Figure 3a shows the variation in the measured fundamental resonance frequency of the micro-beam, while changing the input current ITh and for 0 magnetic field (BZ= 0 mT). It can be observed that the resonance frequency decreases with the increase in ITh and reaches a minimum value around 0.27 mA, which corresponds to the buckling point. By increasing ITh, 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 (VDC= 16 V) is applied between the lower driving electrode and micro-beam, which creates a small deflection in Y-axis, and thus causes decrease in the dip (above zero) of the first resonance frequency27. Note that the thermal time constant of the proposed magnetic sensor is around 176 μs8. For faster operation, the micro-sensor needs to be placed in vacuum.
The variation in the resonance frequency, while changing ITh and for varying magnetic field strength (BZ), is shown in Fig. 3b. By increasing BZ, the Lorentz-force (FY) increases, which also causes increase in initial deflection, and thus increases the resonant frequency of the resonator. The results show that the resonance frequency dip is almost eliminated for high value of BZ (400 mT), which explains that the micro-beam experiences a high perturbed bifurcation due to the existence of magnetic field. In addition, we observe that the frequency responses can be measured with wide range of BZ strengths from 4 mT to 400 mT at atmospheric pressure. However, our proposed micro-sensor can detect a magnetic field lower than 4 mT.
Next, we discuss the sensitivity of the micro-sensor S (1/T), which is defined as the relative change in the resonant frequency (Δf/f0) over the variation of an input magnetic (∂BZ)20. The frequency shift (Δf) is defined as (f -f0), where f0 and f, are respectively, the frequency of the micro-beam at 0 mT and during the measurement with BZ. We first measured Δf with BZ as fixing ITh at 0.14 mA (before buckling) and 0.27 mA (at the buckling point), respectively, Fig. 4a. For both values of ITh, the results show two linear trends, which separate the magnetic strengths in two ranges; low (BZ ≤ 8 mT) and high range (BZ ≥8 mT). As shown, the frequency shift (resulting from the same field strength) is found higher at ITh =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 coefficient of the relative change in the resonant frequency as a function of increasing magnetic field can represent the sensitivity (S) of the micro-sensor. Figs. 4b and 4c present Δf/f0 measurements against BZ at a bias current of 0.14 mA, away from the buckling point, for both low and high magnetic field 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 fitting, 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 field. These values are four and seven times larger than the values at 0.14 mA. This confirms 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 fields. The results indicate that the dependence of S on BZ becomes more significant 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 ITh to have an understanding of the relationship between sensitivity and power consumption. Figs. 5(a,b) plot the sensitivity versus ITh for the two ranges. The results show that S increases as a cubic polynomial with ITh. 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 confirms 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 significant 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 field micro-sensors. As listed in Table 1, the sensitivity of the device is significantly 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 field 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.
Table 1. Summary of the performance of some of the recent MEMS magnetic field sensors based on Lorentz force.
Reference
|
Magnetic range (mT)
|
Power
(mW)
|
Surface
(mm2)
|
Sensitivity
(mA-1.T-1)
|
Sensing method
|
[Zhang et al. [20]]
|
≤ 100
|
40
|
0.48
|
33.9 ppm
|
Capacitive
|
[Bahreyni et al. [34]]
|
[2.5-25]
|
0.1-10
|
0.27
|
332.3 ppm
|
Capacitive
|
[Herrera-May et al. [35]]
|
≤ 7
|
10
|
0.06
|
922.7 ppm
|
Piezoresistive
|
[Laghi et al. [36]]
|
[-5 to 5]
|
10
|
0.308
|
2800000 ppm
|
Capacitive
|
This work
|
≤ 8
|
0.2
|
0.0144
|
125000000 ppm
|
Optical
|
This work
|
[8-400]
|
0.2
|
0.0144
|
9500000 ppm
|
Optical
|