the following equation and boundary conditions:
where \(w\left( {x,t} \right)\) denotes the inplane (along ydirection) displacement in space x and time t, ρ is mass density, E is the Young’s modulus, A = bh and I = bh^{3}/12 are the crosssectional area and second moment of inertia, \(\tilde{c}\) represents the overall viscous damping coefficient, α is the thermal expansion coefficient, \({T}_{amb}\) is the ambient temperature, and \({\varepsilon }_{0}\) is the dielectric constant of air. The spacedependent temperature profile \(T\left( x \right)\) can be obtained through solving the steadystate heat transfer equation as detailed in^{40}.
The resultant equations and boundary conditions system, Eq. (1), are numerically discretized using a Galerkin’s modal based expansion technique^{41}. Accordingly, the microbeam deflection is expanded as follows:
where \(\psi_{i} \left( x \right)\) represents the modeshape of an unforced beam and \(u_{i} \left( t \right)\) is the corresponding unknown timedependent modal coordinate amplitude. Next, Eq. (2) is substituted into Eq. (1). After that, the resultant outcome is multiplied by the modeshape functions \(\psi_{j} \, (1 \le j \le N)\) and the subsequent Nequations are then integrated from 0–L. This produces N OrdinaryDifferential Equations (ODEs) in \(u_{j} \left( t \right) \, (1 \le k \le N)\) forming a ReducedOrderModel (ROM). A convergence analysis was carried out and it was found that five modes (N = 5) is sufficient for convergence. The derived ROM can be used to calculate the microresonator natural frequencies. Toward this, the static deflection of the beam is first calculated. For this, we evaluate the stationary deflection by setting all time dependent terms in the ROM equal to zero. Then the modal amplitudes \(u_{i} \left( t \right)\) are replaced by unknown constant coefficients a_{k}. This leads to a system of nonlinear algebraic equations that can be solved numerically using Newton–Raphson algorithm. Finally, the eigenvalue problem of the microresonator is solved by perturbing the response around the calculated static position and considering only the linear part of the ROM. Further detail of the eigenvalue problem can be found in^{42}.
Results and discussion
To perform the measurements, the magnetometer is placed inside a vacuum chamber with a pressure of 6 mT. The DC voltage V_{DC} and the electrostatic driving rms amplitude V_{AC} are set to 30 V and 9.988 mVrms (− 27 dBm). Figure 3a shows the experimental and analytical results of the variation of the fundamental resonance frequency of the microbeam while changing the electrothermal current I_{Th}. Good agreement is shown among the results. It can be seen that the resonance frequency decreases with the increase of I_{Th} until it reaches the buckling zone (I_{Th} = 0.91 mA). Within this range, the resonance frequency varies from 52 to 35 kHz, confirming that the presence of V_{DC} causes perturbation around the buckling zone (imperfect buckling). Thus, the system shows a midpoint displacement of 0.25 μm (initial curvature), see the inset of Fig. 1a. Hence, operating around this zone, the microsensor is very sensitive to any stiffness change (for instance from the magnetic distributed forces). After the imperfect buckling zone, the resonance frequency increases due to the induced curvature of the microbeam.
To further investigate the dynamic response of the system with I_{Th}, we plot the mode shapes of the microbeam for V_{DC} = 30 V and for various I_{Th}, Fig. 3b. It is clear that, before the buckling point, there is no effect of the compressive axial load on the mode shapes.
Here, the buckling bifurcation will be used to sense the outofplane magnetic field in the low ranges. The Lorentzforce alters the microbeam stiffness, which increases its resonance frequency. This frequency shift forms the basis of the proposed microsensor. Figure 4a shows the variation of the normalized resonance frequency (Δf/f_{0}) variation versus input magnetic field (B) for I_{Th} = 0.81 mA (around buckling point) where Δf is the frequency shift, which is defined as (f–f_{0}). f and f_{0} are the resonance frequency of the microbeam with and without B. In Fig. 4a, the sensitivity can be evaluated as the slop of the variation Δf/f_{0} with B. As seen in Fig. 4a, the device capable of sense a wide range of B from 100 μT to 4 mT. Also, it can be seen that the sensor has high sensitivity (S) of 11.6 T^{−1}. Further, we check the linearity of the proposed microsensor. Figure 4a demonstrates that the microsensor has high linearity less than 0.5% in the entire magnetic field range. Next, we plot the results for I_{Th} = 0.67 mA, inset of Fig. 4a. The result shows a measured S of 10.8/T with 2% of nonlinearity. This value is lower than the value near the buckling point 0.81 mA. The analytical results of the variation Δf/f_{0} with B while changing I_{Th} are shown in Fig. 4b. The results agree with the experimental data, showing a high sensitivity of 11.92 T^{−1} at the buckling point (I_{Th} = 0.91 mA).
To further analyze the effect of the operating current on the sensitivity, Fig. 4c illustrates the analytical and experimental results of S with I_{Th} before the buckling zone. The theoretical results match well with the experimental data. As expected, S increases with the increase of I_{Th}. At 0.91 mA, the sensor achieves a sensitivity that is 14 times higher than the sensitivity of 0.2 mA. The results confirm well the concept that by operating at the buckling point, the Lorentzforce magnetometer is extremely sensitive to outofplane magnetic fields. According to the results presented in Fig. 4b,c, the optimized sensitivity of the device can be achieved by carefully choosing the operating current I_{Th} to reach the bifurcation point.
Next, we discuss the total power consumption of the proposed magnetic sensor. It results from the combination of the electrostatic drive power of the resonator (in order of nW) and the electrothermal power^{43} (in order of mW). Thus, the overall power consumption is dominated by the electrothermal voltage. It is estimated based on resistive heating of the resonator (V_{Th}^{2}/R) where R is the microbeam resistance. For 600 µm microbeam, R is found to be 1.49 KΩ and with I_{Th} of 0.91 mA, the magnetometer power consumption (P) is around 1.2 mW. Moreover, I_{Th} can be decreased to decrease the power dissipation by Joule’s effect. This will be on the expense of decreasing the sensitivity, i.e., S ∝ I_{Th}. It is important to note that there are many ways to improve the microsensor performance, including designing devices with low R, which reduces the dissipation and enhances the sensitivity.
Note here that the current detection method using capacitive sensing may suffer from parasitic capacitances and other sources of noise. Moreover, this sensing method can have an impact on the resolution of measurements. The measured quality factor Q of the sensor is found to be around 200. Thus, for the noise analysis of the system, we assume only the thermomechanical white noise (Q > 100 for frequencymodulated sensors) since the system shows a low noise voltage related to the LNA instrumentation amplifier around 4 nV.Hz^{−1/2}^{44}. Thus, the evaluated thermomechanical noise (Brownian limited resolution) of the proposed magnetic sensor can be expressed as^{15,45}:
where k_{B} is Boltzmann’s constant, T is absolute temperature in Kelvin, and b is the damping coefficient (b = 0.0025). The theoretical Brownian noise is found to be 13.6 µT.Hz^{−1/2} for a 0.81 mA input current. Clearly, a longer microbeam can decrease the noise limit. The Brownian limited resolution may be decreased further by increasing the power or by lowering the damping of the system. Hence, the microsensor design can be adjusted to improve the noise performance.
As we notice above, the thermomechanical noise is dependent on Q, and its magnitude can be affected by the input current. Hence, we measured the Q change versus I_{Th} for V_{DC} = 30 V and 40 V, see Fig. 5a. The results show an increase of Q as increasing I_{Th} or V_{DC}. It is clearly noticeable that the sensor has the highest Q at the buckling zone. Thus, operating around the buckling zone, the proposed magnetometer exhibits high sensitivity, good linearity, and also high quality factor.
Also, highperformance magnetic sensors should have high sensitivity over a wide temperature range. To investigate the effect of the temperature on the sensor, we simulate the temperature dependence of the resonance frequency from 0 to 45 °C, see Fig. 5b. As can be seen in Fig. 5b, the results show a temperature coefficient of frequency (TCF) of − 0.0055/°C. For the temperature compensation, we can use a highly ndoped silicon when the temperature coefficient of the electrical stiffness becomes large enough to compensate or supersede the effect of the temperature coefficient of the mechanical stiffness^{46}. However, at high doping, the increase of electrical stiffness may increase the operating point to higher bias currents.
Further, analytical simulations are performed to predict the device performance. Figure 6 illustrates the effect of V_{DC} on the power consumption and sensitivity. Figure 6a shows the variation of the resonance frequency while varying I_{Th} for different values of V_{DC}. The inset depicts a comparison between analytical and experimental results for V_{DC} = 30 and V_{DC} = 25 V.
As seen in the figure, as V_{DC} decreases, the corresponding resonance frequency drops until reaching a value of 17 kHz. In addition, its corresponding operating point increases to 1.05 mA. For low V_{DC} value (@ 10 V)_{,} the frequency response has the sharpest curve at the perturbed buckling zone (low stiffness), which can be extremely sensitive to any output forces. Figure 3b shows the results of Δf/f_{0} variation versus B for different bias V_{DC.}
As can be seen in Fig. 6b, the increase in Δf/f_{0} follows a linear function with B for different values V_{DC}. For further study of the effect of V_{DC} on the sensor sensitivity, we plot S as increasing V_{DC}, Fig. 6c. The results show that as the actuation V_{DC} increases, the S decreases as a quadratic polynomial. Subsequently, first S drops rapidly, and then slowly decreases, as V_{DC} increases. Decreasing V_{DC} from 30 to 10 V, the sensitivity can be further improved by 920%. Indeed, these results demonstrate that the sensitivity is highly dependent on the applied V_{DC}. In the proposed sensor with high airgap, adding high value of V_{DC} is necessary to improve the motional current on the sensing electrode. Accordingly, future efforts should minimize electrostatic driving V_{DC} with ultrasmall airgap size to make the output signal large enough to be sensed.
As we mention above, the device sensitivity is defined as the fractional change in frequency per unit of magnetic field density, which can be expressed as^{14,15,30}^{:}
where k is the microbeam’s stiffness. As can be seen from Eq. (4), S can be further enhanced by modifying the physical dimensions of the device more carefully or taking advantages from more complaint microbeams. On the other hand, as implied by Eq. (4), increasing L or reducing h increases S. Hence, it responds more to the Lorentzforce for the same given magnetic flux. This explains that the device performance depends strongly on the geometrical parameter of the microbeam. In the following, we investigate the influence of the microbeam length on the resonance frequency and sensitivity. Based on the analytical model, we increase L from 700 μm to 900 μm and compute the resulting resonance frequencies. Figure 7 displays the results as tuning I_{Th} and V_{DC} for microbeams with different lengths L. This figure shows that the initial natural frequency at I_{Th} = 0 mA decreases as L increases. In addition, we observe that at I_{Th} = 0 mA, the frequency responses increase with V_{DC} for microbeam lengths 800 μm and 900 μm, respectively. This is due to the midplane stretching that is dominate over electrostatic force for a longer microbeam and large capacitive gap. As noted from the figure, increasing L shifts the position of the buckling point from high to low values of I_{Th}. This can be explained from the increase of the microbeam’s resistance and the decrease of its stiffness. Hence, these results can be useful to reduce the energy consumption. Increasing the length to 900 μm can reduce the power consumption by 94% (P = 0.07 mW). Operating around the buckling point, Fig. 8 presents the analytical results of the variation of S as a function of V_{DC} and for different lengths L. As shown, increasing L results in the decrease of the sensor sensitivity. S decreases to the minimum of 2.4 T^{−1} for L of 900 μm and V_{DC} of 30 V. We conclude here that to improve the sensor sensitivity, the microbeam length L and the input current I_{Th} should be increased (S ∝ I_{Th} L). However, in both cases there is the cost of increased power dissipation by Joule’s effect. It also shows that the operating current of the magnetic sensor and V_{DC} can be tuned to achieve higher sensitivity.
These results can be used as guidelines to choose the geometric parameters to improve the sensor performance. The power consumption can be reduced by a factor of 2 if a 900 μm long beam is used. Moreover, the device sensitivity is expected to improve after reducing V_{DC}. More analytical studies can be conducted to investigate the effects of the lengthtothickness (L/h) ratio on the energy dissipation and sensitivity. One should mention that the proposed sensor design is flexible and can be optimized to measure ultralow biomagnetic fields’ applications.
Table. 1 shows the performance comparisons of the proposed microsensor with some of the reported Lorentzforcebased magnetometers. For a fair comparison, only MEMS magnetometers with frequencymodulation (FM) output, which the proposed sensor belongs to, have been included. It can be seen from Table 1 that the proposed miniature magnetic sensor has higher sensitivity compared with the state of the art. Moreover, the high sensitivity, lowpower consumption, and good linearity, make the sensor promising for wide range of applications.
Conclusions
In this paper, we proposed an outofplane Lorentzforce magnetic microsensor by exploiting the buckling point bifurcation of a straight microbeam. The concept is based on the frequency shift measured at different magnetic field strengths. We showed that the sensitivity of the proposed magnetic sensor is very high. In addition, we demonstrated analytically an optimal sensor design whose sensitivity can be further enhanced to around 920%. Also, the proposed sensor shows good linearity. These features, combined with lowpower consumption, simplicity of fabrication, lowcost, and scalability, make the sensor promising for variety of applications. We demonstrated experimentally the capability of the magnetometer to sense a low magnetic field around 100 μT using open loop control system. Future research work will include a closed loop control system to improve the noise performance. It was found that the performance of the proposed magnetometer depends strongly on the microbeam length, operating point, and the electrostatic driving voltage. One should mention that the microsensor design is flexible and can be optimized to measure ultralow biomagnetic fields’ applications.
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We acknowledge financial support from King Abdullah University of Science and Technology.
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S.B.M. and N.A. contributed equally in processing the experimental data, drafting the manuscript. H.M.O. developed the analytic model and performed the numerical simulations. M.I.Y. supervised the project.
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Mbarek, S.B., Alcheikh, N., Ouakad, H.M. et al. Highly sensitive low field Lorentzforce MEMS magnetometer. Sci Rep 11, 21634 (2021). https://doi.org/10.1038/s4159802101171z
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DOI: https://doi.org/10.1038/s4159802101171z
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Nonlinear mode saturation in a Ushaped microresonator
Scientific Reports (2022)
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